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Nuclear Data Sheets 110 (2009) 3107–3214 www.elsevier.com/locate/nds

RIPL – Reference Input Parameter Library for Calculation of Nuclear Reactions and Nuclear Data Evaluations R. Capote,1∗ M. Herman,1,2 P. Obloˇzinsk´ y,1,2 P.G. Young,3 S. Goriely,4 T. Belgya,5 A.V. Ignatyuk,6 7 8 9 A.J. Koning, S. Hilaire, V.A. Plujko, M. Avrigeanu,10 O. Bersillon,8 M.B. Chadwick,3 T. Fukahori,11 Zhigang Ge,12 Yinlu Han,12 S. Kailas,13 J. Kopecky,14 V.M. Maslov,15 G. Reffo,16 M. Sin,17 E.Sh. Soukhovitskii,15 P. Talou3 1

NAPC–Nuclear Data Section, International Atomic Energy Agency, A-1400 Vienna, Austria National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973, USA 3 Los Alamos National Laboratory, Los Alamos, NM 87544, USA 4 Université Libre de Bruxelles, BE 1050 Brussels, Belgium 5 Institute of Isotope and Surface Chemistry, Chemical Research Center, H-1525 Budapest, Hungary 6 Institute of Physics and Power Engineering, 249033 Obninsk, Russia 7 Fuels Actinides and Isotopes NRG Nuclear Research and Consultance Group, NL-1755 Petten, The Netherlands 8 CEA, DAM, DIF, F-91297 Arpajon, France 9 Taras Shevchenko National University, 03022 Kiev, Ukraine 10 National Institute of Physics and Nuclear Engineering “Horia Hulubei”, 077125 Bucharest-Magurele, Romania 11 Japan Atomic Energy Agency, Tokai-mura, Naka-gun, Ibaraki-ken, 319-1195 Japan 12 China Institute of Atomic Energy, Beijing 102413 China 13 Bhabha Atomic Research Center, Trombay, 400085 Mumbai, India 14 JUKO Research, NL-1817 Alkmaar, The Netherlands 15 Joint Institute for Power and Nuclear Research – Sosny, BY-220109 Minsk, Belarus 16 Retired in 1998, Ente Nuove Tecnologie, Energia e Ambiente (ENEA), 40129 Bologna, Italy and 17 Nuclear Physics Department, Bucharest University, 077125 Bucharest-Magurele, Romania 2

(Received July 20, 2009) We describe the physics and data included in the Reference Input Parameter Library, which is devoted to input parameters needed in calculations of nuclear reactions and nuclear data evaluations. Advanced modelling codes require substantial numerical input, therefore the International Atomic Energy Agency (IAEA) has worked extensively since 1993 on a library of validated nuclear-model input parameters, referred to as the Reference Input Parameter Library (RIPL). A final RIPL coordinated research project (RIPL-3) was brought to a successful conclusion in December 2008, after 15 years of challenging work carried out through three consecutive IAEA projects. The RIPL-3 library was released in January 2009, and is available on the Web through http://www-nds.iaea.org/RIPL-3/ . This work and the resulting database are extremely important to theoreticians involved in the development and use of nuclear reaction modelling (ALICE, EMPIRE, GNASH, UNF, TALYS) both for theoretical research and nuclear data evaluations. The numerical data and computer codes included in RIPL-3 are arranged in seven segments: MASSES contains ground-state properties of nuclei for about 9000 nuclei, including three theoretical predictions of masses and the evaluated experimental masses of Audi et al. (2003). DISCRETE LEVELS contains 117 datasets (one for each element) with all known level schemes, electromagnetic and γ-ray decay probabilities available from ENSDF in October 2007. NEUTRON RESONANCES contains average resonance parameters prepared on the basis of the evaluations performed by Ignatyuk and Mughabghab. OPTICAL MODEL contains 495 sets of phenomenological optical model parameters defined in a wide energy range. When there are insufficient experimental data, the evaluator has to resort to either global parameterizations or microscopic approaches. Radial density distributions to be used as input for microscopic calculations are stored in the MASSES segment. LEVEL DENSITIES contains phenomenological parameterizations based on the modified Fermi gas and superfluid models and microscopic calculations which are based on a realistic microscopic single-particle level scheme. Partial level densities formulae are also recommended. All tabulated total level densities are consistent with both the recommended average neutron resonance parameters and discrete levels. GAMMA contains parameters that quantify giant resonances, experimental gamma-ray strength functions and methods for calculating gamma emission in statistical model codes. The experimental GDR parameters are represented by Lorentzian fits to the photo-absorption cross sections for 102 nuclides ranging from 51 V to 239 Pu. FISSION includes global prescriptions for fission barriers and nuclear level densities at fission saddle points based on microscopic HFB calculations constrained by experimental fission cross sections. *) Corresponding author, electronic address: [email protected]; [email protected]

0090-3752/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.nds.2009.10.004

RIPL: Reference Input Parameter Lib...

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Contents

I. INTRODUCTION II. ATOMIC MASSES A. Experimental masses B. Finite-Range-Droplet-Model masses C. Hartree-Fock-Bogoliubov masses D. Hartree-Fock-Bogoliubov density distributions and multipoles E. Duflo-Zuker mass formula based on shell model F. Shell correction energy G. Uncertainties in the mass predictions H. Relative isotopic abundances I. Future developments and outlook

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VI. NUCLEAR LEVEL DENSITIES A. Phenomenological total level densities 1. Basic relations of Fermi Gas Model 2. Back-Shifted Fermi Gas Model 3. Composite Gilbert-Cameron Model 4. Generalized Superfluid Model 5. Enhanced Generalized Superfluid Model B. Microscopic total level densities 1. Statistical microscopic approaches 2. Microscopic combinatorial level densities C. Partial level densities D. Recommendations

3113 3113 VII. GAMMA-RAY STRENGTH 3113 FUNCTIONS 3114 A. Experimental γ-ray strength functions 3114 B. Closed-form models for the E1 strength function III. DISCRETE LEVEL SCHEMES 3114 1. Standard Lorentzian model (SLO) A. RIPL-2 sublibrary 3115 2. Kadmenskij-Markushev-Furman model 1. Constant-temperature fit parameters 3116 (KMF) 2. Construction of the Discrete Level 3. Enhanced Generalized Lorentzian model Scheme 3117 (EGLO) 3. RIPL-2 consistency tests 3117 4. Hybrid model (GH) B. RIPL-3 sublibrary 3119 5. Generalized Fermi Liquid model (GFL) 1. RIPL-3 2005 upgrade 3119 6. Modified Lorentzian model (MLO) 7. E1 strength functions in deformed nuclei 2. RIPL-3 2007 upgrade 3119 8. Comparison with experimental data 3. RIPL-3 consistency tests 3119 C. Microscopic approach to E1 strength 4. Validation of the RIPL-3 sublibrary 3120 function C. Maintenance 3120 D. Giant dipole resonance parameters E. M1 and E2 transitions IV. AVERAGE NEUTRON RESONANCE F. Recommendations PARAMETERS 3120 A. Evaluation methods 3121 B. Roadmap from RIPL-1 to RIPL-3 3122 VIII. NUCLEAR FISSION A. Basic relations C. RIPL-3 sublibrary of average neutron B. Phenomenological fission barriers and level resonance parameters 3124 densities 1. Fission input data for pre-actinides V. OPTICAL MODEL 3126 2. Fission input data for actinides A. Phenomenological optical model potential C. Microscopic fission barriers and level and extensions 3126 densities B. Optical model potentials developed during 1. Fission path RIPL-2 and RIPL-3 projects 3127 2. Nuclear level densities at saddle points 1. Complex-particle optical model D. Fission cross section calculations potentials 3127 1. Phenomenological fission parameters 2. Dispersive relations and optical model 2. Global HFB fission parameters potentials 3129 E. Prompt Fission Neutrons C. RIPL-2 and RIPL-3 sublibraries 3135 1. Los Alamos model 1. Nucleon-nucleus potentials: Individual 2. Center-of-mass neutron energy spectrum nucleus potentials 3135 3. Laboratory neutron energy spectrum 2. Nucleon-nucleus potentials: Global 4. Average prompt neutron multiplicity potentials 3136 5. Multiple-chance fission 3. Complex-particle potentials 3138 F. Recommendations and future developments 4. Microscopic optical model 3138 D. Validation 3139 IX. SUMMARY AND CONCLUSIONS E. Uncertainties in optical model potential References parameters 3140 F. Recommendations 3142

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RIPL: Reference Input Parameter Lib... I.

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INTRODUCTION

The International Nuclear Data Committee (INDC) of the International Atomic Energy Agency (IAEA) is committed to improving the accuracy and availability of the nuclear database required for nuclear technology. By the late 1980s and early 1990s, the increasing sophistication of nuclear model codes and their extensive use in support of evaluated nuclear data libraries was widely recognized. Theoretical understanding of nuclear phenomena had become more and more reliable, and nuclear model codes based on the optical model and the statistical HauserFeshbach model with preequilibrium emission provided an increasingly important tool for nuclear data evaluations for energies above the resonance region up to the pion threshold. It was recognized that model codes offer important advantages in that they are capable of providing full sets of cross sections with complete description of secondary nuclear radiations, which are well suited for radiation transport calculations as required in many nuclear applications. At the same time the INDC was fully aware that many nuclear data measurement facilities were in danger of closing down or were already closed, and the question of nuclear data supply for future nuclear applications was becoming increasingly important. Within the 1990s time period, many advances had been made in the development of deformed and spherical optical model codes, Hauser-Feshbach statistical/preequilibrium, and quantum-mechanical multistep direct and multistep compound codes. Great success had been achieved in optimizing nuclear model parameters so that the codes were able to represent much of the experimental database and provide an extrapolative tool. At the same time such fundamental tenets as energy conservation, flux conservation, and consistency among partial, reaction, and total cross sections were automatically guaranteed in the model calculations. The difficulty was that practical use of nuclear theory and models in supplying nuclear data for applications required considerable input of model parameter data and nuclear constants. Much of this nuclear input information resided at major nuclear research laboratories, and often the model input data were undocumented and generally unavailable to many researchers. With the trend toward reduced funding for nuclear data evaluations, the possibility existed that much of the nuclear parameter input database might be lost to future generations responsible for supplying data for nuclear applications. The INDC recognized the vacuum that could develop in nuclear data capabilities, and consequently the IAEA recommended that a Consultants’ meeting be held with the charge of providing technical advice on assembling a reference library of nuclear model input parameters, and advising on the availability, validation, and documentation of modern nuclear reaction model codes. The Consultants’ meeting was held in Vienna in November 1991. The meeting focused on the most important model code input data, including (1) atomic masses; (2) shell correc-

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tions; (3) deformations; (4) discrete level properties; (5) average neutron resonance parameters; (6) optical model parameters; (7) fission barrier parameters; (8) level density parameters; and (9) gamma-ray strength function parameters. Both short-term and long-term solutions were addressed, and a number of practical actions were proposed for use under the sponsorship of the IAEA [1]. A follow-on IAEA Consultants’ meeting was held in 1993 [2] to further delineate the actions required for each of the model parameter data types listed above and to make general recommendations to the IAEA for organizing the effort. A number of specific technical recommendations were made for various data types envisioned for the library, and two major general recommendations were: (1) the IAEA Nuclear Data Section should be responsible for providing central services required for the activity, such as online storage and retrieval of data, format conversion codes, and appropriate documentation; and (2) the IAEA should organize a series of Coordinated Research Programs (CRP) to carry out development of the library under the title “Reference Nuclear Input Parameter Library” (RIPL). The meeting further recommended that a standing technical committee of 8-10 members be formed, with oversight responsibility for each of the technical sublibraries to be included in RIPL [2]. The result of the 1993 Consultants’ meeting was the formation of a Coordinated Research Program (CRP) to carry out development of the RIPL library. The title of the CRP was “Development of Reference Input Parameter Library for Nuclear Model Calculations of Nuclear Data”. A series of three Research Coordination Meetings (RCM) were held over the time period September 1994 to May 1997, aimed at developing the library [3]–[5]. Coordinators were assigned for each of the major code input data segments, and they served over the lifetime of the CRP. The library was targeted at users of nuclear reaction codes interested in low-energy nuclear applications. Incident and outgoing radiation types included in the library were neutrons, protons, deuterons, tritons, 3 He, 4 He, and gamma rays. The major input data segments and associated data directories were: No Directory

Contents

1 2 3 4 5 6 7

Atomic Masses Discrete Energy Level Schemes Neutron Resonance Parameters Optical Model Parameters Level Densities γ-ray Strength Functions Angular Distributions

MASSES LEVELS RESONANCES OPTICAL DENSITIES GAMMA ANGULAR

It was realized early on that the magnitude of the effort required to develop a comprehensive and reliable RIPL, that is, a library that had been thoroughly validated and tested, was such that it could not be achieved on such a short time scale with the resources available. It was therefore decided that a second follow-on CRP would be

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required. For the first CRP, the goal was to develop formats for representing the various model parameters and to proceed as far as possible in assembling data for the initial RIPL library, with emphasis on making the library as comprehensive as possible. For the second CRP, it was recommended that necessary extensions to the library be carried out and that the database be thoroughly tested and validated. The first version of the RIPL library was recognized as a “Phase 1 Starter file” [5] and is referred to as the RIPL-1 library [6]. At the conclusion of the CRP, the RIPL-1 data library was made available for worldwide use through the IAEA Nuclear Data Section in Vienna (see http://www-nds.iaea.org/ripl/ ). A technical document or handbook was prepared that describes information in the database, and several auxiliary codes were provided for retrieval and use of parts of the database. Finally, it was recommended that the necessary extensions and thorough testing and validation of the library be accomplished in a second CRP, and this was approved by the IAEA. The second RIPL CRP was entitled “Nuclear Model Parameter Testing for Nuclear Data Evaluation”, and the library was labeled “Reference Input Parameter Library: Phase II,” or simply RIPL-2. A series of three RCM meetings [7]–[9], were held between November 1998 and April 2002. The formal objectives for the RIPL-2 CRP were: • Test and improve nuclear model parameters for theoretical calculations of nuclear reaction cross sections at incident energies below 100 MeV. • Produce a well-tested Reference Input Parameter Library for calculations of nuclear reactions using nuclear reaction codes.

The RIPL-2 database was well received and is used extensively at many laboratories and facilities. Limitations did exist, however, and several of these were pointed out in the summary report from the final RIPL-2 meeting [9]. For example, information on uncertainties in nuclear reaction parameters was largely absent. Perhaps the most serious limitation was that the RIPL-2 file was constructed mainly to cover the energy range important for nuclear reactors. That is, although RIPL-2 constitutes a comprehensive and consistent set of nuclear reaction input parameters, its scope is largely limited to neutron-induced reactions up to 20 MeV. Addressing the needs of emerging nuclear technologies requires extension of the RIPL-2 database to handle model parameterizations to higher energies. Examples of such technologies are charged-particle beam therapy, accelerator-driven nuclear waste incineration, production of radioisotopes in particle accelerators for radiotherapy and medical diagnostics, and materials analysis. To address these and other limitations, a third RIPL CRP was authorized by the IAEA in 2003. The title of the CRP is “Parameters for Calculation of Nuclear Reactions of Relevance to Non-Energy Nuclear Applications (Reference Input Parameter Library: Phase III)”. Three RCM meetings [11]–[13] directed at developing a RIPL-3 library were carried out from August 2004 to February 2008. The same structure and data format that were utilized in the RIPL-2 database were followed for RIPL-3. The overall objective of the RIPL-3 CRP was to improve the methodology of nuclear data evaluations by increasing predictive power, accuracy, and reliability of theoretical calculations with nuclear reaction model codes. Specific goals of the CRP were: • Extend the RIPL-2 parameter library for theoretical calculations of nuclear reaction data up to 200 MeV incident-particle energy both for energy and non-energy applications.

• Develop user-oriented retrieval tools and interfaces to established codes for nuclear reaction calculations.

• Establish well-defined and documented procedures for maintenance and future updates of RIPL-3.

• Publish a technical report and make the library and tools available on-line and on a CD-ROM. These goals were fulfilled in the course of the second CRP, which included extensive testing of the RIPL-2 library and codes. In addition, the CRP was significantly extended by the addition of a large volume of new parameter data as well as extensive results from microscopic calculations on a large number of nuclei. The same segments or directories as were present in RIPL-1 were included in RIPL-2, except Segment 7 (ANGULAR) was replaced in RIPL-2 by a FISSION directory, covering fission barriers and level densities. Similarly to RIPL-1, the RIPL-2 database was made available for worldwide use through a Web server operated by the IAEA Nuclear Data Section in Vienna (see http://www-nds.iaea.org/RIPL-2/ ). A technical handbook aimed at describing and facilitating use of RIPL-2 for nuclear reaction data calculations was published in 2006 [10].

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• Validate the RIPL-3 database with calculations across the periodic table for major reaction model codes, including comparison with experimental data. • Provide estimates of uncertainties and/or ranges of parameter variation for RIPL-3. • Develop a library of modules that can be interfaced with nuclear model codes. By the conclusion of this third CRP in 2008, all major segments of the RIPL-3 database had been extended to facilitate calculations at incident energies up to 200 MeV, and the libraries in several segments were substantially broadened. In addition to data expansion, procedures were established for maintaining and updating the RIPL3 data, and estimates of uncertainty were provided for

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key quantities in the database and robust optical model and level density modules were developed. The final product is an expanded and updated RIPL3 electronic database that is derived from RIPL-2 but which supersedes it. The database is available online at http://www-nds.iaea.org/RIPL-3/ . A technical document that describes all facets of the RIPL-3 database including all technical details of the format is in preparation [14]. Format issues are not discussed in the present paper. In the chapters that follow here, a complete description of the physics and data embodied in the RIPL3 database is given. Additionally, we will show selected applications of RIPL data in calculations of nuclear reactions and evaluations of neutron-induced nuclear data.

II.

ATOMIC MASSES

Mass, or binding energy, is one of the most fundamental properties of atomic nuclei. A large number of processes in nuclear physics require for their description an accurate knowledge of nuclear masses. Mass predictions are, for example, a critical element of many calculations required in nuclear astrophysics. Mass values also define nuclear reaction thresholds and Q-values. Many different mass formulae are available nowadays. However, we consider in RIPL only global approaches which provide predictions of nuclear masses for all nuclei up to the super-heavy region Z ≥ 120 lying between the proton and the neutron drip lines. The RIPL mass segment consists of a combined set of experimental values [15], along with three theoretical predictions of masses and deformations: Finite Range Droplet Model (FRDM) [16], Hartree-Fock-Bogoliubov (HFB) mass-model HFB-14 [17, 18], and algebraic DufloZuker formula [19]. A detailed description of each method is given in the following sections.

A.

Experimental masses

The mass Mnuc (N, Z) of a nucleus with N neutrons (of mass Mn ) and Z protons (of mass Mp ) is measurably different from the sum of the masses of the free nucleons, and provides a direct determination of the internal energy Enuc (negative of the binding energy) of the nucleus: Enuc = (Mnuc (N, Z) − N Mn − ZMp ) c2 .

(1)

The atomic mass can be calculated from the nuclear mass by the relationship: Mat = Mnuc (N, Z) + ZMe − Be (Z),

(2)

where Me is the electron mass and Be is the total atomic binding energy of all electrons. The latest compilation of experimental atomic masses corresponds to the 2003 publication of Audi et al. [15] and includes 2228 nuclei.

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They also estimated a set of 951 additional masses from trends in systematics based on the regularity of the mass surface. The final set of Audi et al. best-recommended masses includes 3179 nuclei. The uncertainty associated with each experimental or best-recommended mass is also provided by Audi et al., and is included in the present library.

B.

Finite-Range-Droplet-Model masses

Attempts to develop formula or more generally algorithms representing the variation in Enuc from one nucleus to another go back to the 1935 “semi-empirical mass formula” of von Weizsäcker [20]. This approach corresponds to the widely used liquid-drop model (LDM) of the nucleus, i.e. macroscopic mass formula which accounts for all but a small part of the variation in the binding energy. Improvements have been gradually made to the original liquid-drop mass formula, leading to the development of macroscopic-microscopic mass formula, where microscopic corrections to account for the shell and pairing correlation effects are added to the liquid drop part. The macroscopic and microscopic features are treated independently, both being connected exclusively by a parameter fit to the experimental masses. Later developments include macroscopic properties of infinite and semi-infinite [21] nuclear matter and the finite range character of nuclear forces. The most sophisticated version of this macroscopic-microscopic mass formula is the “finite-range droplet model” [16]. The atomic mass excesses and nuclear ground-state deformations are tabulated for 8979 nuclei ranging from 16 O to (Z = 136, A = 339) nucleus. Calculations are based on the finite-range droplet macroscopic model and the folded-Yukawa singleparticle microscopic correction. Relative to the 1981 version, improvements are found mainly in the macroscopic model, pairing model with a new form for the effectiveinteraction pairing gap, and minimization of the groundstate energy with respect to additional shape degrees of freedom. Parameters are determined directly from a least-squares adjustment to the ground-state masses of 1654 nuclei ranging from 16 O to 263 Sg. The rms deviation is 0.656 MeV for the 2149 Z, N ≥ 8 nuclei with experimental masses. Data file masses/mass-frdm95.dat includes the Audi et al. (2003) experimental and best recommended masses when available [15], the FRDM calculated masses, microscopic corrections and deformation parameters in the βparameterization [16]. The microscopic correction Emic corresponds to the difference between the total binding energy and the spherical macroscopic energy (see section II F below). Additionally, FDRM single-particle level schemes corresponding to the ground-state deformation of the nucleus are also included in the database. These single-particle levels could be used in level-density combinatorial calculations.

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FIG. 1: Comparison of HFB-14 charge densities (solid line) vs experimental data (red dots with error bars) C.

Hartree-Fock-Bogoliubov masses

Microscopic theories based on nucleonic interactions have also been used to estimate the binding energies. One of the most promising microscopic approaches is the non-relativistic Hartree-Fock-Bogoliubov (HFB) method based on an effective nucleon-nucleon interaction of Skyrme type. HFB calculations in which a Skyrme force is fitted to essentially all the mass data [15] are not only feasible, but can also compete with the most accurate droplet-like formulae available as shown in Refs. [17, 18] (and references therein). The Skyrme HFB mass model not only seeks optimized fits to the mass data, but also allows constructing a universal effective interaction capable of reproducing most of the observables of relevance in nuclear applications. This behavior is achieved by imposing extra physical constraints on the HFB mass models. In RIPL-2, the HFB-2 mass model [22] was recommended. It represents one of the very first microscopic mass models based on the Skyrme-HFB method. HFB2 reproduces the 2149 experimental masses with an rms deviation of 0.659 MeV, but with the drawback of being characterized by a rather strong pairing force and a rather poor description of nuclear and neutron matter properties. Since development of the HFB-2 mass model, improvements have been made to reproduce additional observables, including in particular infinite nuclear matter properties, as well as nuclear level densities and fission barriers. The HFB-14 model [17, 18], included in the RIPL-3 library, was subjected to the following constraints: (1) energy-density curve of neutron matter was fitted, a requirement that is relevant not only to neutron-star applications, but also to the reliability of finite-nucleus extrapolations out towards the neutron drip line; (2) the strength of the pairing force was held considerably below the value that would emerge from an optimal fit to the mass data, thereby improving considerably the predictions for level densities (see chapter VI); (3) a vibrational term was added to the phenomenological collective correction, fitting this parameter to measured fission-barrier heights. 52 primary fission barriers lower than 9 MeV, which were compiled in the RIPL-2 database, are reproduced with rms deviation of 0.67 MeV and the 45

secondary barriers with rms deviation of 0.65 MeV (see chapter VIII). The resulting HFB-14 mass model is characterized by rms deviation of 0.729 MeV with respect to the Audi et al. (2003) mass data [15]. In addition, the HFB-14 quadrupole moments, charge radii and chargedensity distributions have been shown to be in excellent agreement with experiment. Details of the HFB-14 mass model were published in [18] and references therein. The complete HFB-14 mass table is available in the masses/mass-hfb14.dat file. Along with the Audi et al. (2003) experimental and best recommended masses (when available), this table also includes the HFB masses, deformation parameters in the β-parameterization, and the parameters describing the n and p density distribution for all 8508 nuclei lying between the two drip lines over the range Z, N ≥ 8 and Z ≤ 110. The density-distribution parameters are the amplitude ρq,0 , radius rq and diffuseness aq (where q = n, p) of the spherical nucleon-density distribution. These are determined for each type of nucleon separately by fitting the calculated HFB distribution by a simple spherical Fermi function: ρq (r) =

ρq,0 . [1 + exp(−(r − rq )/aq )]

(3)

The amplitude ρq,0 has been determined so that the nucleon number is conserved in the spherical approximation.

D.

Hartree-Fock-Bogoliubov density distributions and multipoles

The Skyrme HFB density distributions (assuming spherical symmetry) based on the HFB-14 mass model are tabulated in the masses/density-hfb14/ directory for nuclear radius up to 20 fm in steps of 0.1 fm. The parameters of the density distribution as described above are also tabulated in the corresponding HFB mass file. A very good description of the measured charge density distribution is achieved as can be seen for 32 S and 208 Pb nuclei in Fig. 1. In addition to the Skyrme HFB model, the deformed density distributions are also provided as derived from

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large-scale HFB calculations [23] based on the D1S Gogny effective interaction [24, 25] assuming axial symmetry. The axial densities are expanded in terms of spherical harmonics Yλ0 (θ, φ): ρD1S (r) q

=

ρD1S q,λ (r)Yλ0 (θ, φ),

(4)

λ=0,2,4,6,8

where the multipoles ρD1S q,λ (r) of deformed q-type densities are listed in the masses/density-d1s/ directory up to a maximum radius of 20 fm in 0.1 fm steps. Both HFB models provide densities for all nuclei lying between the proton and neutron drip lines over the range 8 ≤ Z ≤ 110. These densities provide the input required for semi-microscopic optical model potentials calculated by folding the target radial matter density with an optical potential in nuclear matter based on the Br¨ ucknerHartree-Fock work of Jeukenne et al. [26]–[29].

E.

attributed to the pairing interaction). Defining a macroscopic deformation energy by the difference in the macroscopic energy between equilibrium and spherical shape: Edef (Z, A, β) = Emac (Z, A, β) − Emac (Z, A, β = 0), (6) the total nuclear binding energy can now be expressed as Etot (Z, A, β) = Emac (Z, A, β = 0) + Emic (Z, A, β), (7) with the microscopic correction Emic (Z, A, β) = Eshell (Z, A, β) + Epair (Z, A, β) +Edef (Z, A, β), (8) including all the shell, pairing and deformation effects. Another frequent definition of the microscopic energy considers the experimental energy Eexp (Z, A), when available, instead of the total theoretical binding energy Etot and can be expressed as follows: exp = Eexp (Z, A) − Emac (Z, A, β) Emic Eshell (Z, A, β) + Epair (Z, A, β) = Es+p (Z, A, β).

Duflo-Zuker mass formula based on shell model

Another microscopically rooted approach worth considering is the development by Duflo and Zuker [19] of a mass formula based on the shell model. The nuclear Hamiltonian is separated into a monopole term and a residual multipole term: the monopole term is responsible for saturation and single-particle properties, and fitted phenomenologically; multipole part is derived from realistic interactions. The latest version of the mass formula defined in terms of 10 free parameters reproduces the 2149 Z, N ≥ 8 experimental masses with an outstanding rms uncertainty of 0.564 MeV. A simple 120-lines FORTRAN subroutine in the masses/duflozuker96.f file computes the mass of any nucleus with ease, especially when relevant data are not available in the above-mentioned tables.

F.

Shell correction energy

Microscopic corrections to the binding energy are quantities of fundamental importance in the derivation of many physical properties affected by shell, pairing or deformation effects. However, great confusion exists in the literature concerning what is referred to as the shell correction energy. Different definitions exist. The most common one defines the various microscopic corrections (e.g. Ref. [16]) as follows: Total nuclear binding energy is written as Etot (Z, A, β) = Emac (Z, A, β) + Es+p (Z, A, β),

(9)

exp = Es+p (Z, A, β). Should the mass formula be exact, Emic exp Emic is often referred to incorrectly as an “experimental” microscopic correction, but this term remains modeldependent through the use of the model-dependent Emac quantity. Each mass model calls for specific theoretical calculations to estimate the macroscopic part, as well as the shell, pairing and deformation energies. The most common approaches to derive the macroscopic part are the Finite-Range Droplet or Liquid Drop model [16], the Thomas-Fermi approach [30], or the Extended-ThomasFermi approach [31]. Depending on the approach followed to derive the smooth macroscopic part of the binding energy and the parameter set adopted for the macroscopic part, the microscopic corrections can take relatively different values. The FRDM microscopic correction Emic can be found in the masses/mass-frdm95.dat file. When shell, pairing, or deformation corrections are introduced to a given quantity (for example the nuclear level density) by means of the corresponding energy correction, special attention should be paid to the prescription adopted. Depending on the level density formula considered, the “microscopic” correction to the level density a-parameter can include very different effects, so that different energy corrections should be considered (see Ref. [8] for more details).

(5)

where β characterizes the nuclear shape at equilibrium, i.e., the shape which minimizes the total binding energy. Es+p = Eshell + Epair is the shell-plus-pairing correction energy (we define the pairing correction for eveneven nuclei and do not consider the odd-even effect also

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G.

Uncertainties in the mass predictions

The uncertainties associated with the mass predictions come from uncertainties affecting either the model parameters or the model itself. Within the macroscopicmicroscopic model, the deviations related to the model

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parameters are known to be potentially very large. An example comes from the well-known mass models of Hilf et al. [32] and von Groote et al. [33] based exactly on the same droplet description, but making use of two possible minimizations relative to the parameter set. Their respective parameters led to very similar rms deviations from experimental masses (typically 750 keV on the 1440 experimental masses known at that time), but led to very different predictions for neutron-rich nuclei. More specifically, the two fits agree within 1 MeV for stable nuclei and within 6 MeV for known unstable neutron-rich nuclei. They show complete disagreement for unknown masses. Mass differences as large as 35 MeV appear for nuclei close to the neutron drip line predicted by Hilf et al. [32]. The location of the neutron drip line predicted by the two liquid-drop model calculations is also quite different. However, since then, many new experimental data have been obtained far away from stability leading more certainly to a significant reduction of such differences in mass predictions, at least if new fits are performed. Such an updated analysis of the parameter sensitivity is unfortunately not available. As far as the HFB mass models are concerned, they have been shown to be relatively stable with respect to changes of the force parameters, at least if the force is fitted to essentially all experimental mass data [18]. Globally, the extrapolations out to the neutron dripline of the different HFB mass models are essentially equivalent, with deviations up to typically 5 MeV being found. However, major differences between the droplet-like, the Duflo-Zuker and HFB models still exist, not only in the prediction of nuclear masses located some distance from the experimentally known region, but also in the strength of the shell and pairing effects. The model uncertainties still affecting the nuclear mass predictions are therefore well illustrated by the mass differences between the three models recommended as shown in Figs. 2-3. Despite the close similarity in the quality of the fits to the data given by these different models, large differences emerge as the neutron-drip line is approached.

H.

Relative isotopic abundances

For practical applications, the relative isotopic abundances (expressed in percent) for each stable nucleus found on earth are given in the masses/abundance.dat file. The data originates from the latest update of the Nuclear Wallet Cards, as retrieved from Brookhaven National Laboratory [34].

I.

Future developments and outlook

Due to the importance of further improving predictions of nuclear structure properties, microscopic mass models need to be further developed. Since the construction of

R. Capote et al.

the HFB-14 mass model, two major breakthroughs have been accomplished. The first one concerns the new Skyrme-HFB nuclearmass model, HFB-17 [35], in which the contact pairing force is constructed from microscopic pairing gaps of symmetric nuclear matter and neutron matter calculated from realistic two- and three-body forces, with mediumpolarization effects included. In this way the rms deviation with respect to essentially all the 2149 available mass data has been reduced for the first time with a mean-field model, to 0.581 MeV. A second major accomplishment has been made with the first Gogny-HFB nuclear-mass model based on the newly developed D1M interaction [36]. The rms deviation with respect to essentially all the available mass data has been reduced from typically a few MeV with previous Gogny interactions to less than 0.8 MeV. Furthermore, for the first time, the mass formula takes an explicit and self-consistent account of all the quadrupole correlations affecting the binding energy. The quadrupole corrections are estimated microscopically on the basis of a 5-dimensional collective Hamiltonian with the same D1M interaction. The D1M interaction, like the Skyrme interactions at the origin of the HFB-17 mass model, has also been constrained on microscopic calculations of neutron matter and symmetric nuclear matter. In contrast to HFB-14, both the HFB-17 and D1M mass models still need to be tested with respect to nuclear level densities and fission barriers.

III.

DISCRETE LEVEL SCHEMES

Nuclear reaction and statistical model calculations require complete knowledge of nuclear level schemes for specifying all possible outgoing reaction channels and for calculating partial (isomeric) cross sections. Knowledge of discrete levels is also important for adjusting level densities, which replace unknown discrete level schemes at higher excitation energies where the discrete levels become too dense to be resolved. For this purpose the estimate of the completeness of the level scheme is of crucial importance. The term “completeness ” means that up to a certain excitation energy (usually called cut-off energy) all discrete levels in a given nucleus are observed and are characterized by unique energy, spin and parity values. The knowledge of particle and γ-ray decay branchings from each level is also required, especially when population of isomeric states is of interest. Complete level schemes can be obtained only from complete spectroscopy using nonselective reactions. Purely statistical reactions, such as (n, n γ) reaction and averaged resonance capture (n, γ), are particularly suitable due to their non-selective excitation mechanism and completeness of information obtained with the rich arsenal of gamma-ray spectroscopy [37]. However, the vast majority of nuclei cannot be experimentally studied; hence the degree of knowledge of the experimentally

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NUCLEAR DATA SHEETS

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30

M(FRDM)-M(HFB-14)

25

M [MeV]

20 15 10 5 0 -5 -10 0

40

80

120

160

16

200

12

8

4

0

S [MeV]

N

n

FIG. 2: Differences between the FRDM and HFB-14 mass predictions as a function of neutron number N (left) and the HFB-14 neutron separation energy Sn (right).

determined discrete level schemes varies widely throughout the nuclear chart. While this knowledge is compiled and available from the Evaluated Nuclear Structure Data File (ENSDF) [38], its format is too cumbersome to be directly used as input data in the modelling of nuclear reactions. The original purpose of the ENSDF was to serve as a database source for the preparation of Nuclear Data Sheets; extracting data from ENSDF is by no means simple. In addition, ENSDF contains a great deal of information in a format that can not be easily decoded by computer codes. Therefore, the data have to be extracted and reformatted to prepare the discretelevel input for nuclear reaction calculations. The first attempt to create a sublibrary of discrete levels for applications was undertaken in the RIPL-1 project [3]–[5]. The RIPL-1 starter file [6], however, suffered from a number of deficiencies related to the use of the retrieval code NUDAT and a too restrictive format. Therefore, a new, extended Discrete Level Schemes (DLS) sublibrary was created during the RIPL-2 project [10] and formatted following the recommendations of the RIPL2 co-ordination meetings [7, 8, 9]. Finally, the RIPL-3 DLS sublibrary [14] was released based on the October 2007 update of ENSDF as a source of data. The RIPL-3 DLS sublibrary used the same format as in the RIPL2 database. In the following sections we review RIPL-2 and RIPL-3 DLS sublibraries.

10

M(FRDM)-M(DZ) M [MeV]

5 0 -5

-10 -15

M(DZ)-M(HFB-14)

M [MeV]

40 30 20 10 0 -10 0

50

100

150

200

N FIG. 3: Differences between the FRDM and Duflo-Zuker masses (top panel) and Duflo-Zuker and HFB-14 masses (bottom panel) as a function of neutron number N .

A.

RIPL-2 sublibrary

The RIPL-2 Discrete Level Scheme sublibrary was created by the Budapest group using the ENSDF-II data set of 1998 [38, 39] as a source. This data set is a slight mod-

3115


NUCLEAR DATA SHEETS 6

Number

Spin ranges from gamma transitions

3560

Spins from spin distribution

3551

Temperature T (MeV)

TABLE I: Type and number of spin estimations. Type of method

R. Capote et al.

Spins chosen from a list using spin distribution 6280

ification of the original ENSDF [38] library and contains explicit final states for gamma transitions. In the RIPL2 version of the DLS sublibrary there are no limitations on the number of levels or number of transitions, which were identified as deficiencies of the RIPL-1 file [6]. The advantage of this data set over the original Evaluated Nuclear Structure Data File (ENSDF) was direct connections between the gamma transitions and their initial and final states. The 1998 ENSDF-II CD-ROM contains 2637 data sets of nuclear decay schemes (606254 rows of data). There are 2546 nuclear decay schemes, with at least one known level, that cover the range A=1–266, Z=0–109. These 2546 level schemes, called the basic set, were processed to obtain the DLS sublibrary files. The basic set contains 113346 levels out of which 8554 have unknown level energies. These are marked with +X, +Y ,... an ENSDF notation also used in the RIPL-2 DLS sublibrary files. A total number of 12956 spins are unique, and for an additional 8708 levels, the spin and/or parity assignment is considered uncertain (parenthesis around a single spin or parity value). These spin-parity values were adopted and extracted from the ENSDF file. The basic set also reports 159323 γ-transitions between the levels. Some of the data such as level spins, parities or electron conversion coefficients were missing in the basic data set. Since their knowledge is crucial for model calculations, they have been calculated or inferred from other available data using statistical assumptions. Table I shows the number of spins that has been inferred under different assumptions. In order to calculate γ-emission intensities from nuclear reactions, the Internal Conversion Coefficients (ICCs) must be known for all electromagnetic transitions from a given level. Since only some of them are available in the ENSDF, the missing values have been calculated and included in the RIPL-2 file. This brings the number of ICCs in RIPL-2 to 92634 compared to 21595 given in the ENSDF. One of the most difficult tasks was the determination of the maximum level number Nmax , and the corresponding cut-off energy Emax , up to which a level scheme is supposed to be complete. A new fitting method, that eliminates the deficiencies of the earlier fitting procedure [6, 40], has been developed as outlined in the next section.

5 4 3 2 1 0 0

50

100

150

200

250

Mass number

FIG. 4: Nuclear temperature T as a function of mass number A for nuclei near the valley of stability. 1.

Constant-temperature fit parameters

It has been noted [41]–[43] that the observed energy dependence of the cumulative number of levels N (E) as a function of excitation energy E can be described rather well by the empirical function N (E) = exp [(E − E0 )/T ] ,

(10)

where E0 and T are free parameters determined by fitting corresponding data. N (E) is related to the level density ρtot (E) by the equation: ρtot (E) =

1 dN = exp [(E − E0 )/T ] , dE T

(11)

in which the parameter T corresponds simply to a nuclear temperature. Since the value of this parameter is assumed to be constant over the energy range considered, Eq. (11) is called the constant temperature level-density model. When we are using Eq. (10) to fit the low-lying discrete levels we usually skip the first few levels (especially in even-even nuclei), so the fit goes from Nmin to Nmax level number. In each iteration of the fitting process the Nmax and Nmin level numbers for each nucleus are varied independently, in such a way that the fit approaches the part of the level scheme that can be well described by a constant temperature formula Eq. (10). Ten iterations were generally enough to minimize the global χ2 and determine the Nmax and Nmin values. The resulting Nmax values have been identified as the level number up to which the level schemes can be considered complete. The temperature T as a function of the mass number A was obtained from a global least-squares fit for 625 nuclei, which were defined as ±4 mass-unit band around the valley of stability. An additional 503 nuclei that were not used in the global fit have been fitted using the above determined T (A) function in order to estimate Nmax values. The results for the 2546 nuclei are reported in the RIPL-2 DLS sublibrary [10]. The nuclear temperature function T (A) obtained is shown in Fig. 4 as a function of the mass number A. A comparison of this function with the temperatures obtained from

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NUCLEAR DATA SHEETS

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TABLE II: Statistics of energy differences. Frequency of cases Energy range [keV] 289 5 ≤ E < 10 116 10 ≤ E < 20 50 20 ≤ E < 50 21 50 ≤ E < 100 33 100 ≤ E Sum 509

• Additional energy cut-offs Ec , corresponding to the energy of the highest level with unique spin and parity assignment, have been determined for all nuclei on the basis of the ENSDF data alone.

FIG. 5: Shift parameter E0 as a function of mass number A.

• Data retrieved from the ENSDF have been extended in order to obtain unique data values as required for reaction calculations. To this end, unique spin and parity values have been generated from known data up to the cut-off energy Emax . Internal conversion coefficients (ICC) for electromagnetic transitions have been calculated using unique spin and parity values if they were not given in the ENSDF. The extension has included the complete (γ and particle emission) decay pattern of the levels if known from experiments.

the Gilbert-Cameron fits by the Bombay group in RIPL1 [7] is favorable, although there are differences at the shell closures and in the transitional regions. The values of the corresponding shift parameter E0 as a function of mass are shown in Fig. 5. We can see that E0 is positive for even-even and negative for odd-odd nuclei, and fluctuates between positive and negative numbers for odd nuclei. These patterns could be explained by subtracting the pairing energy from the excitation energy E in Eqs. (10) and (11), as done in level-density studies. The results of good fits are plotted for each mass number and can be reviewed at the RIPL-3 homepage. It should be noted that the present fitting procedure differs significantly from the method usually used to estimate the temperature in nuclear level density calculations (see chapter VI, in particular, the section on Composite Gilbert-Cameron level density VI A 3). For the latter, neutron resonance information at the binding energy (if available) was used to further constrain the fits of level density parameters, including the temperature. No such information is used here; therefore, any intercomparison of fitted parameters T and E0 to those derived in level-density studies should be undertaken with care.

