LECTURE NOTES for WORKSHOP ON NUCLEAR

5/21/2013

National Mathematical Centre, Abuja, Nigeria

LECTURE NOTES FOR WORKSHOP ON NUCLEAR REACTION THEORY AND CROSS SECTION DATA DETERMINATION USING COMPUTER CODES

Lectures delivered at National Mathematical Centre (NMC), Abuja, Nigeria, May (19 – 24, 2013)

National Mathematical Centre Kwali, Abuja, Nigeria

WORKSHOP ON NUCLEAR REACTION THEORY AND CROSS SECTION DATA DETERMINATION USING COMPUTER CODES

PROF. S.A. JONAH (FNIP) Centre for Energy Research and Training, Ahmadu Bello University, Zaria,Nigeria And Prof. M .N. AGU Nigerian Atomic Energy Commission Asokoro Abuja, Nigeria

Lectures delivered at National Mathematical Centre (NMC), Abuja, Nigeria, May (19 – 24, 2013)

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Table of Contents WORKSHOP ON NUCLEAR REACTION THEORY AND CROSS SECTION DATA DETERMINATION USING COMPUTER CODES ........................................................................................................................................ 2 Report No.NMC/STP/1/2013 ........................................................................................................................ 5 COURSE CONTENT..................................................................................................................................... 6 Editorial Committee of National Mathematical Centre Abuja ..................................................................... 7 List of Contributors ....................................................................................................................................... 8 Foreword....................................................................................................................................................... 9 Preface ........................................................................................................................................................ 12 NUCLEAR REACTION ................................................................................................................................... 14 The Atomic Nucleus ................................................................................................................................ 14 Nuclear Force .......................................................................................................................................... 14 Nuclear Radius and Nuclear Mass .......................................................................................................... 15 Nuclear Mass and Binding Energy........................................................................................................... 15 NUCLEAR REACTION ................................................................................................................................... 17 Why do we study nuclear reaction? ....................................................................................................... 18 TYPES OF NUCLEAR REACTIONS .............................................................................................................. 18 Elastic Scattering ..................................................................................................................................... 19 Inelastic Scattering .................................................................................................................................. 19 Radiative Capture.................................................................................................................................... 19 Breakup Reaction ........................................................................................................................................ 19 Direct Reaction........................................................................................................................................ 20 Many Body Reactions and Spallation Reactions ......................................................................................... 20 Fission ..................................................................................................................................................... 20 Photodisintegration ................................................................................................................................ 20 High Energy Reaction .............................................................................................................................. 21 CONSERVATION LAWS IN NUCLEAR REACTIONS ........................................................................................ 21 Conservation of total energy .................................................................................................................. 21 Conservation of Linear Momentum ........................................................................................................ 21 Conservation of Mass Number ............................................................................................................... 21 (i) Conservation of Proton and Neutron Numbers.............................................................................. 22 (ii)Conservation of Angular Momentum ................................................................................................. 22 3

ENERGETICS OF NUCLEAR REACTIONS ................................................................................................... 23 Analysis in the Centre of Mass Frame of Reference ............................................................................... 26 NUCLEAR REACTION CROSS-SECTION ..................................................................................................... 27 INTRODUCTION TO THE THEORY OF NUCLEAR REACTIONS ....................................................................... 29 Scattering and Reaction Cross-Sections...................................................................................................... 29 MODELS OF NUCLEAR REACTION ............................................................................................................... 32 Part II ........................................................................................................................................................... 35 1.0 INTRODUCTION ..................................................................................................................................... 35 2. THE OPTICAL MODEL (OM) .................................................................................................................... 37 DIRECT REACTION ....................................................................................................................................... 40 2. THE STATISTICAL COMPOUND NUCLEUS REACTION MODEL ................................................................. 40 PRE-EQUILIBRIUM REACTION MODELS .................................................................................................. 43 DETERMINATION OF CROSS SECTIONS ....................................................................................................... 48 COMPUTER CODES FOR THEORETICAL MODEL CALCULATIONS ................................................................ 48 2. EXAMPLES WITH THE EXIFON CODE ....................................................................................................... 49 APPLICATIONS OF NUCLEAR REACTION DATA ............................................................................................ 53 2. REFERENCES ............................................................................................................................................ 58 RECOMMENDED BOOKS ............................................................................................................................. 59 EXECUTION AND RETRIEVAL EXERCISES ..................................................................................................... 60 PART 111 ..................................................................................................................................................... 63 Study Papers ............................................................................................................................................... 63

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Report No.NMC/STP/1/2013

National Mathematical Centre, Abuja, Nigeria WORKSHOP ON NUCLEAR REACTION THEORY AND CROSS SECTION DATA DETERMINATION USING COMPUTER CODES

Edited by PROF. S.A. JONAH (FNIP) Centre for Energy Research and Training, Ahmadu Bello University, Zaria, Nigeria And Prof. B O. Oyelami National Mathematical Centre,Abuja, Nigeria

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WORKSHOP ON NUCLEAR REACTION THEORY AND CROSS SECTION DATA DETERMINATION USING COMPUTER CODES COURSE CONTENT

Introduction: The atomic nucleus, types of reactions (i.e. mechanisms); kinematics of nuclear reactions; elastic collisions (relativistic and nonrelativistic); non-elastic collisions; threshold energy, Q-value; nuclear reaction cross sections and nomenclature: Models of nuclear reactions; optical model; compound nucleus model; direct reaction model; preequilibrium model: Determination of nuclear reaction cross sections: experimental techniques; calculational methods and data retrieval from IAEA Nuclear Data Centre; computer codes (EXIFON 2.0, EXFOR data library): Applications of nuclear reactions

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Editorial Committee of National Mathematical Centre Abuja

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List of Contributors 1. PROF. S.A. JONAH

Centre for Energy Research and Training, Ahmadu Bello University, Zaria, Nigeria

2. Prof. M .N. AGU, Nigerian Atomic Energy Commission Asokoro Abuja, Nigeria 3. Prof. H.Kalka (Studied papers) United Nation Atomic Energy Commission.

