ISSN 10628738, Bulletin of the Russian Academy of Sciences. Physics, 2015, Vol. 79, No. 7, pp. 883–889. © Allerton Press, Inc., 2015. Original Russian Text © V.A. Rubchenya, 2015, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2015, Vol. 79, No. 7, pp. 980–987.
A Consistent Model for Describing Prompt Fission Neutrons V. A. Rubchenya Khlopin Radium Institute, St. Petersburg, 194021 Russia University of Jyväskylä, 40014 Finland St. Petersburg State University, St. Petersburg, 199034 Russia email:
[email protected] Abstract—A theoretical model for a consistent description of prompt neutron emission for spontaneous and neutron and protoninduced fission over a wide range of energies is presented. Prompt fission neutron parameters are formed at the four main fission process stages: preequilibrium particle emission, neutron evaporation upon crossing the fission barrier, neutron evaporation upon descending from the saddle point to the scission point, and neutron emission from pairs of accelerated fragments. The Monte Carlo method is used to simulate the evolution of a nucleus undergoing fission down to the scission point. The parameters of an ensemble of sources after fragmentation are calculated using the scission point model and the multimodal fission model. Calculated neutron spectra and multiplicities are given for spontaneous 244–257Fm and 252Cf fission and 235U fission induced by thermal and 14.7 MeV neutrons. DOI: 10.3103/S1062873815070199
INTRODUCTION Prompt fission neutrons are important probe parti cles for investigating nuclear fission dynamics at dif ferent stages: (i) the preequilibrium stage of com pound nucleus formation; (ii) formation of collective flow over the fission barrier; (iii) neutron evaporation during the saddle stage; (iv) nucleon emission during the descent time from the saddlepoint to the scission configuration; (v) neutron evaporation from com pletely accelerated fragments. Comparing the experi mental data and theoretical predictions allows us to obtain important dynamic and static parameters of the extreme states of nuclear matter in the course of nuclear fission. Reliable predictions of the prompt fis sion neutrons characteristics for superheavy nuclei are also important. Future nuclear energetics based on the GENERATIONIV nuclear reactors demands the high precision neutron data for the large number of heavy nuclei involved in the fuel cycle. In different models [1–12], the parameters of prompt fission neutrons are described using experi mental values for the required parameters of neutron sources. In this work, we develop a theoretical model for describing prompt fission neutrons with consistent calculations of mass, charge, and energy distributions of primary fission fragments upon the spontaneous and induced fission of heavy nuclei. The Monte Carlo method is used for simulating preequilirium nucleon emission processes and fission down to the scission point. An ensemble of compound nuclei is formed near the scission point with wide charge, mass num ber, spin, and excitation energy distributions resulting
from nucleon emission in the preequilibrium state, crossing the fission barrier, and descending to the scis sion point. The mass, charge, and energy distributions of primary fragments are calculated for each member of this ensemble, and differential prompt neutron multiplicity distributions are calculated for the obtained ensemble of fragments. The basics of the model were formulated in [13]. THEORETICAL MODEL Simulating the fission process induced by a particle with spin s and impact parameter l interacting with a nucleus with spin I begins with calculating partial cross sections for the initial composite nucleus with the mass number ACF , charge Z CF , excitation energy ex , and spin J = I + j ( j = l + s ), E CF CF ex J σljIJ ( ACF , Z CF , E CF ) = π (2J + 1)TljI , 2I + 1 2
(1)
where the transmission coefficients TlIJ are calculated using the optical model. In the case of spontaneous fis sion, the simulation begins with the stage of descend ing to the scission point at which neutron emission is possible. At low neutron energies in the input channel, the compound nucleus parameters are similar to those of the initial composite system. For projectile nucleus energies higher than approximately 10 MeV/nucleon, however, the ensemble of compound nuclei is formed as a result of preequilibrium nucleon emission. The twocomponent exciton model [13, 14] is used to cal
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culate the parameters of the compound nuclei ensem ble. The Monte Carlo method allows the introduction of a time criterion for the transition from preequilib rium particle emission to the equilibrium decay of the compound nucleus, including the fission channel. After the preequilibrium stage is complete, the evolution of each member of the compound nucleus ensemble with the specified parameters ex ( ACN , Z CN , E CN , J CN ) is simulated using the Monte Carlo method at the stages of crossing the fission bar rier and upon descending from the saddle point to the scission point with inclusion of the dynamic effects in the fission channel. It is assumed that gamma and par ticle emission starts immediately after the end of the preequilibrium stage with total decay widths. The sta tionary probability current through the fission barrier is formed with some delay τ d , and the influence of nuclear friction or collective motion dissipation reduces the fission width relative to the standard fis sion width ΓBW obtained by Bohr and Wheeler in the f transitionstate method [15]. If the excitation energy of the compound nucleus exceeds the fission barrier by several MeV, the following expression is used for the fission width [16, 17]:
((
2 Γ f (t ) = Γ BW f [1 − exp ( − t τ d )] 1 + γ
)
12
)
−γ .