• Data have been tested for internal inconsistencies, which could be due to misprints, logical errors, or use of improper algorithms. • For nuclei that had at least 10 levels with spin assignment below Nmax , the spin cutoff factors have been calculated from the spin assignments provided in ENSDF. • Missing spins and parities have been inferred up to Nmax .

3. 2.

Construction of the Discrete Level Scheme

RIPL-2 consistency tests

The following tests have been performed with the RIPL-2 DLS sublibrary:

The major steps in the construction of the Discrete Level Scheme (DLS) sublibrary are • Adopted or available discrete nuclear levels and γray transitions from ENSDF have been retrieved and converted into RIPL format using FORTRAN programs developed within the CRP. • Cut-off energies Emax , and the corresponding cumulative numbers of levels Nmax , have been determined from a constant-temperature fitting procedure to the staircase plots for nuclei with at least 20 known levels as described in the previous section. 3117

• The difference in energy between initial and final level energies has been compared with the deexciting γ-ray energy corrected for the nuclear recoil. The number of cases with energy differences larger then 5 keV are listed in Table II. While small differences might well be within the quoted uncertainties, the larger ones indicate internal inconsistencies in the ENSDF library. Therefore, they have been collected and transferred to the ENSDF manager at the National Nuclear Data Center at Brookhaven. As a result, some of the revealed problems have been corrected in the original ENSDF data set.


NUCLEAR DATA SHEETS

R. Capote et al.

TABLE III: Cases of very large multipole order. Symbol Ei [keV] Jπi [] Jπf [] Comment 58 53 53 67 90

Mn

6+ Spin selection to be improved

3040.400 19/2− 9/2− M5 transition from isomer

Fe

3040.400 19/2− 7/2− E6 transition from isomer

Zn

2434.930 11/2− 1/2− M5 non-isomer

Y

117 123 125 133 184 192 206 207 202 204 211

b

1+

Fe

113

a

728.060

682.030

7+

2− E5 transition from isomer

Cd

263.590 11/2− 1/2+ E5 transition from isomer

Sn


Te


Te


Ba

288.247 11/2− 1/2+ E5 transition from isomer +

−

Re

188.010

8

3

Ir

155.160

9+

4+ E5 transition from isomera

Tl

2643.110

12−

6+ No multipolarity in ENSDF

E5 transition from isomer

−

1/2+ E5 transition from isomer

Tl

1348.100 11/2

Pb

2169.830

9−

4+ E5 transition from isomer

Pb

2185.790

9−

4+ E5 transition from isomer

Po

+

1462.000 25/2

FIG. 6: Distribution of relative differences between calculated and ENSDF provided ICCs.

7/2− Spin selection to be improvedb

The level energy is changed in the recent evaluation.

The final spin is given in the recent evaluation as 17/2+ ; this transition is E4 isomeric.

• Deviations from 100% of the sum of decay probabilities from each level have been reviewed. There are cases in which the decay probabilities sum up to much less than 100%. Typically, they correspond to one known partial width and known total width for a level when the ENSDF evaluators were not able to distribute the difference among other possible decay modes such as neutron or α. There are also cases, when the sum of decay probabilities is substantially larger than 100%, which correspond to β-delayed neutron or proton decay. • Multipole orders of γ-ray transitions have been checked yielding 17 cases with unusually high multipolarities (see Table III). Some of them are actually real, while others result from deficiencies in the original ENSDF file or from the spin assignment made in the present work since the unique spin selection makes use only of the final state spin. Cases in which γ-cascade between two states with known spins involves an intermediate state of unknown spin were not considered. Fortunately, there are not too many such cases and they can be treated individually. • 21595 calculated ICC values were compared with those given in the ENSDF file. The distribution of relative differences ((ICCcal − ICCexp )/ICCexp ) is shown in Fig. 6. Out of the 21595 relative differences, about half (10493) are within 1% and follow a normal distribution resulting partly from round3118

ing errors. About 6000 relative differences form a double peak structure centered at −0.05. To pinpoint the reason for these was far from trivial. When analyzing absolute and relative differences it turned out that part of this structure is due to rounding or perhaps to the differences in the ICC theory and/or measured values for small ICC values. The remaining 5000 values show very large differences (with very low frequency) due to various reasons. One of them is the rule used for calculating ICCs in the case of unknown mixing ratios. The present ICC values for M1+E2 mixed transitions were calculated assuming pure E2 in even-even nuclei and pure M1 otherwise, while ENSDF evaluations may involve additional efforts to quantify the mixing ratios of these transitions in order to calculate more precise ICCs (although such efforts are heavily dependent on the information available). In summary, based on the above comparisons, it can be concluded that the applied subroutine provides satisfactory ICC values for the current purpose, although the origin of the differences between present calculations and the ENSDF values should be further investigated. • Electromagnetic transition rates have been successfully tested – all of the calculated rates satisfy the Recommended Upper Limit (RUL) [44]. • Formal correctness of the DLS sublibrary (z000.dat, z001.dat, ..., z118.dat files in directory levels/ ) has been tested and no formatting problems have been encountered. • A new method to extract nuclear temperature T has been developed (see Annex 2.E (RIPL-2) [10]). The temperatures provided by this procedure have been compared to the T (A) function obtained in the global fit. The results are shown in Fig. 7. Most of the T values obtained with the new method are close to the T (A) curve. Only a few percent of cases


NUCLEAR DATA SHEETS

R. Capote et al. 2.

FIG. 7: Comparison of temperature values obtained from two independent methods. Red points are obtained using the method described in Annex 2.E (RIPL-2) [10]; blue points (with error bars) are the results of the global fit procedure.

are discrepant due to bad fits and/or a low number of available levels.

B.

RIPL-3 sublibrary

1.

RIPL-3 2005 upgrade

A subsequent second update of DLS sublibrary was undertaken in late 2007, just before the final RIPL-3 research coordination meeting was held. Minor modifications of the software were necessary to handle some new features found in the ENSDF library. The RIPL-3 levels/ directory contains 117 files (one for each element except Z = 111 and 117) with all known level schemes available from ENSDF in 2007. All excited levels that include +X or +Y notations were eliminated from the DLS sublibrary in this update, as they can not be used in nuclear reaction calculations. These files are arranged and preprocessed into an easy-to-read format for nuclear reaction codes. During preprocessing all missing spins up to the Nmax level were inferred uniquely for each level from spin distributions extracted from the existing data. Electromagnetic and γ-ray decay probabilities were estimated. Missing internal conversion coefficients (ICC) were calculated using the inferred or existing spin information, in which existing multipole mixing ratios were also taken into account. Particle decay modes are also given whenever measured. In all cases, the total decay probability was normalized to unity, including particle decay channels.

3.

The first update of the RIPL-2 DLS sublibrary was undertaken in 2005. Because updates of the Table of Isotopes were slow, we had to revert to the original ENSDF data set that is regularly maintained by the International Network of Nuclear Structure and Decay Data Evaluators and distributed by the Nuclear Data Center at Brookhaven National Laboratory, USA. The aim was to produce an easy to use set of programs which can use ENSDF and create the DLS sublibrary in the format that was defined in RIPL-2 [10]. The developed programs were also intended to be used in future maintenance releases from ENSDF updates to be automatically produced at the IAEA. The RIPL-2 DLS sublibrary was created from an ENSDF data set that was used to update the Table of Isotopes in 1998 [39]. In the past few years the ENSDF data sets were updated considerably. However, most of its update seems to stem from a large amount of experimental results related to high-spin nuclear physics. A quite common feature of these new levels is the lack of connection of the deformed bands to the ground state, which makes these updates not suitable for the purpose of a low-energy discrete level libraries to be used in nuclear reaction calculations. Setup of the library generating codes and files [10] for an easy upgrade of the discrete level sublibrary was undertaken at the IAEA Nuclear Data Section (NDS) in 2005. The NDS setup was used to produce the first RIPL-3 discrete level scheme sublibrary [14] using the October 2004 update of ENSDF as a source of data.

RIPL-3 2007 upgrade

RIPL-3 consistency tests

The same consistency tests used in the validation of the RIPL-2 DLS sublibrary were employed to verify the consistency of the newest RIPL-3 DLS release. Direct comparison of the two datasets has been undertaken. 323 new data sets have appeared in the version of ENSDF dated October 2007 – most of them only possess a well-defined ground state and are not important for the DLS sublibrary. Only a handful have more than nine levels that appear or disappear from the new evaluation. Changes in number of levels common in both evaluations can be represented as a histogram plot (see Fig. 8 (left)). Most of the decreases are due to reductions in the number of uncertain levels, but in some cases the ENSDF evaluators decided to include fewer levels, e.g. 87 Kr. More new levels were successfully placed in the decay schemes over the past six years than the numbers defined as uncertain levels. Figure 8 shows how the maximum level number Nmax (left) and corresponding level energy Emax (right) has changed. No change has occurred in the majority of cases (1942 out of 2547); the Emax values have increased for about 400 nuclei, as a consequence of the higher number of new levels. A comparison of the nuclear temperature function T (A) is shown in Fig. 9, obtained from fits to the cumulative number of levels of a selected set of nuclei for both the 2007 and the 2005 data sets. The temperature fits are reasonably stable and do not change significantly for the new dataset. Most changes occurred near to the closed shells, and modifications to the locations of the

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NUCLEAR DATA SHEETS

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FIG. 8: Changes in the number of levels (left) and Emax (right) in 2005 DLS sublibrary relative to RIPL-2 [10].

maxima occurred closer to the magic numbers for the newer data. The main purpose of the temperature fit is to provide cut-off energies Emax and Ec estimating the completeness of levels and spins in each level scheme. Furthermore, the nuclear temperature inferred from the discrete levels is also provided as a consistency check of the estimated cut-off energy. In our experience we found that the estimate of the completeness of the scheme Nmax is too optimistic, especially for odd and odd-odd nuclei. Fifty percent decrease in the estimated Nmax is often acceptable, so the tabulated Nmax in the DLS sublibrary is recommended to be taken as the upper limit. This observation has been confirmed by independent Nmax estimates obtained in level density studies (see chapter VI).

4.

C.

Maintenance

The developed ENSDF processing codes are scheduled to be used annually at the IAEA Nuclear Data Section to update the DLS sublibrary starting from the updated ENSDF evaluations. There are also on-going efforts led by Firestone at Lawrence Berkeley National Laboratory, USA, to use the IAEA EGAF database [47] (based on measured capture cross sections in a neutron thermal beam at the Budapest reactor) to improve the accuracy of emission probabilities and spin assignments contained in the DLS sublibrary by applying statistical modelling methods. Such improvement is relevant for a better description of the very important neutron capture reaction and for calculations of isomer production cross sections and cross sections on isomeric states.

Validation of the RIPL-3 sublibrary

The RIPL-2 and RIPL-3 Discrete Level Scheme sublibraries have been tested in a huge number of reaction calculations using statistical model codes EMPIRE [45] and TALYS [46] as both codes use the DLS sublibrary as the only input source of all discrete level information and transition probabilities. These routine tests helped to disclose and correct some additional deficiencies. The reliability of the database is considered to be high.

FIG. 9: Comparison of the nuclear temperature fits in the RIPL-3 2007 DLS sublibrary (red) relative to the 2005 DLS sublibrary (blue).

IV.

AVERAGE NEUTRON RESONANCE PARAMETERS

Neutron resonance properties required as input for nuclear reaction calculations and nuclear data evaluation are neutron strength functions, average radiative widths and the average spacing of resonances. These quantities are generally obtained from the analysis of parameter sets for the resolved resonances. The experimental resolution and sensitivity limits create incomplete (missing resonances) or distorted (errors on width determination) information on the resonance parameters. Therefore, the average widths and resonance spacings cannot be directly deduced from available resonance sequences, and should always be estimated by taking into account missing resonances. Various methods for statistical analysis of missing resonances have been developed, and most of them were applied to evaluate average resonance parameters during the RIPL project. Advantages and shortcomings of such methods are briefly discussed below in order to obtain some objective estimation of the accuracy of the recommended parameters. Before describing the statistical methods for resonance analysis, it is necessary to point out that such methods

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NUCLEAR DATA SHEETS

work correctly only when applied to samples of neutron resonances that are undistorted or only slightly distorted. The results become less and less accurate for increasingly incomplete or more distorted samples. Undistorted sample cases are very rare. Therefore, to improve the sample, some selection of analyzed resonances should be applied to reduce the relative contribution of missing or spurious resonances. Thus, an evaluation of average resonance properties is an iterative procedure in which one tries to reach convergence of results from various approaches. A.

Evaluation methods

The neutron strength functions for a given orbital angular momentum l are defined by the relationship Sl =

gΓln 1 = gr Γlnr , (2l + 1)Dl (2l + 1)ΔE r

(12)

where the summation is performed over N neutron resonances in the energy interval ΔE; Γlnr are the reduced neutron widths of resonances; gr = 12 (2Jr + 1)(2I0 + 1) is the statistical weight factor that depends on the angular momentum Jr of a resonance and the spin I0 of the target nucleus, and Dl is the average resonance spacing defined as Dl = ΔE/(N − 1).

(13)

The reliable identification of s- and p-wave resonances is most important for an accurate evaluation of the average parameters. If such an identification can be made, the neutron strength functions can be simply evaluated from a linear approximation to the cumulative sum of the products gr Γlnr . Departure from linearity may indicate missing resonances. The relative uncertainty of the evaluation can be defined from the equation δSl (14) = 2/N, Sl which is based on an asymptotic estimate for the variance of the sum of neutron widths distributed in accordance with the Porter-Thomas law [48]. Dominant contributions to the sum in Eq. (12) for swave neutrons are given by resonances with large neutron widths; missing weak resonances or admixtures of p-wave resonances have a rather small effect on the evaluation of the strength functions. The situation is not so favorable for p-wave neutrons, for which the strength function can be strongly distorted by any admixture of incorrectly identified s-wave resonances. This is the main reason that the relative accuracies of p-wave strength functions for many nuclei are much lower than for the s-waves. For a rather full set of resonances the relative statistical uncertainty of the resonance spacing can be determined by the relationship √ δDl 1 0.45 ln N + 2.18 ≈ , (15) = Dl N N

R. Capote et al.

which was obtained by Dyson and Mehta for the Gaussian orthogonal ensemble [49, 50]. Missing resonances in the analyzed set result in an error that essentially exceeds the statistical error. Thus, an estimation of the missing resonances is crucial for an accurate evaluation of the average resonance spacings. Three approaches were developed to account statistically for the missing or erroneously identified resonances: (1) methods that exploit the statistics of level spacings; (2) methods based on the fit of the reduced neutron width distribution by the Porter-Thomas law; and (3) methods that use combined simulation of the level and width statistics. Both the advantages and weaknesses of various methods have been broadly discussed in Refs. [51]– [57]. Some new developments related to the third type of method were proposed recently [58]–[61]. The simplest method of resonance analysis is the staircase plot of the cumulative number N (E) of resonances as a function of energy. It is usually assumed that at low energies there are no missing resonances, and a linear approximation of this part of the plot gives a direct estimation of Dl . A variation of this method is the approach based on Δ3 statistics given by Dyson and Mehta [49, 50]. The best fit of N (E) is determined through a leastsquares study of the parameter ΔE 1 2 Δ = min [N (E) − AE − B] dE , (16) ΔE 0 where A and B are the fitting parameters. For a complete set of levels, the average value Δ3 and the standard uncertainty σ of this estimator are defined by the relationships: 1 Δ3 = Δ = 2 (ln N − 0.0687), (17) π 7 1 4π 2 + = 0.11 . (18) σΔ = 2 π 45 24 The absence of levels or presence of spurious levels from another sequence affects Δ3 . Therefore, if the fitted value of Δ satisfies the condition Δ3 − σΔ < Δ < Δ3 + σΔ , the analyzed set of resonances may be considered as a pure and complete set. Unfortunately, Δ3 statistics provide no means of correcting an inadequate set of resonance parameters. Besides, one finds in practice that the Δ3 test criteria are often satisfied for samples that are known to be neither pure nor complete [55, 57]. In contrast to the spacing distribution, the neutron width distribution is only slightly affected by missing or spurious weak resonances. The upper part of the Porter-Thomas distribution, corresponding to strong resonances, can be regarded as virtually unperturbed. The number of resonances that have the reduced neutron width above a given value is described by the function ∞ exp(−x/2)dx √ N (Γ) = N0 2πx Γ/Γ

(19) = N0 1 − erf Γ/2Γ + ... .

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Thus, by fitting the corresponding distribution of neutron widths with a maximum likelihood approach, we can find both the average reduced neutron width Γ and total number of resonances N0 in the considered energy interval. Various versions of this method were developed to take into account energy variations of the measurement threshold and other experimental conditions [57, 58]. The third class of methods attempts to account simultaneously for limitations imposed on estimations of a mean width and resonance spacing by the Wigner and Porter-Thomas laws. Simulations of the neutron cross sections by the Monte Carlo method that take into account the experimental resolution and other conditions have been undertaken. Analytical treatments that could replace the Monte Carlo simulation for some calculations were discussed in Ref. [58]. The conclusion that can be formulated on the basis of the resonance parameter analyses performed by different groups is that none of the methods guarantees an unambiguous identification of missed or spurious resonances. Critical analyses of the experimental conditions and approaches used to obtain individual resonance parameters are very important in many cases. Priority in average parameter evaluations should be given to the quality of the selected resonance set rather than to the total number of resonances considered [58, 59]. To show the main problems of the average parameter estimation, one can consider, as an example, the analysis of the resonance data for 235 U. The present-day version of resonance parameters in the energy range up to 2.25 keV was obtained from a least-squares fit of a large set of experimental data using the Reich-Moore formalism in the code SAMMY [62]. The energy dependencies of the cumulative number of resonances N and the corresponding sum of reduced neutron widths are shown in Fig. 10 for the energies below 500 eV. The results of the Porter-Thomas estimation of missing resonances based on Eq. (19) are shown in the upper panel for the energy intervals from 0 up to maximum energy of 100, 200, and 500 eV, respectively. A deviation of the cumulative number of resonances N from a linear dependence, obvious from the middle panel, indicates directly that there is an increasing missing number of resonances with an increase of the energy interval. A growth of discrepancies with an increase of the interval, particularly a large value of the χ2 for the interval of 200 eV, indicates that an estimation of missing resonances is not quite reliable even for this interval. So, only the energy range below 100 eV is suitable for the accurate determination of the average resonance spacing. On the other hand, the sum of reduced neutron widths determining the strength function in Eq. (12) depends much less on the missing of weak resonances, and a linear dependence of the sum is observed for very broad energy intervals (see the bottom panel of Fig. 10 in the previous page). Local deviations from the linear increase indicate some structural effects in the distribution of neutron resonances. The above conclusion that a relatively small energy

R. Capote et al.

interval should be used for an acceptable and reasonable evaluation of resonance spacings had been confirmed by many authors of previous resonance analyses [54]–[62] and it affects crucially the estimated uncertainties of average resonance parameters. Only a rather small number of nuclei have accuracies of the resonance spacing better than 10%. Beside 235 U, the best examples of such nuclei are 238 U, where the relative error of the recommended D0 is better than 2%, and 239 Pu, 240 Pu, and 242 Pu, for which such errors are about 5% [58, 60].

B.

Roadmap from RIPL-1 to RIPL-3

Complete tables of average resonance parameters provided by the Beijing, Bologna and Obninsk [63] groups were collected during the course of the RIPL-1 project [6].

FIG. 10: The Porter-Thomas analysis based on Eq. (19) of the cumulative number of resonances N (solid blue line) is shown in the upper panel for three energy intervals: up to 500 eV (top plot), up to 200 eV (middle plot), and up to 100 eV (bottom plot); the energy dependence of N (solid blue line) is shown in the middle panel. The sum of reduced neutron widths (solid blue line) as a function of the incident neutron energy is shown in the bottom panel. Dashed black lines are used for the results of the fit in all panels. Derived average neutron properties N0 , D0 (eV) and S0 x10−4 are shown in the legends. The observed number of resonances Nres and the χ2 of the fit are also shown in the upper panel.

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All these parameters are mainly based on the analysis of the resolved resonance parameters presented in the wellknown BNL compilation [64, 65]. Despite the common base, many discrepancies were found between the average parameter estimations. These discrepancies were rather large when compared to the parameter uncertainties, especially for cases with less than 20 resonances. After consideration of the existing discrepancies, the Beijing and Obninsk groups re-analyzed some of their previous evaluations, and prepared updated versions of the average resonance parameters. Agreement between the updated parameters has been significantly improved for most of the nuclei, although the uncertainties quoted for the Obninsk evaluations were systematically higher than the uncertainties given in the original BNL evaluations [64, 65] and the revised Beijing data [6]. This difference was shown to be mainly related to the reduction of the energy interval that the Obninsk group used to improve the quality of the analyzed sets of resonances in accord with the statistical methods described above. Taking into account substantial differences between the uncertainties obtained by the various methods, those of the Obninsk group seem more reliable than others. Accordingly, the Obninsk evaluation of the average resonance parameters was included in RIPL-1 as the recommended file, and the Beijing evaluation as the alternative [6]. The Minsk evaluation of the average resonance parameters for the actinides was also included as an alternative file, and the additional compilations of the average resonance spacings (mengoni gc.dat and iljinov gc.dat files) were included in Segment 5 of the RIPL-1 Starter File [6], because of their relevance to nuclear level densities. The recommended data sets have been extended to include the average parameters for p-wave neutron resonances along with the s-wave resonance parameters considered previously. Although the accuracy of the data for p-wave resonances is certainly not as good as for swave resonances, such improved data are important in the optical and statistical models. A reliable separation of p-wave resonances from the background of stronger s-wave resonances plays a crucial role in estimating the average resonance parameters. Thus, trustworthy results for the p-wave resonance parameters can only be obtained under a simultaneous analysis of the resolved resonance data for both s- and p-wave neutrons. Such an analysis for nuclei included in the RIPL-1 list of recommended parameters was performed by the Obninsk group [66], and the corresponding resonance spacings D1 , the neutron strength functions S1 , and the average radiative widths of p-wave resonances were added to the RIPL-2 database [10]. Another task is to produce a complete list of recommended parameters. In the original BNL compilation [64, 65], the average resonance parameters were obtained for about 230 nuclei; Belanova et al. [63] compiled the resonance spacings for 264 nuclei; Iljinov et al. [67] compiled data for 284 nuclei; the Beijing compilation [3] re-

R. Capote et al.

FIG. 11: Ratios of the RIPL-2 spacings for the s-wave resonances to the spacings recommended by Mughabghab [70].

ports data for 344 nuclei; however, the recommended RIPL-1 file is limited to only 281 nuclei [6]. A careful review of the Beijing compilation shows that 35 out of 344 nuclei contain no data quantifying the neutron strength functions and a number of resonances are set to zero. This observation indicates that the resonance spacings for these nuclei were not obtained from the analysis of experimental data, but rather from the systematics of neighboring nuclei. Such systematics could be useful for many applications, but the results should not be mixed with the direct experimental data and therefore have been removed from the RIPL-2 database [10]. All issues related to the resonance spacing systematics that are connected with nuclear level densities are considered in chapter VI. Some skepticism can arise with respect to the recommended parameters of nuclei for which data are available for a rather small number of resonances, particularly for about 45 nuclei in RIPL-1 in which the number of resonances is equal to or less than five. Any statistical analysis of such data is doubtful. Nevertheless, we decided to include such cases in the recommended file to provide an estimate that is certainly better than nothing. So all nuclei available in the alternative files of RIPL-1 [6] but not in the recommended file were re-analyzed on the basis of the updated compilation of the resolved resonance parameters [68]. As a result, 16 nuclei were added to the list of average resonance parameters included in the RIPL-2 file [10]. This file was tested by the Brussels group in their microscopic calculations of nuclear level densities, and some misprints were corrected in the final version of the RIPL-2 file [10]. Independent analyses of the resonance parameters for about 20 nuclei have been performed by the BNL group [69]. They re-evaluated the resonance spacings, the neutron strength functions and the radiative widths for the most important fission products. In most cases, the new BNL results overlap with the RIPL-1 recommended parameters within the accepted uncertainties. The “Atlas of Neutron Resonances” that was published recently by Mughabghab [70] includes both an extensive

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NUCLEAR DATA SHEETS

list of individual resonance parameters and the updated average resonance parameters. The ratios of the s-wave resonance spacings recommended by RIPL-2 and the new Atlas are shown in Fig. 11. Although for most nuclei the ratios are close to unity, nevertheless the deviations from unity exceed two sigma uncertainty for more than 30 nuclei. Nuclides with the largest deviations are directly indicated in Fig. 11. It is necessary to note that Table 2.3 of the resonance spacings in Ref. [70] contains misprinted values for 32 S, 65 Cu and 84 Sr, which contradict subsequent author evaluations. The corrected values were used for these nuclides in Fig. 11. All differences between the RIPL-2 and Mughabghab data were carefully analyzed during the RIPL-3 project. For the nuclides of 29 Si, 31 P, 84,86 Kr, 106,110 Pd , 120,122 Sn, 206 Pb and 232 Pa, such differences can be related to new measurements of individual resonance parameters, which were not considered in the RIPL-2 evaluations [10]. References to the corresponding experimental data can be found in [70]. Another reason for the above differences is connected with a preference by Mughabghab to use the p-wave analysis results for the s-wave spacing estimation, instead of using the direct results of the s-wave resonance analyses. The case of 50 Ti is an striking example of this situation. The new Atlas [70] includes the same set of individual parameters as the old one [65]. There are three s-wave resonances with the energies 56.5, 185.6 and 307.0 keV and about 20 p-wave resonances in the energy interval below 300 keV. An estimation of the swave spacing is 125±70 keV, which was made in Ref. [65] and was adopted in RIPL-2. The p-wave resonance spacing was estimated as 10 ± 3 keV [65] and this value was conserved also in RIPL-2. In the new analysis using the same old data Mughabghab has estimated the p-wave spacing as D1 = 8.30 ± 0.53 keV [70] and has adopted the s-wave spacing (and uncertainty) in accordance with the statistical distribution of nuclear levels as D0 = 3 · D1 = 24.9 ± 1.5 keV. So, the five-fold difference between the RIPL-2 and Mughabghab evaluations of D0 in Fig. 11 is an immediate consequence of preferring more accurate indirect data over direct data of a worse quality. Unfortunately, there is no guarantee concerning a reliable identification of all p-wave resonances and a change of identification for two or three of them could change significantly the results for both D0 and D1 . Taking into account a strong contradiction between the direct and indirect estimations, it is difficult to believe the high accuracy of spacings estimated in Ref. [70]. The previous uncertainty for D1 of about 30% [65], looks more reasonable and a similar uncertainty was confirmed by our additional analysis of the available data under the RIPL-3 project. Finally, D1 = 8.5 ± 2.5 keV and D0 = 26 ± 8 keV were chosen as the updated values for RIPL-3, eliminating the discussed difference shown in Fig. 12. Many discrepancies between the RIPL-2 average parameters and the Atlas evaluations [70] relate to nuclei

R. Capote et al.

FIG. 12: Ratios of the RIPL-3 spacings for the s-wave resonances to the spacings recommended by Mughabghab [70].

for which data are available for less than 20 resonances. Fitting such data by means of Eq. (19) gives a rather bad description of the experimental data in many cases. Uncertainties of such fittings should certainly be much larger than the statistical uncertainty of Eq. (15) that relates to a pure sample of resonances. RIPL-2 and RIPL-3 uncertainties include also some additional uncertainties reflecting a spread between various approximations to analyzed data and, therefore, possible systematic uncertainties of analysis methods. All deviations between the RIPL-2 and Mughabghab estimations shown in Fig. 11 were re-analyzed on the basis of the individual resonance parameters included in the Atlas [70]. In the case of agreement between the reevaluated spacings and Atlas data, we adopted the Atlas data in the RIPL-3 files. However, for contradictory cases the results of re-evaluated spacings were adopted in RIPL-3. Ratios between the s-wave resonance spacings in RIPL-3 and the Atlas data are shown in Fig. 12. The spread of the ratios around unity is certainly reduced in this figure relative to the spread for the RIPL-2 data, but some deviations present in Fig. 11 remain in Fig. 12.

C.

RIPL-3 sublibrary of average neutron resonance parameters

The s-wave resonance spacings included in RIPL-3 are shown in Fig. 13 (left), along with the previous RIPL2 and Mughabghab data. Uncertainties for most of the data do not exceed the sizes of the symbols, showing that the results of all evaluations agree rather well for nuclei with more than 30 known resonances. On the whole, the data selected for RIPL-3 seem preferable for two reasons: (1) a more accurate choice of the energy interval used for the average parameter evaluations; and (2) more reliable estimation of quoted uncertainties. The resonance spacings of the p-wave resonances are shown in Fig. 13 (right). These data were obtained for a smaller number of nuclei than for the s-wave resonances.

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NUCLEAR DATA SHEETS

R. Capote et al.

FIG. 13: Average resonance spacings for the s-wave (left) and p-wave (right) neutron resonances included in the RIPL-2 [10], Mughabghab [70], and RIPL-3 [14] evaluations.

Nevertheless, the p-resonance spacings data are very important in testing the consistency of the results. Based on the general statistical properties of nuclear levels, the spacings of the s- and p-wave resonances should be related by D1 ≈ D0 /3, at least for nuclei far from magic nuclei. This relationship is also useful for a prediction of the s-resonance spacing when only the p-resonance data are available. Data included in RIPL-3 for the s-wave neutron strength functions are shown in Fig. 14 (top panel), and compared with the RIPL-2 and Mughabghab evaluations [70]. As a rule, there are no essential contradictions be-

tween the RIPL-3 and Mughabghab estimations. Nevertheless, it should be remarked that the RIPL-3 values were obtained from a consistent fit of the Porter-Thomas distribution of resonance widths in Eq. (19) in a carefully selected energy interval, while the Mughabghab estimations are based mainly on the analysis of widths over wide energy intervals including in some cases the unresolved resonance region. Similar results for the p-wave neutron strength functions are presented in Fig. 14 (bottom panel). The Atlas [70] includes some additional data on the p-wave strength functions that were obtained from the neutron cross-section analysis in the unresolved resonance region. However, such analyses require additional, model dependent approximations, and therefore we decided to limit the RIPL-3 resonance files to the data based on the resolved resonance parameters observed directly. The average radiative widths included in the RIPL-3 files are shown in Fig. 15 (see next page) for the s- and p-wave resonances, respectively. As before, the updated data are compared with previous evaluations. The main differences between the evaluations relate to the uncertainties of radiative widths. For many nuclei the Atlas uncertainties [70] are smaller than uncertainties adopted in RIPL-2 and RIPL-3. However, taking into account the available width uncertainties for individual resonances, the RIPL-3 uncertainties seem more reasonable, at least for the majority of analyzed nuclei. In summary, all previous evaluations presented by the Beijing, Minsk, Obninsk, and Troitsk groups have been included in the RIPL-1 database [6], and the recommended average resonance parameters were tested and revised during the RIPL-2 project [10]. After publication of the new Mughabghab compilation of resonance parameters [70], discrepant resonance data were reanalyzed and the updated average resonance parameters have been included in the RIPL-3 library [14].

FIG. 14: The s-wave (top) and p-wave neutron (bottom) strength functions included in the RIPL-2 [10], RIPL-3 [14], and Mughabghab [70] evaluations.

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NUCLEAR DATA SHEETS

R. Capote et al.

FIG. 15: The s-wave (left) and p-wave neutron (right) average radiative widths included in the RIPL-2 [10], RIPL-3 [14], and Mughabghab [70] evaluations. V.

OPTMAN08 [73, 74] – are provided. The RIPL-2 OMP sublibrary is organized in two parts:

OPTICAL MODEL

The optical model provides the basis for many theoretical analyses and evaluations of nuclear cross sections that are used in providing nuclear data for applied purposes. As well as offering a convenient tool for the calculation of reaction, elastic and (neutron) total cross sections, optical model potentials (OMPs) are widely used in quantum-mechanical pre-equilibrium and direct-reaction theory calculations, and in supplying particle transmission coefficients for Hauser-Feshbach statistical theory calculations as used in nuclear data evaluations. The importance of optical model parameterizations is made even more apparent by the worldwide diminution of experimental facilities for low-energy nuclear physics measurements and the consequent increased reliance on theoretical methods for providing nuclear data for applications. Therefore the preservation of past work aimed at describing experimental results with optical model potentials is vital for the future development of nuclear databases. Additionally, the availability and use of microscopic optical model codes is important for predicting data for target nuclei far from the line of stability, where phenomenological models might not be valid. The optical model segment of RIPL-3 is aimed at extending the library of optical model parameterizations developed under the RIPL-1 [6] and RIPL-2 [10] projects, and performing validation of the library. Notable additions for RIPL-3 concern dispersive OMPs and special emphasis on complex particle (deuteron and alpha) OMPs. The product of this activity for phenomenological optical potentials is an optical model potential sublibrary that contains reliable state-of-the-art parameterizations for the conventional optical model codes used in calculations of nuclear data. As well as preserving optical model parameterizations for future activities, the sublibrary offers a convenient means for evaluators to access a wide body of information on the optical model. Subroutines have been developed for reading and writing the data sublibrary and for creating convenient summaries of the sublibrary. Processing codes which permit direct interfacing of the sublibrary with the selected optical model codes – ECIS06 [71], SCAT2000 [72] and

• an archival file containing all potentials compiled at that time, totalling some 533 entries, and • a user file, containing a subset of 406 of the most useful optical potentials in the archival file, intended for calculating data for applications. During the RIPL-3 project only the user file has been updated. For older potentials defined at very narrow energy ranges, the interested reader is referred to the RIPL2 TECDOC [10] and corresponding archival file. The RIPL-3 user file contains 495 potentials. Details of the phenomenological OMP sublibrary and supporting codes are given below, followed by descriptions of specific additions provided by RIPL-2 and RIPL-3 partners. An overview of RIPL-2 and RIPL-3 OMP sublibraries is then given. Finally some recommendations (or guidelines) for OMP selection are given.

A.

Phenomenological optical model potential and extensions

One of the primary aims is to provide a format for optical model parameterizations that is general enough to cover all commonly used phenomenological potential representations and that is easily expanded for additional types of optical potentials. We have focused on standard Schr¨ odinger-type forms of optical model potentials, which may include dispersive relations between the imaginary and the real parts. The structure of the target nucleus can either be neglected (as done in spherical optical model), or may be considered by a different nuclear-structure model within the coupled-channel method [75] including rotational, vibrational and nonaxial (soft-rotor) models. As presently formulated, at a given incident laboratory

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NUCLEAR DATA SHEETS

energy, spherical potentials of the form V (r) = − VV fV V (r) − i WV fW V (r) d d fV S (r) + i WS fW S (r) + VS dr dr λ2 d d ˆ + 2 π [Vso fvso (r) + i Wso fwso (r)] Sˆ · L r dr dr + VCoul (r), (20) are allowed, in which VV and WV are the real and imaginary volume potential well depths, VS and WS are the real and imaginary well depths for the surface derivative term, Vso and Wso are the real and imaginary well depths for the spin-orbit potential, VCoul is the Coulomb potential for incident charged-particles, and λ2π is the pion Compton wavelength squared (λ2π = ( mπ c )2 2 fm2 ). ˆ is the scalar product of the intrinThe quantity Sˆ · L ˆ angular momentum operators, and sic Sˆ and orbital L for incident nucleons is given by l for j = l + 21 , ˆ ˆ 2 S ·L = (21) −(l + 1) for j = l − 12 , where fi (r) are geometric form factors which usually are of Woods-Saxon form [76], but other functional forms are also accepted (see more details in Ref. [10]). A similar formulation holds for coupled-channel potentials [10], but the general form of a coupled-channel OMP contains the full Thomas form for the spin-orbit part [77, 78]. Any incident particle is permitted by the format, but we have so far limited our compilation activities to incident neutrons, protons, deuterons, tritons, 3 He and 4 He particles. Our approach is to supply a general form for optical model potentials that is an extension of the representation implemented in the SCAT2000 optical model code [72] and that describes most of the parameterizations that have been commonly used in the past. Additionally, specialized formats are formulated that describe less common forms of potentials, but which offer promise as being important for applied purposes. One important recent addition is the possibility to couple to isobaric-analogue states (IAS) of the target nucleus, which are excited in (p,n) quasi-elastic scattering reactions. Both soft- and rigid-rotor potentials including IAS as coupled levels can be calculated using the OPTMAN08 code [73, 74], which is available and distributed from the RIPL-3 website. OPTMAN08 allows consideration of level-coupling schemes based either on a non-axial soft-rotor model for even-even nuclei or on a rigid-rotor model for a deformed nucleus. The softrotor model allows for the stretching of soft nuclei due to centrifugal forces during rotation. This results in a change of the equilibrium deformation for excited collective states relative to that of the ground state, and may be a critical point for reliable predictions [79]–[81] of some low-energy scattering observables based on the coupled-channel method.

R. Capote et al.

Nucleon coupled-channel potentials based on a softrotor description of the target nucleus structure have been included in the current optical model database for nuclides with mass number A from 22 to 122 [82] (RIPL 2602/5602 for neutrons/protons), and a dispersive potential based on the soft-rotor description of Zr isotopes (RIPL 609/4609 for neutrons/protons) was also added. Those OMPs incorporate nuclear Hamiltonian parameters that describe experimentally known low-lying collective levels, usually from no less than four rotational bands, including the negative-parity ones. We will review the activities undertaken in RIPL-2 and RIPL-3 to develop complex-particle (deuteron and alpha) potentials in section V B 1, and dispersive optical model potentials in section V B 2.

B.

Optical model potentials developed during RIPL-2 and RIPL-3 projects 1.

Complex-particle optical model potentials

a. Global alpha-nucleus OMP by Demetriou et al. The potential of Demetriou et al. [83] was developed during the RIPL-2 project. It corresponds to a global alpha-optical potential which takes into account the strong energy dependence and nuclear structure effects that characterize the alpha-nucleus interaction. The real part of the potential is calculated using a double-folding procedure using the M3Y effective nucleon-nucleon interaction. Three different types of imaginary potential are constructed by assuming volume or surface absorption, or by adopting the dispersion relations linking the real and imaginary parts of the OMP. In all three cases, the imaginary potentials make use of a Fermi-type parameterization introduced to describe the strong energy dependence of the imaginary depth at low energies. All three potentials make use of the HFB-14 nuclear structure properties described in chapter II. The three corresponding OMPs are constrained in order to optimize reproduction of scattering and reaction data. In the case of the OMP with a purely volume imaginary term (OMP I of Ref. [83]), this is done by fitting 19 free parameters. This number is decreased to 10 in the case of a volume plus surface imaginary potential (OMP II of Ref. [83]), with a concomitant reduction of the ambiguities in deriving the OMP from the data. This is further improved by using the dispersive relation (OMP III of Ref. [83] ). As shown in Ref. [83], the three potentials are able to reproduce equally well the bulk of experimental data on (α,γ), (α,n), (α,p) and (n,α) reactions, as well as the elastic scattering data. This good agreement with data occurs at energies both above and below the Coulomb barrier over a wide range of target nuclei. In particular, the three potentials are among the very few capable of correctly reproducing the 144 Sm(α,γ)148 Gd cross sections at sub-Coulomb energies (E ∼ 12 MeV). The three potentials are provided in the

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RIPL: Reference Input Parameter Lib... 1 0.9

Ni(α,α)

17.6

-1

10

-1

10

Cd(α,α)

2

x10

2

x10 -3

10

-3

10

Kiss+ (2006)

50

Cr(α,α)

-1

Avrigeanu+ (2009) Kumar+ (2006)

-5

10 14.6

1

x10

19.7

19.4

Gasques+ (2003)

0.4 12.8

σ/σRutherford

106

16.1

58

0.6 9.6 1 0.8 0.6

10

Y(α,α)

Kiss+ (2007)

0.8

10

R. Capote et al.

89

16.2

8.1

0.8 9.1 1

10

NUCLEAR DATA SHEETS

90

Zr(α,α)

6

x10

x10

15

2

-5

107

Ag(α,α)

5

15

x10

10

Watson+ (1971)

Bredbacka+ (1994) -3

-5

-7

10

12.8

x10

5

Ni(α,α)

10

-7

10

-9

14.6

x10

8

13.8

x10

16.4

x10

-9

62

14.4

Watson+ (1971)

-7

10 92

Mo(α,α)

19.5

8

x10

-9

10 19.4 x1010

19.5 x1010 124

Sn(α,α)

7

-11

-11

10

10

Galaviz+ (2005)

-13

30

Sn(α,α)

10

10

Fulop+ (2001)

0

112 7

x10

60

90

120

ϑc.m. [deg]

150

10

0

-13

30

60

90

120 150

ϑc.m. [deg]

10

0

30

60

90

120 150

ϑc.m. [deg]

FIG. 16: Experimental (cyan symbols) and calculated angular distributions of low-energy elastic scattering of α-particles on nuclei from 50 Cr to 124 Sn, using the global OMP of Kumar et al. [84] (black dotted curve), and the regional OMP of Avrigeanu et al. [92, 93] (blue solid curve).

form of a FORTRAN program, which is available from the RIPL-3 website. b. Global alpha-nucleus OMP by Kumar et al. As part of the RIPL-2 project, a prescription for the determination of the alpha-nucleus optical potential based on synthesizing microscopic and phenomenological approaches was developed. This program is continued during the RIPL-3 project, and a global alpha-nucleus optical potential [84] is proposed for use in model calculations of elastic and reaction cross sections for a range of nuclei (A=12–209) and α-particle energies (from around the Coulomb barrier up to about 140 MeV). The global potential has been found to yield reasonable agreement with a large body of experimental data in the domain of application. However, at incident energies, close to or below the Coulomb barrier, the calculations (black dotted curve) and the data differ considerably as can be seen in Fig. 16. A FORTRAN code was written to obtain the optical model parameters for various nuclei and energies and is available from the RIPL-3 website. c. OMPs for α-particle elastic scattering and alphainduced reactions by Avrigeanu et al. As part of the RIPL-3 project, a semi-microscopic analysis of α-particle elastic scattering on A ∼ 100 nuclei

around the Coulomb barrier was undertaken [85]. However, a major overestimation of measured (α, γ) cross sections for 106 Cd [86] and 112 Sn [87] was found when using the OMP derived from α-particle elastic scattering data to calculate the α-particle capture. Recent highprecision measurements of α-particle elastic scattering, e.g. Refs [88]–[90], have made it possible to improve the OMP parameters for A ≤ 124 target nuclei [91]. The new OMP [92] describes equally well both the lowenergy elastic scattering (see Fig. 16) and α-particle induced reactions (e.g. Fig. 17 on the next page), and yields good agreement with a large body of experimental data [93]. A FORTRAN code was written to calculate the Avrigeanu et al. OMP parameters [92], and is available from the RIPL-3 website. d. Global deuteron-nucleus OMPs by Han et al., and An and Cai Global and local optical model potential parameters for deuteron induced reactions derived within the RIPL3 project by Han et al. [94] and An and Cai [95] are given in Table IV. The OMP parameterization considered for deuteron induced reactions corresponds to the general OMP form given by Eq. (20) (VV =VR ). The energy de-

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NUCLEAR DATA SHEETS

R. Capote et al. -2

-3

10

10

64

-3

10

67

Ni(α,n) Zn

σ(b)

Zyskind+ (1979) -3

10

-4

-4

10

10

62 -5

66

62

-5

Ni(α,γ) Zn

10

65

Ni(α,p) Cu

Zyskind+ (1979)

10

58

-4

10

Sevior+ (1986) 62

61

Ni(α,p) Cu

-2

65

Ni(α,n) Zn

-5

10

Zyskind+ (1979)

σ(b)

10

Avrigeanu+ 2009 58

61

Ni(α,n) Zn

-4

10

-2

10

-6

10

58 -6

10

McGowan+ (1964) Vlieks+ (1974) Rios+ (1974)

-8

10

McGowan+(1964) Rios+ (1974) -7

-3

0.4

0.6

1.0 10

0.8

62

Ni(α,γ) Zn

0.5

0.6

0.7

0.8

10

0.4

0.5

0.6

0.7

c.m.