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Foreword

The National Mathematical Centre Abuja, Nigeria is building a critical mass for Nigeria in the area of Theoretical Physics and training of scientists on nuclear reaction theory which is one of the fertile grounds for research in the Theoretical Physics in the recent times. The Centre has been organizing workshops to contribute towards meeting the increasing needs of researchers, scientists and engineers for application of knowledge in diverse areas of human Endeavour’s. The Centre, under the ambit of its mandates, is trying as much as possible to provide conducive learning environment to participants and allow interactions between them and the resource persons. The Centre also offers facilities to researchers in the Mathematical Science Library and the Computer laboratory The objectives of the Workshop on the use of computer

codes for nuclear reaction cross section determination and data retrieval exercises (19-24 May, 2013) are threefold. Firstly, to expose the participants to basic concepts and models for determination of nuclear cross section. Secondly, to expose them to the use of Exifor software and applications to nuclear reaction and finally, the participants were given research papers published by Prof. 9

H. Kalka to study. The aim of this is enable them to know how to use codes to analyze data. Moreover, learn how to generate questions and proffer solution to problems in area of cross section determination. Furthermore, at the end of the workshop we hoped that value will be added to the research competence of the participants. The Centre will continue to organize more workshops on nuclear research for some years until we have so many young scientists nurtured to research maturity. Professor A.R.T. Solarin The Director & CEO National Mathematical Centre,Abuja,Nigeria

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Preface This booklet contains the notes of lectures given at the workshop entitled “Workshop on Nuclear Reaction Theory and Cross section Data Determination using Computer Codes” held at the National Mathematical Centre (NMC), Abuja, Nigeria, May 19 to 24, 2013. In the first part, the workshop focused on theoretical background of nuclear reaction and covered most and fundamental physics including the mechanisms of the subject matter. The second part of the workshop was devoted to an exhaustive survey on calculational codes based on the theoretical background provided in the first part. Furthermore, relevant experimental data and evaluated data were retrieved from nuclear data libraries and were used to validate calculated data with applications in wide variety of fields. This collection should present a useful reference for researchers interested in this topic. Specifically, graduate students of nuclear physics, nuclear engineering and computational physics/ engineering will find the collection useful for their works. The workshop was organized by the National Mathematical Centre (NMC), Abuja to provide a platform for enriching the knowledge base of upcoming nuclear scientists and engineers in Nigerian Universities, vis-à-vis the intention to add nuclear energy to the energy mix of the country in the near future. The editors are grateful to the management and staff of the NMC. Thanks are also due to the resource persons for their excellent lectures and preparation of lecture notes presented here.

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Part 1 NUCLEAR REACTION Prof. M .N. AGU Nigerian Atomic Energy Commission Asokoro Abuja The Atomic Nucleus The atom was thought of as an indivisible part of an element. Its discovery, in 1803, is associated with a British Chemist known as Dalton. This definition was much later (almost a century) discovered to be false. It was discovered by Rutherford in 1911, that an atom consists of a nucleus surrounded by a cloud of electrons. The nucleus is very dense and contains the protons and neutrons. Since the mass of an electron is small compared to the masses of proton and neutron, then we can say that the mass of an atom is approximately equal to the mass of the nucleus. Electrons carry negative charge while protons carry positive charge; the neutrons are neutral. Since the number of protons in an atom equals the number of electrons, thus the charge of an atom is neutral.

Nuclear Force Since the protons and neutrons are contained in the nucleus and the nucleus is not turn apart by the repulsive forces acting between the protons in the nucleus, then there must be a strong force holding the nucleus together. This force which holds the protons and neutrons together in the nucleus is called the nuclear force. This force has the following properties:(i) (ii) (iii)

It has short range It is charge independent It saturates

The protons and neutrons are sometimes called the nucleons because they are seen as two different quantum states of the same particle.

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Nuclear Radius and Nuclear Mass The radius of the nucleus is one of the important parameters in the description of the nucleus of any particle. Measurements performed show that the radius of the nucleus is proportional to the cube root of the mass number (A) of the nucleus. 1 where R is the radius of the nucleus, A the mass number and = 1.25 fm ( In arriving at the above equation, an assumption was made that shape of the nucleus is spherical. Thus we refer to the radius above as the radius of the mass distribution. The atomic number, Z, which defines the nuclear charge parameter, has been found to be approximately proportional (linearly) to the mass number A. The nuclear charge density is also approximately the same throughout the volume of the nucleus. Thus the nuclear charge distribution +Ze follows the pattern of the mass distribution. Thus the mass distribution and the charge distribution are approximately the same. We define the mean square radius as 2 For a nucleus of uniform charge distribution, ρ= constant, and of radius R, then , note that for r>R, ρ=0 =

3 4

Nuclear Mass and Binding Energy As we mentioned earlier, the nucleus consists of protons and neutrons called the nucleons. Thus the mass of the nucleus ordinarily should be the sum of the masses of the constituent nucleons. However, measurements made revealed that these two differ by an amount called mass defect, where 5 Where M (A, Z) is the measured atomic mass.