Eα /BC FIG. 17: Comparison of measured (α, γ), (α, n) and (α, p) cross sections on nickel isotopes, and calculated values using the regional OMP of Avrigeanu et al. [92, 93] (blue solid curve)

pendence of the potential depths is expressed as (N − Z) Z + V4 1/3 , A A N −Z , (22) WS (E) = Wo + W1 E + W2 A N −Z WV (E) = Uo + U1 E + U2 E 2 + U3 . A

be constructed from a knowledge of the imaginary part W on the whole real axis through the dispersion relation:

VR (E) = Vo + V1 E + V2 E 2 + V3

The parameterization of the radius and diffuseness parameters (i = R, S, V, so, C) is: + ri A−1/3 , (0) (1) (2) N − Z . ai = ai + ai A1/3 + ai A (0)

Ri = ri A1/3 , ri = ri

2.

(1)

(23)

Dispersive relations and optical model potentials

The dispersive optical model is a natural result of the causality principle: a scattered wave cannot be emitted before the arrival of the incident wave. This introduces an integral relationship [96] which links the real (polarization) potential V (r, E) and imaginary W (r, E ) parts of the nuclear potential. Under favorable conditions of analyticity in the complex E-plane, the real part ΔV can

ΔV (r, E) =

P π

∞

W (r, E ) dE , −∞ E − E

(24)

where we have now explicitly indicated the radial and energy dependence of these quantities, and P means that the principal value of the integral should be taken. The imaginary potential W (r, E ) is assumed to be symmetric about E =EF , according to equation W (r, 2EF − E)=W (r, E). For simplicity the geometry of the imaginary terms of the OMP is usually assumed to be energy independent. In this case the radial functions factorize out of the integrals and the energy dependence is completely accounted for by two overall multiplicative strengths, ΔV (E) and W (E). Both these factors contain volume and surface contributions. The methods currently used to calculate dispersive integrals ΔVV (E), ΔVS (E) and ΔVso (E) have been described in detail in the literature [97]–[99], and they have been implemented in the RIPL-3 optical module optical/om-get/ [10, 14]. In dispersion relation, the real potential strength consists of the Hartree-Fock (HF) term VHF (r, E), which varies slowly with energy, plus a correction (polarization) term V (r, E) which is calculated using a dispersion in-

3129


NUCLEAR DATA SHEETS

R. Capote et al.

TABLE IV: Deuteron OMP developed by Han et al. [94] and An and Cai [95] within the RIPL-3 project. Potential depths V0 , W0 , W0 , V3 , W2 , U3 , Vso , and Wso are in MeV; V1 , W1 , U1 have no units; V2 and U2 are in MeV−1 ; and radii and diffuseness are in fm. Functional forms of the energy and mass dependence of the OMP parameters are given by Eqs. (22) and (23). d+6,7 Li

d+9 Be

RIPL number

6202

6203

6204

V0

47.9904

89.3153

87.1094

V1

0

V2 · 10−2

d+12 C d+40,42,44,48 Ca Global, Han et al. [94] Global, An and Cai [95] -

6201

6200

79.182

82.178

91.85

−0.1799 −0.2298

−0.2678

−0.14809

−0.249

−0.7498 −0.4512 −0.1182

0.005643

−0.088571

0.0116

V3

−39.2608

0.1428

0

0

−34.811

0

V4

0.7593

1.7719

1.4140

0.6765

1.0584

0.642

9.8968

21.2950

20.99

W0

−0.05967 −0.09562 −0.0906

W1 W2

6.7296

0.7734

0

−1.7684 −5.5097 −4.6581

U0 U1 −3

U2 · 10

15.19

20.9677

10.83

−0.07243

−0.0794

−0.0306

0

−43.3977

0

−1.4452

−4.9158

1.104 0.062

0.16

0.0401

0.05284

0.06919

0.05545

0.1085

0.04417

0.04417

0.04417

0.04417

0

U3

0

0

0

0

35

0

Vso

10.5

10.3688

3.9666

3.4382

3.7026

3.557

Wso (0)

rR

(1)

0

0

0

0

−0.2059

0

1.2219

1.1872

1.2005

1.2545

1.1736

1.152

rR

0

0

0

0

0

-0.00776

rS

1.95

1.4024

1.2778

1.5054

1.3275

1.334

(0)

(1) rS (0) rV (1) rV (0) rso

0

0

0

0

0

0.152

1.3023

1.6306

1.5449

1.5454

1.5629

1.305

0

0

0

0

0

0.0997

1.0621

1.4455

0.9

1.1269

1.2335

0.972

1.2248

1.8559

1.7154

1.95

1.6977

1.303

aR

0.6168

0.7791

0.7456

0.7813

0.8092

0.719

aR

0.0

0.0

0.0

0.0

0

0.0126

aS

0.3565

0.4952

0.4693

0.3931

0.4649

0.531

0.0

0.0

0.045

0.045

0.045

0.062

0.7

0.7

0

0

0.0

0.0

0.4482

0.6127

0.6719

0.5299

0.7

0.855

0

0

0.045

0.045

0.045

−0.1

0.7

0.7

0

0

0

0

0.9946

0.364

0.6765

0.8879

0.8130

1.011

rC (0) (1) (0)

(1) aS (2) aS (0) aV (1) aV (2) aV (0) aso

tegral of Eq. (24). We describe a general OMP formulation, when the dispersive relations between real and imaginary parts of the potential are considered. The dispersive deformed OMP for incident nucleons is [76]: V (r, R(θ , ϕ ), E) = Coul − VHF (E) + ΔVHF (E) fW S (r, RHF (θ , ϕ )) − ΔVV (E) + ΔVVCoul (E) + i WV (E) fW S (r, RV (θ , ϕ )) − ΔVS (E) + ΔVSCoul (E) + i WS (E) gW S (r, RS (θ , ϕ )) + 2λ2π [Vso (E) + ΔVso (E) + i Wso (E)] 1 d ˆ fW S (r, Rso ) Sˆ · L × r dr + VCoul (r, RC (θ , ϕ )),

where the first term is the real smooth volume HF potential VHF (E) and its corresponding Coulomb correcCoul tion ΔVHF (E), which is described below. Successive complex-valued terms are the volume, surface and spinorbit potentials, all containing the dispersive contributions ΔVV (E), ΔVS (E), and ΔVso (E) and corresponding Coulomb-correction terms ΔVVCoul (E) and ΔVSCoul (E). The spin-orbit potential is assumed not to be deformed: Rso = rso A1/3 , A being the target mass number. The spherical term of the Coulomb potential was calculated assuming a diffuse charge density distribution of the form −1 fc = 1 + exp r − Rc0 /ac . The geometric WoodsSaxon form factors [76] for the Hartree-Fock and volume potentials are given as:

(25) 3130

RIPL: Reference Input Parameter Lib... fW S (r, Ri (θ , ϕ )) =

NUCLEAR DATA SHEETS

λ=2,4,6,8

where Yλ0 means spherical harmonics, βλ – deformation parameters, θ and ϕ – angular coordinates in the bodyfixed system, and Ri0 = ri A1/3 (i = HF, V, S, C). The geometric form factor for the surface potential is given as: d gW S (r, RS (θ , ϕ )) = −4aS f W S (r, RS (θ , ϕ )). (28) dr It is known that the energy dependence of the depth VHF (E) is due to the replacement of a microscopic nonlocal Hartree-Fock potential by a local equivalent. For a Gaussian non-locality, VHF (E) is a linear function of E for large negative E and is an exponential for large positive E. Following Mahaux and Sartor [96, 100], the energy dependence of the Hartree-Fock part of the nuclear mean field is taken as that found by Lipperheide [101]: VHF (E) = AHF exp(−λHF (E − EF )),

(29)

where AHF and λHF are undetermined constants with the latter associated with nuclear matter nonlocality range, and EF is the Fermi energy. It should be noted that the geometric parameters ri , ai of the dispersive potentials defined by Eqs. (26) and (27) are energy independent; and the geometric parameters rHF and aHF of the HF potential are in general different from the geometric parameters rV , aV , rS , and aS of the volume and surface absorptive potentials. Therefore, the volume dispersive contribution has different geometry (determined by rV and aV ) from the real smooth volume potential (determined by rHF and aHF ). As a result there are two separate volume contributions to the potential (as can be seen in the second and third line of Eq. (25)). On the other hand, the real and imaginary spin-orbit terms usually share the same rso and aso parameters. It is useful to represent the variation of the surface WS (E) and the volume absorption potential WV (E) depths with energy in functional forms suitable for the dispersive optical model analysis. The following expression for the energy dependence of the imaginary-surface term (with three parameters AS , BS and CS ) suggested by Delaroche et al. [102] is commonly used: (E − EF )ns exp(−CS (E − EF )), (E − EF )ns + BSns (30) where the exponent ns should be an even number. The isospin dependence of the dispersive potential (the Lane term [103]) was considered in real VHF (E) and imaginary surface WS (E) potentials as follow, WS (E) = AS

N −Z , AHF = V0 1 + (−1) V0 A

Cwiso N − Z , AS = W0 1 + (−1)Z +1 W0 A

1 , (26) 1 + exp[(r − Ri (θ , ϕ )) /ai ]

where i = HF, V , and Ri (θ , ϕ ) denotes the deformed instant nuclear shapes described by deformed radii, ⎡ ⎤ Ri (θ , ϕ ) = Ri0 ⎣1 + βλ Yλ0 (θ , ϕ )⎦ , (27)

R. Capote et al. Z +1 Cviso

(31) (32)

where V0 , Cviso , W0 and Cwiso are undetermined constants, and Z is the projectile charge (equal 0/1 for neutron/proton induced reactions, respectively). An energy dependence for the imaginary volume term (with two parameters AV and BV ) has been suggested in studies of nuclear matter theory by Brown and Rho [104]: WV (E) = AV

(E − EF )nv , (E − EF )nv + BVnv

(33)

where the exponent nv should be an even number, and nv = 2, 4 is commonly used. The assumption that the imaginary potential WV (E) of Eq. (33) is symmetric about E=EF no longer holds for large values of |E − EF | [100]. The influence of the nonlocality of the imaginary part of the microscopic mean field produces an increase of the empirical imaginary part WV (E) at large positive energies, and a decrease toward zero at large negative energies [105, 106]. Following Mahaux and Sartor [100], we assume that the volume absorption strength is only modified above some fixed energy Ea , which is expected to be close to the depth of the nuclear potential well ∼40–60 MeV. If the non-local imaginary potential V (E), then we to be used in the dispersive integral is W nonl V (E) = WV (E) + W can write W (E), where the nonV locality correction WVnonl (E) for the volume absorption potential is given by [96]: ⎧ (EF −E−Ea )2 ⎪ −W (E) V 2 2 ⎪ (EF −E−Ea ) +Ea ⎪ ⎪ ⎪ for E < EF − Ea , ⎨ WVnonl (E) = √ ⎪ ⎪ (EF +Ea )3/2 3 ⎪ E + − (E + E ) α ⎪ F a 2E 2 ⎪ ⎩ for E > EF + Ea . (34) The coefficient α can be roughly estimated to be equal to 1.65 MeV1/2 [106]; in practice the α value is rather uncertain [107], and α is taken as a fitting parameter. There are different ways of considering non-localities in a dispersive treatment. Interested readers should look for additional information in [105]–[111] and references therein. The addition of the dispersive (polarization) potential provides a more realistic description of the energy dependence of the optical potential and enables the prediction of single-particle, bound-state quantities (such as binding energies, occupation probabilities, and spectroscopic factors). Pioneering dispersive optical model (DOM) analyses of nucleon scattering have been carried out by Lipperheide [101, 112], Passatore [113] and Lipperheide and Schmidt [114]. Consideration of dispersion effects allows both bound and scattering states to be described by the same nuclear mean field [105]–[109],[115]–[120].

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FIG. 18: Neutron total cross section σtot (E): experimental data vs. calculated values using RIPL 2408 [121, 125, 126] OMP. Calculations using RIPL 2409 [127] OMP are shown for 235 U, 238 U, and 239 Pu. Calculations using RIPL 608 [111] and RIPL 2601 [128] OMPs are also shown for 232 Th (top left panel).

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NUCLEAR DATA SHEETS

An additional constraint imposed by the dispersion relations helps to reduce the ambiguities inherent in deriving phenomenological OMP parameters from the experimental data. Accounting for the dispersion relationships between the imaginary and real parts of the optical potential, allows a significant decrease in the number of parameters in the phenomenological optical model potential. This leads to much better description of the experimental data and also allows the accurate prediction of small differences in the total cross section along the given isotopic chain [121].

0.04 238

232

[σtot( U) - σtot( Th)] _____________________ 238 232 1/2 [σtot( U) + σtot( Th)] 0.03

Total cross section differences

a. Dispersive optical model potentials Before RIPL-3, only a few analyses included dispersive relationships, and only a handful of papers made use of the dispersive approach within the coupled-channel framework [109, 122]. One reason is that the dispersive integral of Eq. (24) is analytical for only a very restricted number of absorption-potential energy dependences (e.g. piecewise linear [72] , Brown and Rho form [104, 123] or Delaroche et al. form [102]). However, the advantages shown by dispersive approaches have resulted in an increase in the number of publications on the use of dispersive theory to derive phenomenological OMPs valid over a very large energy range. During RIPL-3 more than 50 sets of dispersive OMPs were added to the RIPL-2 database, most of them based on the coupled-channel method. One relevant set of dispersive coupled-channel OMPs was derived using a well-tested methodology by Soukhovitskii, Capote, and collaborators within the RIPL-3 project for 90−96 Zr (soft-rotor), 55 Mn, 103 Rh [124], 197 Au, 178,180 Hf, 181 Ta, 180−186 W, 232 Th [111], 238 U [121], and other actinides [125, 126, 127]. Unpublished potential parameters will be given in section V C. A good example of the high quality dispersive OMP obtained for 232 Th and 238 U describing the measured neutron total cross section from 1 keV up to 200 MeV can be seen in Fig. 18 (top panel, see previous page). The calculations used the RIPL 2408 [121, 125, 126] and 2409 [127] OMPs; similar calculations for thorium used the RIPL 608 [111] and RIPL 2601 [128] OMPs by Soukhovistkii et al. The isovector terms of the RIPL 2408 and RIPL-2409 OMPs and the very weak dependence of the geometric parameters on mass number A give the possibility of extending the derived potential parameters to neighboring actinide nuclei with great confidence. As an example of the excellent agreement achieved with experimental data not used in the potential fit, we show the calculated neutron total cross section on major actinides 239 Pu and 235 U nuclei in Fig. 18 (bottom panel, see previous page). The coupling scheme adopted for these nuclei is discussed in Ref. [125]. Additional proof of the reliability of the RIPL 2408 OMP extrapolation is given by using this potential to calculate the difference of the total cross sections (divided by the average total cross section) of 232 Th and 238 U nuclei measured by Abfalterer et al. [129]. Good agreement of calculations with the experimental data

R. Capote et al.

0.02

0.01

0.00

-0.01

Abfalterer et al. data Capote et al. RIPL 2408 Soukhovitskii et al. RIPL 2601 10

100

neutron energy (MeV) FIG. 19: Energy dependence of the measured ˜ratio ˘ˆ ¯ [σtot (238 U ) − σtot (232 T h)]/ σtot (238 U ) + σtot (232 T h) /2 [129] vs. calculated values using RIPL 2601 [128] and RIPL 2408 [121, 126] OMPs.

(almost within experimental uncertainty) can be observed in Fig. 19 for results obtained with the RIPL 2408 OMP [121, 125, 126]. Some disagreement with data, including small phase shift, is observed for results obtained with the RIPL 2601 OMP [128]. Recent developments allowed the derivation of dispersive coupled-channel OMP including soft-rotor couplings to describe up to 10 low-energy discrete levels of near-spherical nucleus 90 Zr. An example of neutron and proton angular distributions calculated using RIPL 609/4609 OMP are shown in Figs. 20(a) and 20(b) (see next page) for the scattering of 24 MeV neutrons and 12.7 MeV protons on 90 Zr. The derived Lane-consistent potential can be used to describe scattering data on all Zr isotopes, allowing for a full re-evaluation of the zirconium nuclear data library. An isospin-dependent optical model potential can be used to predict quasi-elastic (p,n) scattering to the isobaric analogue states (IAS) of the target nucleus; such an exercise is the best test of the quality of the isovector part of the optical potential. Angular distributions of quasi-elastic scattering to the IAS have been calculated for the 55 Mn(p,n) and 197 Au(p,n) reactions using dispersive coupled-channel OMPs RIPL 4434/1484 (55 Mn) and RIPL 4433/1483 (197 Au); results are shown in Figs. 20(c) and 20(d), respectively (see next page).

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NUCLEAR DATA SHEETS

(a)Coupled-channel calculation of the n+90 Zr reaction using the RIPL 609 OMP parameters of Table VI(a).

(c)55 Mn(p,n)(IAS): Coupled-channel calculation using the RIPL 4434/1484 OMP parameters of Table VI(b) includes the contribution of the GS 3/2+ (dashed), the 5/2+ (dotted), and the 7/2+ (dash-dotted) IAS.

R. Capote et al.

(b)Coupled-channel calculation of the p+90 Zr reaction using the RIPL 4609 OMP parameters of Table VI(a).

(d)197 Au(p,n)(IAS): Coupled-channel calculation using the RIPL 4433/1483 OMP parameters of Table VI(d) includes the contribution of the GS 3/2+ (dashed), the 5/2+ (dotted), and the 7/2+ (dash-dotted) IAS.

FIG. 20: Angular distributions for the 24 MeV neutron (top left) and 12.7 MeV proton (top right) scattering on 90 Zr nucleus, and for the quasi-elastic (p,n) scattering to isobaric analogue states (IAS) of 55 Mn (bottom left) and 197 Au (bottom right) nuclei at 18 and 26 MeV proton incident energy, respectively.

b. Coulomb corrections It is well known that the nuclear part of the nucleonnucleus interaction experienced by incident protons is different from that experienced by neutrons, because of the Coulomb field and the energy dependence of the nuclear OMP. A Coulomb correction to the proton potential is applied to account for the change in the interacting proton energy due to Coulomb repulsion of the proton by the nucleus. In earlier optical model analyses, usually applied over a limited energy range, such corrections were a constant value that was added to the real potential. A more general expression assumed that corrections were proportional to the derivative of the nuclear potential as

proposed by Satchler [130]: ΔV Coul (E) = −CCoul

ZZ d (V (E)) , A1/3 dE E=Ep

(35)

where CCoul ZZ /A1/3 is the value of the Coulomb field averaged over the space coordinates, assuming that the protons experience a uniform charge distribution. The constant e2 is included in the constant Ccoul . The constant CCoul is an adjustable parameter to account for the “effective” radius of interaction of protons in the nucleus, which is expected to be around 1.44 MeV (≈ e2 ). In general, ΔV Coul (E) of Eq. (35) is complex because the underlying nuclear potential is complex. However, Coulomb

3134


NUCLEAR DATA SHEETS

corrections for the imaginary potential are expected to be small [28, 29] and are usually ignored. The above definition of Eq. (35) also implies that dispersive Coulomb corrections have to be considered for proton potentials, especially for the proper description of proton scattering near the Coulomb barrier as suggested in Refs. [111, 131], and used in Refs. [121, 125, 126] to derive a dispersive coupled-channel OMP RIPL 5408. A Coulomb correction ΔV Coul (E) to the nuclear potential given by Eq. (35) represents the second term of the Taylor expansion of the proton potential accounting for the Coulomb repulsion as will be shown below. If we assume that the kinetic-energy loss of the incident proton in the interaction region is equal to CCoul ZZ /A1/3 due to Coulomb repulsion, then we can define the “effective energy” of the proton interacting with the nucleus as Ep − CCoul ZZ /A1/3 . The Coulomb energy is a small fraction of the total proton energy Ep ; therefore, the proton OMP at the “effective energy” becomes:

)≈

ZZ d (V (E)) V (Ep ) − CCoul 1/3 + ..., dE A E=Ep

in Refs. [10, 14]. This numbering system was adopted in order to separate the potentials for different incident particles into different reference number regions, and to provide approximate information on the sources of the various potentials by geographical region. For example, the latter information might be used if only potentials from a particular source are adopted for a given set of calculations. For the RIPL-3 sublibrary only the user file was included as we have compiled only those OMPs defined in broad energy regions. There is a grand-total of 494 OMP datasets compiled into the RIPL-3 sublibrary, which comprises information published in 139 references (see the complete reference list in [14]). The RIPL-3 user sublibrary to date includes: • 335 optical model parameterizations for incident neutrons, • 130 parameterizations for incident protons,

V (Ep − CCoul ZZ /A

1/3

• 13 parameterizations for incident deuterons, • 2 parameterization for incident tritons,

(36)

which is the sought Taylor expansion. The left-side of Eq. (36) is a general definition of the Coulomb correction. If we use the “effective energy” Ep −CCoul ZZ /A1/3 to calculate the proton potential, then we consider the Coulomb effects in all orders. Such a definition of the Coulomb correction allows a fully Lane-consistent [103] potential to be obtained [125]; all Coulomb-correction terms ΔViCoul (E) in Eq. (25) vanish, and the nuclear OMP becomes symmetrical with respect to the isospin. Unpublished parameters of a regional Lane-consistent OMP for Hf, Ta and W isotopes are listed in Table V (RIPL 610/4610). Unpublished OMP parameters of local (dispersive Lane-consistent) potentials for 90,92 Zr, 55 Mn, 103 Rh, and 197 Au nuclei are listed in Table VI. A new Lane-consistent global OMP for actinides [127] has been recently derived, and will be published elsewhere (RIPL 2409/5409 OMPs for neutron/proton projectiles, respectively). The (p,n) quasielastic data have been used in the OMP fit to constraint the isovector terms of the Lane-consistent potentials.

C.

R. Capote et al.

RIPL-2 and RIPL-3 sublibraries

The RIPL-2 OMP sublibrary was given in two parts: an archival file and a user file [10]. The archival file contained all potentials compiled so far, totaling 533, of which 287 were potentials for incident neutrons, 146 potentials for protons, 11 for deuterons, 26 for tritons, 53 for 3 He particles, and 10 for incident α-particles. The RIPL2 user file is a subset of the archival file with all singleenergy potentials eliminated, and contained 406 entries. Each potential included in RIPL is given a unique reference number, according to a system that is described

• 3 parameterizations for 3 He particles, and • 11 parameterizations for incident alpha particles. From the grand-total of 494 compiled datasets, 383 sets correspond to spherical optical model potentials, and only 29 sets are defined for incident complex particles. There are 155 high-energy potentials obtained with the use of relativistic kinematics (112 of them for neutrons). The relativistic OMPs given in the sublibrary may include the Madland γ relativistic factor [132], which multiplies the potential depth. This factor results from the reduction of the full Dirac equation to the Schr¨ odinger equation describing the scattering of a relativistic particle incident on a non-relativistic target nucleus. The factor γ has been used in the Madland potentials RIPL 2001 (neutrons) and 5001 (protons).

1.

Nucleon-nucleus potentials: Individual nucleus potentials

The majority of the nucleon-nucleus potentials in the OMP sublibrary are for single-target nuclei (or perhaps for a very narrow range of neighboring targets/isotopes) and cover a range of incident energies. These potentials are usually the most accurate for specific targets and should be considered whenever accuracy is imperative. a. Spherical potentials There is one important series of spherical potentials which were derived as part of a comprehensive analysis covering targets from 24 Mg to 209 Bi. These are the spherical OMPs of Koning and Delaroche [133]. Other significant individual nuclide potentials include those of Arthur [134], and the extensive lists of potentials used in China [135] and Japan [136] for fission products.

3135


NUCLEAR DATA SHEETS

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TABLE V: Dispersive coupled-channel OMP parameters for nucleons incident on 178,180 Hf, 181 Ta, and 180,182,183,184,186 W nuclei (RIPL 610/4610 for neutron/protons, respectively). Notation follows Eqs. (25)–(34), where ns = 2 and nv = 2 are used in Eqs. (30) and (33), respectively. The OMP is Lane-consistent. “Effective energy” Ep − CCoul ZZ /A1/3 should be used for proton calculations, all ΔViCoul (E) ≡ 0 in Eq. (25). Coulomb potential parameters are CCoul = 1.33 MeV, rC = 1.0626 fm and aC = 0.64 fm. 5 levels of the corresponding ground-state (GS) rotational band are coupled (4 levels for 181 Ta). The nonlocality parameter α = 1.51 MeV1/2 . All depth parameters are in MeV, except λHF , λso and CS which are in MeV−1 . Hf, Ta, W isotopes Volume Real depth

Surface

V0 = 50.03

Spin–Orbit Vso = 6.04

λHF = 0.00913

dispersive

λso = 0.005

Cviso = 21.0 Imag. depth

AV = 7.9

W0 = 21.91 + 0.0725(A–178) Wso = −3.1

BV = 84.24

BS = 14.69

Ea = 52

CS = 0.00961

Bso = 160

Cwiso = 29.4 Geometry [fm]

rHF = 1.2447 − 0.00175(A–178) rS = 1.240

rso = 1.1293

aHF = 0.631 − 0.0005(A–178)

aso = 0.59

aS = 0.5 + 0.0025(A–178)

rV = 1.0797 aV = 0.714 The following deformation parameters are used for rigid-rotor potentials: β2 = 0.240, β4 = −0.055, β6 = +0.0056 for

178

Hf, β2 = 0.225, β4 = −0.060, β6 = +0.0059 for

180

β2 = 0.251, β4 = −0.079, β6 = −0.0500 for

181

Ta, β2 = 0.245, β4 = −0.056, β6 = −0.0030 for

180

β2 = 0.229, β4 = −0.064, β6 = +0.0080 for

182

W, β2 = 0.235, β4 = −0.068, β6 = +0.0120 for

183

β2 = 0.242, β4 = −0.073, β6 = +0.0170 for

184

W, β2 = 0.200, β4 = −0.095, β6 = −0.0011 for

186

b. Coupled-channel potentials The RIPL-3 OMP sublibrary contains 65 neutron and 32 proton rigid-rotor coupled-channel potentials for ground-state rotational bands. These potentials have been necessarily developed for more limited target Z and A ranges, as they depend on the structure of each target nucleus. The targets for incident neutrons include 55 Mn, 103 Rh, 151−153 Eu, 150,152 Sm, 165 Ho, 169 Tm, 174−181 Hf, 180−182 Ta, 180−186 W, 185,187 Re, 197 Au and a number of actinides between 229 Th and 252 Cf. There are also 10 neutron potentials based on the harmonicvibrational model for 54,56 Fe, 63,65 Cu, 58,60 Ni, 148,150 Sm and 208 Pb. Finally, there is a neutron potential based on soft-rotor (non-axial) couplings for 24,26 Mg, 28,30 Si, 32 S, 40 Ar, 40 Ca, 48 Ti, 52 Cr, 54,56 Fe, 58,62 Ni, 90−96 Zr, 92−98 Mo and 116−124 Sn, and a dispersive potential based on softrotor couplings for 90,92,94,96 Zr nuclei.

W,

W, W.

for rigid-rotor and soft-rotor potentials. These potentials were determined with neutron potentials using the Lane model [103]. There are no proton potentials in the database based on the vibrational model.

2.

Nucleon-nucleus potentials: Global potentials

The use of the term “global” refers to OMPs that cover a wide range of incident energy and target nuclei. A few important potentials that are global for both incident neutrons and protons are those of • Koning and Delaroche [133] (RIPL 2405/5405, spherical, Z=12–83, A=27–209, E=0.001– 200 MeV), • Madland [132] (RIPL 2001/5001, spherical, Z=6– 82, A=12–208, E=50–400 MeV),

Structure information is provided in the OMP sublibrary for the target nuclei for which the potential was determined. These potentials can be used over a broader range of target nuclei, most probably with reduced accuracy, although the user must provide the required structure information, in particular the ground-state-band deformation for rigid-rotor potentials or the dynamical deformation for the vibrational potentials. The soft-rotor potentials can not be used for other nuclei as the required structure information (soft-rotor Hamiltonian parameters) is difficult to extrapolate to different nuclei. Coupled-channel potentials for incident protons are often provided for the same target nuclei as neutrons

Hf,

• Soukhovitskii et al. [128] (RIPL 2601/5601, coupled-channel (CC), Z ≥90, A ≥230, E=0.001– 200 MeV) and • Capote et al. [121, 125, 126] (RIPL 2408/5408, dispersive CC, Z ≥90, A ≥230, E=0.001–200 MeV), • Capote et al. [127] (RIPL 2409/5409, Lane consistent, dispersive CC, Z ≥90, A ≥230, E=0.001– 200 MeV). New global dispersive potentials have been also proposed

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NUCLEAR DATA SHEETS

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TABLE VI: Dispersive coupled-channel OMP parameters for nucleons incident on 90−96 Zr – soft-rotor, 55 Mn–, 103 Rh–, and 197 Au – rigid-rotor nuclei. Notation follows Eqs. (25)–(34), where ns = 2 and nv = 2 are used in Eqs. (30) and (33), respectively. “Effective energy” Ep − CCoul ZZ /A1/3 should be used for proton calculations, all ΔViCoul (E) ≡ 0 in Eq. (25). All depth parameters are in MeV, except λHF , λso and CS which are in MeV−1 . (b)Lane-consistent OMP RIPL 1484(n)/4434(p) for nucleon scattering on 55 Mn nucleus. Coupled-levels energies of the 5/2− (a)Lane-consistent OMP RIPL 609(n)/4609(p) for nucleon GS rotational band are 0, 0.126, 0.984 and 1.292 MeV. The scattering on 90−96 Zr nuclei. Soft-rotor Hamiltonian parameters following deformation parameters are used: β2 = 0.202, are listed in Ref. [14]. Coulomb parameters are CCoul = 1.1 MeV, β4 = 0.044, β6 = 0.0010. Coulomb parameters are rC = 1.1559 fm, and aC = 0.36 fm. CCoul = 1.1 MeV, rC = 1.156 fm and aC = 0.36 fm. The nonlocality parameter α = 0.38 MeV1/2 . The nonlocality parameter α = 0.374 MeV1/2 . 90,92,94,96

Zr

Volume

Surface

Real depth V0 = 53.66

55

Spin-Orbit Vso = 7.9

λHF = 0.01013 dispersive

Mn

Volume


W0 = 11.77

Wso = −3.1

BV = 79.58

BS = 12.74

Bso = 160

Ea = 52

CS = 0.01396

Imag. depth AV = 11.79

W0 = 19.99

Wso = −3.1

BV = 80.62

BS = 18.62

Bso = 160

Ea = 52

CS = 0.01334

Cwiso = 27.3

Cwiso = 29.4

Geometry

rHF = 1.2384

rS = 1.2014

rso = 1.1214

[fm]

aHF = 0.586

aS = 0.566

aso = 0.59

Geometry

rHF = 1.2178

rS = 1.2256

rso = 1.1214

[fm]

aHF = 0.625

aS = 0.452

aso = 0.59

rV = 1.1658

rV = 1.2420

aV = 0.927

aV = 0.725

(c)Lane-consistent OMP RIPL 1485(n)/4435(p) for nucleon scattering on 103 Rh nucleus. Coupled-levels energies of the 1/2− GS rotational band are 0, 0.295, 0.357, 0.847, 0.920, 1.411 and 1.638 MeV. Coulomb parameters are CCoul = 1.1 MeV, rC = 1.1974 fm and aC = 0.4 fm. The following deformation parameters are used: β2 = 0.195, β4 = −0.0134, β6 = 0.011. The nonlocality parameter α = 0.374 MeV1/2 .

Volume

Surface


(d)Lane-consistent OMP RIPL 1483(n)/4433(p) OMP for nucleon scattering on 197 Au nucleus. Coupled-levels energies of the 3/2+ GS rotational band are 0, 0.279, 0.547, and 0.855 MeV. Coulomb parameters are CCoul =0.93 MeV, rC =1.199 fm and aC = 0.38 fm. The following deformation parameters are used: β2 = −0.14, β4 = −.0173, β6 =0.0023. The nonlocality parameter α = 0.374 MeV1/2 . 197



Au

Volume

Surface


Cviso = 21.0



λso = 0.005

λso = 0.005

Cviso = 18.25


W0 = 16.21

Wso = −3.1

BV = 80.76

BS = 13.19

Bso = 160

Ea = 52

λso = 0.005

Cviso = 21.0


Rh



λso = 0.005

Cviso = 15.30

103

Surface


W0 = 17.09

Wso = −3.1

BV = 80.93

BS = 12.97

Bso = 160

CS = 0.01334

Ea = 52

Cwiso = 29.4

CS = 0.01334 Cwiso = 29.4

Geometry

rHF = 1.2503

rS = 1.1815

rso = 1.1214

[fm]

aHF = 0.617

aS = 0.607

aso = 0.59

Geometry [fm]

rHF = 1.2497

rS = 1.1796

rso = 1.1214

aHF = 0.619

aS = 0.577

aso = 0.59

rV = 1.2571

rV = 1.2546

aV = 0.716

aV = 0.695

• Morillon and Romain [137, 138] for neutrons (RIPL 2407, dispersive, spherical, Z=12–83, A=27–209, E=0.001–200 MeV) and • Li and Cai [139] for protons (RIPL 5501, dispersive, spherical, Z=12–92, A=27–238, E=0.001– 200 MeV).

mental data than the older potentials. Particularly nice features of dispersive potentials are the reduced number of potential parameters, the energy-independent geometry and a reliable extrapolation to the low-energy region. Other older global spherical potentials that cover both incident neutrons and protons are those of

By using improved methodology, these seven potentials are the most recently developed global potentials, and their analyses have drawn upon a wider range of experi3137

• Becchetti and Greenlees [140] (RIPL 100/4101, Z=20–92, A=40–238, E=10–50 MeV), • Walter and Guss [141] (RIPL 2101/5101, Z=26–82,


NUCLEAR DATA SHEETS

A=54–208, E=10–80 MeV), and • Varner et al. [142] (RIPL 2100/5100, Z=20–83, A=40–209, E=16–65 MeV). Older global spherical potentials developed exclusively for incident neutrons are those of • Moldauer [143] (RIPL 116, Z=20–83, A=40–209, E=0.001–5 MeV), • Wilmore and Hodgson [144] (RIPL 401, Z=20–92, A=40–238, E=0.01–25 MeV), • Engelbrecht and Fiedeldey [145] (RIPL 800, Z=20– 83, A=40–210, E=0.001–155 MeV),

Global potentials of Becchetti and Greenlees [154] for incident tritons (RIPL 7100, Z=20–82, A=40–208, E=1–40 MeV) and 3 He particles (RIPL 8100, Z=20–82, A=40–208, E=1–40 MeV) are also included. The energy dependence of the triton potential is based on the 3 He potential, which covers the same energy range for the same target charge and mass range. Additionally, a new triton global potential recently derived by Li et al. [155] has also been compiled (RIPL 7200, Z=13–90, A=27–232, E=1–40 MeV). There are three new global OMPs for incident αparticles: • Demetriou et al. [83],

• Strohmaier et al. [146] (RIPL 404, Z=23–41, A=50–95, E=0.001–30 MeV). For incident protons alone, there are the global spherical potentials of

R. Capote et al.

• Kumar et al. [84] (A=12–209, E 220 actinide targets Strongly deformed actinide nuclei – we recommend that coupled-channel rotational potentials are used for the individual nucleus involved if they are available. Again, new regional parameterizations for the actinide region by Soukhovistskii et al. [128] (RIPL 2601/5601 for neutrons/protons) and Capote et al. [121, 125, 126, 127] (RIPL 2408/5408 and RIPL 2409/5409 for neutrons/protons) potentials have been completed, and are good alternatives. The latter is also a Lane-consistent dispersive potential. 3142

1. We recommend the individual-nucleus potentials of Koning and Delaroche [133] (RIPL 4416–4429). For some selected odd-A nuclei where dispersive coupled-channel potentials have been derived, those may be preferred (see Table VI for 55 Mn – RIPL 4434, 103 Rh – RIPL 4435 and 197 Au – RIPL 4433 potentials). 2. For selected even-even nuclei (from 24 Mg to 122 Sn), one may consider the use of the soft-rotor coupledchannel potential [82] (RIPL 5602). However, the OPTMAN code has to be used for optical model calculations in this situation. A high-quality softrotor potential has been developed for Zr isotopes (RIPL 4609) and was presented in Table VI. 3. For cases where the above options are not possible but the target is still close to the line of stability, we recommend the global potential of Koning and Delaroche [133] (RIPL 5405). For 50 ≤ E ≤ 400 MeV, a possible alternative is the potential of Madland [132] (RIPL 5001). 4.

If the target lies far from the line of stability, we recommend use of the microscopic optical model code MOM with radial densities tabulated in chapter II.


NUCLEAR DATA SHEETS

Incident deuterons 1. We recommend the individual-nucleus potentials of Daehnick [153] (RIPL 6112–6116) or Han et al. (see Table IV) for light nuclides (RIPL 6202–6204), where applicable. 2. If option 1 is not possible, we recommend the global potential of An and Cai [95] (RIPL 6200), Han et al. [94] (RIPL 6201) or Bojowald et al. [150] (RIPL 6400). Incident tritons 1.

We recommend the global potential of Becchetti and Greenlees [154] (RIPL 7100); the global potential recently developed by Li et al. [155] (RIPL 7200) is an alternative.

Incident 3 He 1. We recommend the global potential of Becchetti and Greenlees [154] (RIPL 8100). Incident α-particles 1. We recommend the available individual-nucleus potentials. 2.

If the above option is not possible, we recommend the global potential of Strohmaier et al. [146] (RIPL 9400 – Z=20–45, E=1–30 MeV) or Avrigeanu et al. [91, 92] (Z=22–50, E=1–50 MeV). The double-folding potential by Demetriou et al. [83] or Kumar et al. OMP [84] are alternatives.

VI.