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The mass defect could also be expressed in terms of the Packing Fraction, f, for each nuclide as 6 7 Thus the mass defect is the packing fraction per nucleon.

Fig.1: Graph of packing fraction Vs Mass number To understand better the strength of nuclear interaction, let us consider the so called binding energy of the nucleus. This is defined as the energy required to separate the nucleus into its constituents. Consider a nucleus of mass M and consisting of A nucleons (Z protons and N (= (A-Z)) neutrons), the binding energy is

=931.5 The average binding energy per nucleon,

MeV

8

, gives a good indication of the stability of a

given nucleus. Calculate the binding energy and average binding energy per nucleon for the following nuclei: . Plot the graph of the binding energy and binding energy per nucleon against the atomic mass and comment on your results.

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Fig.2: Binding Energy per nucleon Vs Atomic mass

From the graph of the binding energy per nucleon against atomic mass number, we observe that: (i) (ii)

(iii)

The binding energies of some light nuclei are small For the nuclides with 4> ℓ =

57 58

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Using this in the expression for eikz gives 59 Here, the first term e-ikr represents an incoming spherical wave and the second term e+ikr represents the outgoing spherical wave. The above equation therefore contains the incoming as well as the outgoing spherical waves. However, in nuclear reactions and scattering, the amplitude of the outgoing spherical wave component of the plane wave is modified thus,

60 This wave, ψ, represents superposition of incident and scattered waves: Ψ = Ψin + Ψsc where ηℓ is complex amplitude which accounts for changes in the ℓth outgoing partial wave. = where

ℓ

61 is the phase shift of the ℓth partial wave.

If

, only elastic scattering takes place

If

, both elastic scattering and nuclear reaction take place.

Note: If the incident particle is a charged particle, the exponential term in equation (60) must be replaced by appropriate coulomb wave function. We define the elastically scattered wave function Ψs as . Thus, 62 The scattering cross section,

, is defined as 63

Where

no. scattered into dΩ

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64 where m = mass of the particle r0 = radius enclosing the scattered. Using eqn. (64) in (63) gives 65 Using eqn. (64) in eqn. (65) gives 66 Since Yℓ0 is orthonormal, the total scattering cross section is given as

67 [Note:

]

If elastic scattering were the only process that could occur, then where δ ℓ is the phase shift of the ℓth partial wave. In this case

and and 68

Note: If there are other processes in addition to scattering, then eqn. (68) is incorrect ( ). All other processes are grouped under reaction cross section, . This is determined by the number of particles removed from the beam by all processes other than scattering. This is defined as: 69 )

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70

Thus the total cross section becomes ) The maximum value of

is obtained when

71

= -1 72

  whereas the maximum value of

is when ηℓ = 0 73

From eqns. (72) and (73) it follows that it is possible to have elastic scattering in the absence of other process. However, it is not possible to have reaction without also having elastic scattering.

MODELS OF NUCLEAR REACTION a) The Optical Model This is the simple model used to account for elastic scattering in the presence of absorption. This model is called optical model because it resembles the case of light incident on an opaque glass sphere. In this model, we assume the potential to be the sum of the real and imaginary parts, i.e. 74 where the real part and W is is the imaginary part. The values of and W are chosen to give the potential its proper radial dependence. The real part is responsible for elastic scattering and W, the imaginary part is responsible for absorption. The simplest form of this potential is of the form – ; for rR 75 Let define the cross section at low energy as 76 Here is the total cross section, the elastic cross section and the reaction cross section. In the asymptotic region, the solution of the Schrödinger equation for scattering is 76 In equ(76), the first term represents the ingoing spherical wave while the second term represents the outgoing wave modified by 32

The amplitude of the outgoing scattered wave is given by 77 The differential cross section for elastic scattering is given by 78 while the cross sections are given by 79 80 81 If we now define the shape elastic cross section fluctuation cross section as

and the compound elastic 82 83

The shape elastic scattering cross section corresponds to direct elastic scattering without the formation of compound nucleus. The cross section for the formation of compound nucleus, given by

, (reaction cross section) is

84 From equations (82)-(84), it could be shown that

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Part II

Jonah S.A.

1.0 INTRODUCTION Nuclear Physics is said to be composed of nuclear reactions and nuclear structure. Nuclear reactions have close relationship with fundamental properties of nuclear structure, such as: size, shape, density, spin, parity and nuclear levels. The first observation of an induced nuclear reaction was in 1917, when Lord Ernest Rutherford was able to accomplish transmutation of nitrogen into oxygen at the University of Manchester, using alpha (α) particles directed at nitrogen. Eventually, in 1932 at Cambridge University a fully artificial nuclear reaction and nuclear transmutation was achieved by Rutherford’s colleagues John Cockcroft and Ernest Walton who used artificially accelerated protons against lithium-7, to split the nucleus into two alpha particles. In both nuclear reaction and nuclear structure, models based on theory are formulated due to the complexity via solving the Schrodinger’s Wave Equations (SWE) of the nucleon –nucleon interactions of nuclei involved. Models are used to reduce the many-body potentials to a one-body potential in order to solve the SWE. Models serve to provide a convenient, admittedly over-simplified structure from which quantitative results can be computed with the desire that they compare well with experiment. A comparison of measured and calculated data is often necessary in order to test the suitability of the model. The probability of a projectile to “hit” a target nucleus (i.e. interact with it, such as scatter from it or break it up) may be described by an analogous “cross section” (but not the actual, physical cross sectional area of the nucleus). The different processes (reaction channels) possible for a given particle incident on a nucleus have different cross sections. Cross sections depend on a variety of reaction variables and cross section measurements are 35

some of the most important (and most common?) measurements made in a nuclear physics laboratory experiment. Other important parameters in nuclear reactions include Q-value, threshold energy, energy and angular distributions of emitted particles, excitation function as well as differential and integral data. Why is a Cross Section Important? It is the meeting ground between theory and experiment. Nuclear theory, using quantum mechanics (QM) is used to predict the probability (likelihood) that a specific nuclear process will occur under certain conditions (e.g. incident energy, angle of observation, etc.).The quantitative measure of this prediction is the cross section of the process. That is, nuclear theory is used to predict the specific cross section of a process, which may be measured in the laboratory, Comparison between theory and measurement is used to evaluate the significance of the underlying theory. The sentence (above) describes the essence of “doing science”. Nuclear reactions are described by several models, which are linked together to calculate nuclear reaction cross sections. In this process, the basic model is the Optical Model (OM), which enables us to separate total cross section into various components. Therefore, particle induced nuclear reaction processes are classified according to the time scale of interaction as direct reaction (DR) and compound nucleus reaction (CN). DR occurs on a time scale of 10-22 s, while CN occurs on a slower time scale of 10-16 s or more. In between these two is the pre equilibrium reaction (PER). All three processes contribute to calculating particle-induced cross sections. A schematic diagram depicting nucleon-nucleus interaction is displayed in Fig. 1.

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Form elastic Elastic scattering

(n,n) Target nucleus Compound elastic

Compound Nucleus

Equilibration

Pre-compound

Evaporation

Incident neutron Inelastic scattering

Pick-up; knock-out etc.

Direct reactions

(n,n´) (n,2n);(n,)

Charge exchange

(n,p)

Fig. 1 Schematic diagram of neutron-nucleus interaction processes

The Models on which the formula for calculating nuclear reaction cross section are based include: 1. Optical Model for DR, inelastic DR is treated by the DWBA 2. The Statistical Hauser-Feshbach Model for CN 3. Several Classical & QM Models for PER

2. THE OPTICAL MODEL (OM) The OM is also known as the cloud crystal ball model, which means that nuclei absorb and scatter incoming particles. This would mean that the reaction is independent of the internal structure of the nucleus and behaves much like the scattering of light from a crystal ball. The model is based on the mathematical techniques used in optics in which the nucleus is described by a potential well containing neutrons and protons. Thus, equation for the nuclear potential consists of terms for scattering and absorption. This potential can be used to calculate the probability for 37

elastic scattering of incident particles and the angular distribution of the scattering. The model is in excellent agreement for experiments for elastic scattering. Unfortunately, it does not allow us to obtain much information about the consequences of the absorption of the particles, which leads to inelastic scattering and transmutation. The optical model of the nucleus employs a model of the nucleus that that has a complex part to its potential. Calling this generalized potential, U(r), we have the definition: U(r) = V(r) − iW(r)

(1)

where V (r) is the usual attractive potential (treated as a central potential in the optical model), and its imaginary part, W(r), where W(r) is real and positive. The real part is responsible for elastic scattering, while the imaginary part is responsible for absorption. The Schrodinger’s Wave Equation to be solved is given as (2) Where  is the reduced mass of the system In solving the SWE, the optical potentials are difficult to derive and instead phenomenological optical model potentials are normally used to compare and fit to experimental data. Over the years, a standard form of the phenomenological optical potential has evolved, which permits the parametrization of the scattering of a light particle (such as neutron, proton, deuteron, triton helium-3 or alpha) from a given nucleus. It is defined as Uopt (r) = VC(r ) –(V +iW)fV,W(r) +(VS-iWS)fVS,WS(r )- dso l.s(Vso+iWSO)h (3) 1st term on the RHS is a Coulomb term 2nd term is a complex volume term 3rd term is complex surface term 4th term is complex spin – orbit term 38

F (r 0 is the Wood-Saxon form factor often described as a smoothed step function of the form f(r)=

(4)

r and R are radii, a is the diffuseness of the surface The phenomenological optical potential has been parametrized in terms of potential strengths and geometrical parameters, which have been adjusted to experimental data for many systems and many values of relative energies. The potentials obtained using such sets of parameters are called global optical potentials. Many individuals have compiled and include Perey and Perey compilations and Reference Input Parameters Library (RIPL). These data are available online via IAEA NDS (www-nds.iaea.org) For nucleons, typical values of the potential strengths are V  (45 – 55) MeV – (0.2 – 0.3) E, (5) WS  (2 – 7) MeV – (0.3 – 0.5) E (6) VSO  (4 – 10) MeV (7) These are valid for E  8 – 10 MeV Cross sections are calculated by the OM using the following formula

(8)

(9)