NUCLEAR LEVEL DENSITIES

The statistical properties of excited nuclear levels have been a matter of concern and study for many years. One of the basic statistical properties of these levels is their density, for which the Fermi gas [163] and constant temperature models are frequently used with empirical input parameters obtained from fitting certain experimental data. However, the physical assumptions underlying these models are not sufficiently sophisticated to account properly for variations of level densities over a wide energy interval from the ground state to well above the neutron separation energy. This is not surprising as these models were formulated more than sixty years ago, in the infancy of nuclear physics. Nuclear level densities (NLD) are also needed in the statistical model of nuclear reactions when discrete level information is considered to be incomplete. The optical model and nuclear level density are perhaps the most crucial ingredients for a reliable theoretical analysis of

R. Capote et al.

scattering observables. NLDs have been extensively studied, leading to more sophisticated approaches, both phenomenological [43],[164]–[190], as well as microscopic or semi-microscopic [191]–[203]. Some of the most important concepts upon which our current understanding of the structure of low-lying nuclear levels is based are shell effects, pairing correlations and collective phenomena. All these concepts have been incorporated into the generalized superfluid model developed by many authors over the last 40 years. Most consistently all these properties are taken into account in microscopic versions of the model, but phenomenological versions of the model – convenient for the analysis of experimental data – have also been developed during the previous decades. All the modern models discussed in this section explicitly deal with pairing effects, but collective and shell effects are not always explicitly treated. This will be discussed later. For practical applications of the statistical models, it is very important to obtain parameters of the level densities from reliable experimental data. The cumulative numbers of low-lying levels and the average distances between neutron resonances are usually used for this purpose. These mean spacings (see chapter IV) have been analyzed by many authors [41, 43, 67, 165, 167, 170],[204]–[206]. For the majority of nuclei the observed resonances correspond to s-wave neutrons so that the value of the average resonance spacings D0 is related to the level density ρ of the compound nucleus by the relationships: ⎧ ρ(Bn + ΔE/2, 1/2, π0 ) for I0 = 0, ⎪ ⎨ D0−1 = ⎪ ⎩ ρ(Bn + ΔE/2, I0 + 1/2, π0 ) + for I0 = 0, ρ(Bn + ΔE/2, I0 − 1/2, π0 ) where Bn is the neutron binding energy, ΔE is the energy interval for which the resonances are being examined, and I0 and π0 are the target-nucleus spin and parity, respectively. If necessary, resonances for p-wave neutrons can also be considered in a similar way. The main goal of the present study is to provide the user with a complete database of phenomenological and microscopic NLDs. By complete, we mean that NLDs can be calculated for any nucleus of interest, provided that this nucleus exists, i.e. is not beyond the neutron or the proton drip-line. In addition to the NLD sublibrary, we also developed a FORTRAN90 module to calculate level densities, starting from NLD parameters or numbers tabulated in the NLD sublibrary. This module could be rather simply inserted into modern codes for the nuclear cross-section calculations. The various level density approaches developed under the RIPL project are discussed below together with recommendations regarding their application to different tasks. Section VI A is devoted to descriptions of the various phenomenological models that have been revisited, starting from the basic relations of the Fermi gas model. Microscopic level densities, which can nowadays reach a degree of accuracy similar to that of the best global analytical approaches, are described in section VI B. Finally,

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section VI C discusses the partial level densities widely used within pre-equilibrium model approaches. Recommendations have been assembled in the last section.

A.

Phenomenological total level densities

The best known analytical level density expression is that of the Fermi Gas Model (FGM) proposed by Bethe in 1937 [163]. All the phenomenological analytical (closed-form) expressions follow a Fermi gas expression at high excitation energies and mainly depend on the level density parameter a. The FGM will be reviewed in section VI A 1. We may distinguish between two types of phenomenological approaches depending upon whether collective effects are explicitly accounted for or not. The first two cases – the Back-Shifted Fermi Gas Model (BSFGM) and the Composite Gilbert-Cameron Model (CGCM) – are described in sections VI A 2 and VI A 3 respectively, and collective effects are not explicitly included in those models. These models are based on simple but easy to use analytical expressions with few parameters, which hopefully reduces the uncertainty of the model. On the contrary, both the Generalized Superfluid Model (GSM) and the Enhanced Generalized Superfluid Model (EGSM), described in sections VI A 4 and VI A 5 respectively, explicitly deal with collective effects but at the price of introducing more complicated expressions. Different expressions for the collective enhancement are encountered in the literature, reflecting our current (lack of) knowledge of such physical phenomena. This fact introduces an uncertainty in the GSM-like models. In RIPL-3, different strategies have been employed to improve the RIPL-1 and RIPL-2 systematics for the four analytical models considered below. The first strategy, employed for the CGC, BSFG and GS models, consists in simultaneously fitting the s-wave mean spacings and low energy discrete levels as described below, thereby providing global predictions for any existing nucleus. A different approach was used for the EGSM. The EGSM global systematics does not account for discrete levels. Instead, the adjustment is performed automatically by the EGSM code (provided in RIPL-3 as a module) when the level densities are calculated.

1.

Basic relations of Fermi Gas Model

The Fermi gas model was proposed by Bethe in 1937 [163]. This model is based on the physical assumption that the single particle states from which the excited levels of the nucleus are constructed are equally spaced, and that collective levels are absent. These assumptions allow the derivation of a closed-form equation for the density of states. For a two-fermion system that distinguishes between neutrons and protons, the total Fermi gas state

R. Capote et al.

density is given by the equation:

√ exp 2√aU π ωFtot (E) = , (37) 12 a1/4 U 5/4 where a is the level density parameter, given by a = π2 6 (gπ + gν ), where gπ (gν ) denote the spacing of the proton (neutron) single particle states near the Fermi energy. The single-particle level density g is defined as g = gπ + gν . Equation (37) is used to determine a from experimental data quantifying the specific nucleus under consideration or from global systematics. The level density parameter a determined by fitting from experimental data at the binding energy is referred as “experimental” in this chapter. Contemporary analytical models have been developed that show a to be energydependent, as assumed in RIPL-3 but not in RIPL-1 and RIPL-2 [6, 10]. Equation (37) also contains the effective excitation energy U = E − Δ,

(38)

where the energy shift Δ is an empirical parameter equal to, or closely related to, the pairing energy included to simulate the known odd-even effects in nuclei. The underlying proposal is that Δ accounts for the fact that pairs of nucleons coupled to spin 0 must be broken before each component can be excited individually. Δ plays an important role as an adjustable parameter to reproduce available observables, and can be defined differently for the various models discussed below. Throughout this chapter we adopt both the true excitation energy E as a running variable for expressions related to discrete levels, and U for expressions related to the continuum. When implementing level density models, care is required for E < Δ, when Eq. (37) and some other related quantities become undefined – the state density prescription (37) is usually replaced by alternatives (e.g. expressions calculated directly from the number of discrete levels), or appropriate corrective actions are taken. Such cases are explicitly described to avoid any confusion or uncertainty in the way they are handled. Under the assumption that the projections of the total angular momentum are randomly coupled, the Fermi gas level density for a two-component fermion system can be derived [42, 163]:

(J + 12 )2 1 2J + 1 √ ρF (E, J, Π) = × exp − 2 2 2πσ 3 2σ 2 √ √ exp 2 aU π , (39) 12 a1/4 U 5/4 where J and Π are the nucleus spin and parity, the 12 factor arises from the assumption of equal positive and negative parity distributions, and σ 2 is the spin cutoff parameter, which represents the Gaussian width (dispersion) of the z-projection of the angular momentum distribution J (depends on excitation energy and will be discussed in more detail below).

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Summing ρF (E, J, Π) over all spins and parities yields the total Fermi gas level density: √ exp 2√aU 1 π ρtot , (40) F (E) = √ 2πσ 12 a1/4 U 5/4

and the corresponding moment of inertia, which coincides with the rigid-body value 0 = g(F )m2F =

which is related through Eq. (37) to the total Fermi gas state density: ωFtot (E) ρtot . F (E) = √ 2πσ

(41)

These equations demonstrate that nuclear level densities ρF in the Fermi gas model are determined by three parameters, namely a, σ and Δ. The energy and mass dependence of those parameters are discussed in the sections below dedicated to specific level density models. For the Fermi gas model, the state equations determining the dependence of the excitation energy U , the entropy S and other thermodynamic functions of a nucleus on temperature t have simple forms: U = at2 ,

S = 2at,

2.

144 π

g(F ) =

4m0 r02 A(1 + βs A−1/3 ), 2 (3π 2 )1/3

(47)

and the temperature t is usually added to the excitation energy U in the denominator of Eqs. (39) and (40) [167]. Such an approach was followed in RIPL-1 and RIPL-2. For the RIPL-3 [190] we followed the Grossjean and Feldmeier recommendations [176] as implemented by Demetriou and Goriely [188]. The expression for the total BSFGM level density is defined as ρtot BFM (E) =

1 1 + tot ρF (E) ρ0 (t)

−1 ,

(48)

a exp(a2 t2 ), (49) 12σ where t is given by Eq. (42) (t = U/a). The expression (48) suppresses the divergence of Eqs. (40) and (41) at low energies, when the temperature definition of Eq. (42) is used, and coincides with the Fermi gas formula Eq. (40) after a few hundreds of keV. These equations show that nuclear level densities in the BSFGM are determined by three parameters, namely a, σ and Δ whose mass and energy dependencies are now discussed. ρ0 (t) = e

If we neglect the surface contribution βs we obtain a ≈ A/13.5. Differences between various semi-classical determinations of the level density parameters (44) are mainly due to the large uncertainties in the existing evaluations for βs . Similarly, we find the average value of the square of the projection of angular momentum for single-particle states on the Fermi surface (3π 2 )1/3 2/3 A , 10

Back-Shifted Fermi Gas Model

where ρ0 is given by the equation [188]: (43)

where m0 is the nucleon mass, r0 is the nuclear radius parameter, A is the mass number, and βs defines the surface component of the single-particle level density. We write the corresponding level density parameter a, in MeV−1 , in the form π 4/3 m r2 π2 0 0 a= g(F ) = 2 A(1 + βs A−1/3 ). (44) 6 3 2

m2F =

(46)

U = at2 − t,

a 3 t5 ,

where m2 is the mean square value of the angular momentum projections for the single-particle states around the Fermi energy, which may also be associated with the moment of inertia of a heated nucleus = g m2 . There are obvious connections between the thermodynamic functions of Eq. (42) and the state Eq. (37) and level densities Eq. (39). The main parameters of the Fermi gas model may be estimated rather simply using the semi-classical approximation. For the density of states g(F ) on the Fermi surface, we obtain

2 m0 r02 5/3 A . 5 2

An approach to the problem of a simultaneous description of neutron resonance densities and low-lying levels was proposed in Ref. [167]. Both sets of experimental data are assumed to be described on the basis of the Fermi gas relations if the level density parameter a and the excitation energy shift Δ are considered as free parameters for each nucleus. Since for odd-odd nuclei the resulting displacement Δ is negative, such an approach was named as the back-shifted Fermi gas model. Since in this approach the Fermi gas formulae are applied at rather low excitation energies, a more accurate estimation of a temperature t was used in the original work of Dilg et al. [167],

(42) σ 2 = m2 π62 at, Det =

R. Capote et al.

(45)

a. Energy shift Δ As mentioned above, the pairing energy Δ usually accounts for the fact that pairs of nucleons must be broken before each component can be excited individually. Its behavior is therefore on average similar to that of the quantity used in the analytical mass formulae. However, following the spirit of the back-shifted Fermi gas model,

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we introduce a parameter δ resulting in the following expression 12 Δ = n √ + δ, A

(50)

where n = −1, 0 and 1 for odd-odd, odd-A, and eveneven nuclei, respectively, and adjust δ to fit experimental data for each nucleus. b. Energy-dependent level density parameter a One of the deficiencies of RIPL-1 and RIPL-2 systematics for the BSFGM is the assumed energy independence of the a-parameters. Indeed, all microscopic calculations of nuclear level densities display a damping of the shell effects at high excitation energies [166, 183, 207]. To account for this damping, the BSFGM level density parameter a in the RIPL-3 project is approximated following Ignatyuk [169]: a ≡ a(E, Z, A) = a ˜(A) 1 + δW (Z,A) [1 − exp(−γU )] , U

where Mexp (Z, A) is the experimental value of the nuclear mass taken from Ref. [15] (see chapter II), and MLDM (Z, A, β) is the liquid drop component of the mass formula [209]. Differences between the various approximations are mainly related to the liquid drop component; if the liquid-drop model formula for spherical nuclei is adopted in Eq. (55), the shell corrections coincide with the microscopic energies considered in chapter II. However, some authors prefer to include the deformation energies within the liquid-drop component of Eq. (55) [209]. Such a consistent definition of the shell corrections is important for the analysis of equilibrium deformations of highly excited nuclei [210], but differences in the shell correction estimations are less essential for the systematics of the level density parameters. In the current BSFG model, we adopt the parameterization of Mengoni and Nakajima [182] for the liquid-drop model mass formula MLDM (Z, A, β ≡ 0) of Myers and Swiatecki (MS) [209], which is, Mn N + MH Z + Evol + Esur + Ecoul + Δm , 8.07144 MeV, 7.28899 MeV, −c1 A, c2 A2/3 , Z2 Z2 Ecoul = c3 1/3 − c4 , A A 2 N −Z , i = 1, 2 (56) ci = a i 1 − κ A

MLDM Mn MH Evol Esur

(51)

where δW (Z, A) is the shell correction energy and a ˜(A) is the asymptotic level density value obtained when all shell effects are damped, given by the smooth formulation: a ˜ = αA + βA2/3 ,

(52)

where A is the mass number, and α and β parameters determine the volume and surface contributions. The damping parameter γ determines how rapidly a(E, Z, A) approaches a ˜(A), and is γ=

γ0 1

A3

.

lim a(E) = a ˜ [1 + γδW ] ,

U→0

(54)

which is computationally convenient if E ≤ Δ. Systematics based on an energy-dependent level density parameter a were proposed in Refs. [169, 173, 182, 208], and the differences between the corresponding level density parameters are mainly related to different shell corrections. Shell corrections are usually determined as the difference δW (Z, A) = Mexp (Z, A) − MLDM (Z, A, β),

(55)

= = = = =

a1 = 15.677 MeV, a2 = 18.56 MeV, κ = 1.79, c3 = 0.717 MeV, c4 = 1.21129 MeV, ⎧ 11 ⎨ − √A even-even, 0 odd, Δm = ⎩ √11 odd-odd. A

(53)

α, β and γ0 are global parameters that need to be fitted to the experimental data to give the best average level density description over a whole range of nuclides. The absolute magnitude of δW (Z, A) determines how different a(E, Z, A) is from a ˜(A) at low energies, while the sign of δW (Z, A) determines whether a(E, Z, A) decreases or increases as a function of E. Equation (51) should be applied at all excitation energies. Therefore, the limiting value of Eq. (51) for small excitation energies is given by the first order Taylor expansion:

R. Capote et al.

Other prescriptions are available for the MS formula with slightly modified coefficients. However, the shell corrections that result do not differ much from the original MS parameterization. Figure 22 shows the shell corrections corresponding to the most widely used mass formulae [16, 43, 182, 209]. c. Spin cutoff parameter σ 2 The spin cutoff parameter σ 2 characterizes the angular momentum distribution of the level density. Spin cutoff parameters are usually determined on the basis of the evaluation of the moment of inertia as either the rigidbody value 0 or half of this value. The results of the spin distribution analysis of low-lying levels considered in chapter III are shown in Fig. 23. The half-rigid-body value of the moment of inertia 12 0 is in reasonable agreement with the available data and was used in the RIPL-2 BSFGM analysis, but because of the large uncertainties more accurate spin cutoff parameters are difficult to estimate.

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NUCLEAR DATA SHEETS

FIG. 22: Shell corrections to the nuclear binding energies estimated by different authors [16, 43, 182, 209].

A general expression for the spin cutoff parameter in the continuum can be formulated based on the observation that a nucleus possesses collective rotational energy that can not be used to excite the individual nucleons. One can relate σ 2 to the moment of inertia of the nucleus I0 and the thermodynamic temperature t through the thermodynamic functions (42). However, observations from microscopic level density studies show that the quantity σ 2 /t is not constant [183, 184], but exhibits marked shell effects similar to the level density parameter a. This effect can be taken into account by adopting the following expression [183, 186]: a U σF2 (E) = 0 , (57) a ˜ a with a ˜ from Eq. (54) and 0 being the rigid body moment of inertia given by Eq. (46). This gives σF2 (E) = 0.01389

A5/3 √ aU. a ˜

(58)

√ a term has the same energyOn average, the aU /˜ and mass-dependent behavior as the temperature U/a. However, marked differences between these two √ terms occur in the regions with large shell effects. A aU dependence for σF2 has already been adopted by Gilbert and Cameron [43], although the mass dependence through shell effects present in a and the functional form of a˜ differ. Analogous to the level density parameter, we have to account for low excitation energies for which Eq. (58) is not defined in the energy range E ≤ Δ. An alternative method can be applied to determine the spin cutoff parameter from the spins of the low-lying discrete levels. This method has been implemented in the RIPL-3 NLD retrieval module. Assuming we wish to determine this discrete spin cutoff parameter σd2 in the energy range where the total level density agrees well with the discrete level sequence, i.e. from a lower discrete level NL with

R. Capote et al.

FIG. 23: Spin cutoff parameters obtained from the analysis of low-lying levels (in blue) in comparison with calculations using the half-rigid-body value 12 0 of the moment of inertia (red points).

energy EL to an upper level NU with energy EU . The discrete spin cutoff parameter [190] can be determined by the equation: σd2

=

3

!NU

1

i=NL (2Ji

NU

+ 1) i=NL

Ji (Ji + 1)(2Ji + 1), (59)

where Ji is the spin of discrete level i. These spins can be taken from the discrete level sublibrary (see chapter III) to derive the value for σd2 , and σd2 can be determined on a nucleus-by-nucleus basis when discrete levels are known, or from the global systematics given by the equation: 2 2 σd,global = 0.83A0.26 .

(60)

The functional form that we use in the RIPL-3 leveldensity module for σ 2 (E) is a combination of Eqs. (58) and (59). Defining Ed = 12 (EL + EU ) as the energy at the mid-point of the NL − NU region, we assume σd2 is constant up to this energy and can then be linearly interpolated to the expression given by Eq. (58). The matching point is chosen to be the neutron separation energy Sn of the nucleus under consideration: ⎧ 2 σd for 0 ≤ E < Ed , ⎪ ⎪ ⎪ ⎪ ⎨ 2 d σF (Sn ) − σd2 for Ed ≤ E < Sn , σ 2 (E)= σd2 + SE−E −E n d ⎪ ⎪ ⎪ ⎪ ⎩ 2 σF (E) for E ≥ Sn . When no discrete levels are available, we adopt a systematic formula, see Eq. (60) for σd2 , while we take Ed equal to zero. d. BSFGM results Results of the analysis of the neutron resonance densities and low-lying nuclear levels are shown in Fig. 24 (next page). The complete set of the RIPL-3 level density parameters obtained for BSFGM is given in the densities/level-density-bsfg.dat file, and the format is described in the densities/level-density-bsfg.readme file.

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a(B n

(a -a )/a


FIG. 25: Deviations between the RIPL-3 “experimental” level density parameter a(Bn ) derived from the BSFGM resonance spacing analysis and BSFGM global systematics defined by Eqs. (51)–(53) and (61).

parameters of BSFGM describe the experimental data with the deviation frms = 1.68.

3.

FIG. 24: Level density parameters a(Bn ) and Δ in the BSFG model.

Similar results for the previous analysis can be found in the RIPL Handbooks [6, 10] and the corresponding alternative and recommended data files of the RIPL project. Differences between parameters obtained in RIPL-2 and RIPL-3 reflect improvements achieved for both sets of experimental data (the neutron resonance spacings and the cumulative numbers of low-lying levels) as well as the different fitting strategy adopted for RIPL-3. The BSFGM parameter systematics obtained through Eqs. (50)–(53) is given by the following coefficients: α = 0.0722396 MeV−1 , β = 0.195267 MeV−1 , γ0 = 0.410289 MeV−1 ,

δ = 0.173015 MeV.

(61)

Deviations of the a-parameters in Fig. 24 from the parameters calculated on the basis of the systematics of Eq. (61) are shown in Fig. 25. The root mean square deviation factor frms can be used to estimate the overall deviation with respect to the experimental data for the neutron resonance spacings:

frms

Ne Di 1 ln2 ith = exp Ne i=1 Dexp

1/2 ,

(62)

where Ne is the number of nuclei considered. The global

Composite Gilbert-Cameron Model

Low-lying nuclear levels N (U ) are very important for level density analysis. It was noted many years ago [42, 43] that the observed energy dependence of the cumulative number of levels can be described rather well by an exponential law of the form: N (E) = exp [(E − E0 )/T ] ,

(63)

where E0 and T are free parameters determined by fitting experimental data. N (E) is related to the level density by the equation: ρT (E) =

1 dN = exp [(E − E0 )/T ] , dE T

(64)

in which the parameter T corresponds simply to a nuclear temperature. Since the value of this parameter is assumed to be constant over the energy range considered, Eq. (64) is called the constant temperature model. To obtain the level density for the whole range of excitation energies, Gilbert and Cameron [43] proposed to combine the low-energy region (64) with the high-energy dependence predicted by the previously described Fermi gas model. Such an approach is usually called the Composite Gilbert-Cameron model (CGCM). Many authors have carried out such an analysis [43, 165, 205]. The regular differences of the level densities for even-even, odd and odd-odd nuclei, analogous to the even-odd differences of the nuclear masses, have been already noted in the early systematics of the experimental data. The effective excitation energy U = E − Δ is normally introduced to take this effect into account [43], Δ being the

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corresponding pairing energy in this case: ⎧ δ + δN for even-even, ⎪ ⎨ Z δZ for even Z, Δ= δ for even N , ⎪ N ⎩ 0 for odd-odd,

(65)

where δI is the corresponding phenomenological correction for even-odd differences of nuclear binding energies. Such a definition of the pairing shift was used in the RIPL-1 and RIPL-2 CGCM analyses. However, in the RIPL-3 CGCM [190] we defined the pairing shift based on the average pairing shift √12A as follows, 12 Δ = n√ , A

(66)

where n = 0, 1 and 2 for odd-odd, odd-A, and even-even nuclei, respectively. Unlike the BSFGM, no adjustable pairing shift parameter is used, and the definition of n in Eq. (66) is different from the one used in the BSFGM. This definition was used in some mass formulae and the previous analysis of NLDs [67], and reproduces rather well the averaged behavior of the pairing shifts considered in Refs. [43, 211]. The link between the parameters of the constant temperature and Fermi gas models in the CGCM can be found by imposing the continuity of the level density function and its first derivative at some matching energy EM . This leads directly to the conditions:

and

E0 = EM − T ln [T ρF (EM )] ,

(67)

d ln ρF (E) 1 = . T dE E=EM

(68)

Equation (68) could be solved analytically for all Fermi gas type expressions (including the energy dependent expressions for a, σ 2 , etc.), but we use a numerical method of solution. We determine the inverse temperature of Eq. (68) numerically by calculating ρF on a sufficiently dense energy grid. The matching problem results in two conditions given by Eqs. (67) and (68), with three unknowns: T, E0 and EM . Hence, another constraint is required that is obtained by demanding that the constant-temperature law reproduces the experimental discrete levels from a lower level NL with energy EL to an upper level NU with energy EU . This condition can be written as follows: EU NU = NL + ρT (E)dE, (69) EL

or after inserting Eq. (64): EU − E0 NU = NL + exp T

− exp

R. Capote et al.

a(B


EL − E0 T

. (70)

FIG. 26: Level density parameter a(Bn ) of the Composite Gilbert and Cameron model.

The combination of Eqs. (67), (68) and (70) determines T, E0 and EM . Replacing Eq. (67) in the Eq. (70) yields: NU =NL +

T ρF (EM ) exp

EU − EM T

− exp

(71)

EL − EM T

from which EM can be solved by an iterative procedure with the simultaneous adoption of the tabulated values given by Eq. (68). Nuclear temperatures T and energy shifts E0 in chapter III (Discrete levels) are evaluated for a much larger number of nuclei than included in this chapter. When available, these parameters derived from discrete level information are certainly preferable for level density calculations at low excitation energies well below the binding energy. However, for intermediate energies, values of T and E0 should always be re-estimated in accordance with the matching conditions of Eqs. (67)–(68) and corresponding systematics of the a-parameters for highly excited nuclei. a. CGCM results Analyses of the experimental data within this phenomenological approach was carried out initially in Ref. [43], and the parameters obtained by different authors in the subsequent analyses of experimental data are reported in the RIPL-1 Handbook [6]. The complete set of level density parameters obtained for the Composite Gilbert-Cameron model is given in the RIPL3 density/level-density-gc.dat file, and the format is described in the RIPL-3 level-density-gc.readme file. Values of the level density a-parameters depend to some extent on the determination of the spin cutoff parameter. The early systematics used a value of m2 = 0.146A2/3 , which corresponds to mean-square averaging of the proton and neutron angular momenta projections over all single-particle levels occupied in the ground state

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R. Capote et al.

matching equations for the level densities – Eqs. (67) and (68) – set some limitations on the temperatures and the energy shifts, resulting in differences between the values derived from the matching conditions and the same parameters estimated from the analysis of low-lying levels only. These differences can be seen in the middle plot of Fig. 27, where the temperatures obtained from the analysis of low-lying discrete levels in chapter (III) are shown together with the temperatures derived from the matching equations (67)–(68) for the same nuclei. The CGCM parameter systematics can be expressed using Eqs. (51)– (53). The following values were obtained (in MeV−1 ) α = 0.0692559, β = 0.282769, γ0 = 0.433090.

(72)

As already mentioned above, in the RIPL-3 CGCM approach a systematic (i.e not adjustable) pairing shift Δ given by Eq. (66) has been used. For nuclides with negligible or zero number of discrete levels, we use the empirical expression for the temperature T (in MeV): 9.4 T = −0.22 + , A (1 + γ δW )

FIG. 27: Level density parameters EM , T and E0 for the Composite Gilbert and Cameron model.

of a nucleus [43, 205]. Later analyses used different values such as m2 = 0.24A2/3 . In the RIPL-3 analysis the spin cutoff values were calculated as described in the previous section (in the BSFGM discussion). These differences in the choice of the spin cutoff parameters, as well as some variations in the even-odd corrections to the excitation energies, should be borne in mind while comparing the a-parameters obtained by different authors. The a-parameters derived in the RIPL-3 project are shown in Fig. 26 in comparison with the previous RIPL2 data. Agreement of the RIPL-3 and RIPL-2 results is reasonably good for most nuclei and the observable differences relate mainly to the updated values of the resonance spacings and to the differences in the fitting procedure. Due to the change in the determination of the effective excitation energies, the values obtained for the a-parameter of the BSFGM are somewhat lower then those for CGCM (compare Figs. 24 and 26). However, shell effects in the mass dependence of the a-parameter remain essentially the same. Figure 27 (next page) shows the matching energies EM , the nuclear temperatures T , and the energy shifts E0 . Generally, the RIPL-3 results are close to previous RIPL-1 and RIPL-2 analyses [6, 10]. However, the

(73)

which takes into account the observable shell effects (through δW ) present in the A-dependent plot of the temperature as seen in Fig. 27. We then directly obtain E0 from Eq. (67) and EM from Eq. (68). In a few rare cases, the global expression of Eq. (73) may lead to a matching energy EM that is too high. In such situations we rely for the matching energy EM (in MeV) on the empirical expression EM = 2.33 + 253/A + Δ,

(74)

and then obtain T from Eq. (68). With these global parameters, the frms deviation with respect to experimental D0 is frms = 1.76. Deviations of the “experimental” a-parameters included in the RIPL-3 densities/level-densities-gc.dat file from the global a-parameters calculated on the basis of Eqs. (51)–(53) and systematics of Eq. (72) are shown in Fig. 28. Standard deviation between the “experimental” and global parameters is equal to 0.0613. This means that the global systematics of Eq. (72) describes the available a-parameters with one-sigma uncertainty of 6.1%. Similar uncertainty could be expected in evaluations of the a-parameters for unknown nuclei.

4.

Generalized Superfluid Model

On the basis of all results considered above, we can conclude that the Fermi gas and constant temperature models provide us with comparatively simple and convenient formulae for parametrizing experimental data on nuclear level densities. However, these models do not give any explanation for excitation-energy shifts and shell

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(a -a )/a


FIG. 28: Deviations between the RIPL-3 “experimental” level density parameter a(Bn ) derived from the CGCM resonance spacing analysis and the CGCM global systematics defined by Eqs. (51)–(53) and (72).

dependencies of the level density parameters. An interpretation of these effects must be obtained by more rigorous models that take into consideration shell effects in the single-particle level spectra, and the superfluid and collective effects produced by the residual interaction of nucleons. A detailed discussion of such models can be found in Ref. [210]. However, rigorous microscopic methods for calculating level densities are extremely laborious. This fact limits their application to experimental data analyses. Thus there is a need to develop a consistent phenomenological description of nuclear level densities that takes into account the basic ideas of microscopic approaches to the structure of highly excited nuclear levels, while being sufficiently simple and convenient for broad application. If collective effects are included into consideration of excited level structure [168], the nuclear level density may be expressed as: ρ(E, J) = ρqp (U, J)Kvibr (U )Krot (U ),

(75)

where ρqp (U, J) is the level density due to quasi-particle excitations only, and Kvibr (U ) and Krot (U ) are the corresponding enhancement coefficients due to vibrational and rotational excitations, respectively. To take into account possible shortcomings of the global systematics of the pairing correlation functions and collective enhancement coefficients, an additional shift of the excitation energy δshif t is introduced in the GSM. The effective excitation energy U is related to the excitation energy E by means of the relationship: U = E + nΔ0 + δshif t ,

(76)

√ where Δ0 = 12/ A is taken as the average correlation function of the ground state, and n = 0, 1 and 2 for even-even, odd-A and odd-odd nuclei, respectively.

R. Capote et al.

Quasi-particle excitations are usually described by the relations of superconductivity theory [212]. The phenomenological version of the GSM is characterized by a phase transition from superfluid behavior at low energy [169, 210], where pairing correlations strongly influence the level density to a high energy region which is described by the FGM. Thus, the GSM resembles the CGCM to the extent that the model distinguishes between a low energy and a high energy region, although for the GSM this distinction follows naturally from the theory and does not depend on specific discrete levels that determine a matching energy. Instead, the GSM provides constant temperature-like behavior at low energies. The level density of quasi-particle excitations can be written in the form # " S − (J + 12 )2 2J + 1 √ exp ρqp (E, J) = √ , (77) 2 3 2σef 2 2πσef f f D where the thermodynamic functions S and D have two forms (see below) depending upon whether the excitation energy U is above or below the critical energy Uc , and the 2 effective spin cutoff parameter σef f = ef f t is defined as a function of the temperature t, and the effective moment of inertia ef f . The latter is given as ⎧ for spherical nuclei, ⎨ (78) ef f = ⎩ 1/3 2/3 ( ⊥ ) for deformed nuclei, where parallel and perpendicular ⊥ moments of inertia are defined as 6 at(1 − π2 6 ⊥ = m2 2 at(1 + π = m2

2 β), 3 1 β), 3

(79) (80)

respectively, and β is the quadrupole deformation parameter; where m2 is the average value of the square of the projection of angular momentum for single-particle states on the Fermi surface, which is calculated following the parameterization m2 = 0.24A2/3 , and a is the level density parameter a. The shell inhomogeneities of single-particle level spectra result in a particular form of energy dependence of the level density parameter a(U ): shell effects become weaker with increasing excitation energy, and at sufficiently high energies the dependence of parameter a on the mass number tends to the semi-classical value (44). This important behavior of the level density parameter was already used in defining the systematics of the Fermi gas model parameters (see Eq. (51)). The similar equation for the GSM model [171] can be written as

a ˜(A) [1 + δW f (U ))/U ] for U ≥ Uc , a(U ) = (81) ac (Uc ) for U < Uc , where f (U ) = 1−exp(−γU ), the damping parameter γ is defined by Eq. (53), and ac is the level density parameter

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defined at the critical energy Uc . The spherical liquiddrop model mass MLDM (Z, A, β ≡ 0) expressed in MeV is given by Myers and Swiatecki (MS) [209]. This definition was used to obtain the shell corrections δW for the GSM used to calculate the level density parameter a(U ) following Eq. (81). The derived shell correction values δW are listed in the RIPL-3 density/shellcorr-ms.dat file. The influence of the superconducting pairing correlations on nuclear properties can be characterized by the value of the correlation function Δ0 , which determines directly the even-odd differences in the nuclear binding energies and the energy gap of 2Δ0 in the spectrum of quasi-particle excitations in even-even nuclei. The critical temperature tc of the phase transition from a superfluid to a normal state is connected with the correlation function through the relation tc = 0.567Δ0 .

Uc = ac t2c + Econd ,

that obeys the superfluid equation of state [169]: tc ϕ = tanh ϕ , t

−1 1+ϕ t = 2tc ϕ ln . 1−ϕ

3 ac Δ20 . 2π 2

S = Sc

For energies below Uc , the level density is described in terms of thermodynamic functions defined at Uc , Eq. (83), where the critical level density parameter ac is given by the implicit equation

1 − exp(−γac t2c ) ac = a , (85) ˜(A) 1 + δW ac t2c ˜(A), δW and which is easily solved by iterations, once tc , a γ are given. Equation (85) indicates that shell effects are again appropriately taken into account. The level density ρqp (Uc , J) at the critical energy Uc is determined by invoking the expressions for the critical entropy Sc : S c = 2 a c tc ,

(86)

and the critical determinant Dc : Dc =

144 3 5 a t . π c c

(87)

The parallel crit and perpendicular ⊥crit moments of inertia at the critical temperature tc are calculated using Eqs. (79) and (80), where a = ac . Everything is now specified at Uc , and we can use the superfluid Equation of State (EOS) to define the level density below Uc . To describe the thermodynamic functions for U ≤ Uc we use the subsidiary function ϕ2 = 1 −

U , Uc

(88)

tc tc U 1 − ϕ2 = Sc , t t Uc

(91)

the determinant D:

D = Dc 1 − ϕ

2

2 2

1+ϕ

U = Dc Uc

U 2− Uc

2

, (92)

and the moments of inertia that define the effective moment of inertia ef f (see Eq. (78)) calculated as

(83)

(84)

(90)

The other functions for U ≤ Uc are the entropy S:

where Econd is the condensation energy for the eveneven nucleus that determines a reduction of the nuclear ground-state energy due to the pairing correlations Econd =

(89)

which is equivalent to

(82)

The excitation energy corresponding to the critical temperature, i.e., the critical energy Uc , may be expressed as

R. Capote et al.

tc (1 − ϕ2 ), t

2tc 1 2 (1 − ϕ ) . = ⊥crit 1 + 3 t

= crit

(93)

⊥

(94)

It is necessary to note that the superfluid model predicts the different energy dependencies of the orthogonal and parallel moments of inertia for U < Uc , and this difference is important for calculations of the rotational enhancement of the level densities. Above the critical energy Uc , the level density and other nuclear thermodynamic functions are described by the Fermi gas model Eq. (77) with an energy shift that differs from the pairing correction of the CGCM and BSFGM, and is equal to the condensation energy. Uef f = U − Econd .

(95)

The other functions defining the level density of Eq. (77) for U ≥ Uc are those of the Fermi gas model (see Eqs. (42)), and the moments of inertia are given by Eqs. (79). As result, two adjustable parameters, a ˜ and δshif t , are used in the GSM for each nucleus. In the adiabatic approximation, the rotational enhancement of the level density depends on the nuclear shape symmetry and can be written as [168]

1 for spherical nuclei, Krot (U ) = (96) ⊥ t for deformed nuclei, where the thermodynamic temperature t is calculated following Eqs. (90) or (42), depending upon whether the excitation energy U is below or above the critical energy Uc . Equation (96) should be multiplied by Qdamp rot (U ) if the damping of the rotational enhancement is considered. This formula is valid if the mirror and axial symmetry of a deformed nuclei is assumed. The most stable nuclei

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of the rare-earth elements (150 ≤ A ≤ 190) and the actinides A ≥ 230 are of this shape. For non-axial forms the rotational enhancement of the level density becomes greater [168] (see fission level densities in chapter VIII). The vibrational enhancement coefficient is determined in the microscopic approach [210] by the relationship Kvibr (U ) =

$ i

g exp(−ωi0 /t) i

1− 1 − exp(−ωi /t)

,

1 , 1 + exp[(U − Ur )/dr ]

(97)

(98)

that describes the damping of rotational enhancement factors. Originally it was supposed that the damping parameters could strongly depend upon nuclear deformations. However, the analysis of heavy nuclei fissilities [179, 213] shows that damping is independent of deformation with the corresponding parameters Ur = 40 MeV and dr = 10 MeV. Of course, uncertainties of these parameters are rather large and additional confirmations are required. Nevertheless, it seems that the damping of the rotational enhancement is comparable with the damping of the shell effects in magic nuclei. The vibrational enhancement of the level density can be approximated by the equation Kvibr (U ) = exp[δS − (δU/t)],

modes, and t is the thermodynamic temperature. These changes are described by the Bose gas relationships: (2λi + 1)[(1 + ni ) ln(1 + ni ) − ni ln ni ], δS = i

δU =

(2λi + 1)ωi ni ,

(100)

i

where ωi is the energy of the vibrational excitations, ωi0 is the energy of the corresponding quasi-particle excitation, and gi is the degeneracy of such excitations. The presence of quasi-particle energies in Eq. (97) accounts to some extent for non-adiabatic effects in excited nuclei. Due to symmetry constraints imposed on the nuclear Hamiltonian, the rotational and vibrational excitations become connected in a consistent microscopic approach [210]. As a result, the calculated collective enhancement coefficients are always reduced in comparison to the adiabatic estimation. The adiabatic estimation of Krot (t) increases the nuclear level densities by a factor of 50–100 compared to the calculations based on quasi-particle excitations alone. The increase of level densities due to vibrational excitations will only be appreciable for low-energy excitations with ωi ∼1–2 MeV. Over the previous twenty years, some microscopic models have been developed to consider collective effects in highly excited nuclei. The results of all these models demonstrate the damping of level density enhancement factors with increase of excitation energy. On the basis of the level density calculations within the SU(3) model (oscillator mean field with quadrupole-quadrupole interaction between particles), Hansen and Jensen [175] obtained the empirical function Qdamp rot (U ) =

R. Capote et al.

(99)

where δS and δU are changes in the entropy and excitation energy, respectively, that result from the vibrational

where ωi are the energies, λi are the multipolarities, and ni are the occupation numbers for vibrational excitations at a given temperature. The disappearance of collective enhancement of the level density at high temperatures can be taken into account by defining the occupation numbers in terms of the equation: ni =

exp(−γi /2ωi ) , exp(ωi /t) − 1

(101)

where γi are the spreading widths of the vibrational excitations. This spreading of collective excitations in nuclei should be similar to the zero-sound damping in a Fermi liquid, and the corresponding width can be written as γi = C(ωi2 + 4π 2 t2 ).

(102)

A value of C = 0.0075A1/3 MeV−1 was obtained from the systematics of the neutron resonance densities of medium-weight nuclei [214]. This analysis adopted experimental values for the ω2 energies of the first 2+ excitation when available, otherwise the parameterization ω2 = 30A−2/3 MeV was used. Energies ω3 = 50A−2/3 MeV were adopted for the octupole excitations. Due to the higher energies, the influence of the octupole enhancement is much weaker than for the quadrupole excitations. The set of GSM parameters a ˜(A) and δshif t was obtained from the simultaneous fitting of the cumulative numbers of low-lying levels and neutron resonance spacings, using the same data set as for the CGC and BSFG models. The parameters obtained under the RIPL-2 and RIPL-3 projects are compared in Fig. 29 and can also be found in the corresponding densities/level-densitiesgsfm.dat file. At first glance, the systematics of the level density parameters in terms of the Fermi gas and GSM models appear to be equally valid, since they give almost identical descriptions of the level densities at excitation energies close to the neutron binding energy. However, these descriptions correspond to different absolute values of the level density parameter a, because the inclusion of collective effects decreases the resulting a-parameters. The reduced a-values agree reasonably well with both the experimental data derived from the spectra of inelastically scattered neutrons with energies up to 7 MeV, and with theoretical calculations of the a-parameters for the singleparticle level schemes of a Woods-Saxon potential [171]. This agreement is very encouraging because the evaporation spectra are most sensitive to the level density parameter a. It is impossible to explain the differences

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FIG. 30: Deviations between the RIPL-3 “experimental” level density parameter a(Bn ) derived from the GSM resonance spacing analysis and GSM global systematics defined by Eqs. (81).

FIG. 29: Level density parameters a(Bn ) and δshif t for the generalized superfluid model.

between the a-parameter obtained from resonance data and from evaporation spectra in terms of the Fermi gas model without accounting for the collective effects. Consideration of the level density collective enhancement is also very important for a consistent description of the observed fissile behaviour of highly-excited nuclei [172]. Nowadays, there is overwhelming evidence that the description of the level densities of excited nuclei should use more consistent models than those of the Fermi gas model, albeit inevitably more complex. The success of the generalized superfluid model is attributed to the inclusion of the well-known major components of nuclear theory: pairing correlations, shell effects and collective excitations. Some complexity in the model seems to be justified by the mutual consistency of the parameters obtained from the various experimental data, and by the close relationship between the theoretical concepts used to describe the structure of low-lying nuclear levels and the statistical properties of highly excited nuclei. The individual parameters are preferable for all practical applications. Parametric uncertainties are not important for prediction of the level densities in an intermediate energy region if the experimental data for neutron resonances and low-lying levels have been chosen correctly. Analyses of the evaporation spectra of different particles is of great interest in studies of the nuclear level densities. The energy dependencies of the level densities obtained from the spectrum analysis of various threshold reactions are in good agreement with the calculations based on the individual parameters of the GSM model [214].