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DIRECT REACTION

A “direct reaction” (DR) involves a projectile that is energetic enough to have a reduced wavelength, λ, of the order of 1 fm (i.e. a 20MeV nucleon, for example), that interacts in the periphery of the nucleus (where the nuclear density starts to fall off), and interacts with single valence nucleon. That single nucleon interacts with the projectile leaving them both in bound, but unstable orbits. This state typically lives for about 10−22 s, which is long enough for the valence nucleon and projectile to (in classical terms) make several round trips around the nucleus, before one of them finds a way to escape, possible encountering a Coulomb barrier along the way. Since angular and linear momentum must be conserved, the ejected particle is generally ejected into the forward direction. The direct reaction (DR) theory accepts the OM description of elastic scattering by the complex potential well as first approximation as described above. However, an additional direct interaction capable of giving rise to nonelastic processes is introduced as a perturbation. The perturbation treatment takes into account the distortion of the waves through an interaction between particle and target. It is known as the Distorted –Wave Born Approximation (DWBA). The DWBA cross section is given below as:

(10)

Where,

2. THE STATISTICAL COMPOUND NUCLEUS REACTION MODEL

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Nuclear reactions leading to the formation of compound nucleus are best described by the statistical models of which the Hauser-Feshbach (HF) model is the most widely used. In the HF model, apart from the consideration given to conservation of energy, mass and charge, angular momentum and parity conservation are also taken into account. The compound nucleus (CN) mechanism conceives of a captured particle trapped within a complex system for a long time, such that the decay mode is independent of its mode of formation (i.e. the Bohr independence hypothesis). Unlike DR, which occurs on the surface of the nucleus CN involves the whole nucleus. Thus, a projectile enters the nucleus and interacts many times inside the nucleus, boosting individual nucleons into excited states, until it comes to rest inside the nucleus. This “compound nucleus” has too much energy to stay bound, and one method it may employ is to “boil off” nucleons, to reach stability. One, two, or more nucleons can be shed. The nucleons that are boiled off, are usually neutrons, because protons are reflected back inside, by the Coulomb barrier. Symbolically, the reaction is: a + X → C → Yi + b

(11)

The resultant light particle, b can represent one or more particles. In this model, the reaction products lose track of how the compound nucleus was formed. The consequences and restrictions of this model are: 1. Different initial reactants, a + X can form the same C with the same set of decays. Once the projectile enters the nucleus it loses identity and shares its nucleons with C . It should not matter how C is formed. In Classical Mechanics (CM) based on the Rutherford Scattering, the cross section is given in terms of the impact parameter, b, as (12) In the realm of Quantum Mechanics (QM), for a particle with orbital angular momentum, l., and wave length ƛ, then the cross section is given as 41

(13) The statistical model describes the emission of the flux that is absorbed into the long-lived compound nucleus during a collision. The contribution of this flux to the average cross section is given by the average of the fluctuation term in form of the Scattering Matrix (S-matrix) as follows:







(14)



Where  is the amplitude of the wave with an entrance channel and exit channel, . According to the HF statistics, the cross section for the formation of CN is a product of factors consisting of the probability for the formation of the CN and the probability that it decays in a given way. Thus the cross section for an entrance channel b and exit channel a, is written as follows: (15) Where, (16) Pa is the decay probability via exit channel a, and Ta is the transmission coefficients for channel a. The denominator gives the sum of transmission coefficients of all possible exit channels. The transmission coefficients can be defined as in terms of the S-matrix as follows (17) In general, the angle-integrated cross section for a given reaction (a,b) averaged over compound nucleus formation is given below as;

a,b

2J  1 = 2  J , ( 2 I  1)( 2i  1)

T T J asl

sl

s l 

J a s l 

(18)

D( J )

42

where, I and i are the spin values of the incident particles, J and total angular momentum and parity of the compound nucleus.

are the J and Tasl

TaJsl  are the average absorption cross sections into the compound nucleus

a, via the exit channel a  . They are also known as the transmission coefficients, which can be specified by various physical models for different exit channels. For the neutron, proton and other nuclear channels, the Optical Model (OM) is frequently used to calculate the transmission coefficients. The denominator, D(J, ) is given as:

Ed  Em  D(J, )=     E a , I , TaJsl  dE   TaJsl   a s l   0   l   Ed

(19)

Where,  E a , I ,  is the nuclear level density of the intermediate nucleus having quantum number ( E a , I , ), E d is the lower excitation bound of the continuum region and E m is the maximum excitation energy. Given the angular momenta and parities involved in the transition sequence, numerical calculation of cross section requires only the knowledge of the respective transition coefficients. Consequently, equations (18) and (19) are the basic expressions upon which majority of the nuclear model computational codes are derived. The transmission coefficients are calculated in the frame of the OM by solving the scattering problem with the OM potential.

PRE-EQUILIBRIUM REACTION MODELS

Up to this point, nuclear reactions have been considered to occur on two distinct time scales. This point of view is valid at low energies but becomes less so as the incident energy increases. Due to the two-body nature of the nuclear interaction, a nucleon-nucleus interaction may be decomposed as a 43

series of nucleon-nucleon interactions. Taking the entrance channel of a nucleon-nucleus composite system to a 1(projectile) particle -0 (target) hole (i.e.1p-0h) state, one can consider the result of a collision as leading to another 1p-0h state or to a set of 2p-1h states. At energies up to 20 to 30 MeV, the wavelength of nucleon is still about 5 to 6 fm, so that the projectile nucleon – target nucleus interaction would not excite an individual particle-hole state, but a complicated linear combination of particle-hole states. However, in all the cases one can classify the complex configuration in terms of the number of particles h and number of holes h. one notes that, since the number of particles is conserved, the difference between the number of particles p and the number of holes h, p – h , remains constant throughout a collision. The quantity n = p + h, is called the exciton number. There are several fundamental theories developed for accounting for pre-equilibrium emission. The models include the exciton, the hybrid, the multistep direct and multistep compound and others. In this workshop, the EXIFON code will be used to perform nucleoninduced cross section calculations. The code is based on the formalism of statistical multistep direct and compound reaction models (i.e. MSC and MSD). Before discussing the MSC and MSD models, it is worthwhile to present an overview of the original exciton model proposed by Griffin. The exciton model is a time evolution of occupation probability of n-exciton state in energy space described by the Figures below

44

45

(20)

(21)

The multistep reaction concept was formulated by Feshbach, Kerman and Koonin in 1980 and is known as the FKK theory. In this theory, the interaction of projectile with the nucleons of target nucleus takes place in stages, which can lead to particle-hole excitation analogous to the exciton model. In a statistical MSD reaction, at least one particle is in a continuum energy state, whereas in the MSC reaction all nucleons remain bound. At low energies multistep reaction takes place by MSC process and as the energy increases, the MSD process becomes increasingly important and eventually dominates. Others PER models include the intranuclear cascade model and the hybrid Monte Carlo simulation, which are applicable at even higher incident energies from about 100 MeV to 200 MeV to calculate angular distribution and energy spectrum of emitted particles.

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In the EXIFON code, which is based on the MSD and MSD processes, a simple two-body interaction is assumed and the single particle state density g of the particles (i.e. n, p, ) with reduced mass c is given by;

g = 4  E F 

(22)

where, the factor 4 takes into cognizance the spin and iso-spin degeneracy and

 E F  = 4.8  10 3 fm 3 MeV 3

2

r

3 o

AE 1

2

(23)

The shell structure effects are considered in the multi-step compound (MSC) process, thus the single-particle state density g, is multiplied by the following factor,

  W 1  exp  E X  1   EX 

(24)

where,  = 0.005 MeV-1 and W is the shell correction energy.

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The quantity E X = E and denotes the excitation energy of the composite or residual systems. In principles, the shell correction energy includes only the fluctuating, microscopic contribution to the binding energy. DETERMINATION OF CROSS SECTIONS Like all other nuclear reaction data, cross sections can be determined by experimental techniques and theoretical calculations. The statistical methods are also used to process the experimental and theoretical data into evaluated data files. Experimental techniques are based on ‘on-line” and “off-line” methods. The “on-line” methods include the use of neutron spectrometers in combination with time-of-flight (TOF), telescope counter, quadrupole spectrometers and gamma-ray spectrometers. The “off line” methods are basically based on the neutron activation technique using nuclear research reactors and other neutron sources, the use of mass spectrometers (MS) and accelerator mass spectrometers (AMS). In this workshop, the theoretical methods and statistical evaluation of nuclear reaction cross section data will be dealt with. Theoretical model calculations are important for producing ‘complete’ data found in the evaluated data libraries. They are useful for the predictions, where no measured data exist and to extrapolate/interpolate from measured to unmeasured regimes. COMPUTER CODES FOR THEORETICAL MODEL CALCULATIONS Over the years, computer codes have been developed for the determination of cross sections based on the nuclear reaction mechanisms that were described above. Some of the computer codes based on reaction mechanisms (i.e. DR, CN and PER) are:  SCAT-2 STAPRE  ALICE

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 EXIFON  EMPIRE The experimental data are compiled and regularly updated by the IAEA as the EXFOR data library, which can be accessed at the website, wwwnds.iaea.org. Furthermore, a number of the evaluated data libraries from different countries and organizations, which are accessible via the same IAEA websites include:  ENDF (Evaluated Nuclear Data Library-USA) http://www.nndc.bnl.gov/exfor/endf00.htm  JEFF (Joint Evaluated File- Europe) http://www.nea.fr/html/dbdata/eva/evaret.cgi  JENDL (Japanese Evaluated Data Nuclear Data Library- Japan) http://wwwndc.tokai-sc.jaea.go.jp/jendl/jendl.html  CENDL (Chinese Evaluated Nuclear Data Library-China) http://159.226.2.40/ In this workshop, the EXIFON code will be used to calculate particleinduced cross sections of some reactions of interest, which will be compared with experimental data from the EXFOR data library and evaluated data retrieved from JENDL, ENDF and JEFF data libraries. 2. EXAMPLES WITH THE EXIFON CODE The EXIFON code version 2.0 is a calculational code based on both manybody theory and random matrix physics. In this example, it has been used to calculate neutron induced reaction cross section data from 0 to 20 MeV on an even-even, magic number nuclide 52Cr with neutron number, N=28. Specifically, the (n,p), (n,) and (n,2n) reaction cross section data were calculated as functions of incident energy of neutrons. Data obtained from 49

the experimental data in the IAEA, EXFOR data library and recommended data libraries around the globe, JENDL, ENDF and JEFF were used to validate the calculated data. The data indicate that the calculated data without shell corrections are in good agreement with experimental data as well as the recommended data from the evaluated data libraries. The calculated results could provide useful insight into the choice of some input parameters near closed shells using the EXIFON code.