On the other hand, many tasks require nuclear level density parameters for which no experimental data are available. Under such circumstances, global parameters may be used. Also, certain localized systematics may be proposed for these parameters that are based on extrapolations of the isotopic or isotonic changes. Experimental data on the cumulative number of low-lying levels can be fitted to one of the individual parameters, which may be advantageous in keeping the global systematics for other parameters. We performed the analysis to obtain global systematics of the GSM parameters. Results compared to experimental data are shown in Fig. 30. Using the Myers-Swiatecki shell corrections [209], the collective enhancement with damping as described above, and Eqs. (51)–(53) for the asymptotic a-parameters, the following coefficients in MeV−1 have been obtained from a GSM least-squares fit of data: α = 0.093 ± 0.004, β = 0.105 ± 0.014, γ0 = 0.375 ± 0.015,

(103)

and δshif t = 0.617 − 0.00164A MeV. Deviations between the a-parameters calculated on the basis of the above global systematics and the resonance spacing analysis are shown in Fig. 30. The standard deviation for the a-parameters is equal to 0.169 and the equivalent factor frms = 1.98. Deviations for the GSM are generally not much larger than equivalent ones for the CGCM or BSFGM. From a more careful analysis of significant deviations an impression arises that the main deviations occur for near-magic nuclei for which the shell effects are so strong that their consistent description is only possible by using microscopic models. Important deviations exist also for transitional nuclei in which the structure of collective excitations is intermediate between vibrational and rotational. A more accurate estimation of collective enhancements is certainly required for such nuclei to avoid the adiabatic separation of vibrational and rotational effects.

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Enhanced Generalized Superfluid Model

The Enhanced Generalized Superfluid Model [215] includes a more accurate treatment of high angular momenta, which are important for heavy-ion induced reactions. The properly parametrized EGSM (including adjustment to discrete levels) is the default level density formulation in the EMPIRE code [45] for calculations of nucleon induced reactions at fairly low energies; therefore, it is also referred to as “EMPIRE-specific level densities”. Despite the strong link to heavy-ion physics, EGSM has been extensively used in the evaluation of low-energy neutron-induced reactions for the ENDF/BVII.0 nuclear data library [216]. Similar to other phenomenological approaches, application to nuclei far from the stability valley is questionable. Enhancement compared to GSM relates to the treatment of the spin distribution in the Fermi gas model. The rotational energy in the EGSM is subtracted from the intrinsic excitation energy. This contrasts with the treatment in the models discussed earlier, in which the spin dependence is treated as a separate factor characterized by a spin cutoff parameter. The collective enhancement of the level densities arising from nuclear rotation is taken into account in the non-adiabatic form. The vibrational collective enhancement is calculated in the adiabatic approach. Vibrational energies are estimated from the liquid drop model assuming surface oscillations of the liquid drop. The formalism, inspired by the GSM, uses the superfluid model below the critical excitation energy Uc of Eq. (83), and the Fermi gas model above. The excitation energy U in the EGSM is related to the excitation energy E by means of the relationship: U = E + nΔ0 ,

(104)

√ where Δ0 = 12/ A is taken as the average correlation function of the ground state, and n = 0, 1 and 2 for eveneven, odd-A and odd-odd nuclei, respectively. Differing from other formulations, the EGSM accounts explicitly for the rotation-induced deformation of the nucleus which becomes spin dependent. Deformation enters the level density formula through the moments of inertia and level density parameter a that increases with increase in the surface of the nucleus [45]. However, such an increase of the level density parameter is negligible for the relatively small deformation typical of the ground state and will not be discussed here. Assuming that prolate nuclei, being quantum objects, rotate along the axis perpendicular to the symmetry axis, the explicit level density formula is ρ(E, J, π) =

1 √ 16 6π

J 2 K 2 U− 2 ef f

K=−J

2

−5/4

1/2

a−1/4 Qdamp rot (U )× (105)

' % & 2 K 2 , exp 2 a U − 2 ef f

R. Capote et al.

where a is the level density parameter, J is the nucleus spin, and K is the projection of the nucleus spin on the symmetry axis. In the case of oblate nuclei, we assume rotation parallel to the symmetry axis although, strictly speaking, it is not allowed in quantum mechanics. However, it was shown in Ref. [217], that for a Fermi gas the moment of inertia calculated with respect to the axis of symmetry is appropriate for calculating the rotational energy. The same approach has also been adopted in Ref. [218]. Thus for oblate nuclei we have 2 1/2 1 √ ρ(E, J, π) = a−1/4 Qdamp rot (U ) × (106) 16 6π " #−5/4 J 2 J (J + 1) − K 2 U− × 2| ef f | K=−J ' % & 2 [J (J + 1) − K 2 ] . exp 2 a U − 2| ef f | Above the critical energy Uc , the level density is described by the equations above, with an energy shift that is equal to the condensation energy Uef f = U − Econd ,

(107)

in which the condensation energy Econd is given by Eq. (84). The effective moment of inertia ef f is defined in terms of the perpendicular and parallel ⊥ moments through the difference of their inverses: 1 1 1 = − . ef f ⊥

(108)

Equations (105) and (106) include a summation over projection of the angular momentum K, and thus account automatically for the rotational enhancement. The rotational enhancement in the EGSM (as in the GSM) includes the damping parameterization Qdamp rot (U ) given by Eq. (98). The yrast line is obtained, setting level densities to 0 whenever the rotational energy becomes larger than U . The model also accounts for the vibrational enhancement; Eqs. (105) and (106) are multiplied by Kvib . The vibrational enhancement is calculated using the liquid drop parameterization of vibrational modes (see p.75 of Ref. [210]). In this model the surface oscillation energy of the nucleus is determined by the relation ωλ2 =

2 c2 α λ(λ − 1)(λ + 2), ρ0 R03

(109)

where λ is the multipolarity of the surface oscillations, 4πr 3 ρ0 = m0 /( 3 0 ) is the density of the nuclear matter, m0 = 939 MeV is the nucleon mass, R0 = r0 A1/3 is the nuclear radius, and r0 = 1.26 fm. The coefficient of surface tension α is obtained as a2 , (110) α= 4πr02

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FIG. 32: Ratio of the experimental s-wave resonance spacings D0 to the predictions of the global EGSM systematics as a function of the neutron number N .

parametrized in the empirical form: Qvib (t) =

FIG. 31: Mass dependence of the asymptotic level density parameter a ˜ for the EGSM vs the GSM results. The EGSM values were obtained by normalizing the localized EGSM systematics to reproduce each experimental a(Bn ).

where a2 ≈ 17 MeV is the assumed phenomenological surface parameter of the MS liquid drop mass formula [209]. To simplify the analysis, we have omitted in Eq. (109) the terms associated with the Coulomb energy of the charged drop. From the statistical sum of the surface oscillations, Eq. (99), we find [210] ' % 2/3 ρ0 R03 LDM 4/3 Kvib (U ) = exp C4/3 Qvibr (t), t 2 c2 α (111) where C4/3 ≈ 1.694, t is the thermodynamic temperature obtained from the corresponding equation of state, and Qvibr (t) defines the damping of the vibrational enhancement with an increase of the temperature. The LDM above liquid drop expression Kvib (U ) was preferred over the seemingly more advanced formulations, such as Eqs. (100)–(102), since the liquid drop approach provides lower values of the vibrational enhancement factor, supported by semimicroscopic QRPA calculations of the level density using the quasiparticle-phonon model [219]. LDM However, the Kvib estimate of Eq. (111) may be on the low side since consideration of the Coulomb energy of the drop reduces the surface oscillation energy (109) LDM and consequently increases Kvib [210]. Further investigation of the vibrational enhancement and damping is very desirable. The vibrational damping Qvibr (t) was

1 , 1 + exp [(t − tvibr )/dvibr ]

(112)

with tvibr = 1 MeV and dvibr = 0.1 MeV. These constants are to a certain extent arbitrary, since there are no reliable global estimates deduced from experiments or theory. The parameters α, β and γ0 defining a ˜(A), and therefore the level density parameter at the binding energy a(Bn ), were obtained by fitting average s-wave neutron resonance spacings Dobs compiled in RIPL-3 with the level density formulae Eqs. (105) through (112). The original Myers-Swiatecki shell-corrections [209], listed in the RIPL-3 density/shellcorr-ms.dat file, were adopted. The derived a ˜ are shown in Fig. 31 (see previous page) in comparison with similar results of the GSM analysis. Differences between the a-parameters obtained are small and related to the different parameterization of the vibrational enhancement of the level densities. It should be noted that the CERNLIB code MINUIT was employed to minimize the frms as defined by Eq. (62). We prefer frms over the typical χ2 minimization since the latter tends to be oversensitive to the outliers. This approach, however, makes parameter uncertainties provided by the MINUIT meaningless. The resulting global EGSM systematics is represented by the following set of parameters (in MeV−1 ): α = 0.0741, β = 0.0003, γ0 = 0.5725.

(113)

These parameters yield frms = 1.70. The corresponding value of χ2 is 36 per degree of freedom and could be reduced by a factor of two if χ2 were minimized instead of frms . In the latter case, however, the resulting Dobs , driven by a few reportedly very accurate measurements, would be considerably higher. By minimizing frms , we

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a

a


FIG. 33: Level density parameter a(Bn ) at the neutron binding energy. Predictions of the global EGSM systematics (blue squares) are compared with the experimental data (red points with error bars).

choose to rely on the bulk of experiments rather than on a few with very small uncertainties. Figure 32 shows the ratio of the experimental Dobs to the s-wave spacings predicted by the global EGSM systematics. One notes only slight indications of the shell closure effects. No discernible trends were observed when the same ratio was plotted as a function of the target spin, neutron binding energy, shell correction, or deformation. Values of the level density parameter at neutron binding energy a(Bn ) are compared against experimental data in Fig. 33. The systematics describes adequately the shell structure, but tends to underestimate scatter of the experimental points for the deformed nuclei, especially for the actinides. Notable features of the EGSM parameterization are the vanishing role of the nuclear surface term (β parameter is negligible compared to α in Eq. (52)), and the linear dependence of “experimental” asymptotic a˜ values on mass number A (˜ a ≈ 0.0741A = A/13.5). The derived asymptotic value of the level density parameter is very close to the theoretical value of the Fermi gas model of Eq. (44); the complete absence of the shell effects in the mass dependence of a ˜ is a strong argument in favor of the collective enhancements and shell corrections adopted in the EGSM. Contrary to other level density models considered in RIPL-3, the EGSM global systematics does not account for discrete levels. Instead, the adjustment is performed automatically by the EGSM code (provided in RIPL-3) when level densities are calculated. A shift is applied to the excitation energy to reproduce the cumulative number of levels at the energy corresponding to the highest level considered in the calculations. This shift is linearly decreased with increasing energy in such a way as to reach zero at the neutron binding energy. Therefore, adjustment to discrete levels never changes level densities at and above the neutron binding energy, ensuring that the global EGSM systematics of Eq. (113) is independent of the number of adopted

FIG. 34: Level density parameters a(Bn ) for selected isotopic chains. The results of the global (blue lines) and localized (green lines) EGSM systematics are compared with the experimental values (red dots).

discrete levels. We stress, however, that level densities below the neutron binding energy strongly depend on the selection of discrete levels; thus the user is advised to inspect carefully the cumulative plots generated by the EGSM code to ensure that a proper number of discrete levels is included in the calculations. In a default run, the EGSM code included in RIPL-3 searches an internal file (densities/level-densities-egsm.dat) for the “experimental ” values of the a-parameter derived from Dobs during the fit. These results are given priority over the systematics. The default treatment of the remaining nuclei in the EGSM code also goes beyond the global systematics. It has been observed that a(Bn ) for a given element reveals rather smooth dependence on the neutron number as shown in Fig. 34. These shapes for individual elements are usually quite well approximated by the global systematics while the absolute values are predicted with lesser accuracy. The global systematics can be improved by applying normalization factors defined for each element for which experimental a(Bn ) are available. The localized level-density systematics obtained for EGSM is given in the densities/leveldensities-egsm-norm.dat file. Results obtained with such localized EGSM systematics are plotted in Fig. 35 showing an improvement in comparison with Fig. 33 (previous page), especially for actinides (N > 126), and near the extremes of the distribution at neutron numbers N ∼ 50, 64, 82 and 94–110. The localized systematics accounts for an unknown Z-dependence that is either not considered in the shell corrections or included improperly. This improvement comes at the cost of an additional 83 parameters, but maximizes experimental input in estimating level densities for nuclei for which experimental resonance spacings are not available. a. EGSM parameter uncertainties The uncertainties in Dobs as given in the resonance segment of RIPL-3 are, for convenience, reproduced in the densities/level-densities-egsm.dat file. Due to the

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a


FIG. 35: Level density parameter a(Bn ) at the neutron binding energy. Predictions of the localized EGSM systematics (blue squares) are compared with the experimental data (red points with error bars).

strongly non-linear (actually exponential) dependence of level densities on the level density parameter a, the initially symmetric uncertainty in the Dobs becomes asymmetric when propagated into the level density parameters (see densities/level-densities-egsm.data file). The average value for the “plus” uncertainties of a(Bn ) (3.1%) is about 30% larger than the corresponding “minus” value (2.3%). The individual a(Bn ) uncertainties are scattered between 0.25% and 16%. The average ratio of the experimental a(Bn ) to the global EGSM systematics is 0.993 with a standard deviation of 7.5%. The maximum discrepancy between the experimental a(Bn ) and global systematics is 30%. The predictions are somewhat more accurate with the localized EGSM systematics, which actually involves 86 instead of 3 fitted parameters. The average ratio of the experimental a(Bn ) to the localized EGSM systematics is 0.99993 with the standard deviation of 5.6%. The overall improvement of 2% compared to the global systematics is not an impressive gain considering the dramatic increase of the number of fitting parameters. However, the main advantage of the localized systematics is the reduction of the artificial discontinuities between neighboring isotopes if some of them are taken from experiment and other are calculated with the systematics. The statistical model calculations are highly sensitive to the ratios of level densities. The maximum deviation of the localized systematics from the experimental a(Bn ) reaches 21% for the two silicon isotopes (29 Si and 30 Si). For the lighter of the two isotopes this discrepancy is practically within the the experimental uncertainty of Dobs , while for the heavier isotope it is outside the limit suggested by the 46% uncertainty of the Dobs for 30 Si. When calculating cross section covariances with nuclear reaction codes, it is recommended to use for the uncertainty of the level density parameter the lesser of the experimental uncertainty and the uncertainty of the systematics (5.6% for localized and 7.5% for global). In

R. Capote et al.

principle, the asymmetry of the experimental uncertainty should be considered. For nuclei for which Dobs are not available, use of the systematics uncertainties is suggested as long as the nuclei are reasonably close to the valley of stability. Even in such a case, one can argue that the systematics uncertainties should be increased to reflect the actual extrapolation of the results. A value of 10% might be reasonable if the extrapolation is two or three mass units outside of the experimentally known region. Further extrapolation outside the stability valley is questionable and any uncertainty guidance would be unreliable. An additional rather difficult to quantify source of uncertainty in the level densities is the adjustment to the discrete levels. The results below the neutron binding energy are strongly dependent on the number of discrete levels used in the calculations. Obviously, this number is a discrete quantity, which complicates application of standard statistical methods. Monte Carlo calculations accounting for several acceptable choices of level schemes are recommended to deal with this problem.

B.

Microscopic total level densities

Most semi-empirical approaches to level densities such as those described in the previous section are based on various simplifying approximations. Such approaches often account inadequately for shell effects, pairing effects and parity distributions. More involved microscopic methods have been developed to address these deficiencies and calculate realistic level densities using the singleparticle level scheme of the shell model, taking into account both short-range (pairing) and long-range (collective) interactions. The advent of high speed computers has encouraged the use of numerical calculation methods which do not depend on the analytical closed-form expressions. An exact method of computing nuclear level densities directly from a set of single-particle states has been proposed by Williams [220], which is based on the repetitive use of recursion relations to expand the grand partition function and does not rely on the saddle-point approximation, unlike the traditional statistical method. Unfortunately, this method cannot treat residual interactions and therefore is only useful in the non-interacting Fermi gas model. A completely different approach for level density calculations is the shell model Monte Carlo method (SMMC) [221, 222], which potentially could consider in an exact manner all possible correlation effects. Correlation effects are taken into account in this method by including fluctuations of the mean field through the Hubbard-Stratonovich transformation, which permits decomposition of the many body operator exp(−βH), considered as an imaginary-time propagator, in a set of onebody propagators of non-interacting particles moving in fluctuating auxiliary fields. The SMMC method has been

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extensively applied by Alhassid and coworkers [202] to the calculation of level densities of nuclei in the mass and energy range that can be described in a model space limited to the f pg9/2 major shell, i.e. nuclei in the 50 < A < 70 region, up to excitation energies of the order of 30 MeV. This is accomplished by using a residual fermion interaction made of an isovector pairing interaction and a number of isoscalar multipole-multipole interactions, which are all attractive, thus yielding a good-sign Hamiltonian. The SMMC method has been extended recently to treat even-even nuclei in the rare-earth region A ∼ 150 (i.e., the model space extended to the f pgh11/2 shell), but at the expense of significantly increased computation times. Unfortunately, the SMMC method in its current formulation can not be applied to odd nuclei. At present, SMMC methods for level density calculations for practical purposes are still at an early stage of development [202, 223]. During the RIPL project we studied several microscopic methods for level density calculations. The Microscopic Generalized Superfluid Model is a statistical approach to calculate level densities based on realistic single particle-level schemes. It was considered during the RIPL-1 and RIPL-2 projects and is described in section VI B 1 a below. Another realistic statistical approach based on the Extended-Thomas-Fermi plus StrutinskyIntegral model [183] was also introduced within the RIPL-2 project and used to produce level density tables for nearly 8000 nuclei [183] as described in VI B 1 b. Finally a new combinatorial approach [199, 200, 203] was developed within RIPL-3 which goes beyond usual statistical approximations such as the Gaussian spin distribution or the parity equipartition of excited states. The new combinatorial method has also been used to generate tables of level densities for more than 8000 nuclei and will be reviewed in section VI B 2.

R. Capote et al.

Fermi gas model) may be written in the form S= gi [βEi ni + ln(1 + exp(−βEi )] , i

1 U= gi 2 i

gi Ei (1 − ni ) +

1 1 − 2ni gi , 4 i Ei 1 i − λτ Nτ = gi [1 − (1 − 2ni )], 2 i Ei

(115) (116)

where Nτ is the number of protons or neutrons in a nucleus, and Gτ is the pairing force constant. For t = 0 Eqs. (115) and (116) determine the proton and neutron correlation functions Δ0τ for the ground state of a nucleus. For given schemes of single-particle levels Eqs. (114)– (116) enable calculation of the thermodynamic functions and the nuclear level densities without any additional parameter. Differences between the behavior of the thermodynamic (114) and the Fermi gas functions (42) can be traced by determining the following functions:

m2 =

a. Microscopic Generalized Superfluid Model The description of the level densities and other statistical characteristics of excited nuclei can be obtained from calculations performed with realistic schemes of the single-particle levels. Such calculations are considered in detail in Refs. [183, 207, 210]. The thermodynamic functions of an excited nucleus (similar to Eq. (42) of the

Δ20τ − Δ2τ , Gτ

G−1 τ =

Statistical microscopic approaches

As mentioned above, two statistical microscopic models were considered during the RIPL-1 and RIPL-2 projects – the Microscopic Generalized Superfluid Model explained in section VI B 1 a and the HFBCS statistical approach in section VI B 1 b.

(114)

where Δ0τ (Δτ ) is the correlation function of the ground (excited) state, λτ is the corresponding chemical potential, β = 1/t is the inverse temperature, Ei are the energies of quasiparticle nuclear excitations, ni = [1 + exp(−βEi )]−1 are the occupation numbers for the corresponding single-particle levels, gi is the degeneracy of these levels, and the sums for i run over all singleparticle levels for both protons and neutrons (index τ ). The quasiparticle energies Ei = [(i − λτ )2 + Δ20τ ]1/2 are related to the single-particle energies i and the correlation function Δτ by the BCS equations:

a= 1.

(i − λτ )2 + Δ20τ −

π2 β gi ni (1 − ni ), 6 i

π2 2 β mi gi ni (1 − ni ). 6a i

(117) (118)

These relations are so chosen that in the continuousspectrum approximation all the quantities in Eqs. (117) and (118) are transformed into the Fermi gas model level density parameters: a into a of Eq. (44), and m2 into m2F of Eq. (45). Calculated level density parameters, Eqs. (117) and (118), for low excitation energies reproduce rather well the shell variations of the Fermi gas model parameters observed in experimental data [210]. At high excitation energies (above 50 MeV) the mass number dependence of the calculated parameters is very close to the semiclassical one of Eq. (44). Codes for microscopic calculations of the nuclear level densities based on the statistical model are included

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• deformation effects taken into account in the singleparticle spectra,

in the RIPL-1 and RIPL-2 libraries. Collective effects are also included in these codes using the same approximations as for the phenomenological generalized superfluid model. The single-particle level schemes of M¨ oller et al. [10, 16] are recommended for such calculations because they were also used to determine the recommended nuclear binding energies, shell corrections and deformations (see chapter II). Therefore, their application to the level densities provides the desired consistency between the ground and excited states. b. The HFBCS statistical approach An alternative description of nuclear level densities has been proposed in Ref. [183], based on the ExtendedThomas-Fermi plus Strutinsky-Integral model for the ground-state properties (single-particle level schemes and pairing strengths). Although this approach represents the first global microscopic formula that could reasonably reproduce the experimental neutron resonance spacings, some large deviations have been found (for example, in the Sn region). Additionally, this approach uses statistical theory so fluctuations typical of the level density at low energies could not be described. Some of those deficiencies have been removed in the RIPL-2 HFBCS-based model [188], which predicts all the experimental resonance spacings with a accuracy comparable to the equivalent data obtained by the phenomenological RIPL-2 BSFGM formula (frms = 2.14 for the ratios Dth /Dexp). This microscopic model is based on the HFBCS ground-state description [188] as characterized by a nucleon effective mass close to the real mass, a property of particular importance for reliable level density evaluations that provides a good description of the single-particle level density near the Fermi surface. HFBCS quantities relevant to the level density calculations can be found in the RIPL-2 web site (densities/singleparticle-levels/spl-hfbcs/ directory), including the deformation parameters, the single-particle level schemes (energy, parity, and spin), pairing strengths and the corresponding cut-off energies for both the neutron and proton systems. The HFBCS single-particle schemes and deformations are used in addition to the renormalized pairing strength to estimate the spin-dependent level density within the statistical approach. All details of the level density calculations and predictions can be found in Refs. [10, 188]. This model includes • BCS pairing (constant-G approximation) with a renormalized strength and blocking effect for oddmass and doubly odd nuclei, • Gaussian-type spin dependence with microscopic shell and pairing effects on the spin cutoff parameter, • collective contribution of the rotational band on top of each intrinsic state, and the disappearance of collective enhancements at increasing excitation energies,

R. Capote et al.

• improved description of the cumulative number of nuclear levels at very low energies, • reliable description of the level densities at high energies.

2.

Microscopic combinatorial level densities

Two previously described microscopic approaches are limited at low excitation energy, where the saddle-point approximation breaks down and non-statistical features are known to be significant. A truly combinatorial method can provide NLDs as a function of the excitation energy, spin and parity without any a-priori assumption on the spin and parity distributions for all the nuclei of the periodic table. Many combinatorial approaches involving exhaustive counting of particle-hole configurations have been published [191]–[196, 200, 201]. A Combinatorial Monte Carlo technique has also been proposed by Cerf [197, 198] to avoid exhaustive counting of the excited levels. However these combinatorial methods were focused on obtaining the level density of intrinsic excitations. A microscopic combinatorial approach initially described in Refs. [199, 200] was developed further during the RIPL-3 project to include both collective effects and improved pairing correlations and led to the improved results of Ref. [203] through the use of a better treatment of the residual pairing interaction. This method consists of using single-particle level schemes obtained from the constrained axially symmetric Hartree-Fock-Bogoliubov (HFB) method based on the BSk14 effective Skyrme force [18] to construct incoherent particle-hole (p − h) state densities ωph (U, M, π) as functions of the excitation energy U , the spin projection M (on the intrinsic symmetry axis of the nucleus), and the parity π. It is worth mentioning that this HFB+BSk14 method is also used to provide the microscopic masses, the fission barriers and fission level densities in the whole RIPL-3 project, thus ensuring a global coherence for the microscopic ingredients employed for nuclear reaction calculations. Once these incoherent p − h state densities are determined, we must account for collective effects. The choice of multiplying the level densities by the phenomenological vibrational enhancement factor was made in Ref. [203], after accounting for rotational motion, if necessary (i.e. for deformed nuclei). The resulting NLDs were found to reproduce rather well the available experimental data. However, the phenomenological treatment of vibrational effects had to be improved, and feedback from fission cross-section calculations [224] also suggested the need to address the same issues. Indeed, the phenomenological vibrational enhancement factor is simply defined as a multiplicative factor; theoretically, a folding between incoherent and co-

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ω2 [MeV] = 65A−5/6 /(1 + 0.05Eshell ),

R. Capote et al.

8 Beta Gamma Sph Approx.

7 6 5 4

2

herent state densities should be performed. Therefore, the formalism of Ref. [203] has been modified [225] by introducing the boson partition function as described in the Ref. [200]. This boson partition function provides us with vibrational state densities ωvib (U, M, π) which depend on the various phonon energies that we account for. Whereas in Ref. [200] the use of the boson partition function was restricted to nuclei for which enough experimental data on the quadrupole and octupole vibrational modes were available, we now deal with such modes even when they are not experimentally known. Consequently, we have adjusted three analytical expressions based on sets of experimentally tabulated vibrational levels in order systematically to provide our boson partition function with quadrupole, octupole, as well as hexadecapole phonon energies. These expressions are as follows:

[MeV]


3 2 1 0

8

μ=0 μ=1 μ=2 μ=3

6

Approx.

(119)

(120)

for the octupole excitations. The hexadecapole mode can be expressed relative to the quadrupole mode [210], leading to a similar expression: ω4 [MeV] = 160A−5/6 /(1 + 0.05Eshell ).

3

ω3 [MeV] = 100A−5/6 /(1 + 0.05Eshell ),

[MeV]

for the quadrupole vibrations, and

4

(121)

The shell correction energy Eshell is determined as in Ref. [226], i.e., Eshell = Etot − ELDM where Etot is the experimental total binding energy [15], or theoretically as described in Ref. [18] if not available experimentally and ELDM is the phenomenological binding energy of the spherical liquid drop given by the equation:

2

0 0

20

40

60

80

100 120 140 160

Neutron number N

ELDM = av A + as A2/3 + 1/3

(asym + ass A

)A

N −Z 2 A

(122) 2

1/3

+ ac Z A

.

An optimized fit to the N, Z ≥ 8 experimental masses of Audi et al. [15] leads to a final rms deviation of 3 MeV with liquid drop parameters (expressed in MeV) av = −15.6428, as = 17.5418, asym = 27.9418, ass = −25.3440 and ac = 0.70. As shown in Fig. 36, the expressions for ω2 and ω3 describe the experimental values quite reasonably. Using these expressions, we can compute the vibrational state density for any nucleus, which is then folded in with the incoherent particle-hole state density, as suggested in Ref. [189]. This folding procedure corresponds to the well known adiabatic approximation and implies that no coupling occurs between the vibrational excitations and the incoherent particle-hole excitations. It also presents the advantage of enabling the introduction of purely vibrational states in the pairing gap of even-even nuclei (i.e. between the nucleus ground-state and the first particle-hole excitation), a feature which cannot be obtained with a simple multiplicative enhancement factor Kvib . However, the introduction of the boson partition

FIG. 36: Comparison between the analytical expressions (red dots) of Eq. (119) for the quadrupole 2+ (top panel) and Eq. (120) for the octupole 3− (bottom panel) phonon energies and the corresponding experimental values (black symbols) taken from Ref. [39]. In the top panel, the labels Beta and Gamma are used for quadrupole 2+ phonons in deformed nuclei; while the label Sph corresponds to quadrupole 2+ phonons in spherical nuclei. In the bottom panel, the labels μ corresponds to octupole 3− phonons; μ = 0, 1, 2, 3 is the spin projection of the phonons.

function [200] to calculate the vibrational state density has a disadvantage: the limitations imposed on the number of multiphonon states by the Pauli principle. Because of violation of the Pauli principle, spurious states occur among the multiphonon excitations, and their number rapidly increases with the number of excited phonons. Moreover, in complex multiphonon states it is important to take into account the changes in phonon structure due to the weakening of the pair correlations, and the increasing admixture of particle-hole states eventually leading to the damping of the collective motion. Rigorous consideration of all these effects is an extremely complex problem, to which no satisfactory solution has yet been found. An

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102

The level density for deformed nuclei is then ρdef (U, J, π) =

1

10

1 2

J

J,K ωint (U − Erot , K, π) +

K=−J,K =0

J,0 [δJ,even δπ,+ + δJ,odd δπ,− ] ωint (U − Erot , 0, π). (124)

th exp

D /D

R. Capote et al.

100

10-1

10-2 50

100

150

200

250

A FIG. 37: Ratio of the HFB combinatorial (Dth ) to the experimental ( Dexp ) s-wave (open squares) and p-wave (full circles) neutron resonance spacings compiled in RIPL-2 [10].

empirical solution is to restrict the number of phonons that can be coupled to one another. Several tests have been performed to optimize the reproduction of experimental mean s-wave resonance spacing D0 . A maximum number of three coupled phonons folded in with p − h configurations with a total number of particles lower than or equal to three results in a fairly good reproduction of D0 (Bn ), as shown in Fig. 37. However, it has to be noted that the chosen maximum number of phonons (3), and the corresponding cutoff in the number of folded p − h configurations is arbitrary. This is certainly a weakness of the method which needs improvement. Further investigations of the vibrational enhancement and its damping are certainly needed. Once the folding is performed, one has to compute level densities (as a function of the nucleus spin J) out of the resulting intrinsic state densities. If the nucleus under consideration displays spherical symmetry, the level density is trivially obtained through the relation ρsph (U, J, π) = ωint (U, M = J, π)−ωint (U, M = J +1, π), (123) where ωint is the state density obtained after performing the folding. For deformed nuclei, rotational motion has to be explicitly treated. For an axially and mirror symmetric nucleus, rotation takes place around an axis perpendicular to the nucleus symmetry axis. In this case, any intrinsic state of specified spin projection K and parity π is the band head of a set of levels having the same parity and spins J = K, K + 1, K + 2, ... if K = 0, and J = 0, 2, 4, ... or 1, 3, 5, ... if K π = 0+ or 0− , respectively. These sequences of levels form rotational bands in which the energy of each member can be deduced from J,K the bandhead energy, if the difference Erot between the π energy of the level J and that of the bandhead state K π is known.

Within the right-hand side term of Eq. (124), the factor 1/2 accounts for the fact that in mirror axially symmetric nuclei, the intrinsic states with spin projections +K or −K give rise to the same rotational levels. Moreover, in the second and third terms of the summation, the symbol δi, j (defined by δi, j = 1 if i = j and 0 otherwise) restricts the rotational bands built on intrinsic states with spin projection K = 0 and parity π to the levels sequences 0, 2, 4, ... for π = + and 1, 3, 5, ... for π = −. Finally, the rotational energy is obtained with the well-known expression J,K Erot =

J(J + 1) − K 2 , 2J⊥

(125)

where J⊥ is the moment of inertia of a nucleus rotating around an axis perpendicular to the symmetry axis. In the present approach, J⊥ is approximated by the rigidbody value J⊥rigid which is " # 2 5 rigid 2 β2 , (126) = mR 1 + J⊥ 5 16π for an ellipsoidal shape with axial quadrupole deformation parameter β2 . Globally, the D0 values are predicted within a factor of two. As illustrated in Fig. 37, the D1 values are also fairly well described. The HFB combinatorial model also gives satisfactory extrapolations to low energies. As an example, the predicted cumulative number of levels N (U ) are compared in Fig. 38 (next page) with the experimental data on low-lying levels for the same 15 nuclei as presented in Ref. [203], including light as well as heavy species, and spherical as well as deformed species. The value of the factor frms given by Eq. (62) found with the present approach is equal to 2.3 including both the s- and p-wave resonance spacing data. This result is to be compared to the frms = 1.68 deviation of the global BSFG model described in section VI A 2, with the frms = 1.76 deviation of the global Composite Gilbert and Cameron model described in section VI A 3, with the frms = 1.98 deviation of the global generalized superfluid model described in section VI A 4 , or with the frms = 1.70 deviation of the global EGSM described in section VI A 5. The present approach therefore gives rather comparable predictions with respect to the other existing global models. The LD overestimations for deformed nuclei seen in Fig. 38 at low energies (see rareearth and actinide nuclei) are a direct consequence of using the above J⊥rigid moment of inertia of Eq. (126) for calculations of the rotational enhancement of the level

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FIG. 39: Parameters p and c of Eq. (127) obtained in the fitting of the HFB+BSk14 model to the experimental data on the low-lying levels and the neutron resonance spacings. FIG. 38: Comparison of the cumulative number of observed levels (thin red staircase) with the HFB combinatorial predictions (thick black lines) as a function of the excitation energy U for a sample of 15 nuclei. Only for 208 Pb, both curves have been shifted by 2.5 MeV, the energy range corresponding consequently to [2.5-6.5] MeV instead of [0-4] MeV.

density. The experimental values of the moment of inertia for low-lying rotational bands correspond to about three times lower values than the rigid-body value; a more accurate approximation of the moments of inertia could substantially improve the description of the experimental data for low-lying levels using the HFB combinatorial model. For the phenomenological models of section VI A, the fit to the available experimental data was made by the adjustment of several model parameters. Similar fits can be achieved with the microscopic level densities if the tables provided are renormalized using the scaling function ρ(E, J, π) = exp(c E − p)ρHFB (E − p, J, π), (127) where ρHFB is the original HFB level density, and by default c = 0 and p = 0 (i.e., ρ(E, J, π) = ρHFB (E−p, J, π)). The “pairing” shift p simply means retrieving the level density from the table at a different energy while the constant c plays a role similar to that of the level density parameter a of phenomenological models. Adjusting p and c together gives flexibility at both low and higher energies, so that, e.g., both discrete levels and experimen-

tal mean resonance spacings can be reproduced as well as possible. For nuclei with unknown resonance spacings, c is set to 0 and p is the only parameter enabling a fit to the discrete levels. The values of these parameters obtained for nuclei contained in the RIPL-2 database of neutron resonance spacings are shown in Fig. 39. For some nuclei very large deviations occur, which are probably due to an insufficiently accurate description of the single-particle level schemes and the properties of calculated ground states, but on the average the description is reasonable. Moreover, the fact that c (on average) remains close to zero (as shown in Fig. 39) indicates that the normalization does not significantly modify the energy dependence of the combinatorial method. The microscopic model has been renormalized to experimental data (nearly 300 neutron resonance spacings and 1200 low-lying level schemes) to account for the available experimental information and can consequently be used for practical applications. The HFB combinatorial level density tables are provided in the RIPL-3 directory densities/level-densities-hfb, and are not normalized. They are contained in zXXX.tab files, and the normalization c and p parameters are listed in corresponding zXXX.cor files. Such structure offers the possibility to modify normalization parameters (c and p), as is usually done with analytical level-density parameters to fit some experimental reaction data. As already mentioned, interesting features of these mi-

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181

Ta(,n)180Ta

[mb]

100

10 181

1

m

Ta(,n)180Ta

IAEA: Lee et al. 1998 Utsunomyia et al. (2003) Goko et al. (2006)

0.1 7

8

9

10

11

12

13

14

E [MeV] FIG. 40: Isomeric (red lines) and total (blue line) production cross sections of 180 Ta with a statistical spin distribution (red dotted line), and with the HFB combinatorial approach (red solid line).

croscopic level densities are their non-statistical nature. As explained in Ref. [227], accounting for deviations from a Gaussian spin distribution or from parity equipartition can have a large impact on cross section calculation. For instance, the impact of the non Gaussian spin distribution on production cross sections of the meta-stable state of 180 Ta is illustrated in Fig. 40. It is clear that using the microscopic combinatorial results for the spin distribution (full line), which significantly differs at low energy from the statistical Gaussian spin approximation (dotted line), gives a much better agreement for the isomeric production cross sections. It should be noted that the total (isomer+gs) cross section is not modified. This result is explained by the fact that the combinatorial treatment predicts more high-spin states above the 9− spin of the isomeric level of 180 Ta thus favoring the decay to this level with respect to the equiparity distribution assumed in the conventional statistical treatment. The HFB microscopic level densities are provided in table format for about 8000 nuclei with 8 ≤ Z ≤ 110 lying between the proton and the neutron drip lines, and can be found in the RIPL-3 densities/level-densities-hfb directory. Each table includes the spin-dependent level densities at energies up to U = 200 MeV and spin up to J=29 (59/2) for each isotope considered. The nuclear temperature, cumulative number of levels and total level and state densities are also included in these tables. Level densities at the inner and outer fission saddle points are also estimated using this method for some 2300 nuclei with 78 ≤ Z ≤ 120; details of the corresponding calculations can be found in chapter VIII.

C.

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numerous theoretical methods have been developed to determine partial level densities (PLD), and a variety of approaches have been used in pre-equilibrium calculations. Some of these studies involve theoretical methods for incorporating physical phenomena such as shell effects and residual pairing interactions [229, 231]–[236]. The most widely used approach to PLD is an equidistant single-particle model using closed-form expressions, as proposed by Williams [237], and further refined by many authors [238]–[248]. Despite extensive research, even the most sophisticated theoretical predictions can significantly deviate from reality. Difficulties arise in testing the validity of determining partial densities through comparison of calculated and measured pre-equilibrium spectra because of the uncertainties in our understanding of the pre-equilibrium reaction mechanisms. Moreover the partial level density is not a directly measurable quantity. A useful collection of articles on PLD can be found in the Conference Proceedings of the Nuclear Level Densities in Upton, New York in 1983 [174] and at Bologna in 1989 [177]. State-of-the-art methods to calculate PLD were reviewed by Bˇeták and Hodgson [249]. RIPL-2 [10] includes a code written by M. Avrigeanu and V. Avrigeanu to calculate partial level densities using various models as described in their extensive paper [250]. During the RIPL-3 projects some attempts have been made to calculate particle-hole level densities using the combinatorial HFB+BSk14 method [225]. However, the resulting partial level densities have not been validated in pre-equilibrium model calculations, as the use of particlehole level densities containing collective effects in current pre-equilibrium models is not straightforward. The density of p-particle h-hole states with residual nucleus energy U can be factorized into the energydependent state density ω(p, h, U ) and the spin distribution Rn (J), i.e., ρ(p, h, U, J) = ω(p, h, U ) Rn (J). A Gaussian spin distribution is usually adopted:

2J + 1 (J + 1/2)2 , (128) exp − Rn (J) = √ 2σn2 2 2πσn3 where J is the spin and σn is the spin cutoff parameter (often taken as σn2 = 0.24nA2/3 , n = p + h [245]). Adopting the equidistant model expression for the energy dependent one-component state density (no neutron-proton distinction) with finite hole-depth and binding-energy restrictions yields a simple and commonly used formula derived by Bˇeták and Dobeˇs [239] and Obloˇzinsk´ y [242]: p h g p+h h i+j p ω(p, h, U ) = × (−1) p!h!(n − 1)! i=0 j=0 i j (U − A(p, h) − iB − jEF )p+h−1 × Θ(U − EP (p, h) − iB − jEF ), (129)

Partial level densities

The partial level density is used in pre-equilibrium reaction calculations to describe the statistical properties of particle-hole excitations [228]. Since the pioneering studies by Strutinsky [229], Ericson [42] and Blann [230],

where g is the single-particle density, B is the binding energy, EF is the Fermi energy, A(p, h) is the Pauli correction, and EP (p, h) is the Pauli energy (i.e., the minimum energy required for the p − h configuration by the

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Pauli exclusion principle). The Θ(x) function is unity if x > 0, and zero otherwise. If we define the Fermi level to be halfway between the last filled and first vacant single particle states, we can obtain a symmetric (in p and h) Pauli correction function as given by Kalbach [234]: A(p, h) = EP (p, h) −

p(p + 1) + h(h + 1) , 4g

(130)

account shell and pairing effects. However, the results should always be treated with caution, as the predictive capabilities of all partial level density theories are limited. The analytical formula of Eq. (129) [239, 242] is recommended for pre-equilibrium calculations. However, the limitations of such partial level density predictions should be kept in mind, especially considering the uncertainties in our understanding of pre-equilibrium reaction mechanisms.

where EP (p, h) = pm /g, and pm = max(p, h). We can implement the pairing corrections by replacing the Pauli energy EP (p, h) in Eqs. (129) and (130) with the new ˜P (p, h) [234, 236]: threshold energy E 2 2 ˜P (p, h) = g(Δ0 − Δ ) +pm (pm /g)2 + Δ2 , E 4

(131)

where Δ0 and Δ are the ground and excited state pairing gaps, respectively. The ratios of the pairing gaps Δ/Δ0 were recently refitted by Rejmund et al. [236] In summary, the closed-form expression of Eq. (129) derived in the equidistant spacing model [239, 242] is adequate for many pre-equilibrium calculations, and easier to implement than the RIPL-2 recommendation [10, 247].

D.