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APPLICATIONS OF NUCLEAR REACTION DATA Nuclear data in general and cross section data are of great importance with respect to applications in a wide variety of fields from the earth crust to the upper atmosphere. The starting point of any reliable computational modeling of any nuclear systems is the accurate knowledge of reliable nuclear information often referred to as nuclear data. The specific areas of applications of particle-induced cross section data include but limited to:  Energy  Fission reactors The design, commissioning and optimal utilization of fission reactors (i.e. nuclear power plants, nuclear research reactors) depend on well tested cross section data. The basic data required in this respect are fission cross sections, capture cross 53

sections, scattering cross sections etc. Accurate knowledge of these data means better safety and economics of reactor construction, nuclear fuel cycle, waste management, radiation damage etc.  Fusion reactor technology The fusion reactor technology (FRT) also known as thermonuclear reactor is based on the fusion of light nuclei, which leads to the release of energy. Because of its high positive Q-value, the D-T reaction is the most promising of all possible nuclear reactions. The main problem with this technology is the search for suitable materials in the design, the tritium fuel cycle and the plasma confinement. Therefore, data are needed areas such as the plasma fuel cycle, tritium breeding, transmutation products and radiation protection.  Generation IV reactors Generation IV reactors (Gen IV) are a set of theoretical nuclear reactor designs currently being researched. Most of these designs are generally not expected to be available for commercial construction before 2030. Current reactors in operation around the world are generally considered second- or third-generation systems, with most of the first-generation systems having been retired some time ago. Relative to current nuclear power plant technology, the claimed benefits for 4th generation reactors include:   

Nuclear waste that remains dangerously radioactive for a few centuries instead of millennia 100-300 times more energy yield from the same amount of nuclear fuel The ability to consume existing nuclear waste in the production of electricity 54



Improved operating safety

Therefore, the design and fabrication would require high energy nuclear reaction data.  Energy Amplifiers/Accelerator Driven Systems Energy Amplifiers (EA) and Accelerator Driven Systems (ADS) are the focus of R & D in nuclear energy so as to eliminate the opposition to the present nuclear energy programmes across the globe. The main reasons for the opposition are due a) association of nuclear energy with military applications and the fear of proliferation; b) the fear of accidents as witnessed in in the Three Mile Island accident in Pennsylvania, USA in 1979, Chernobyl, in 1986, and Fukushima in 2011; c) the issue of the back-end of the fuel cycle, which has to do with waste management of spent fuel. The EA/ADS is a sub-critical nuclear system based on U – Th fuel cycle, driven by a high intensity proton accelerator to produce energy with minimum amount waste. Therefore, the design, fabrication and utilization of these systems would require high energy nuclear reaction data in the energy region of GeV.  Biomedical  Radioisotope production Radioisotopes are often used in medicine for diagnosis, treatment and research. The radioactive tracers which emit gamma radiation can offer a large amount of information about the anatomy and the proper functioning of different organs in the human body, as they are often used for tomography investigations (single Photon Emission Computed Tomography, PET scanning). Also, the radionuclides (gamma and beta emitters) have become a promising method for the treatment of some tumors. The radioisotopes are the result of the nuclear reactions, the interaction between a projectile particle (neutron, proton, deuteron, alpha particle, and photon) and an atomic 55

nucleus. Therefore the knowledge of reaction cross section is of great importance with regards to maximizing production yields and minimization of radioactive impurities.  Dosimetry Dose calculations are very important in nuclear medicine because of the amount of radioactivity required for the specific diagnostic and therapeutic applications. Specifically, neutron therapy demands an accurate knowledge of activation cross sections, threshold reaction cross sections as well as scattering cross sections.  General  Astrophysics Nuclear astrophysics aims at understanding of energy production in sun and stars as well as the nucleosynthesis of chemical elements. The sources of energy of stars are based on the thermonuclear processes, which rely on the interaction of light nuclei. For a good understanding of energy production in stars and the thermonuclear processes leading to the formation of elements heavier than Oxygen require an exact knowledge of all low energy fusion reaction cross sections.  Cosmochemistry Cosmochemistry involves the investigation of formation of chemical elements under the influence of cosmic radiation. The process demands the accurate knowledge of nuclear processes, which lead to cosmogenic nuclides. Two important components of cosmic radiation are the solar cosmic radiation (SCR) and galactic cosmic radiation (GCR). SCR consists of charged particles emitted in solar eruptions and are accelerated in magnetic fields. On the other hand GCR consists of particle radiation which is produced outside the solar system in the 56

galaxies, presumably in supernova explosions. Therefore, the knowledge of reaction cross sections in the energy range of 100 to 30000 MeV is needed in connection with the production of cosmogenic radionuclides.  Geosciences Several nuclear methods are used in geology for multielemental analysis, age determination, bore hole logging etc. An accurate knowledge of nuclear reaction cross section data is needed to optimize the applications of the nuclear methods.

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2. REFERENCES i.

ii.

iii.

Buczko, Cs. M., Csikai, J S. Sudar, A. Grallert, S.A. Jonah, B.W. Jimba, T Chimoye and M Wagner 1995: 'Excitation functions and isomeric cross-section ratio of 58Ni(n,p)Co58m,g reaction from 2 to 15 MeV' Phys. Rev. C. 52, 1940-1946 Gopych, P.M., Demchenko M.N., Zalyubovskiy, I.I., Kizim, P.S., Sotnikov, V.V., Schchus A.F., 1987: Neutron cross section for tin, tellerium, silicon and phosphorus nuclei at neutron energy 14.6 MeV, Proceedings of Int. Conf. On Nuclear Physics, Kiev, 14-18, Sept. 1987, C 87.KIEV, 3, 295, EXFOR #41032 J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, John Wiley & Sons, 1958

iv.