Recommendations

For any application of the statistical theory of nuclear reactions, the parameters describing the level density must be strongly constrained by reliable experimental data. Both the cumulative numbers of low-lying levels and the average spacings of neutron resonances are usually used for this purpose. The level density parameters fitted to such data were compiled for the four phenomenological models most frequently used in practical calculations: i) Gilbert-Cameron approach; ii) back-shifted Fermi gas model; iii) generalized superfluid model and iv) enhanced generalized superfluid model. Furthermore, the systematics of the level density parameters were developed for each model in terms of the shell correction approach. Such global systematics are recommended for the level density calculations involving nuclei close to the valley of stability that have no experimental data on resonance spacings or for low-energy level densities. RIPL-3 contains large quantities of data from the microscopic calculations of nuclear level densities based on the HFB+BSk14 model. The microscopic model takes into account the shell, pairing and blocking effects, the deformation effects in the single-particle spectra, the collective enhancement of level densities at low excitation energies and damping at high excitations. Tables of level densities for about 8000 nuclei are provided. Singleparticle level schemes for the microscopic calculations of the nuclear level densities are also given. Microscopic calculations can be useful for detailed studies of the role of partial level densities in preequilibrium emission since such calculations take into

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VII.

GAMMA-RAY STRENGTH FUNCTIONS

Gamma emission is one of the most significant channels for nuclear de-excitation at energies below 1 MeV, and accompanies most nuclear reactions. Both gamma decay and photo-absorption can be described through radiative (photon) strength functions (RSF) [251, 252]. Electron-positron pair emission also depends on the radiative strength functions [253, 254]. There are two types of radiative strength functions: ← − (1) ‘downward’ strength function ( f ), which determines the average radiative width of the γ-decay; and → − (2) photo-excitation (upward) strength function ( f ) related to the cross-section for γ-ray absorption. The γ-decay strength function for γ-ray emission of multipole type XL is defined as the average reduced par−(2L+1) tial radiation width γ ΓXL (γ ) divided by the average level spacing Dl : ← − ΓXL (γ ) , f XL (γ ) = −(2L+1) γ Dl

(132)

where γ is the γ-ray energy. The photo-excitation → − strength function f EL is determined by the average photo-absorption cross section σXL (γ ) summed over all possible spins of the final states [252]: −2L+1 − → σXL (γ ) γ . f EL (γ ) = (πc)2 2L + 1

(133)

In general, gamma-decay strength functions depend on the temperature Tf of the final states, which is a function of the excitation energy of deexcited nucleus, Uf = Ui − γ , in contrast to the initial state temperature Ti calculated for the excitation energy Ui . The transmission coefficient TXL (γ ) of the γ-ray emission is given by the relationship: ← − TXL (γ ) = 2π2L+1 f XL (γ ). γ

(134)

Gamma-ray strength functions are important constituents of the compound nucleus model calculations of capture cross sections, γ-ray production spectra, isomeric state populations, and competition between γ-ray

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and particle emission. Relevant multipolarities in this context are E1, M1 and E2. The γ-ray strength functions include information on nuclear structure, and are widely used to study the mechanisms of nuclear reactions as well as nuclear structure. Widths and energies of the giant multipole resonances and nuclear deformation parameters in heated nuclei are extracted from experimental data by comparing the shape of the experimental γ-ray strength function with theoretical calculations [255]–[260]. Since γ-ray strength functions sometimes require time-intensive calculations, simple closedform expressions or ready-to-use tables are most convenient. The more detailed approaches based on recent theoretical achievements are useful in improving the reliability of the closed-form expressions. Lorentzian parameters of giant resonances are important quantities for the calculation of γ-ray strength functions within closed-form methods, derived traditionally from the analysis of the photo-absorption cross sections for the E1 and E2 giant resonances. However, this experimental database is rather sparse, and measurements have not been undertaken for many important target nuclei. Therefore, several global systematic parameterizations have been derived for the multipolarities of primary importance. Finally, experimental γ-ray strength functions are extremely useful to check theoretical approaches. A.

Experimental γ-ray strength functions

Experimental γ-ray strength functions have been collected over a period of about forty years, based on measurements of partial radiative widths Γγi by three different types of experiment. Most of the data are derived from discrete resonance-capture measurements using the method of slow neutron time-of-flight spectrometry to determine the incident neutron energy. Thermal neutroncapture data can be used (with some restrictions) in certain cases to derive γ-decay strength functions. Finally, another source of data is provided by photonuclear reactions. Analysis of all these experiments involves averaging over Porter-Thomas fluctuations, which governs the distribution of partial radiative widths. The most extensive compilation of experimental γ-ray strength functions at fixed gamma-ray and excitation energies was presented in RIPL-1 [261]. These data were also adopted for RIPL-3. E1 and M1 strength functions are given for nuclei from 20 F up to 239 U, and some of the original values were corrected (typically for nonstatistical effects). Both original and recommended values are listed in units of 10−8 MeV−3 (see RIPL-1 report for more details regarding this compilation and data reduction [261]). The sum of the experimental γ-decay strength functions for the E1+M1 transitions in the energy interval up to Bn was determined in Refs. [262]–[267] from the analysis of two-step γ cascades after thermal neutron capture. The low-energy averaged sum of the γ-decay RSF for the E1+M1 transitions from the analysis of γ-emission

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after the reactions (3 He,3 He γ) and (3 He,4 He γ) was obtained in Refs. [268]–[274]. High sensitivity studies of the low-energy behavior of the RSF were performed in photon scattering experiments [275]-[281]. Experimental dipole radiative strengths for nuclei with 50 ≤ A ≤ 90 and 6 ≤ γ ≤ 10 MeV were obtained by studying γ-emission in (p, γ)-reactions (see Refs. [282, 283], and references therein). One of the main experimental problems in determining the gamma-transition characteristics is a correct estimation of the initial state excitation energy, since the RSF is generally dependent on this energy.

B.

Closed-form models for the E1 strength function

The absorption and emission of dipole γ rays from atomic nuclei in the energy region up to γ ≈ 20 MeV is mainly governed by excitation of the giant isovector dipole resonance (GDR). At energies around the GDR, the imaginary part of the nuclear response function has a Lorentzian-like shape, and realistic phenomenological models of the dipole RSF assume Lorentzian-like shapes. The scaling parameter (or width Γ(γ , T )) of such shapes in a heated nucleus at temperature T is given by different expressions governed by the damping of the collective states. Its behavior depends on assumptions concerning the damping mechanism for the collective states (see [284] for details and references).

1.

Standard Lorentzian model (SLO)

The Brink hypothesis in the standard Lorentzian model (SLO [285, 286]) is widely used to calculate the dipole γ-ray strength. Dipole RSF in the SLO model is a single Lorentzian (for spherical nuclei) with an energyindependent width Γ taken to be equal to the GDR width Γr : ← − → − f (γ ) = f (γ ) ≡ fSLO (γ ) = 8.674 · 10−8 × γ Γr MeV−3 , (135) σr Γr 2 2 γ − Er2 + 2γ Γ2r where the Lorentzian parameters σr , Er are the peak cross section and the GDR energy, respectively; the energies and widths are given in units of MeV, and σr in mb. Width Γ(γ , T ) = Γr = const is similar in the SLO model to the fragmentation component of the collective state damping width that corresponds to redistribution of the γ-strength in a self-consistent mean-field when nucleon collisions in the nuclear interior are not taken into account. Fragmentation of the damping width in the semiclassical approach is governed by nucleon collisions with a moving surface of the nucleus (one-body dissipation),

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and is practically independent of the energy (see [284] and references therein). The SLO approach is probably the most appropriate method for describing the photo-absorption data of medium-weight and heavy nuclei [251, 287, 288]. However, the SLO model for γ-emission significantly underestimates the γ-decay spectra at low energies γ ≤ 1 or 2 MeV [289]. A global description of the γ-decay spectra by the Lorentzian approach can be obtained over the energy range 1 or 2 ≤ γ ≤ 8 MeV, but the parameters become inconsistent with those derived from the photo-absorption data near the GDR peak. Generally, SLO with the GDR parameters overestimates experimental data around the neutron separation energy, such as capture cross sections and the average radiative widths of heavy nuclei [251], [290]–[293].

2.

where Γr is the GDR width in cold nuclei (T = 0). The KMF approach serves as a basis of the enhanced generalized Lorentzian (EGLO) model [261, 291, 299], the hybrid model [300], and the generalized Fermi-liquid (GFL) model [301]. 3.

← − ← − f (γ = 0) ≡ f KMF = 8.674 · 10−8 × Γc (γ = 0, Ti ) σr Γr K MeV−3 , Er3

← − ← − f (γ ) ≡ f EGLO (γ ) = 8.674 · 10−8 × γ ΓK (γ , Tf ) + σr Γr (2γ − Er2 )2 + 2γ Γ2K (γ , Tf )

0.7ΓK (γ = 0, Ti ) MeV−3 , Er3

1/2

/ (1 + F0 /3)

1/2

=

Er /E0 ,

(136)

(137)

where E0 is the average energy of the one-particle onehole states forming the GDR. The value K = KKMF = 0.7 is adopted in the KMF model. The width Γc (γ , T ) was assumed in Ref. [294] to have the form of the collisional component of the damping width of the zero sound in the infinite Fermi liquid. The width is obtained by replacing the GDR energy with the gamma-ray energy γ , i.e., Γc (γ , T ) = Ccoll 2γ + 4πT 2 .

Ccoll =

Γr ≡ CKMF , Er2

Γr · χ(γ ) = CKMF · χ(γ ) ≡ CK (γ ). (141) Er2

The quantity χ(γ ) is a function obtained from a fit to experimental data: χ(γ ) = k + (1 − k)(γ − ε0 )/(Er − ε0 ),

(142)

where k reproduces the experimental E1 strength around a reference energy ε0 ; χ(γ = Er ) = 1 and CK (Er ) = CKMF . The value of k depends on the model used to describe the nuclear state density, and was obtained from the average resonance capture data. The reference energy ε0 was chosen as 4.5 MeV. If the Fermi gas model is used, k is given by [261]:

1 A < 148, k= 1 + 0.09(A–148)2 exp[−0.18(A–148)] A ≥ 148. (143)

(138)

According to a semi-classical approach based on the Landau-Vlasov equation, the energy dependence of the collisional width Γc (γ ) results from the non-Markovian form of the collision integral with retardation effects [295]–[298]. The constant Ccoll in Eq. (138) is derived from the normalization condition Γc (γ = Er , T = 0) = Γr , and determined to be:

(140)

where Ti (Tf ) is the temperature of the initial (final) states. The EGLO width ΓK (γ , T ) has the form represented by Eq. (138), but with the energy-dependent parameter Ccoll : Ccoll (γ ) =

where the quantity K is determined by the Landau parameters F0 and F1 of the quasi-particle interaction in the isovector channel of the Fermi system: K = (1 + F1 /3)

Enhanced Generalized Lorentzian model (EGLO)

RSF in the EGLO model for spherical nuclei consists of two components [6, 10, 291, 299]: (1) a Lorentzian with an energy- and temperature-dependent empirical width; and (2) a term for the shape corresponding to the KMF limit (136) of the RSF at zero for the γ-ray energy:

Kadmenskij-Markushev-Furman model (KMF)

The first model with a correct description of the E1 strengths at energies γ close to zero was developed using the Fermi-liquid theory of finite systems [294] (KMF model). Consider the KMF expression of RSF in the limiting case of γ =0:

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4.

Hybrid model (GH)

A hybrid formula for the E1 strength function was proposed in Ref. [300], and defined as the GH model. The general form of this hybrid approach coincides with Eq. (135) with Γh (γ , T )Γr adopted in the numerator and denominator instead of Γ2r :

(139)

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← − ← − f (γ ) ≡ f GH (γ ) = 8.674 · 10−8 σr Γr × γ Γh (γ , Tf ) MeV−3 . (2γ − Er2 )2 + 2γ Γh (γ , Tf )Γr

(144)


NUCLEAR DATA SHEETS

The energy-dependent width Γh (γ , T ) is: Γh (γ , T ) = KGH Γr

2γ + 4π 2 T 2 , γ Er

(145)

and at zero temperature depends linearly on the γ-ray energy, and rises to infinite at zero energy (γ → 0) in heated nuclei. The factor KGH is given by Eq. (137). The value KGH = 0.63 is adopted in the GH model (see also KGF L below). 5.

Generalized Fermi Liquid model (GFL)

Another approach in the determination of the dipole γray strength with non-zero limit at γ → 0 is the Generalized Fermi Liquid (GFL) model as proposed in Ref. [301]. The general low-energy shape of the dipole strength function is similar to that obtained by applying the Fermiliquid theory to finite systems [294], but with an energydependent width which includes fragmentation damping from the dipole-quadrupole interaction. The GFL dipole strength function in spherical nuclei is defined by the equation: ← − ← − f (γ ) ≡ f GF L (γ ) = 8.674 · 10−8 σr Γr KGF L × γ Γm (γ , Tf ) MeV−3 , (146) 2 2 2γ − Er2 + KGF L 2γ (Γm (γ , Tf )) where KGF L = K = 0.63 as given by Eq. (137) [301]. Equation (146) is an extension from Ref. [302] of the original expression of Ref. [301], in which the term KGF L (Γm γ )2 has been added to the denominator to avoid singularity of the GFL approach near the GDR energy. The factor KGF L is included to maintain the standard relationship between the strength function at the GDR energy and the peak value σr of the photoabsorption cross section. The energy- and temperature-dependent width Γm is taken to be the sum of a collisional damping width Γc of Eq. (138), and an additional term Γdq that simulates the fragmentation width: Γm (γ , T ) = Γc (γ , T ) + Γdq (γ ) .

(147)

The collisional component corresponds to an extension of the zero sound damping width in the infinite Fermi-liquid model (Eq. (138)) but with the constant Ccoll ≡ CGF L , which is determined by normalizing the total width Eq. (147) at γ = Er and T = 0 to the GDR width of a cold nucleus, i.e., Γm (γ = Er , T = 0) = Γr . The Γdq component results from the damping of the nuclear response due to quadrupole vibrations of the nuclear surface: Γdq (γ ) = Cdq 2γ β¯22 + γ s2 MeV, (148) where Cdq = 5 ln 2/π = 1.05; s2 = E2+ β¯22 ≈ 2 217.16/A MeV, with E2+ representing the energy of

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the first vibrational quadrupole state, and β¯2 the effective deformation parameter characterizing the nuclear stiffness with respect to surface vibrations. β¯2 is determined from the reduced electric photo-absorption rate B(E2) ↑≡ B(E2, 0+ → 2+ ) for the transition between the ground and 2+ states [303, 304]: 5 E2 3 = B(E2) ↑ /( eR02 )2 , β¯22 ≡ 2 C2 4π

(149)

where C2 is the stiffness factor of the restoring force, R0 = r0 A1/3 is the nuclear radius, and e is the proton charge. Equation (148) coincides with the expression for the GDR damping width in Ref. [305] (see Eq. (16) of [305]), after substituting the gamma-ray energy γ with the GDR energy Er . The photoexcitation strength function fE1 can also be calculated within the EGLO, GH and GFL models. Corresponding expressions of fE1 for cold nuclei are determined by Eqs. (140), (144) and (138) by setting Ti = Tf = 0. The low-energy behavior of the γ-decay RSF in the KMF, EGLO and GFL models are similar in that the RSF fE1 (γ → 0) → const, ← − f α (γ = 0) = 8.674 · 10−8 × Kα Γα (γ = 0, T ) σr Γr MeV−3 , Er3

(150)

but the values of fE1 (0) differ due to differences between the contributions of the temperature-dependent component in the widths Γc (γ , T ), ΓK (γ , T ), Γm (γ , T ) at γ = 0 and the adopted values for the Landau parameters of the quasi-particle interaction. Index α in Eq. (150) specifies the corresponding model. 6.

Modified Lorentzian model (MLO)

All above RSF expressions for the gamma-decay strength function of heated nuclei are in fact parameterizations of experimental data. They are in contradiction with some aspects of microscopic theoretical studies - specifically, shapes of the RSF within these models are inconsistent with the general relationship between the γray strength function of heated nuclei and the imaginary part of the nuclear response function to the electromagnetic field [306]–[309]. This shortcoming can be approximately avoided as proposed in Refs. [310]–[312]. The suggested model was renamed as the Modified Lorentzian approach (MLO) [313], and is based on general relations between the RSF and the imaginary part of the nuclear response function [309]. An expression for the dipole γ-ray strength function ← − ← − within the MLO model ( f E1 ≡ f MLO ) is obtained by calculating the average radiative width of nuclei with microcanonically distributed initial states. This function

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has the following form for spherical nuclei [310]–[314]: ← − ← − f (γ ) ≡ f MLO (γ ) = 8.674 · 10−8 Λ(γ , Tf )σr Γr × γ Γ(γ , Tf ) MeV−3 , (151) (2γ − Er2 )2 + 2γ (Γ(γ , Tf ))2 1 Λ(γ , Tf ) ≡ , (152) 1 − exp(−γ /Tf ) where Γ(γ , Tf ) is the strength function width that depends on the γ-ray energy and temperature Tf of the final state; the width Γ(γ = Er , T ) can be identified with the GDR width Γr (T ) in the heated nucleus with temperature T . The scaling factor Λ(γ , Tf ) in Eq. (151) determines the enhancement of the radiative strength function in a heated nucleus as compared to a cold nucleus. This quantity can be interpreted as the average number of 1p − 1h states excited by an electromagnetic field with frequency ω = γ /, and is only important for low-energy radiations. It should be noted that Λ also appears in the Fermi-liquid approach with explicit allowance for singleparticle occupation numbers of compound nuclear states (in Ref. [315], see Eq. (22) for the strength function at γ Er ). An extension of this method to γ-ray energies near GDR resonances slightly violates the detailed balance principle [282, 283]. However, Λ is essential for the consistency of Eq. (151) with the detailed balance in constant temperature systems (see Refs. [306]–[309]). The Lorentzian term appears in Eq. (151) as the imaginary part of the nuclear linear response function to the electric dipole field. This shape is predicted by the extended hydrodynamic model of Steinwedel-Jensen (ESJ) [312, 316] for heated nuclei, with friction between the proton and neutron fluids, and also by a semi-classical Landau-Vlasov equation with a memory-dependent collision term, providing the γ-transition strength is concentrated near the giant resonance [313, 314]. The Lorentzian shape stems also from the random-phase approximation in cold nuclei [317]. Different semi-empirical expressions for the width Γ(γ , T ) have been previously used in the MLO approach (MLO1, MLO2, MLO3), but as a rule the resulting RSF are in rather close agreement. The MLO1 model is based on the semi-classical approach that involves the use of the Landau-Vlasov equation with a collision term [314, 318]. As a result, the following expression for the width was obtained: β(γ , T ) Γ (γ , T ) = , τc (γ , T )

2α ˜ (Er2 + E02 ) , − E02 )2 + 4(γ /τc )2 2 E2 E02 1 − r2 , α ˜ = 2 E0

β(γ ; τc ) ∼ =

(Er2

with τc ≡ τc (γ = ω, T ) for a collisional relaxation time of the collective motion in a heated Fermi system with the temperature T under an external field with frequency ω = γ /; E0 = 41A−1/3 . Equation (154) at γ = Er with τc (γ = Er ; T = 0) corresponds to the collisional relaxation time of the GDR in cold nuclei, and has the same form as the expression obtained in Ref. [319]. The dependence of τc (γ = ω; T ) on the γ-ray energy results from retardation effects in the collision integral which is used to describe the scattering of particles in the nuclear interior [295]–[298]. Relaxation time in the MLO1 approach is evaluated by analogy with the relaxation times of states in the exciton nuclear reaction model, and is defined as [297]: = b (γ + U ) , τc (γ ; T )

(154) (155)

(156)

where U is the excitation energy that corresponds to the temperature T ; b is a constant determined by the in-medium cross section of neutron-proton scattering. The magnitude of b is determined from the condition Γr = Γ(γ = Er , T = 0) in cold nuclei, i.e. from the relationship β(Er ; τc (Er , T = 0))/τc (Er , T = 0) = Γr /. A linear energy dependence of the collisional relaxation time in Eq. (156) results from the inverse proportionality of the effective mean square matrix element to the excitation energy for transitions between the exciton (phonon) states [320]. The linear energy dependence of the collisional width (which is proportional to /τc ) was also obtained when nucleon collisions were considered as s-wave scattering between pseudo-particles [321, 322]. A linear dependence on gamma-ray energy of the damping width was also adopted for cold nuclei in the hybrid model (see Eq. (145)). The MLO strength function fE1 describing the photoexcitation of cold nuclei is given by the expression: ← − f (γ ) = fMLO (γ ) = 8.674 · 10−8 σr Γr × γ Γ(γ , T = 0) MeV−3 , 2 2 2 2 γ − Er + 2γ (Γ(γ , T = 0))

(157)

with Γ(γ , T = 0) of the MLO1 model determined by Eq. (153) with the following relaxation time: = bγ . τc (γ ; T = 0)

(153)

where β(γ , T ) ≡ β(γ ; τc (γ , T )) is a function of the form

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(158)

Compared with Eq. (158), the relaxation time (156) used for the calculation of the γ-decay strength function within MLO1 is independent on the γ-ray energy, and depends only on the initial excitation energy Ui = γ + Uf . Consider excitation energies that are not high where γ ranges from zero up to the GDR energy: the function β in Eq. (153) depends weakly on the energy, and the

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NUCLEAR DATA SHEETS

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MLO1 width can be expressed as follows: ⎧ ⎨ Γ (γ , Tf ) = a(γ + Uf ) = aUi for γ-decay, Γ= ⎩ Γ ( , 0) = a for photoabsorption, γ γ (159) where a = Γr /Er = CKMF Er if the normalization condition Γr = Γ(γ = Er , T = 0) is adopted for cold nuclei. The modified Lorentzian model, given by Eqs. (151) and (157) with the simplified expression in Eq. (159) for Γ (γ , T ), is defined as the Simplified Modified Lorentzian (SMLO) model [314] . The RSF of the MLO2 and MLO3 models also takes the form of Eq. (157), but includes an approximation of independent dissipation sources for the widths [286]. These widths are taken as the sum of the collisional component /τc and a term ks /τw which simulates the fragmentation contribution to the width: Γ (γ , T ) = + ks (γ ) . (160) τc (γ , T ) τw Collisional relaxation time is determined by Eq. (156) in the MLO2 model; collective relaxation time in the MLO3 model takes the form of Eq. (138), with Ccoll determined by the neutron-proton cross section σin (n, p) in the nuclear medium near the Fermi surface: σin (n,p) , (161) F = Ccoll ≡ CMLO = F · c, σf ree (n,p) 4 m c = 2 2 σf ree (n, p) = 0.05386 MeV−1 , (162) 9π where σf ree (n,p)=5 fm2 is adopted for the free space cross section and 2 /m = 41.80349 MeV·fm2 . The magnitude of the second fragmentation component in Eq. (160) is taken to be proportional to the wall formula [323, 324]: 3 vF 32.846 = Γw = = MeV , τw 4 R0 A 1/3

(163)

with εF = m vF2 /2 = 37 MeV for the Fermi energy and R0 = r0 A1/3 with r0 = 1.27 fm. A scaling factor ks in Eq. (160) is included in the energy-dependent power approximation:

kr + (k0 − kr )|(γ − Er )|ns , γ < 2Er , ks (γ ) = k0 , γ ≥ 2Er , (164) where k0 = 0.3, ns = 1 and F = 1 were adopted from comparisons of calculations with experimental γ-decay strengths. The constant kr is determined from the condition of equivalence of the MLO2 and MLO3 widths Γ (γ = Er , T = 0) at the GDR energy with the GDR width Γr in cold nuclei: kr = (Γr − Ccoll Er2 ) τw /. The zero frequency limit of the gamma-decay RSF in the MLO models has the following form: ← − f MLO (γ = 0) = 8.674 · 10−8 σr Γr × Γ(γ = 0, Ti ) Ti MeV−3 . (165) Er3 Er

FIG. 41: E1 γ-decay strength functions plotted against mass number; experimental data are taken from the gamma/gamma-strength-exp.dat file [10].

This expression differs from Eq. (150) for the KMF, EGLO and GFL models by a factor of: Ti Γ(γ = 0, Ti ) · . Γα (γ = 0, Ti ) Er Kα 7.

(166)

E1 strength functions in deformed nuclei

An approximation of axially deformed nuclei is usually adopted for the calculation of E1 strength functions in deformed nuclei, with the radius defined by:

R(θ) = R0 (1 + α2 P2 (cos θ)) = R0 (1 + β2 Y20 ) , 3 2 λ3 = 1 + α22 + α32 , R0 = R0 /λ, 5 35

(167) (168)

where R0 is the nuclear radius of a spherical nucleus with the same volume of the deformed nucleus, P2 (cos θ) is a Legendre polynomial, and Y20 = 5/4πP 2 is the 4π/5α2 are spherical harmonic. Both α2 and β2 = quadrupole deformation parameters chosen to reproduce the ground-state quadrupole nuclear moments Q. The E1 strength function in axially deformed nuclei is taken as the sum of two components [316], each with corresponding energy Er,j , damping width Γr,j and peak value for the photoabsorption cross section σr,j . Parameters Er,j , Γr,j and σr,j (j = 1, 2) correspond to collective vibrations along (j = 1) and perpendicular to (j = 2) the axis of symmetry, and σr,1 = σr,2 /2. Fragmentation damping widths ΓF,1 , ΓF,2 of the collective vibrations along two principal axes of a spheroid are assumed in the MLO2 and MLO3 methods to be proportional to the dipole widths (Γs,1 and Γs,2 ) of the surface dissipative model [325]:

3170

ΓF, j (γ ) = ks (γ )Γs, j , Γs,1 = Γw /aδ0 ,

Γs,2 = Γw /bδ0 ,

δ = 1.6,

(169)


NUCLEAR DATA SHEETS

R. Capote et al.

where Γw is defined by Eq. (163), and a0 and b0 are relative semi-axes of a spheroid: a0 ≡ R(θ = 0)/R0 = (1 + α2 ) /λ, b0 ≡ R(θ = π/2)/R0 = (1 − 0.5α2 ) /λ. (170) Parameters kr (Eq. (164)) and CGF L appear in expressions that define the damping widths for MLO and GFL in deformed nuclei, and are determined by fitting theoretical damping widths (Γr,j ) of the normal modes of the giant dipole resonance in cold nuclei to the corresponding experimental values. CGF L can become negative. It means that the fragmentation component Γdq of the width Eq. (147) is overestimated. However the GFL model can be used as a good parameterization of the dipole γ-decay strength functions in deformed nuclei too.

8.

Comparison with experimental data

Variations in the dipole γ-decay strength functions ← − f E1 with mass number are shown in Fig. 41 for 50 nuclei included in the RIPL experimental database. The back-shifted Fermi gas model (BSFGM) was used to define the thermal excitation energy Uf = Ui − US − γ of the final state in terms of the temperature Tf [167]. This approach relates the temperatures Ti and Tf to each other and to the thermal excitation energy Ui of the ini tial state, with Tf = (1 + 1 + 4a(aTi2 − Ti − γ ))/2a and Ti = (1 + 1 + 4a(Ui − US )/2a, where US is the energy pairing parameter and a is the level density parameter. Values for the level density parameters a and energy shifts US were taken from the beijing bs1.dat file of RIPL-1 [6] with rigid-body moments of inertia, or from global systematics [326] when no experimental data were available. Quadrupole deformation parameters β2 were calculated from the ground-state deformation parameters given in the RIPL-1 mass file and using Eqs. (178) and (177). Nuclei with β2 ≤ 0.01 were considered to be spherical. The effective quadrupole deformation parameters β¯2 and energies E2 of the first 2+ state for even-even nuclei in the GFL model were taken from table I of Ref. [303]. When experimental data on the lower quadrupole vibrational states were unavailable for even-even nuclei and for all odd-A and odd-odd nuclei, |β¯2 | were used for β¯2 and a global parameterization was adopted for s2 [303] s2 ≡ E2 β¯22 = 217.16/A2 MeV.

FIG. 42: E1 γ-decay strength function plotted against energy γ for 90 Zr; experimental data are taken from Ref. [327].

Ui = Sn better than the EGLO and SLO models for A ≤ 220. GFL and MLO calculations are in very close agreement. ← − Figure 42 shows calculated γ-decay strengths f E1 for 90 Zr; experimental data are taken from Ref. [327], and GFL, MLO and EGLO data are calculated for the experimental energies Ui and γ . The MLO and SLO models for 90 Zr describe the experimental data better than GFL and EGLO, and the MLO representation is closer to the experimental data than that of the SLO model. Figure 43 compares experimental strength functions taken from Ref. [289] with the calculated strength func← − tions f E1 for 144 Nd, with the initial excitation energy Ui equal to the neutron separation energy Sn (≈ 7.8 MeV). EGLO, GFL and MLO results are characterized by a non-zero limit and temperature dependence at low γ-ray energies. All these models are in reasonable agreement for γ ≤ 2 MeV, and describe the experimental data much better than the SLO model (which predicts a vanishing strength function at zero γ-ray energy) [314, 328]. The photo-excitation strength function Eq. (133) is of the same form as the γ-decay strength function, except that the temperature of the initial state (Ti ) is adopted instead of the final state temperature (Tf ). E1 photoexcitation strength functions calculated by means of the MLO(SMLO), GFL and SLO models are in good agreement for cold nuclei over a wide range of gamma-ray energies near the GDR peak energy [314, 329, 330].

C.

Microscopic approach to E1 strength function

(171)

The results shown in Fig. 41 were calculated for γ-ray energies that correspond to the mean energy ¯γ of E1 transitions in the gamma/gamma-strength-exp.dat file. Experimental data were taken also from this file and they were extracted by Kopecky from the average resonance capture data at low energies. Plots show that the GFL and MLO(ML02 variant) models describe the experimental γ-decay data with γ ≈

The Lorentzian and previously described closed-form expressions for the γ-ray strength suffer from various shortcomings: (1) they are unable to predict the resonance-like enhancement of the E1 strength at energies below the neutron separation energy as demonstrated, for example, by nuclear resonance fluorescence experiments. This departure from a Lorentzian profile

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NUCLEAR DATA SHEETS

FIG. 43: E1 γ-decay strength function of 144 Nd for Ui = Bn : experimental data are taken from Ref. [289].

may occur in various ways, such as a pygmy E1 resonance [275, 281, 331]–[335], which is observed in f p-shell and heavy spherical nuclei near closed shells (Zr, Mo, Ba, Ce, Sn and Pb);

εγ

FIG. 44: E1 strength function fγ (E1) for the Sn isotopic chain as predicted by QRPA with the SLy4 Skyrme force; only isotopes between A=110 (I = 0.09) and A=155 (I = 0.35) are displayed in steps of ΔA=5.

shown in Fig. 44 for the E1 photo-excitation strength function fγ (E1) in units of mb/MeV:

(2) they are unable to describe isospin structure of the RSF, specifically observed isospin splitting of the GDR in light- and middle-weight atomic nuclei [336]–[339]; (3) even if a Lorentzian function provides a suitable representation of the E1 strength, the location of the maximum and width still need to be predicted from some underlying model for each nucleus, as described in the previous sections. This approach lacks reliability when dealing with exotic nuclei. Therefore, microscopic models have been developed with the aims of providing predictive power and reasonably reliable E1 strength functions. Attempts in this direction have been specifically conducted within the quasi-particle random-phase-approximation (QRPA). The spherical QRPA model (as well as the microscopic Hartree-Fock-Bogoliubov plus quasi-particle random phase approximation model (HFB–QRPA)) includes a realistic Skyrme interaction, and has been used for large-scale derivations of the E1 strength function [340]–[342]. The final E1 strength functions obtained by folding the density of QRPA excitations for a given spin and parity (QRPA strength) with a Lorentzian function (similar as the general form of Eq. 135) also reproduce photoabsorption satisfactorily, as well as the average resonance capture data at low energies [340]. These aforementioned QRPA calculations have been performed for all 8 ≤ Z ≤ 110 nuclei lying between the two drip lines. QRPA distributions in the neutron-deficient region and along the valley of β-stability are very close to a Lorentzian profile in the MLO model. Significant departures from the Lorentzian shape are found for neutron−Z rich nuclei with large asymmetry coefficients NA , as

R. Capote et al.

→ − σE1 (mb) = 3(πc)2 f E1 (γ ) γ (MeV) → − ≡ 1.15 · 106 f E1 (γ ) MeV−3 . (172)

fγ (E1) ≡

RPA-like calculations [341, 343]–[346] as well as the semiclassical secondary RPA models [347]–[350] show that neutron excess affects the spreading of the isovector dipole strength, as well as the centroid of the strength function. The energy shift is found to be larger than predicted by means of the usual A−1/6 or A−1/3 dependence given by the phenomenological liquid-drop approximations [323]. Some extra strength is also predicted to exist at sub-GDR energies, and to increase with neutron excess (Fig. 44). Even if this behavior represents only a few percent of the total E1 strength, an increase by up to an order of magnitude of the radiative capture cross section can occur for some exotic neutron-rich nuclei [340]. Microscopic predictions of the E1-strength functions have been determined by the QRPA model based on the BSk-14 Skyrme force, and are included in RIPL-3 for 3317 nuclei with 8 ≤ Z ≤ 84, lying between the proton and neutron drip lines. The QRPA equations were solved in configuration space so as to exhaust the energy-weighted sum rule, and all calculations were performed within the spherical approximation. A folding procedure was applied to the QRPA strength distribution in order to account for the damping of the collective motion. A phenomenological splitting of the QRPA resonance strength was performed within the same folding procedure for the deformed nuclei. All modelling details and a comparison with experimental data can be found in Refs. [340, 342]. The E1 strength has been tabulated on an energy grid of 0.1 MeV between 0 and 30 MeV, and these data can be found in a series of gamma/gammastrength-micro/zXXX.dat files in RIPL-3 [14].

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Note that the QRPA method does not use a thermodynamic description of the ensemble of initial highly-excited states. A rather simple semi-classical approach was proposed in Refs. [347, 348]. This method is based on solving the kinetic Landau-Vlasov equation for finite systems with a moving surface [351] (semi-classical approach with moving surface, MSA). Calculations of the E1 RSF using the MSA model for medium and heavy atomic nuclei are in rather close agreement with QRPA predictions [314].

D.

Giant dipole resonance parameters

The parameters for giant resonances with E1, M1 and E2 multipolarities were collected in RIPL-1 [6], and also presented in RIPL-3 [14]. A compilation of experimental giant dipole resonance parameters from fits to the total photoneutron cross-section data by the SLO expression (reformatted from the RIPL-1 database [6]) is contained in the RIPL-3 gamma/gdrparameters-exp.dat file. New compilations of experimental giant dipole resonance parameters and their uncertainties from fits to an extended database of photoreaction cross sections by the SLO and MLO1 expressions are given in files gamma/gdr-parameters&errorsexp-SLO.dat and gamma/gdr-parameters&errors-expMLO.dat. A database of the photonuclear reaction parameters Er,j , σr,j can also be found in Atlas of Giant Dipole Resonances [352]. This database does not contain explicit information on the damping width components Γr,j in deformed nuclei, but provides only full-width at half-maximum data for the largest peaks in the photonuclear cross sections. Unknown GDR parameters can be estimated from various systematics, which are most reliable for nuclei close to the beta-stability line with A ≥ 40. The global systematics for dipole isovector giant resonance parameters are usually adopted from the interpolation of experimental data based on some theoretical description of the GDR. The simplest systematics approach for spherical nuclei is as follows (see also Refs. [287, 353, 354]): Er ≡ E0 = aA−1/3 + bA−1/6 MeV, Γr = cErδ MeV, π Sr ≡ σr Γr = 60 d N Z/A mb·MeV, 2

(173)

where Sr within the SLO model is the total energyintegrated cross section for electric-dipole photon absorption. Analysis of the photoneutron cross-section data for heavy atomic nuclei within the SLO model shows that a = 31.2, b = 20.6, d = 1.05 ± 0.07 ([287]) and c = 0.026 ± 0.005, δ = 1.9 ± 0.1 ([353]). A new fit to the RIPL-3 experimental database (using a fixed value of δ = 1.9 in Eq. (173)) leads to the following values of the parameters [355, 356]:

R. Capote et al.

SLO : a = 27.47 ± 0.01, b = 22.063 ± 0.004,

(174)

−4

c = 0.0277 ± 0.4 · 10 , d = 1.222 ± 0.002 ; M LO : a = 28.69 ± 0.01, b = 21.731 ± 0.004, (175) c = 0.02 85 ± 0.4 · 10−4 , d = 1.267 ± 0.002. The value of d = 1 in the expression for Sr of Eqs. (173) corresponds to the classical dipole Thomas-Reiche-Kuhn sum rule Sr ≡ ST RK . The quantity d − 1 is the enhancement factor due to the exchange and velocity dependent components of the nucleon-nucleon interactions [357, 358]; d − 1 ≈ 0.2 − 0.3 is in agreement with theoretical indications [354, 357, 358]. The systematics approach for deformed nuclei (spheroidal approximation) is as follows:

a0 Er,1 = Er,2 / 0.911 + 0.089 , b0 1 1 − 1.51 · 10−2 (a20 − b20 ) , Er,2 = E0 b0 1.91 1.91 , Γr,2 = 0.026Er,2 , Γr,1 = 0.026Er,1 σr,1 = σ0 /3,

σr,2 = 2σ0 /3,

(176)

where indexes 1 and 2 correspond respectively to the collective motion along and perpendicular to the axis of symmetry with relative semi-axes of a spheroid a0 = (1 + α2 ) /λ, b0 = (1 − 0.5α2 ) /λ (see Eqs. (167) and (170)), and σ0 is calculated using Eq. (173): σr ≡ 2Sr /(Γr π). The energy expressions in Eqs. (176) were derived from the hydrodynamic model of Steinwedel-Jensen [359] (see Fig. 2 and Eq. (9) in this reference), and Er,1 E0 /(1+α2 ) and Er,2 E0 /(1−0.5α2) for small deformations, as reported in RIPL-1 [6] (β2 was used in RIPL-1 instead of α2 ). Effective quadrupole deformation parameters α2 or β2 of the equivalent spheroid (Eq. (167)) were determined fromthe ground-state deformation parameters βn ≡ αn / (2n + 1)/4π, with the nuclear radius expansion expressed in spherical harmonics. βn parameters were calculated in Ref. [16] and are listed in the masses/mass-frdm95.dat file. The nuclear quadrupole moment Q was calculated in units of (3/4π)ZeR02 for every nucleus by the equation (see also Eqs. (1.22) and (6.19) in Ref. [360]): 4 2 1 3 94 4 ¯ − α ¯ − α ¯ + Q = α ¯2 + α 7 2 7 2 231 2 72 2 200 2 8 α ¯2α ¯ α α ¯ , ¯4 + α ¯4 + (177) 7 77 2 693 4 where α ¯ n ≡ βn (2n + 1)/4π, with the ground-state deformation parameters βn taken from Ref. [16]. The quadrupole deformation α2 of an effective spheroidal nucleus was subsequently determined from the calculated quadrupole moment Q by solving the equation

3173

1 94 4 4 α =Q, α2 + α22 − α32 − 7 7 231 2

(178)


NUCLEAR DATA SHEETS

where only terms up to the fourth order in α2 from the general expression for Q are retained. Effective quadrupole deformation parameters α2 have been determined this way for all nuclei tabulated in the mass segment. Nuclei at high excitation energy and with high angular momentum can be created in heavy-ion reactions. Static deformation is damped with increasing excitation energy, and the nuclei become spherical. Rotation leads to dynamic deformation, and calculation of the gamma emission in such cases should use a spheroidal shape approximation [257, 258, 361, 362]. Simple expressions were proposed in Ref. [363] based on the liquid-drop model [364], with a rigid-body estimate for the nuclear moment of inertia and a dynamic quadrupole deformation parameter for the rotating nuclei as a function of angular momentum and mass number A. The oblate nucleus at slow rotation transits sharply to a prolate shape when the frequency increases. General expressions are of the form: β2 = β2 (J, A) = ES (J)

(a1 + a2 ES (J)) (1 + a3 ES (J))2

,

(179)

with 2

ai = bi + ci (A + di ) ,

(180)

where: 0 ES (J) = Erot J (J + 1) = 34.5A−5/3 J (J + 1) MeV, (181) is the rotation energy of the spheroidal nucleus with spin J. Slowly rotating spheroidal nuclei have an oblate shape, which changes to a prolate shape at a critical angular momentum Jcr . Coefficients bi , ci and di in Eq. (180) have the following values in the case of oblate nuclei (slow rotation):

b1 = −7.46 · 10−3 , c1 = −1.94 · 10−7 , d1 = −107.1; b2 = −4.20 · 10−5 , c2 = −4.25 · 10−9 , d2 = −93.90; b3 = +5.70 · 10−3 , c3 = +2.44 · 10−7 , d3 = −73.51. (182) Coefficients bi , ci and di for prolate nuclei (fast rotation) are: −3

−7

b1 = −6.36 · 10 , c1 = −6.33 · 10 , d1 = −48.3; b2 = +1.02 · 10−3 , c2 = +1.42 · 10−7 , d2 = −95.9; b3 = +0.02, c3 = +8.59 · 10−7 , d3 = −74.1. (183) The dependence of the critical spin Jcr on mass number A and proton number Z is given by the formula: Jcr = Jcr (A, Z) = q1 + q2 Z 2 ,

(184)

qi = q˜i,1 + q˜i,2 · A + q˜i,3 · A2 ,

(185)

The dynamical deformation parameter at slow rotation (Eqs. (179)-(182)) is practically identical to the value obtained analytically in Refs. [304, 362, 365]: β2 −

q˜1,1 = +55.10, q˜1,2 = −6.30 · 10−2 , q˜1,3 = +5.12 · 10−3 ; q˜2,1 = −0.013, q˜2,2 = +2.84 · 10−6 , q˜2,3 = −2.57 · 10−7 . (186)

5π J(J + 1) 2.1A−7/3 . 4 1 − 0.0205Z 2/A

(187)

Compilation and parameterization of GDR resonances built on excited states are given in Refs. [366, 367]. Macroscopic models that describe the relative motion of protons and neutrons have also been successfully applied to reproduce experimental GDR energies and widths [284, 323, 368]. An effect often neglected by systematics concerns the experimentally-observed shell dependence of the GDR width that can be explained by considering the coupling between dipole oscillations and quadrupole surface vibrations [369]. RIPL-3 provides improved predictions of the GDR energies and widths for about 6000 nuclei from 14 ≤ Z ≤ 110 between the proton and the neutron drip lines. The GDR can be represented by the Goldhaber-Teller model [370] in which the proton sphere vibrates against the neutron sphere. The dynamics of the oscillation is assumed to be dominated by the np-interaction as described in Refs. [300, 368]. Both the nucleon density distribution and ground-state deformation are taken from the Extended Thomas-Fermi plus Strutinsky Integral (ETFSI) compilation [31, 371]. The expression for the shell-dependent GDR width is taken from Ref. [369], and is based on the newly-determined GDR energies and ETFSI shell corrections. Comparisons between predicted and experimental GDR energies and widths are shown in Fig. 45 for the RIPL-3 experimental compilation; more details can be found in Ref. [300]. The GDR in deformed nuclei is split into two peaks for oscillations parallel and perpendicular to the axis of rotational symmetry.

E.

M1 and E2 transitions

The GFL and MLO models can be used to estimate ← − the M1 strength f M1 (γ ) over a broad range of γ-ray energies when either experimental data or systematics for ← − ← − the ratio R = f E1 (Bn )/ f M1 (Bn ) at neutron binding energy Bn are known. Under such conditions, the M1 strength function can be calculated from the following relationship: ← − ← − f E1 (Bn ) φM1 (γ ) f M1 (γ ) = , R φM1 (Bn )

where

and

R. Capote et al.

(188)

where φM1 (γ ) describes the shape of the dipole magnetic radiative strength function, and the dipole electric ← − radiative strength f E1 (Bn ) is calculated using one of the models already discussed. Experimental values of the ratio R can be extracted for some nuclei from the gamma/gamma-strength-exp.dat file. A global parame-

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FIG. 45: Comparison of experimental data in the RIPL-3 gamma/gdr-parameters&errors-exp-SLO.dat with GDR energies and widths given in the gamma/gdr-parameters-theor.dat file.

terization of R is given by [10]: ← − f E1 (Bn ) = 0.0588 · A0.878 , R= ← − f M1 (Bn )

Bn ≈ 7 MeV. (189)

Two models are commonly used for the function φM1 : (1) φM1 (γ ) = const as defined in the single-particle model [372]; and (2) φM1 (γ ) which corresponds to the spin-flip giant resonance mode [304, 373, 374] from the SLO model (Eq. (135)), with global parameterization for the energy and damping width of [10]: Er = 41 · A−1/3 MeV and Γr = 4 MeV. E2 radiation is linked to the excitation of the giant quadrupole iso-scalar resonances, and a Lorentzian is recommended to describe the E2 strength [10]. The singleparticle model with energy-independent strengths is recommended for the M2, E3 and M3 radiations [372]. F.

R. Capote et al.

MLO models give similar results at low γ-ray energies (γ ≤ 3 MeV). These three models describe the experimental data much better than the SLO model at low energies, and also define a non-zero and temperaturedependent limit for the vanishing γ-ray energy. Results from the GFL, MLO and SLO models at γ ≥ 5 MeV are closer to the experimental data than those obtained from EGLO. The E1 photo-excitation strength functions for cold nuclei calculated with the MLO and SLO model agree over a reasonably extensive range of γ-ray energies around the GDR peak. An overall comparison of the calculations within different simple models and experimental data shows that the EGLO, GFL and MLO (SMLO) approaches with asymmetric RSF provide a reasonably reliable and simple method of estimating the dipole RSF for both γ-decay and photoabsorption over a relatively wide energy interval ranging from zero to slightly above the GDR peak, when GDR parameters are known or GDR systematics can be safely applied. The MLO (SMLO) closed-form models with asymmetric RSF are recommended for general use; they can be used to predict the statistical dipole γ-ray emission and extract the GDR parameters from the experimental data for heated nuclei. Different variants of the MLO (SMLO) approach are based on sound relations between the RSF and nuclear response function, leading potentially to more reliable predictions among the simple models. However, the energy and temperature dependence of the width Γ(Eγ , T ) is governed by complex mechanisms of nuclear dissipation, and is still an open problem. Reliable experimental information is needed to improve the determination of the temperature and energy dependence of the RSF, so that the contributions of the different mechanisms responsible for the damping of the collective states can be further investigated. This approach would help discriminate between the various closed-form models describing the dipole RSF, and will lead to enhancement of the reliability of the dipole RSF calculations using simple models. Large-scale HFB+QRPA calculations of the E1 strength have been undertaken in Refs. [340, 342], and give the same degree of accuracy as the MLO model in the energy range from 4 to 8 MeV for nuclei close to the stability line. However, HFB+QRPA calculations reveal broadening of the GDR shape when moving away from the stability line. This effect stems from the microscopic treatment and can not be accounted for by using experimental GDR shapes, which were measured for stable nuclei only. Thus, the use of the HFB+QRPA results should be recommended for calculations on nuclei far from the stability line.

VIII.

Recommendations

Numerical studies indicate that the calculations of the γ-decay strength functions within the EGLO, GFL and

NUCLEAR FISSION

Nuclear fission remains the most complex topic in applied nuclear physics. Since its discovery, nuclear fis-

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sion has always been an active field of research both as a purely theoretical challenge, as well as for practical applications. Fission is a process whereby a complex quantum system (nucleus) goes from an equilibrated shape to such a highly deformed shape that it finally splits into two lighter fragments. The probability that fission occurs when competing with other decay channels, the number and spectrum of neutrons released during the fission process, the fragments’ charge and mass distributions, the energy released, the fragments’ deformations and excitation energies are among many aspects of the fission process which are addressed on the basis of different models and are not yet fully understood. The RIPL project had no intention to cover the whole complexity of fission phenomena. Instead we focus on those input parameters needed in fission cross-section calculations for energy and non-energy applications. The most important of these activities are mentioned below. • The broad scale investigations of fission cross sections for pre-actinides that undergo different charged-particle induced reactions provide the main experimental information on the droplet properties of nuclei. The interest in such data has increased dramatically over the previous decade due to extensive discussions of accelerator-driven power systems for the transmutation of nuclear waste. • The highly accurate studies of actinides that undergo neutron-induced reactions have a significant importance in reactor physics. Such studies are extended to the high energies of neutron and charged-particle induced reactions to address the needs of emerging nuclear technologies such as accelerator-driven systems or production of alpha emitters for radiotherapy. They also allow the investigation of the changes of fission barriers with increasing excitation energy. • Microscopic calculations of fission barriers are required to explain the existence of long-lived superheavy elements with Z > 102 and to provide extensive data on fission for astrophysics, which encompasses stellar nucleosynthesis and the rapid neutron capture process. These fields of interest have been reflected in the way fission was represented in the different stages of the RIPL project. In the Starter File no special segment was dedicated to fission [6]. However, a section entitled “Fission Level Densities” was introduced in the Level Densities chapter. Because fission level densities are strongly correlated with the fission barriers, phenomenological parameters describing the fission barrier and the transition states for the most important actinides were also provided (see section VIII B 2). RIPL-2 brought two novelties : (1) phenomenological fission barriers and transition-state level density functions for pre-actinides (see section VIII B 1); and (2) large-scale microscopic calculations of fission barriers for 2301 nuclei with 78 ≤ Z ≤ 120 derived by means of the Extended Thomas-Fermi plus Strutinsky Integral (ETFSI) method and the corresponding level density at saddle-point(s) based on HFBCS single-particle proper-

R. Capote et al.

ties. In RIPL-3 the focus is on new microscopic determinations that can provide information on the fission path and the corresponding nuclear level densities required for fission calculations. A set of fission paths within the HFB method and corresponding level densities at saddle points within the combinatorial model are provided in a table format. These two nuclear ingredients are determined coherently, the NLD being estimated using the singleparticle scheme and pairing strength of the same mean field model that was used to determine the fission saddle points. The compilation includes fission paths and the corresponding nuclear level densities for about 1000 nuclei with 90 ≤ Z ≤ 102 from the valley of β-stability to the neutron drip line (see section VIII C). An important RIPL-3 milestone is the validation of recommended fission parameters in calculations of neutron-induced fission cross sections. In RIPL-3 both microscopic and phenomenological fission barriers and NLDs at saddle points for actinides have been tested in fission cross section calculations, as shown in section VIII D using the basic relations and methods presented in section VIII A. In addition, in RIPL-3 the estimation of the prompt fission neutron spectrum using the “Los Alamos model” is provided.

A.

Basic relations

The main concepts of nuclear fission theory are based essentially on the liquid-drop model [375, 376]. According to this model, competition between the surface tension of a nuclear liquid drop and the Coulomb repulsion related to the nuclear charge leads to the formation of an energy barrier which prevents spontaneous decay of the nucleus by fission. The penetrability of the barrier determines the half-life for spontaneous fission. In the liquid-drop model, the height of the fission barrier for heavy nuclei decreases rapidly with increase of Z 2 /A, and should disappear when (Z 2 /A)cr ≈ 46−48. The decrease in height results in an exponential increase in barrier penetrability. These barrier changes exhibit good agreement with the behavior of the spontaneous fission lifetimes of the actinide nuclei, ranging from the long-lived isotopes of uranium to the artificially synthesized short-lived isotopes of fermium and mendelevium [377]–[381]. Early studies showed that, despite some successful results, the liquid-drop model cannot explain the major peculiarity of spontaneous and low-energy fission of the actinides, namely the asymmetric mass distribution of fission fragments [382]. Initially, fission mass-asymmetry was explained in terms of some modifications of the liquid-drop model predictions for configurations close to the scission point, although the strong influence of shell effects on fission fragment formation was already foreseen.

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B Eci (Jπ )

εi (KJπ ) transition states

U

ai

Ei (KJπ )

bi Bfi

class II states

β FIG. 46: Double-humped fission barrier and associated parameters: Bf i is the height of the fundamental fission barrier i, i (KJπ) is the energy of the transition state i, and Eci (KJπ) is the cutoff energy, above which the continuum starts for a barrier i. Each transition state has an associated barrier above the fundamental barrier; ai and bi are the classical turning points of Eq. (192).

Newly discovered phenomena in the 1960s, particularly the spontaneously-fissioning americium isomers [383] and the intermediate resonance structures observed in neutron-induced fission cross sections [384, 385], required radical changes in the fission model. Calculations of nuclear deformation energies based on the shell correction method by Strutinsky [386] played a crucial role in explaining the above phenomena. The fission barriers calculated for the actinides consisted of a doublehump curve with a rather deep potential well between the humps. This double-hump shape also identified the spontaneously-fissioning isomers as the lowest states of a fissioning nucleus in the second potential well and the intermediate resonances as excited states of a nucleus in this well [53, 387]. Since the 1970s, many improvements have been made in the liquid-drop-type models, although inherent problems related to the consistency between the macroscopic and microscopic parts still remain. Fission barrier calculations have also been performed in the framework of the Thomas-Fermi model [388]–[390], as well as more recently the Hartree-Fock-Bogolyubov model [18, 391]– [393]. While reflection asymmetry is always required for a proper description of the outer barrier, only a few calculations take the triaxiality of the inner barrier into account (e.g. [392]). Briefly, the concepts most important for fission cross section calculations are: the double- or triple-humped barriers of the actinides, the role of transition states (or fission channels) played by the excited states of the nucleus with deformations corresponding to the maxima of the fission barriers (saddle points), and the influence of the quasi-stationary states of the nucleus in the well(s) (the class II or III states) on the transmission through the barrier [394]. Presently all the codes involved in nuclear data evaluation use static, one-dimensional fission barriers, repre-

R. Capote et al.

senting the energy of a nuclear state along the deformation path. Fission barrier is considered, in most cases, as a function of the quadrupole deformation. Consequently, there is a set of barriers corresponding to the entire spectrum of nuclear states as functions of deformation which, for practical calculations, is divided into: a fundamental barrier corresponding to the lowest state (ground state at a given deformation), several discrete barriers corresponding to the low-lying excited states, and the barriers described by level-density functions corresponding to the states with higher excitation energies. Present knowledge indicates that the pre-actinides have single-hump barriers, while the actinides have double- or triple-humped barriers. Usually, the barriers associated with the discrete transition states are parametrized as a function of the quadrupole deformation β by inverted parabolas (see Fig. 46): 1 Bi (β) = Bf i − μ2 ωi2 (β − βi )2 , i = 1, N, (190) 2 where N is the number of humps, the energies Bf i represent maxima of the deformation potential, βi are the deformations corresponding to these maxima, the harmonic oscillator frequencies ωi define the curvature of the parabolas and μ is the inertial mass parameter, assumed independent of β and approximated by the semiempirical expression μ ≈ 0.054A5/3 MeV−1 , where A is the mass number of the fissioning nucleus. Since comprehensive microscopic calculations have become available, providing all the nuclear ingredients required to describe the fission path from the equilibrium deformation up to the nuclear scission, the fission barriers are available in tabular form. The discrete transition states for all barriers i (i = 1, N ) are obtained by building rotational levels on vibrational or non-collective bandheads, characterized by a given set of quantum numbers (angular momentum J, parity π and angular momentum projection on the nuclear symmetry axis K) with the excitation energies: 2 [J(J + 1) − K(K + 1)], 2 i (191) where i (Kπ) are the bandhead energies and 2 /2 i are the inertial parameters (the decoupling parameter for K = 1/2 bands was neglected). A parabolic barrier with height Ei (KJπ) and curvature ωi is associated with each transition state. Usually, Bf i , ωi , and i (Kπ) are free parameters and their values are extracted from systematics or obtained from a fit to the experimental fission cross section. Since the inner barrier is known to be triaxial (e.g. [392]), K is not a good quantum number, therefore Eq. (191), strictly speaking, is not valid for inner barriers. However, Eq. (191) is used for all barriers throughout this work, triaxiality is considered through the symmetry enhancement factor fsym as will be shown below. The transition state spectrum has a discrete component up to a certain energy Eci , above which it is continEi (KJπ) = Bf i + i (Kπ) +

3177


NUCLEAR DATA SHEETS

uous and described by level density functions, ρi (, J, π), accounting for collective enhancements specific to the nuclear shape asymmetry at each saddle point. The transmission coefficients through each hump are expressed in first-order WKB approximation in terms of the momentum integrals for the humps [394]–[396]: ( ( ( ( bi ( ( Ki (U ) = ± ( 2μ[U − Bi (β)]/2 dβ ( , i = 1, N, ( ( ai (192) where ai and bi are the classical turning points (see Fig. 46) and U is the excitation energy in the fissioning nucleus. The + sign is taken when the excitation energy U is lower than the hump under consideration and the − sign when it is higher. In the latter case, the intercepts are complex conjugate ( bi = a∗i ) and the WKB approximation is valid when their imaginary parts are small, i.e., for energies slightly higher than the hump. The singlehump transmission coefficients Ti turn out to be Ti (U ) =

1 1 + exp[2Ki (U )]

i = 1, N.

(193)

In the case of a single parabolic barrier, Eq. (193) yields the well-known Hill-Wheeler transmission coefficient [394] TiHW (U )

1 = 1 + exp [−(2π/ωi )(U − Bf i )]

i = 1, N,

(194) which is an exact result. To avoid numerical problems due to the failure of the WKB approximation, it is recommended to always use the Hill-Wheeler expression above the barrier. In the case of the single-hump barriers, the total fission transmission coefficient for a given excitation energy U , spin J and parity π is determined by summing the penetrabilities through all the fission barriers i (i = 1, N ) that might be tunneled through, i.e., Ti (U Jπ) = Ti (U Kπ) +

K≤J ∞

Eci

ρ (εJπ)dε i . 1 + exp − 2π ωi (U − Bf i − ε)

(195)

The sum runs over all the discrete transition levels having the same spin J and parity π as those of the decaying compound nucleus, and the integration runs over the continuum of the transition levels described by the NLD ρi (εJπ). In practical calculations, the NLD ρi (εJπ) at the saddle points have usually been determined from various implementations of the Fermi gas model. As developed for the ground-state NLD, many different expressions exist, with or without an explicit account of pairing, shell and deformation effects. An important effort has recently been made to improve the quality of both microscopic [203, 225] and analytical [45, 190] NLD models and parameterizations (see chapter VI). For the ground-state

R. Capote et al.

deformation, both types of NLD models are adjusted in order to obtain simultaneously the best description of the low energy cumulative discrete level schemes as well as the mean s-wave resonance spacing at the neutron separation energy, whenever available. These NLD models are considered global as they reproduce observables for all nuclei in a reasonable way. Since they have been adjusted to experimental data, they should not be modified in the description of the fission cross sections. In contrast, the saddle-point level densities (required to determine fission transmission coefficients) suffer from a much more severe lack of experimental information than those describing nuclei in their equilibrium configuration. The NLD parameters are in this case essentially tuned to reproduce fission cross sections. However, it remains theoretically difficult to apply models used for ground-state NLDs to the estimate of the NLDs at saddle points. The difficulty lies in the large sensitivity of NLD (in particular shell, pairing and collective effects) to the deformation parameters and the breaking of additional symmetries around the saddle points. It is well known [168, 392, 394] that the axial symmetry could be broken at the inner barrier and that the outer barrier is usually mass-asymmetric and sometimes triaxial. Saddle-point symmetry depends on the charge and mass of the fissioning nucleus. The quasi-stationary states of the nucleus in the well(s) between the humps (called class II or class III states for the second and third well, respectively) have a strong modulating effect on the penetrability of the multihumped barriers [53, 395, 397, 398]. These modulations are manifested in the sub-barrier fission cross sections as intermediate structures, examples of which are the resonances observed in the 230 Th(n,f) reaction at neutron energies of about 700 keV [384] and the structure displayed in the neutron-induced fission cross sections of 240 Pu [385]. More comprehensive discussions of the experimental data on intermediate structures in actinide fission cross sections can be found in Refs. [53, 394], and details of the fission models implemented in codes used for evaluation can be found in Refs. [45, 399, 400, 401]. Usually the wells are considered deep enough so that the transmission coefficient can be averaged over the intermediate structures. For a double-humped barrier the fission coefficient becomes Tf (U Jπ) =

T1 (U Jπ)T2 (U Jπ) , (196) T1 (U Jπ) + T2 (U Jπ) + TγII (U Jπ)

where T1 and T2 are the penetrabilities of the inner and the outer humps, respectively, calculated according to Eq. (195), and TγII is the probability for gamma decay in the second well. Because in general, no information about wells is used and in most cases TγII 1 used in all analyses of fission cross sections. However, there are large uncertainties associated with estimates of the coefficient ratio cs /cv [415, 418, 419]. Evaluations of the ratio cs /cv = 1.30 based on microscopic calculations involving the single-particle level schemes of the Woods-Saxon potential [415], or cs /cv = 1.53 derived from phenomenological

3181


NUCLEAR DATA SHEETS

8.0

Outer barrier BB(MeV)

TABLE XI: Fission barrier parameters for actinides: BA (BB ) and Sym–A (Sym–B) are the height and symmetry of the inner (outer) fission barrier, respectively. The nuclear shape symmetries were adopted from FRDM calculations by Howard and M¨ oller [427] as recommended in [6]. Δf is the pairing parameter at saddle points.

Th Pa U Np Pu Am Cm

7.5 7.0 6.5 6.0

R. Capote et al.

5.5 5.0 4.5

Inner barrier BA(MeV)

7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 230 232 234 236 238 240 242 244 246 248 250

Mass number FIG. 48: Outer BB (top panel) and inner BA (bottom panel) fission barriers for actinide nuclei derived from the analysis of neutron-induced fission cross sections [421]–[425].

systematics of the neutron resonance densities seem to be the most reasonable approaches. Both methods give an af /an ratio of between 1.05 and 1.07 which is supported by the direct results of fission cross-section analyses [408].

2.

Fission input data for actinides

Highly accurate evaluations are required for the neutron-induced fission cross sections of the main fissile and fertile nuclei. Full-scale Hauser-Feshbach theory, the coupled-channel optical model, and the double-humped fission barrier parameterization are normally used in such calculations, supported by numerous experimental data that demonstrate the importance of the shell, pairing, and collective effects at both the equilibrium and saddle deformations. Fundamental fission barriers Fission barrier parameters corresponding to the lowest state have been estimated from modelling analyses of the available experimental data for the neutron-induced fission cross sections of the uranium, neptunium, plutonium, americium and curium isotopes [420]–[422]. Barrier parameters for Th and Pa nuclei were also obtained by adopting essentially the same approach [423]–[425];

S – axially symmetric saddle point, GA – axially asymmetric saddle point, MA – mass asymmetric saddle point.

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NUCLEAR DATA SHEETS

however, these specific data were less accurately estimated due to the more complex structure of the fission barriers in light actinides as compared to transuranium nuclei. The inner and outer fission barrier heights are shown in Fig. 48. A comparison between these data and the fission barriers evaluated in Ref. [406] shows that there is good agreement for the outer barriers of the uranium, plutonium and americium isotopes. Some disagreements exist for the inner barrier heights, although the general isotopic dependencies are very similar; furthermore, significant discrepancies occur in the case of the outer barriers of the curium isotopes. Barrier curvature parameters are usually evaluated with relatively large uncertainties (∼ 100−150 keV) from analyses of the sub-barrier behavior of fission cross sections [394, 426]. The following curvature parameters can be recommended (in MeV): ωA ωB

even-even 0.9 − 1.0 0.6

odd 0.8 0.5

odd-odd 0.6 0.4

Even-odd differences in the curvature parameters are dependent on the superconducting pairing correlations and their impact on the stiffness and reduced mass parameters of the fissioning nucleus at the saddle point. An empirical set of double-humped fission barrier parameters for the most important actinides containing the heights, widths and nuclear shape symmetries corresponding to the inner and the outer humps, together with the pairing correlation energies used in level density calculations are given in the fission/empirical-barriers.dat file. The same data, except for the barrier widths, are also presented in Table XI. The nuclear shape symmetries listed in Table XI, and recommended in RIPL-1 [6], were adopted from Howard and M¨ oller macroscopic-microscopic calculations using the FRDM model [427] (the droplet model with a 1973 set of parameters). Recently, a new comprehensive set of fission barriers and corresponding symmetries have been obtained by M¨ oller et al. [428]. However, numerical results of new calculations have not been released yet. The parameters of the fundamental fission barrier are strongly related to the transition-state level densities. Even in the case of fertile nuclei for which the threshold behavior of the fission cross section provides a relatively clear signature of the barrier height, the barrier determination remains sensitive to the NLD adopted. The leveldensity formulation and the associated parameters used in the analysis to provide the fission barrier parameters in Table XI are described in the RIPL-1 Handbook [6]. Many of the analyses of fission cross sections mentioned above did not exceed 20 MeV incident energy. According to the general concepts of the shell correction method, the double-humped structure of any fission barrier should disappear at higher excitation energies, and liquid-drop fission barriers with asymptotic level density parameters should be used to describe both the fission cross section and all competitive reaction

R. Capote et al.

cross sections at excitation energies above 30–50 MeV. This prediction would appear to be supported by the available experimental data [213, 414], although there are difficulties separating unambiguously the first-step fission of an initial nucleus from the multi-chance fission of daughter nuclei that arise after the emission of one or more nucleons. So, we have only theoretical estimates of the excitation energy at which the transition from shell to liquid-drop model behavior should occur, and lack a satisfactory understanding of what approximations should be applied to define the fission barriers and fission level densities in the transitional regime. Discrete transition state spectra The quantum structure of the fission channels at the top of each barrier is extremely important in the formulation of accurate descriptions of the fission cross sections at the sub-barrier and near-barrier energies. Such structure depends strongly on the symmetry of nuclear deformations at the corresponding saddle-point states. Shell model calculations have been undertaken by many authors since the early days of fission studies (e.g., see recently published HFB calculations using a Gogny interaction by Delaroche et al. [392] and references therein). The fissioning actinide nucleus has been calculated to possess axial asymmetric shape (triaxial with mirror symmetry) at the inner saddle, and axialsymmetric and mirror-asymmetric shape at the outer saddle. Mirror-asymmetric even-even nuclei have saddlepoint rotational band levels with K π = 0+ , J= 0, 2, 4. . . , that are almost degenerate in energy with the octupole band levels K π = 0− , J= 1, 3, 5. . . . Therefore, the lowest rotational band at outer saddle deformation includes levels with all possible values of angular momentum and parity. A similar situation arises for triaxial shapes and levels of the γ-vibrational band. The situation is even more complex for odd and odd-odd fissioning nuclei: the quantum number of the corresponding rotational bands should be estimated in accordance with the angular momentum addition rules for unpaired particles and the corresponding rotational bands. As a result, the fission cross section analysis involves a much more complex structure of saddle transition states than the well-studied collective level sequences of the deformed rare-earth nuclei and actinides at low excitation energies. The discrete transition state spectra in odd fissioning nuclei for excitation energies up to 200 keV, can be constructed using one-quasiparticle states calculated by Bolsterli et al. [429]. Each one-quasiparticle state is assumed to have a rotational band built on it with a rotational constant (or moment of inertia) that depends on the saddle-point deformation (see Table XII (a,c)). As a consequence of the axial asymmetry (triaxiality) at the inner saddle point, we additionally assume (2J + 1) rotational levels for each J value. The positive parity bands K π = 1/2+, 3/2+ , 5/2+ . . . at the outer saddle are assumed to be doubly degenerate due to mass asymmetry

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NUCLEAR DATA SHEETS

R. Capote et al.

TABLE XII: Discrete transition spectra bandheads at the top of corresponding fission barriers. (a)Z-odd, N -even fissioning nuclei.

inner saddle

(b)Even-even fissioning nuclei.

outer saddle

inner saddle

K π EK π , MeV K π EK π , MeV 3/2− +

0.0

5/2+ −

0+

0.0

outer saddle

ax nonax K π EK , MeV K π EK π , MeV π , MeV EK π

0.0

0.0

0+

0.0

5/2

0.140

5/2

0.0

2

0.5

0.1

2+

0.5

7/2−

0.180

3/2+

0.08

0−

0.4

0.4

0−

0.2

5/2−

0.180

3/2−

0.08

1−

0.4

0.4

1−

0.5

+

+

+

1/2

0.04

2

0.5

0.5

2

1/2−

0.04

2−

0.4

0.4

2−

1/2+

0.05

0+

0.8

0.8

0+

1/2−

0.05

0+

0.8

0.8

0+

(c)Z-even, N -odd fissioning nuclei.

inner saddle

(d)Odd-odd fissioning nuclei.

outer saddle

inner saddle

K π EK π , MeV K π EK π , MeV 1/2+ +

+

0.0

1/2+

1−

0.0

1−

0.0

0.08

1/2

0.0

0

0.044

0−

0.044

1/2−

0.05

3/2+

0.08

5−

0.049

5−

0.049

3/2

0.0

−

−

0.0

5/2

−

−

outer saddle

K π EK π , MeV K π EK π , MeV

−

−

3/2

0.08

6

0.170

6

5/2+

0.0

1−

0.220

1−

0.220

5/2−

0.0

3−

0.242

3−

0.242

2−

0.288

2−

0.288

(parity is not a good quantum number). Recommended discrete transition states for even-even fissioning nuclei are listed in Table XII (b); they are given up to higher excitation energies U ∼ 1 MeV than similar states in odd fissioning nuclei, because the pairing interaction suppresses single-particle excitations within the pairing gap of the even-even fissioning nucleus. Therefore, discrete states for even-even fissioning nuclei are pure vibrational bandheads (of collective nature) lying within the pairing gap. In the case of axial symmetry at the inner saddle the bandheads spectra are similar to the ones at the equilibrium deformation. In the case of axial asymmetry at the inner saddle, the 2+ bandheads are lowered. The position of the negative parity band K π = 0− at the outer saddle is lowered due to mass asymmetry. The use of transition states in calculations of neutron-induced fission cross sections on 232 Th was extensively discussed by Maslov [426, 430]. The intrinsic two-quasiparticle spectrum of the oddodd fissioning nuclei (such as 238 Np and 240,242,244 Am) at equilibrium deformation can be obtained following Sood and Singh [431]. The expected location of the still unobserved two-quasiparticle states was estimated (see Table XII (d)). Using these intrinsic states as the bandhead energies, the rotational bands can be built in the same way as for Z-odd, N -even nuclei. Level densities at saddle points The level density of a deformed nucleus at equilibrium deformation has been proven to depend on the collective properties, pairing correlations, and the shell structure,

0.170

and these effects can be introduced within the framework of the Generalized Superfluid Model (GSM) [171] or the EGSM extension. It would be natural to use the same model to describe the level densities for the deformations specific to the saddle points, also referred to as the level densities of the transition states. The difference between modelling the densities of the transition states and the normal states is that in the first case there are no experimental data available for normalization. Therefore, the starting values of the parameters entering the transition state density are obtained by using physical arguments to adjust the parameters of the normal states at equilibrium deformation for the saddle point deformation. The final values are deduced from the fit of the experimental data for neutron-induced fission cross sections. The total nuclear level density is represented as a product of quasi-particle and collective contributions. The collective contribution to the level density at a saddle point is defined by the order of symmetry of the saddle point (see Table XI) derived from calculations using the shell correction method [427] : ρsad (EJ) = ρsad int (EJ)Krot (E)Kvib (E) × damp Qdamp rot (E)Qvib (E)fsym .

(208)

The level density ρsad int describing the single-particle excitations, the rotational and the vibrational enhancements Krot and Kvib and the corresponding damping coand Qdamp are calculated following the efficients Qdamp rot vib GSM prescriptions in chapter VI adapted to the deformation specific to each saddle point. It is important to note that Eq. (208) is, strictly speak-

3184

NUCLEAR DATA SHEETS

ing, only valid for the GSM. EGSM eqs. (105) and (106) for the intrinsic level density ρsad int already include a summation over projection of the angular momentum K, and thus account automatically for the rotational enhancement. Therefore, the rotational enhancement factor Krot (E) in Eq. (208) should be set to one for the EGSM. As in the case of the GSM, the EGSM includes the damping parameterization Qdamp rot (U ) given by Eq. (98). The EGSM vibrational enhancement is based on a liqLDM uid drop estimate Kvib given by Eq. (111), instead of Eqs. (100)–(102) used in the GSM. The EGSM vibrational damping Qvibr (t) was calculated using Eq. (112). The enhancement factor fsym (common for both EGSM and GSM) associated with the nuclear shape symmetry is determined in Ref. [432]: ⎧ 1 S–axial symmetry, ⎪ ⎪ ⎨2 MA–mass asymmetry, fsym = (209) π/2σ GA–triaxial+mirror symmetry, ⎪ ⎪ ⎩√ 8πσ triaxial+mass asymmetry, where σ is the spin cutoff factor parallel to the axis of symmetry (or parallel to the longest axis for triaxial shapes). The saddle-point symmetries (SYM–A and SYM–B) depend on Z and N of the fissioning nucleus as listed in Table XI [427]. The factor fsym reflects the enhancement of the level density with respect to the level density of an axially symmetric nucleus. The calculation of the level density parameter af i for the transition states at the hump i requires knowledge of the asymptotic value a ˜f i , of the damping of the shell effects γf i , and of the shell correction δWf i . All these quantities are interrelated and depend on their corresponding values for the equilibrium deformation; therefore, it is difficult to provide a general prescription for their calculation. No specific recommendation for the asymptotic values of the level density parameter at the saddle points for actinides is given in RIPL-2. Calculations show that the recommended a ˜f /˜ an ratio of between 1.05 and 1.07 given for pre-actinides might also be adopted as a starting value for actinides. The shell corrections at saddle points recommended in RIPL-2 were derived from the analysis of fission cross sections in the first plateau region above the fission threshold, using the phenomenological version of GSM (see chapter VI). Their values in MeV for the inner and the outer barriers were calculated according to:

2.6 Z ≤ 97 δWf 1 = 2.6 − 0.1(Z − 97) Z > 97

0.6 + 0.1(Z − 97) + 0.04(N − 143) Z < 97 δWf 2 = 0.6 + 0.04(N − 143) Z ≥ 97 (210) For the pairing correlations, √ RIPL-2 recommends the parameterizations Δf i = 14/ A MeV, or Δf i = Δ0 + 0.2 MeV, where Δ0 is the pairing for the neutron channels [10]. These values were also deduced from the fit to the experimental neutron-induced fission cross sections.

R. Capote et al.

It should be noted that the uncertainty of the above estimates is significant [394], both for the pairing parameter Δf i and shell corrections δWf i at saddle points. The damping of the shell effects at saddle points particularly influences the shape of the first-chance fission cross section at excitation energies above the fission threshold (i.e., in the “fission plateau”). The shell-effect damping parameter γ0 used for equilibrium deformation (see Eqs. (51 and (53)) should be slightly adjusted to fit the experimental value of the fission cross sections, as will be shown below. RIPL-2 recommends the perpendicular moments of inertia to be equal to 1002 /MeV for the inner saddle, 2002/MeV for the outer saddle and approximately 752 /MeV for the ground state [433]. Such information is useful when decoupled humps are considered and the nuclear shape deformation at the saddle points is not known. When the full fission path is used, the deformation at the saddle points can be determined and the moments of inertia can be calculated according to the formulae given in chapter VI. Note that matching the discrete transition states at saddle points (recommended in Table XII) with the level density parameters used at higher energies can be important for a consistent description of the fission cross sections at near-barrier energies. 1014

Level density (1/MeV)


10

235-U ground state 236-U ground state 236-U inner saddle point 236-U outer saddle point

12

1010 108 106 104 102 1

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16

Excitation energy (MeV)

FIG. 49: Total EGSM NLD of 236 U at the ground state (dotted blue), inner saddle (dotted dashed red) and outer saddle (dashed black). Total EGSM NLD of 235 U at the ground state is also shown (solid blue).

The EGSM NLDs at the ground-state and inner (outer) saddle points for the 236 U nucleus (blue and red (black) dotted lines, respectively) are shown in Fig. 49. The NLD is highest at the inner saddle point, in accordance with the strong collective enhancement of the level density at this saddle (due to triaxiality). The NLD at the ground state for the 235 U nucleus (solid blue line) is also shown in Fig. 49. A typical shift to lower excitation energies in the NLD of the odd 235 U nucleus is observed (compared to the NLD of the even-even 236 U nucleus). Analyzing all the recommendations for the calculation of fission level densities even if they do not repre-

3185

sent a unique prescription to be used in fission cross section evaluations, they provide valuable hints. Extensive fission-cross-section calculations for a significant number of actinides have been performed using the EGSM (which is a close version of GSM as shown in chapter VI) for both normal and transition states. For the transition states the above recommendations for the asymptotic value of the level density parameter af i , the shell corrections δWf i , and the pairing correlations Δf i have been adopted. All the remaining parameters, including the moments of inertia, have been calculated according to the formulae presented in chapter VI by replacing the equilibrium deformation with the deformations corresponding to the saddle points. Parameters which are independent of deformation have been initially considered to be the same as for equilibrium deformation. A reasonably good global description of the experimental fission cross section was obtained by adjusting only the parameter γ0 of Eq. (53) describing the damping of the shell correction with a factor of 0.5, and the vibrational enhancement (calculated according to Eq. (111)) with a factor of 2. The fission cross sections obtained using the phenomenological barrier parameters from Table XI together with the discrete transition states given in Table XII and combined with the EGSM used for the level densities of both normal and transition states will be discussed in section VIII D 1.

C.

Microscopic fission barriers and level densities

Among the non-energy nuclear applications which require estimates of a large number of fission data, the most important is the stellar nucleosynthesis of the rapid neutron-capture process (or r-process) invoked to explain the origin of approximately half of the stable nuclides heavier than iron that are observed in nature. Although the exact astrophysical site in which the r-process develops still represents one of the major puzzles in nucleosynthesis theory, the r-process is believed to take place in environments characterized by high neutron densities (Nn ≥ 1020 cm−3 ), such that successive neutron captures can proceed into neutron-rich regions well off the β-stability valley. This process produces nuclei with decreasing neutron binding energies, and consequently faster (γ, n) photodisintegration, which at high temperatures (T ≥ 109 K) is able to compete with the (n, γ) reactions. With timescales usually thought to be much longer than the characteristic timescale of the (n, γ) and (γ, n) reactions, the β-decays finally drive material to higher Z elements. Depending on the strength of the neutron flux as well as the duration of the neutron irradiation, nuclei up to the super-heavy mass region can be produced in timescales of the order of a second. So far, no astrophysical model is capable of providing reliable predictions for the thermodynamic conditions in which the r-process can take place. In the description of the nuclear mechanisms taking place during the r-process, fission processes are

B(exp)-B(th) [MeV]

NUCLEAR DATA SHEETS

R. Capote et al.

4 2 0 -2 -4

B(exp)-B(th) [MeV]


4 2 0 -2 -4 135

140

145

150

155

N FIG. 50: Deviations between empirical (Table XI) and HFB14 highest (top panel) and second highest (bottom panel) barriers for actinide nuclei with 88 ≤ Z ≤ 96.

important in many respects. First, if matter reaches the super-heavy neutron-rich region during the neutron irradiation, fission could recycle the nuclear flow and, consequently, affect the nucleosynthesis of the bulk of the r-process nuclei. Second, all nuclei produced during the r-process irradiation and characterized with A ≥ 208 are the progenitors of the r-abundance peak in the Pb region. The production of these neutron-rich heavy progenitors depends on the fission probabilities, so that a correct treatment of fission processes is needed to predict the production of Pb and Bi during and after neutron irradiation. Finally, the r-process nucleosynthesis is at the origin of the long-lived actinides 232 Th, 235 U, 238 U that are used as cosmochronometers to estimate an upper limit of the age of the Galaxy. An accurate knowledge of both the astrophysics conditions, as well as the nuclear properties affecting the progenitors of these chronometers (and in particular the fission probabilities of their progenitors) is a fundamental prerequisite for a reliable estimate of age. To calculate the changes in abundance of the heavy nuclei as a result of the high neutron fluxes and high temperatures encountered during the r-process nucleosynthesis, a nuclear reaction network including all neutron captures, photodisintegrations, β-decays, as well as βdelayed processes, fission processes and α-decays must be used. The fission processes include spontaneous, βdelayed and neutron-induced fission, the probabilities of which must be estimated for some 2000 nuclei with 80 ≤ Z ≤ 110 mainly located in the neutron-rich region of the nuclear chart. For these specific applications, a compilation was included in RIPL-2 containing the fission barriers for some 2301 nuclei with 78 ≤ Z ≤ 120 derived using the ETFSI method and the corresponding NLDs at both saddle point deformations predicted by the microscopic model of NLD. More details are given in the RIPL-2 Handbook [10]. A set of fission paths obtained by the Hartree-FockBogolyubov (HFB) calculations and corresponding nu-

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NUCLEAR DATA SHEETS

R. Capote et al.

(a)Fission paths for the U (top left panel) and Pu (top right panel) isotopes close to the valley of β stability.

10 Cm252 Cm254 Cm256 Cm258 Cm260 Cm262 Cm264 Cm266 Cm268

E-EGS [MeV]

8

6

Cm270 Cm272 Cm274 Cm276 Cm278 Cm280

4

2

0 0

0.5

1

1.5

2

0

0.5

1

2

1.5

2

2.5

2

(b)Fission paths for the neutron-rich isotopes of Cm.

FIG. 51: Fission paths (i.e., total energy with respect to the ground-state energy EGS ) as a function of the quadrupole deformation parameter β2 for U, Pu and Cm isotopes.

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NUCLEAR DATA SHEETS

clear level densities (NLDs) at the fission saddle points from the HFB combinatorial model are provided in table format in RIPL-3. These two nuclear ingredients are strongly interdependent and are determined coherently, the NLD being estimated from the single-particle scheme and pairing strength of the same mean field model HFB14 that was used to determine the fission saddle points. More details are given in the next sections.

1.

Fission path

The HFB model has proven capacity to estimate the static fission barrier height with a relatively high degree of accuracy. In particular, the barriers determined within the HFB-14 model [18] (see also chapter MASSES) reproduce the 52 primary empirical barriers (i.e., the highest barriers of prime interest in cross-section and decay calculations) of nuclei with 88 ≤ Z ≤ 96, which are always less than 9 MeV in height, with an rms deviation as low as 0.67 MeV. A similar accuracy is obtained (0.65 MeV) for the secondary barriers (i.e., the second highest barriers). The empirical (see Section VIII D 1, Table XI) and HFB-14 barriers are compared in Fig. 50, where differences up to 1 MeV on the highest barrier can be observed. Such a large difference obviously has a significant impact on cross section calculations, but at this stage, no theoretical models can claim to provide predictions of barrier heights with a global accuracy better than 0.5–1 MeV (in the best case). The HFB-14 model usually overestimates the height of the primary barrier, so that a global decrease of the energy surface may be required. The deformation energies relative to the ground-state energy along the fission path predicted by HFB-14 are illustrated in Fig. 51(a) (previous page) for the U and Pu isotopes close to the valley of β stability. The fission path corresponds to the most gently climbing or steepest descending path found and projected along one deformation parameter, namely the quadrupole deformation β2 (for more details, see [18]). Although for some U and Pu isotopes a tiny third barrier clearly appears at large deformations, the fission path for these nuclei appears to be well represented by a traditional doublehumped barrier, at least locally close to the saddle point deformations. The situation can be quite different as soon as we depart from the valley of stability. As shown in Fig. 51(b) [393], the fission path for exotic neutronrich nuclei ( e.g., curium isotopes) cannot, in general, be simply approximated by a double-humped barrier with parabolic shapes. To estimate the transmission coefficients with fission barriers deviating from the simple inverted parabolic picture, the full WKB method as given by Eq. (192) is used. The deformation energies relative to the ground-state energy along the HFB-14 (HFB model with the BSk14 Skyrme force) fission path have been estimated for all nuclei with 90 ≤ Z ≤ 102 lying between the valley of β-stability and the neutron drip lines, i.e., about 1000

R. Capote et al.

nuclei. The heights of the primary, secondary and tertiary barriers (i.e., the highest, second highest, and third highest barrier, respectively) for the 90 ≤ Z ≤ 101 nuclei are illustrated in Fig. 52. Generally, for most of the neutron-rich nuclei, three barriers are found. The calculated HFB deformation energies along the fission path as a function of the quadrupole deformation parameter β2 are available in table format in the fission/hfb-barriers/ directory. For an accurate description of the fission cross section, the barrier heights can be renormalized (see section VIII D).