Hauser, W. and Feshbach, H. 1952: Inelastic scattering of neutrons Phys. Rev., 87, 366-373

v.

IAEA 2003: EXFOR+CINDA, Database and Retrieval System, Version 1.10, March 2003, Nuclear Data Section, IAEA, Vienna, Austria JAERI 2002: Japan Evaluated Nuclear Data Library, Version 3, Revision 3 (JENDL-3.3), \Nuclear Data Center, Japan Atomic Energy Research Institute

vi.

vii.

Kalka, H., Torjman, Lien, H. N., Lopez, R., and Seegler, D. 1990: Description of (n,p) and (n,2n) activation cross section for mediummass nuclei within statistical multistep theory, Z. Phys. A-Atomic Nuclei 335, 163-171

viii.

Kalka, H. 1991: EXIFON – A Statistical Multi-step Reaction Code, Report, Technische Universitat Dresden, Germany

ix.

C.M. Perey and F.G. Perey, At. Data and Nucl. Data Tables 17, 1-101 (1976). 58

RECOMMENDED BOOKS i. ii. iii. iv. v. vi. vii.

viii. ix. x.

Introduction to Atomic and Nuclear Physics. Henry Semat, Chapman & Hall Ltd. Nuclear Physics. An introduction, W.E. Burcham, Longman L. S. Rodberg and R. M. Thaler, Introduction to Quantum Theory of Scattering, Academic Press, 1967. The Atomic Nuclear. R.D. Evans, McGraw-Hill Book Company Nuclear Physics, Irving Kaplan, Addison – Wiley Publication Company Elements of Nuclear Power, Longman Group Ltd. D.J. Bennet Computational Methods in Engineering and Science: With Applications to Fluid dynamics and Nuclear Systems, WileyInterscience Publication, 1977 Shoichiro Akamura Nuclear Fuel Management, John Wiley & Sons Inc., 1979 Graves, Harvey W. Hodgson, P. E. 1971, Nuclear Reactions and Nuclear Structure, Clarendon Press, Oxford. Welton, T. A. 1963, Fast Neutron Physics Part II, ed. by J.B. Marion and J. L. Fowler, Interscience Publisher, New York.

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EXECUTION AND RETRIEVAL EXERCISES This is essentially an introductory exercise which will involve the use of the EXIFON Code version 2.0 for the calculation of cross sections of particleinduced reactions on different target nuclei. Results obtained are to be compared with experimental data retrieved from the EXFOR data library and recommended data libraries (i.e. ENDF, JEFF and JENDL) at the IAEA Nuclear data Centre website (www-nds.iaea.org) a) How to run EXIFON Quit Windows and go to MS-DOS by clicking on All Programs; then Accessories Click on “Command Prompt” Change Directory to C:\EXIFON Run the program by typing ‘Exifon’ You will be prompted to change INPUT and OUTPUT directories Follow the instructions given by the Resource Person b) To retrieve experimental data from the EXFOR data library Go to IAEA Nuclear Data Centre website: on your web browser type http://www-nds.iaea.org to log on to the IAEA nuclear data servces On the Quick Links, scroll down and click on “EXFOR” You will be taken to the page “Experimental Nuclear Reaction Data EXFOR” On this page, enter the retrieval parameters as follows: Click on the box “Target” and select the target nucleus of interest (e.g. 58Ni) Click on the box “Reaction” and select reaction of interest (e.g. n,p) Click on the box “Quality” and select CS for cross section Click on the box “Energy from” and input the energy range of interest (0 to 20 MeV) Click on the box “author” and type ‘Jonah’

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Click on the box “submit” (ensure that nothing is checked under “options” except sort by ‘publication’ and view by ‘basic’ You will be taken to the page “Data Selection” from where you can retrieve the data in output of different formats such as X4; EXFOR; Bibliography etc.,. You may also wish to plot your data using the following options Quick plot (for cross section only); Advanced plot etc., Select data of interest from the list displayed and click on “Retrieve” This will take to the “Output” page containing the retrieved data and the plot. You can copy the plotted data into an Excel worksheet together with the calculated data from EXIFON Wait for further instructions from Resource Person c) If internet connectivity is not available, copy “EXFOR” data files to the desktop from a CD Rom that will be made available d) To retrieve evaluated data from data libraries Go to IAEA Nuclear Data Centre website: on your web browser type http://www-nds.iaea.org to log on to the IAEA nuclear data servces On the Quick Links, scroll down and click on “ENDF retrieval” You will be taken to the page “Evaluated Nuclear Data File (ENDF)” On this page, under “Standard Request” enter the retrieval parameters as follows: Click on the box “Target” and select the target nucleus of interest (e.g. 58Ni) Click on the box “Reaction” and select reaction of interest (e.g. n,p) Click on the box “Quality” and select SIG for cross section Check the boxes of ‘major libraries of interest’ Click on the box “submit” You will be taken to the page “ENDF Data Selection” from where you can retrieve by checking the box “all” You may also wish to plot your data using the following options Quick plot (for cross section only) 61

If you choose to plot the selected data of interest This will take to the “Output data” page containing the plot and plotted data. You can copy the plotted data into an Excel worksheet together with the calculated data from EXIFON and EXFOR data for comparison Wait for further instructions from Resource Person

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PART 111 Study Papers The follow papers by Prof. H. Kalka are study papers given to the participants:

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