2.

Nuclear level densities at saddle points

The HFB combinatorial method developed to estimate the NLD at ground-state deformation [225] (see chapter VI) has been used to calculate the NLD coherently at the saddle points, making use of the corresponding HFB predictions for the single-particle level scheme and pairing strength at the corresponding deformation. However, although the NLD is rather well constrained by the HFB structure properties (though still affected by the complicated determination of the saddle point deformation), the inclusion of phonon excitations is still subject to a rather large uncertainty. Due to the lack of observables, the same prescription is used for the saddle points as for the ground state, i.e., a total of three phonons have been coupled to the excited configurations of a maximum of up to three particle-hole pairs. Quadrupole, octupole, and hexadecapole phonons are included, their energies being assumed identical to those of the ground state. This prescription leads to a damping of the NLD vibrational enhancement factor at a relatively low energy (typically 10 MeV). This damping prescription can have a rather large impact on the first-chance fission cross section at energies above typically 10 MeV [225]. As suggested by mean field calculations [392], the inner barrier may be triaxial. For those cases, the corresponding enhancement factor fsym of Eq. (209) should be taken into account by users. The NLD obtained within the HFB combinatorial model [225] at each of the 2 (or 3) highest saddle point barriers and of the 1 (or 2) shape isomers are given in a tabular format (in an energy, spin and parity grid identical to that for the ground-state NLD given in chapter VI) in the directory fission/hfb-levden/. Each isotopic chain is included in a zXXX.dat file (where XXX corresponds to the atomic number Z) in the subdirectories Max1/,Max2/, Max3/, Min1/, Min2/ corresponding to the first, second and third saddle points (sorted by increasing quadrupole deformations) and first and second minima, respectively (if there are only 2 barriers, the data for the third saddle and second minimum are missing in the respective files). The second and third saddle points as well as the second minimum are found to be left-right asymmetric (mass asymmetric) within the HFB framework. Therefore, tabulated NLDs have al-

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NUCLEAR DATA SHEETS

R. Capote et al.

FIG. 52: Heights of the highest (black circles), second highest (blue squares) and third highest (red triangles) barriers for the 90 ≤ Z ≤ 101 nuclei as a function of the neutron number.

ρ(U, +) − ρ(U, −) . ρ(U, +) + ρ(U, −)

ρ(+) ρ( )]

R(U ) =

ρ(+) + ρ( )]

ready been multiplied by a factor of 2 (see Eq. (209)). In contrast, the inner barrier and first isomer may or may not be triaxial (note that the HFB approach assumes axial symmetry), and hence the corresponding NLD in the tables has not been enhanced by the fsym factor of Eq. (209), by default. The deviation of parity-dependent NLD from the equal parity distribution in statistical NLD models can be estimated by calculating the following quantity: (211)

Calculated values of the R(U ) function for the paritydependent HFB combinatorial NLDs at the ground state (solid black lines) and saddle points (dotted red lines) for the 235,236 U nuclei (bottom and top panels, respectively) are shown in Fig. 53. A significant deviation from parity equipartition can be observed both at the ground state and saddle points, which the statistical NLD models could not describe. For many nuclear physics applications a renormalization procedure of the NLDs based on experimental data is required, in particular for nuclear data evaluation or for

FIG. 53: Parity dependence of the HFB combinatorial NLD for 235,236 U at the ground state (solid black line) and inner saddle point (dotted red line) as a function of the excitation energy.

an accurate estimate of reaction cross sections. Though the HFB combinatorial NLDs at the saddle and isomeric deformations are provided in a tabular format, it

3189


NUCLEAR DATA SHEETS

is possible to renormalize them using Eq. (127), as was also suggested for the ground-state NLDs (see Ref. [225] and chapter VI). The energy shift p and the scaling factor c are free parameters which can be adjusted at each saddle or isomer deformation to optimize the fit to the fission cross section. These parameters can be expected to reach values similar to those derived for the ground-state NLD [225], i.e., typically ±1 MeV for p and ±0.5 MeV−1/2 for c. These values reflect the remaining uncertainties in the NLD predictions on the basis of the mean-field combinatorial approach.

D.

Fission cross section calculations

Both phenomenological and microscopic fission barriers and saddle-point level densities recommended in RIPL-3 have been tested in cross-section calculations for almost all the actinides for which there are experimental data. Some of them have been already published [434]. The calculations were performed both with the EMPIRE and TALYS codes using similar reaction models and input data. The reactions induced by neutrons in the energy range studied (1 keV–20 MeV) proceed through the direct, pre-equilibrium and compound nucleus mechanisms. The fission cross-section calculations presented here are based on a theoretical analysis that utilizes the optical and direct reaction models, the pre-equilibrium exciton model and the full Hauser-Feshbach model. The direct interaction cross sections and the transmission coefficients for the incident channel were obtained from the dispersive coupled-channel optical model potential RIPL 2408 [111, 121, 125] (see chapter V), which is able to reproduce the measured total cross section within 2.0%. The coupled-channel ECIS code [71] incorporated into the evaluation codes was used for all optical model calculations. The pre-equilibrium emission was taken into account by a one-component exciton model with gamma, nucleon and cluster emissions. The modified Lorenzian (MLO1) radiative-strength function was used, and provided an excellent description of the experimental neutron capture database. For the compound nucleus mechanism, the Hauser Feshbach statistical model that accounts for multiple-particle emission and the full gamma cascade was employed. Almost all required input parameters were retrieved from the RIPL-3 database. The fission coefficients entering the fission cross section in Eqs. (200) and (201) were calculated in the full damping approximation according to Eq. (197). Two sets of input data were considered, namely (1) the empirical barriers with the associated NLDs, as recommended in RIPL (section VIII B); and (2) those obtained with the recent HFB model for the fission path and the NLDs determined at the corresponding saddle point within the HFB combinatorial method (section VIII C) as recommended in RIPL-3. The calculated cross sections are presented below.

1.

R. Capote et al. Phenomenological fission parameters

Modelling analyses of the experimental fission cross sections provide interrelated information about the fission barrier parameters and the transition-state level densities without any guarantee about their validity as independent quantities. Therefore it is recommended that both fission barrier parameters and the NLDs at saddle points should be taken from the same source (either the HFB model for both barriers and NLDs, or empirical barriers and GSM/EGSM NLDs). Considering also the model uncertainties, possible code differences and the uncertainty of model parameters, adoption of the recommended parameters is not expected to produce an accurate description of the experimental data. However, recommended input parameters should represent good starting values for further adjustment. The main purpose of the present calculations is to test the empirical barrier parameters recommended in all RIPL versions (Table XI), together with the discrete transition states given in Table XII and combined with the EGSM used for the level densities of both normal and transition states. The calculated neutroninduced fission cross sections compared to EXFOR data for 233,234,235,236,238 U,237 Np, 238,239,240,241,242 Pu, 241 Am and 243,244,245,246,247,248 Cm isotopes are shown in Fig. 54 (previous page). The calculations have been extended up to 20 MeV, but to avoid any complications from other nuclear reaction ingredients (e.g. the treatment of preequilibrium mechanism) the comments about the fission barriers are based on the analysis of the firstchance fission cross section only. The slight overestimation of the experimental data that appears for several fissile nuclei (233,235 U, 239 Pu, 245,247 Cm) in the region of 100 keV–1 MeV could be eliminated by a redistribution of discrete transition states (in excitation energy). The underestimation of the experimental data in the energy range 1–6 MeV that appears for the fissile nuclei starting with 241 Pu and increasing towards 245,247 Cm seems to be related to the overestimation of the height of the outer barriers. Indeed, the impact on the calculated n+247 Cm fission cross-sections of 400 keV outer-barrier reduction (blue dots) is shown in Fig. 55 (right panel). For almost all the studied fertile nuclei there is an underestimation of the experimental data at low energies, especially in the rise and around the threshold. The fission cross section in this region is very sensitive to the height of the highest hump (which dictates the position of the threshold), the width of the barrier, the spectra of the discrete transition states and the level densities of the transition states. A slight adjustment of the barrier parameters was performed for 238 U and 240 Pu. By increasing the curvatures of the two humps by 50–100 keV, and reducing the height of the inner barrier by 50–100 keV, the blue dotted curves in Fig. 55 were obtained (left and center panel for 238 U and 240 Pu, respectively) which are closer to the experimental data. The same procedure

3190


NUCLEAR DATA SHEETS

R. Capote et al.

234U(n,f)

236U(n,f)

237Np(n,f)

238Pu(n,f)

239Pu(n,f)

242Pu(n,f)

240Pu(n,f)

241Am(n,f)

244Cm(n,f)

246Cm(n,f)

248Cm(n,f)

FIG. 54: Neutron-induced fission cross sections of U, Np, Pu, Am and Cm isotopes calculated with empirical ingredients recommended in RIPL: fission barriers from Table XI, discrete transition states from Table XII, and transition-state level densities calculated from EGSM. Experimental data were retrieved from the EXFOR library.

3191


NUCLEAR DATA SHEETS

238U(n,f)

240Pu(n,f)

R. Capote et al. 247Cm(n,f)

FIG. 55: Neutron induced fission cross sections of 238 U, 240 Pu and 247 Cm calculated with empirical ingredients as recommended in RIPL (solid red line), and adjusted as described in text (dotted blue line). Experimental data were retrieved from the EXFOR library.

could be applied for other cases. Adjustment of the discrete transition states and a more sophisticated fission model, able to consider the partial damping of the vibrational states in the isomeric well, are required to further improve the agreement with experimental data [399, 400]. A general comment resulting from the analysis of these calculations is that the empirical fission barrier parameters perform well as starting values for an evaluation. The EGSM (see chapter VI) level densities adjusted to describe the transition-state spectra by using the prescriptions recommended for the GSM in the RIPL-2 Handbook [10] are also supported by these results. Most of the discrepancies between the calculated and the experimental cross sections can be resolved by fine tuning of the parameters defining fission or the competitive channels. A more elaborate fission formalism would also have a positive impact on the accuracy of the calculated fission cross sections, especially for fertile nuclei.

2.

Global HFB fission parameters

Similar calculations to the ones described above have been performed using the full fission path calculated in the Hartree-Fock-Bogolyubov (HFB) approach, together with the microscopic NLD derived using the HFB combinatorial model (see section VI B). There are, however, a few aspects particular to this type of calculation that are worth mentioning. The most important is related to the fact that the full fission paths calculated within the HFB approach are provided numerically as a function of the quadrupole deformation. Using this type of description for the fission barrier requires several specialized subroutines in smoothing, interpolating and finding extrema in order to establish the number of humps and to allow penetrability calculations. Because the humps are not parabolic, the fission coefficients cannot be calculated using the Hill-Wheeler formula of Eq. (194). The WKB approximation Eq. (193) in terms of integral momenta has to be used instead. The second aspect that should be noticed is that at the present time no global prediction for discrete transition states is available. Therefore, the transition state spectra

are considered continuous and can be described by level densities starting at the top of the (fundamental) fission barrier. Selected fission cross-sections of several isotopes of U, Pu, Am and Cm were calculated using both the full fission path and the corresponding parabolic approximation; calculations were compared to available experimental data from the EXFOR library. Some results of these studies have been published [434]. The main conclusion of such sensitivity studies is that the description of the experimental data usually requires a decrease of the barrier heights and a decrease of the transition-state level density at high energies. We also conclude that it is possible to renormalize the relevant fission parameters if experimental cross-section data are available. In particular, the inner and outer barrier heights can be renormalized together or independently [434], and the NLD at each saddle point can be renormalized in a way similar to what was done with the ground-state NLD (see Eq. (127)). The simplest method is to renormalize the HFB energy surface globally by a deformation-independent parameter b adjusted to fit the experimental cross sections and to keep the level densities unchanged. Some systematics can be deduced from such a renormalization procedure. An optimized fit is obtained with a constant renormalization factor amounting to 0.86, 0.89, 0.94 and 1.02 for eveneven, even-odd, odd-even and odd-odd target nuclei, respectively [434]. The overall deviation of a factor of more than 10 is reduced to less than a factor of 3, in particular at low neutron energies. The results of such a procedure are shown by the dotted blue curves in Fig. 56 (next page). As expected, the agreement with the experimental data improves (compared to unnormalized HFB-14 barriers [434]), being even acceptable in some cases (e.g. 234 U and 237 Np) for applications which require a low degree of accuracy, but remains not very satisfactory in general. In many cases the experimental data are: (1) underestimated in the low energy region where the calculated cross sections are more sensitive to the height of the maximum hump; and (2) overestimated at higher energies where the calculations are more sensitive to the height of the lower hump and to the level density of the transition states.

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NUCLEAR DATA SHEETS

234U(n,f)

235U(n,f)

237Np(n,f)

R. Capote et al. 238U(n,f)

239Pu(n,f)

238Pu(n,f)

240Pu(n,f)

242Pu(n,f)

243Cm(n,f)

241Am(n,f)

244Cm(n,f)

245Cm(n,f)

247Cm(n,f)

246Cm(n,f)

FIG. 56: Neutron induced fission cross sections obtained with a renormalized HFB fission path: independently for the inner and outer barriers plus renormalized NLDs (solid red line), and globally plus original HFB NLDs (dotted blue line). Experimental data were retrieved from the EXFOR library.

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NUCLEAR DATA SHEETS

emp TABLE XIII: Fission barrier parameters (in MeV): BA and emp BB are the RIPL-2 barrier heights (from Table XI). The HF B HF B barrier heights BA and BB were obtained when optimizing the inner and outer barriers of the HFB fission path independently (the corresponding HFB combinatorial NLDs were also individually renormalized to optimize the fit [434]).

Z

A

emp BA

emp HF B HF B BB BA BB

92

234

4.80

5.50

5.38

6.15

92

235

5.25

6.00

5.54

5.80

92

236

5.00

5.67

5.52

6.03

92

237

6.40

6.15

5.90

6.03

92

238

6.30

5.50

5.80

6.17

92

239

6.45

6.00

6.07

6.07

93

236

5.90

5.40

5.39

5.54

93

237

6.00

5.40

6.08

5.74

93

238

6.50

5.75

6.20

6.27

94

237

5.10

5.15

5.35

5.07

94

238

5.60

5.10

5.57

5.35

94

239

6.20

5.70

5.96

5.86

94

240

6.05

5.15

5.89

5.73

94

241

6.15

5.50

5.85

5.81

94

242

5.85

5.05

6.02

5.61

94

243

6.05

5.45

5.78

5.59

95

239

6.00

5.40

6.10

4.49

95

240

6.10

6.00

6.00

4.56

95

241

6.00

5.35

6.00

5.35

95

242

6.32

5.78

6.32

5.78

95

243

6.40

5.05

6.20

5.33

95

244

6.25

5.90

6.25

5.90

96

241

7.15

5.50

6.04

4.07

96

242

6.65

5.00

6.20

4.90

96

243

6.33

5.40

6.10

5.40

96

244

6.18

5.10

6.18

5.10

96

245

6.35

5.45

6.35

5.45

96

246

6.00

4.80

6.00

4.80

96

247

6.12

5.10

6.12

5.10

96

248

5.80

4.80

5.80

4.80

96

249

5.63

4.95

5.63

4.95

R. Capote et al.

the deformation axis (i.e., the deformation axis has been multiplied by a constant number, effectively changing the saddle-point deformations) to improve the agreement of the calculated cross sections with data. HF B HF B The optimal values of inner BA and outer BB fission-barrier heights are listed in Table XIII and seen to be in many cases lower (by a few hundred keV) than the empirical barriers of Table XI, i.e., barriers obtained in the Hill-Wheeler approximation using the phenomenological analytical NLD model (GSM) recommended in RIPL-2. The results presented above demonstrate that it is possible to use results from theoretical studies of fission to produce the ingredients for fission cross-section calculations. However, as expected from the calculation with microscopic ingredients, the results also show that the uncertainty in the fission path, and more specifically in the barrier heights, gives rise to a cross section that can hardly be estimated better than within a factor of roughly 10 for energies below a few MeV. This corresponds to a 0.5–1.0 MeV error in the fission barrier height obtained from the HFB-14 model [18] that cannot be expected to be significantly improved in the near future. Therefore, the predictive power of fission cross-section calculations based on HFB inputs remains limited.

E.

A further step to improve the fit to the experimental fission cross sections is to tune the inner and outer barrier heights independently and to renormalize the NLD at each saddle point. Using the barrier heights, widths, and HFB NLD renormalization factors p and c of Eq. (127) for each saddle point from the fission/empirical-hfbbarriers.dat file, a reasonably good description of the experimental data presented by the solid red curves in Fig. 56 is obtained. The barrier width for all U to Pu isotopes is the value predicted by the HFB model and obtained by fitting an inverted parabola along the fission path. For some Am and Cm cases, the fission path has been stretched along

Prompt Fission Neutrons

The Madland-Nix or Los Alamos (LA) model [435] constitutes the basis for the evaluation of prompt fission neutron spectra in most current evaluated nuclear data libraries. Although a relatively simple and compact formalism with only a handful of adjustable input parameters, this model has been very successful in predicting the prompt neutron spectra for neutron-induced as well as spontaneous fission reactions for a wide range of nuclei and incident neutron energies. A large number of evaluations of prompt fission neutron spectra has been undertaken based on the LA model. However, there are very few publications dealing with systematic studies of required input parameters for such calculations. Relevant input parameters for the LA model are (1) Er – the average fission energy release; (2) Sn – the average neutron separation energy of the fission fragments; (3) Eftot – the total kinetic energy of fission fragments (also denoted T KE); and (4) Eγtot – the average total energy carried away through gammaray emission. Average model parameters for prompt fission neutron emission in 235,238 U(nth ,f) and 239 Pu(nth ,f) reactions have been suggested by Madland [436]. A comprehensive review of average input parameters for the LA model was recently published by Tudora [437] (see also references therein). However, available experimental data are often sparse and further studies are under way. More refined versions of the Los Alamos model have been developed in recent years for evaluation purposes,

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NUCLEAR DATA SHEETS

but the main assumptions of the LA model have not changed. It was considered important to develop and make available a reference code for calculations of prompt fission neutron spectra based on the LA model, written in a modern programming language. Such coding can be incorporated easily in any existing or future nuclear reaction code or evaluation tool. 1.

Los Alamos model

The Los Alamos model is described at length in the original reference [435], and we will only summarize the main results here. In this model, the prompt fission neutron spectrum N (E) is calculated as a function of the fissioning nucleus and its excitation energy. Weisskopf statistical evaporation theory [438] is used to predict the emission of neutrons from an excited compound nucleus at a given temperature T , and a triangular distribution of initial fission fragment residual temperatures is assumed.

R. Capote et al.

with k(T ) given by Eq. (213). Considering the most probable fragmentation only, the average center-of-mass neutron energy spectrum φ() is therefore given by an average over the spectra for the light and heavy fragments as 1 (φL () + φH ()) , (216) 2 and similarly for the laboratory neutron energy spectrum N (E) φ() =

1 (NL (E) + NH (E)) . (217) 2 In the last two equations, it is implicitly assumed that half of the emitted neutrons come from the light fragment and the other half from the heavy fragment. N (E) =

4.

Average prompt neutron multiplicity

The average prompt fission neutron multiplicity ν p is simply obtained from energy conservation: 2.

Center-of-mass neutron energy spectrum

The neutron energy spectrum in the center-of-mass of a fission fragment is given by 2σc () Tm φ() = k(T )T exp(−/T )dT, (212) 2 Tm 0 with the temperature-dependent normalization constant k(T ) ∞ −1 σc () exp(−/T )d, (213) k(T ) = 0

σc () is the energy-dependent cross section for the inverse process of compound nucleus formation. Equation (212) was obtained by integrating over a triangular distribution of fission fragment temperatures with a maximum temperature Tm .

E ∗ − Eγtot , (218) Sn + where E ∗ is the average total excitation energy equal to: νp =

E ∗ = Er + Bn + En − Eftot .

Er is the average fission energy release, Bn is the neutron binding energy of the target nucleus, En is the neutron incident energy, Eftot is the average total fissionfragment kinetic energy, Eγtot is the average total energy carried away through gamma-ray emission, Sn is the average neutron separation energy of the fission fragments, and is the average energy of outgoing neutrons in the center-of-mass reference frame. 5.

3.

(219)

Multiple-chance fission

Laboratory neutron energy spectrum

In the laboratory system, the neutron energy spectrum N (E) for a fission fragment moving with a kinetic energy per nucleon Ef is (√E+√Ef )2 1 φ() √ d, N (E) = (214) √ √ 2 4 Ef ( E− Ef ) where E is the laboratory neutron energy. Inserting Eq. (212) into Eq. (214), the laboratory neutron energy spectrum N (E) becomes (√E+√Ef )2 √ 1 N (E) = √ 2 σc () d (215) √ 2 2 Ef Tm ( E− Ef ) Tm × k(T )T exp(−/T )dT, 0

At higher incident neutron energies, above the neutron binding energy, the excited compound system may emit one or more neutrons before undergoing fission. Although the neutrons emitted prior to fission are not directly related to the ones emitted from the fission fragments, in the LA model they are treated in a similar manner. In the case of multiple-chance fission, the prompt fission neutron spectrum in the laboratory system can be written as ! i !i−1 P φ (E) + ν N (E) j i i i f j=1 ! i N (E) = , (220) i Pf (i − 1 + ν i ) where φj (E) is the pre-fission evaporation spectrum for the (j + 1)th -chance fission channel, and Pfi is the ith chance fission probability.

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NUCLEAR DATA SHEETS

R. Capote et al.

- fission/fis-barrier-liquiddrop.for - FORTRAN code for the liquid-drop model calculations of the fission barriers and the moment of inertia of light pre-actinide nuclei (Z < 80) written by Sierk [413]; - fission/hfb-barriers/ - directory including the HFB fission paths; - fission/hfb-levden/ - directory including HFB combinatorial level densities calculated for the saddle points and isomers;

!

- fission/pfns.for - FORTRAN code for the calculation of prompt fission neutron spectra. FIG. 57: Prompt fission neutron spectrum N (E) calculated for the reaction n+235 U at Einc = 0.53 MeV, and compared to the normalized experimental data points of Johansson and Holmqvist [439].

The average prompt neutron multiplicity can be obtained similarly ! i i Pf (i − 1 + ν i ) ! i ν= . (221) i Pf The ith prompt neutron multiplicity ν i is given by ) * Ei∗ − Eγtot i , (222) νi = Sni + i where the average total excitation energy for the ith chance fission is Ei∗ = Eri + Bn (A) + En − Efi − i−1

(Bn (A − j + 1) + j ).

(223)

j=1

Here, j it the mean kinetic energy of the neutron evaporated from the (Z, A − j + 1) nucleus. Figure 57 shows calculated vs. experimental prompt neutron spectra in one of the four test cases provided in the RIPL-3 library. Experimental data were normalized following the procedure described in Appendix B of Ref. [435]. F.

Recommendations and future developments

The FISSION segment in RIPL-3 contains the following data files: - fission/empirical-barriers.dat - fission barrier parameters derived from the analysis of experimental data available for pre-actinides with Z ≥ 80 and actinides; - fission/empirical-hfb-barriers.dat - fission barrier and NLD parameters derived from the analysis of experimental data for actinides using the HFB fission path and HFB combinatorial NLDs;

The description of each data file is included in corresponding readme files. Fission barriers and fission (saddle point) level densities are key ingredients of the statistical description of the fission cross sections induced by different incident particles. Both ingredients are strongly interrelated. The fission barrier parameters derived from analyses of the available experimental data have been compiled for both pre-actinides (Table X) and actinides (Table XI) on the basis of a phenomenological nuclear level density model. A similar evaluation has been performed using the HFB fission path and HFB combinatorial NLDs. The corresponding fission parameters are given for elements between U and Cm. A large amount of data produced from microscopic calculations of the fission path (based on the HFB approach) and fission level densities (coherently determined on the basis of the HFB single-particle level scheme and pairing force) is included in RIPL-3. Tables of the HFB fission paths and NLD for about 1000 nuclei with 90 ≤ Z ≤ 102 are provided. The number of possible sources of uncertainties in the fission cross-section calculations is far larger than for any other competing channel. Both empirical and microscopic fission barriers and transition state densities have been tested in fission crosssection calculations. The tests confirm that it is possible to tune model parameters to reproduce experimental data accurately, but it remains difficult to provide systematics or theoretical estimates of the various nuclear parameters necessary for predicting fission cross sections reliably. It must be emphasized that including the global microscopic predictions in practical calculations represents an important breakthrough. In order to obtain high quality fission data one should continue to introduce in the evaluation procedure the results of the modern tools usually employed to study the purely theoretical aspects of fission. The subject is too vast to be exhausted here; therefore, only a few developments that might be accomplished in the short term will be mentioned. They are interrelated and refer mainly to a better knowledge of the fission barriers and a more accurate calculation of the transmission coefficient for fission. As mentioned previously, the fit to the experimental fission cross sections provides global information on the

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parameters of the fundamental fission barrier and on the discrete and continuous spectra of the transition states. While both phenomenological and microscopic NLDs are relatively reliable, the fission barriers obtained from HFB models are affected by uncertainties up to 1 MeV with small chances of improvement, and therefore represent a major concern. More direct information about the fundamental barrier and the discrete transition states could be extracted from the analysis of experimental fission cross sections at sub-barrier excitation energies, where the influence of the continuous spectra of the transition states described by level density functions is minimum. In this energy range, the full-damping approximation is not valid, and a more elaborate formalism such as the optical model for fission is needed [394, 395, 397, 399]. The optical model for fission was implemented in the EMPIRE code [45] and used for the neutron cross section evaluations of 232 Th and 231,233 Pa [399]. The experimental fission cross section, including the gross vibrational structure, was accurately reproduced by fitting the parameters. But this formalism would be more properly used if based on phenomenological or microscopic predictions for the characteristics of discrete transition states and of the corresponding states in the well(s). There are hopes that this information will be available soon. Another step in improving the treatment of the fission probability at low energies (and to extract more accurate information about the fission barriers) is to exploit the available microscopic information on the wells to estimate the γ-decay in the isomeric well(s) (Eq. (196)). This might be important for the fertile nuclei for which the neutron separation energy is small, and the isomeric well is not very deep, so the γ-decay probability is comparable to the probability of tunneling through the inner and/or the outer hump. Information on the well(s) should be also used to estimate the isomeric (delayed) fission, the resonance structure in sub-barrier fission, and the lifetime of the fission isomers. To apply these procedures to a large number of nuclei, apart from the neutron-induced fission cross-section data, data from the fission of nuclides that are reached via charged-particle transfer reactions and photo-fission must be used. These reactions can give information on fission probabilities at excitation energies well below the neutron separation energy and thus give valuable information on fission barrier parameters of fissile nuclides as well as nuclides for which samples for (n,f) measurements are not readily available. In summary, a consistent and complete modelling of the fission cross section remains an elusive challenge in nuclear physics. The prediction of fission cross sections is far from being satisfactory. Recent studies have aimed at providing a better description of some of the basic nuclear ingredients required to describe fission cross sections. These studies concern in particular fission barriers (or more generally fission paths) and NLDs at the fission saddle points, but also the deformed optical model potential, inertial masses, transition and class II/III states.

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In particular, due to the fundamental property they represent, the fission barriers need to be predicted with a much higher accuracy than achievable today (best global models provide barriers within 10 to 20%, while an accuracy better than 5% would be required for any practical application). As far as energy applications are concerned, it is clear that the model-based evaluations, properly combined with available experimental fission cross sections, could and should replace the simple fits of the experimental data used so far. The improvement of evaluations will be possible if the strong points of both microscopic and phenomenological approaches can be combined. Detailed structure effects can only be predicted by microscopic models, but only phenomenological adjustments will enable one to reach the required accuracy needed for such applications. A mandatory requirement for future developments will be the validation of new fission input parameters in cross-section calculations based on an advanced fission formalism. The most important quantities next to the fission cross section, especially for energy applications, are the average multiplicity and the spectra of the prompt fission neutrons. For the first time in RIPL-3, their estimation by means of the “Los Alamos model” is provided. Through a relatively simple and compact formalism with a small number of adjustable parameters, this model has been very successful in predicting the prompt neutron spectra for neutron-induced as well as spontaneous fission reactions for a wide range of nuclei and incident neutron energies. The reference code can be incorporated easily into any nuclear reaction code leading to more consistent and accurate evaluations.

IX.

SUMMARY AND CONCLUSIONS

Nuclear data for applications constitutes an integral part of the International Atomic Energy Agency (IAEA) programme of activities. When considering low-energy nuclear reactions induced by light particles, such as neutrons, protons, deuterons, alphas and photons, one addresses a broad range of applications, from nuclear power reactors and shielding design through transmutation of nuclear waste to cyclotron production of medical radioisotopes and radiotherapy. In all these and many other applications, one needs a detailed knowledge of cross sections, spectra of emitted particles and their angular distributions, and isotope production. A long-standing problem of how to meet nuclear data needs of the future with limited experimental resources puts a considerable demand upon computational nuclear modelling capabilities. Originally almost all nuclear data were provided by measurement programmes. Over time, theoretical understanding of nuclear phenomena has reached a considerable degree of reliability, and nuclear reaction modelling has become a standard practice in nuclear data evaluation (with measurements remaining

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critical for data testing and benchmarking). Theoretical calculations have become instrumental in obtaining complete and internally consistent nuclear data files. The practical use of nuclear model codes requires considerable numerical input that describes properties of the nuclei and their nuclear reactions. Experts in leading nuclear data centers have used a variety of different input parameter sets, often developed over many years in their own laboratories. Many of these partial databases were poorly documented, or not documented at all, and not always available for other users. With the trend of reduced funds for nuclear data evaluations, there was a real threat that the immense accumulated knowledge of input parameters and related state-of-the-art might be compromised or even lost for future applications. Therefore, the IAEA has undertaken an extensive coordinated effort to develop a library of evaluated and tested nuclear-model input parameters. Considering that such a task is immense, it was decided to proceed in two major steps. First, to summarize the present knowledge on input parameters and to develop a single Starter File of input model parameters [6], and second, to focus on testing, validation and further improvement of the Starter File. The first step was addressed through the IAEA RIPL-1 CRP (19941997); the second step followed immediately through the RIPL-2 CRP (1998-2002). This second CRP revised and extended the original RIPL-1 Starter File to produce a consistent RIPL-2 library containing recommended input parameters, a large amount of theoretical results suitable for nuclear reaction calculations, and a number of computer codes for parameter retrieval, determination and use [10]. Finally, the RIPL-3 CRP (2003-2008) extended the scope of RIPL-2, and brought the programme to a highly successful conclusion [14]. Nuclear data evaluators around the world have emphasized and recognized the importance of the IAEA RIPL databases, and have continued to stress their full reliance on the IAEA RIPL-2 and 3 databases for reaction crosssection calculations in many publications and conference presentations. The RIPL-3 database is arranged in seven segments (or directories), which are summarized below: MASSES: Nuclear ground-state properties are fundamental quantities in many different fields of physics. Available experimental data of relevance to atomic masses are considered in this chapter (Audi et al. [15]). When no experimental data exist, ground-state properties can be derived from local or global theoretical approaches. RIPL-3 provides ground-state properties predicted by three global models: Finite-Range Droplet Model (Möller et al. [16]), Hartree-Fock-Bogoliubov HFB-14 model (Goriely et al. [17, 18]) and shell-model approximation of Duflo and Zuker [19]. The FRDM model also provides microscopic corrections and deformation parameters in addition to nuclear masses, while the HFB model provides density distributions and deformation parameters. As supplementary information, the

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relative isotopic abundances for all stable nuclei found on earth are also available. DISCRETE LEVELS: This segment contains 117 files with all known level schemes available from ENSDF in 2007. These files are arranged and preprocessed into an easy-to-read format for nuclear reaction codes. During preprocessing all missing spins are inferred uniquely for each level from spin distributions extracted from the existing data. Electromagnetic and γ-ray decay probabilities are estimated. Missing internal conversion coefficients (ICC) are calculated using inferred or existing spin information, in which existing multipole mixing ratios are also taken into account. Particle decay modes are also given whenever measured. In all cases, the total decay probability is normalized to unity, including particle decay channels. This chapter also contains the results of constant temperature fits to nuclear level schemes. The main purpose of this file is to provide cut-off excitation energies Uc for completeness of levels and spins in each level scheme. NEUTRON RESONANCES: This segment contains average resonance parameters, which have been updated for RIPL-3 on the basis of the recent “Atlas of Neutron Resonances” compiled by Mughabghab [70]. The differences between the RIPL-2 and Mughabghab recommendations have been analyzed and the optimal values of the averaged parameters estimated in accordance with new identifications of the individual resonance parameters. Uncertainties of the recommended parameters are derived by taking into account systematic uncertainties of the analyzed data in addition to the statistical uncertainties. The number of nuclei included in RIPL-3 is extended to 300 for the s-wave resonances and to 119 for the p-wave resonances. OPTICAL MODEL: Improved phenomenological optical model potentials have been developed over recent years that can enhance significantly the quality of optical model calculations for applied problems. Global as well as nuclide-specific potentials are included that utilize spherical, vibrational, soft-rotor and rigid-rotor models. These include the Koning and Delaroche extensive set of optical model potentials for individual target nuclei covering the nucleon energy range E=1–200 MeV and the target range Z=12–83, A=24–209, with accompanying global potentials covering the same energy and target ranges [133]. These potentials not only span very broad ranges in energy, Z and A, but are also based upon a modern analytical approach that utilizes an extensive experimental database [440]. A valuable extension of these types of potentials has been provided by Soukhovitskii et al. [128] and Capote et al. [121, 125, 126, 127] for the actinides. Global potentials for incident deuteron [94, 95] and triton [155] projectiles have been also derived. Also, many new local dispersive OMPs have been added. Additionally, the use of microscopic optical model calculations has advanced to a state where routine use of such potentials as a supplement to phenomenological models is not only feasible but also very desirable.

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A primary aim of the RIPL-3 optical segment has been to accumulate a large body of information that will be useful in optical model calculations and to provide computer codes for processing the information into inputs for commonly used optical model codes. The OMP database and codes in the RIPL-3 library should permit users to perform a wide range of optical model calculations of high quality. Considerable testing of the OMP sublibrary has been performed, and the reliability of the database is judged to be high. While it is recognized that the information base is by no means exhaustive, these data have been formulated in a manner that permits easy expansion as new information becomes available or as gaps are filled in older data. LEVEL DENSITIES: For any application of the statistical theory of nuclear reactions, the parameters describing the level density must be obtained from reliable experimental data. Both the updated evaluations of the low-lying level systematics and the neutron resonance spacings considered in chapters III and IV have been used for this purpose. The level density parameters fitted to such data are compiled for the three phenomenological models most frequently used in practical calculations: i) Gilbert-Cameron approach; ii) Back-shifted Fermi gas model; iii) generalized superfluid model (as well as EGSM – see section VI A 5) taking consistently into account consideration of the shell, pairing and collective effects (in GSM and EGSM). Furthermore, the systematics of the level density parameters have been developed for each model in terms of the shell correction approach. Such global systematics are recommended for the level density calculations involving nuclei close to the valley of stability that have no experimental data on resonance spacings or low-energy level densities. RIPL-3 contains large quantities of data from the microscopic calculations of nuclear level densities [203, 225] based on the HFB+BSk14 model [18]. The microscopic model takes into account the shell, pairing and blocking effects, the deformation effects in the single-particle spectra, the collective enhancement of level densities at low excitation energies and damping at high excitations. Corresponding tables of level densities for about 8000 nuclei are also provided. GAMMA: Methods and related parameters for modelling γ-ray cascades in highly excited nuclei have been reviewed. This assessment includes experimental radiative strength functions, giant-resonance parameters and various means of calculating γ-ray strength functions in excited nuclei. Emphasis has been placed on the E1 γtransitions, which tend to dominate nuclear reactions. Recent analytical expressions for E1 γ-ray strength functions provide reasonably reliable results over a relatively wide range of γ-ray energies (from zero to above the GDR energy). RIPL-1 data are recommended for other γ-ray multipolarities [6]. RIPL-3 [14] contains tabulated E1 γray strength functions and GDR parameters that result from extensive microscopic calculations and can be used directly by nuclear reaction codes. FISSION: Fission barriers and fission level densities

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are key ingredients of the statistical description of the fission cross sections induced by different incident particles. Both ingredients are strongly interrelated. The fission barrier parameters derived from analyses of the available experimental data have been compiled for both pre-actinides (Z ≥ 80) and actinides using a phenomenological nuclear level density model [6]. For light preactinides (Z < 80), a FORTRAN code for estimating the liquid-drop barrier is also provided. A similar evaluation has been performed on the basis of the HFB fission path and HFB combinatorial NLDs for actinides. The corresponding adjusted fission parameters are given for elements between U and Cm. The RIPL-3 files include a large amount of data produced from microscopic calculations of the fission path (based on the HFB-14 approach) and fission (saddle-point) level densities (coherently determined from the HFB-14 single-particle level scheme and pairing force) [434]. Corresponding tables of the HFB fission paths and NLDs for about 1000 nuclei with 90 ≤ Z ≤ 102 are provided. We would like to conclude with a citation guideline. Most of the results compiled in the RIPL database, and discussed in this contribution, have been published elsewhere. We suggest that the original work be quoted, while stating that the actual data or code used have been taken from the RIPL database (reference to a RIPL index, if relevant, should be made – see the optical model chapter for examples). Last but not least, we would like to stress that the IAEA Nuclear Data Section will maintain in the foreseeable future both the RIPL coordination activities and the compilation of input parameters, including updates to the released RIPL-3 database. There are still large uncertainties in modelling important nuclear reactions, where insufficient experimental data for guidance exist – e.g. fission and nuclear reactions on excited states, and isomer production, to mention a few. These outstanding issues should be the subject of further activity, and could lead to improvements in practical calculations. Acknowledgments Our sincere thanks to all colleagues who have contributed to and worked on this project during the last fifteen years. The preparation of this paper would not have been possible without the support, hard work and endless efforts of a large number of individuals and institutions. First of all, the IAEA is grateful to all participant laboratories for their assistance in the work and for support of the CRP meetings and activities. The work described in this paper would not have been possible without IAEA Member States contributions. The IAEA Nuclear Data Section would like to acknowledge the very important contribution made to the whole project by our chairmen A.V. Ignatyuk (RIPL-1), P.G. Young (RIPL-2) and M. Herman (RIPL-3). We are grateful to the heads of the IAEA Nuclear

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Data Section since 1993, C.L. Dunford, D.W. Muir and A.L. Nichols for their encouragement, advice and continued support of the RIPL projects. Editing of the final RIPL-3 document by A.L. Nichols and R.A. Nichols is especially acknowledged. We would like to thank A.B. Pashchenko who was involved in the IAEA Consultants’ Meetings leading to the start of the project. We are in permanent debt to J. Raynal, who made ECIS the golden standard of all optical model codes. We very much appreciate valuable contributions made by E. Bˇeták, M. Uhl, Zongdi Su and G.L. Moln´ ar during the RIPL-1 stage of the project. We would like to thank RIPL-3 consultants O. Iwamoto, T. Kawano, A. Kumar, S. Kunieda, L. Leal, S. Chiba, J.M. Quesada and A. Tudora, who contributed to the overall success of the project. We would like to thank one of us (GR) and all local ENEA staff for their organization of extremely inspiring RIPL-1 “Italian” meetings held in June 1993 (Sirolo, Ancona), September 1994 (Cervia, Ravenna) and May 1997 (Trieste). We also acknowledge Prof. E. Gadioli for play-

ing host of a RIPL-2 meeting held in June 2000 (Varenna, Italy). We very much appreciate advice and feedback provided by B.V. Carlson, H. Wienke, M. Pigni, D. Rochman, V. Avrigeanu, M. Gupta, A. Konoveyev, K.I. Zolotarev, and many other people who use the RIPL database in nuclear reaction calculations for applications and code development. We would like to thank anonymous referees for the extensive and detailed revision of the paper and valuable comments and suggestions which significantly improved the original manuscript. Finally, we would like to thank A. Trkov, A. Ventura, V. Pronyaev, J.P. Delaroche and E. Bauge for their permanent enthusiasm, advice and inspiration. Last but not least, we would like to thank K. Nathani for her assistance in the preparation of the manuscript, and C. Mattoon and A. Sonzogni for assembling the Nuclear Data Sheets issue. The participants in this study dedicate their work and results to the memory of their coworkers G.L. Moln´ ar (Hungary) and M. Uhl (Austria).

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