(2)
Here, γ is the dimensionless nuclear friction coeffi cient determined by the ratio of reduced friction coef ficient β and oscillatory frequency ωb at the top of the fission barrier: β (3) γ= . 2ωb The intensity of nuclear friction depends on the tem perature and the deformation of the compound nucleus. At present, the available experimental and theoretical data do not allow to make definite conclu sions. In our model, the quadratic dependence of the friction coefficient β on temperature is assumed for internal excitation energies higher than 10 MeV [13]. After crossing the fission barrier, the probability of light particle and gamma emission is determined by the excitation energy and the time interval of the
d 2 M npost = dEd Ω
(
∑ ∫
Asc,J sc , A, Z
)
(
dE sc*W sc E sc*, Asc, J sc
)
descent from the saddle point to the scission point, which depends on the force of nuclear friction coeffi cient [18]
((
τ sdsc = τ 0sdsc 1 + γ 2
)
12
)
+γ ,
(4)
where τ 0sdsc is the descent time when there is no nuclear friction, and was set equal to ~10–20 s in the calcula tions of low energy fission presented below. The mul tiplicity of neutrons emitted at this stage is propor tional to the product of the average descent time and the average neutron width. The nucleus excitation energy at this stage is equal to the sum of the internal energy of the compound nucleus after crossing the fis sion barrier and the energy dissipation upon descend ing to the scission point. The difference between potential energies at the saddle point and the scission point increases along with nucleus fusibility parameter Z 2 A . In the case of strong friction, a notable part of this difference is converted into internal excitation energy, which increases the probability of neutron emission. The Monte Carlo method is used to simu late statistical particle emission during specified time interval (4) of descending to the scission point. The initial internal excitation energy is then considered equal to the sum of excitation energies at the saddle point after crossing the fission barrier and part of the difference of potential energies at the saddle point and the scission point (see expression (34) in [13]). The remaining part of the difference between potential energies at the saddle point and the scission point is converted into the kinetic energy of collective degrees of freedom and the initial kinetic energy of the frag ments. An ensemble of compound nuclei is formed in the scission point as a result of particle and gamma emis sions, and each member of this ensemble is trans formed into an ensemble of primary fragments with the initial mass, charge, spin, excitation, and kinetic energy distributions. The double differential postfis sion neutron spectra in the center of mass system of the compound nucleus are calculated by summing the spectra from many primary fragment nuclei,
M nF ( A, Z ) EA F W F E, A, Z , J F , E kin Y pre ( A, Z ) . F 4π E kin
Here, W sc E sc*, Asc, J sc is the excitation energy, mass number, and spin distribution of compound nuclei at F the scission point; W F (E, A, Z , J F , E kin ) is the neutron spectrum from a fragment with primary mass, charge, F spin, and average kinetic energy E kin ; M nF ( A, Z ) is the multiplicity of neutrons emitted by the fragment;
(
)
(5)
Y pre ( A, Z ) is the primary fission fragment yield. This expression was obtained by assuming isotropic neu tron emissions from fully accelerated fragments in the fragment’s center of mass. The excitation energy and the kinetic energies of the primary fragments are calculated using the model of nascent deformed fragments with inclusion for the
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shell and pairing effects and their temperature depen dence [13]. The kinetic energy of fragments is equal to the sum of Coulomb interaction energy at the mini mum potential energy of the light and heavy fragments and the kinetic energy at the scission point,
(
{ }
F Ekin = VCoul Ah, Z h, εih
min
{}
, Al , Z l , εli
+ Arel L ( L + 1) + Ekin.
min
, dt
)
(6)
sc
{ }
{}
Deformations ε ih and ε li are determined by min min minimizing the potential energy of nascent fragments for a fixed distance between the nearest poles of axially deformed fragments dt ≈ 2.5 fm. The kinetic energy at the scission point is treated as a model parameter, sc E kin ≈ 10 MeV. The orbital angular momentum of nascent fragments L is determined from the condition of equality of the spin of the compound nucleus and the sum of spins of the fragments and their relative angular momentum. The total excitation energy of each fragment is the sum of the deformation energy, rotational energy, and the internal thermal energy at the scission point. The latter is determined from the thermal equilibrium condition. The shell structure of deformed nuclei at the scis sion point to a large extent determines the fragment mass dependences of excitation and kinetic energies [19, 20]. The macromicroscopic Strutinsky method [21] or the microscopic Hartree–Fock–Bogoliubov method with effective nuclear forces [22] can not pro vide quantitative descriptions of nuclear fission char acteristics. In our approach, the semiphenomenolog ical model [13] is used to calculate the deformation energy at the scission point with amended shell cor rections of fragments found using the Strutinsky method [21]. The mass and charge distribution of primary frag ments are presented in factorized form [23], (7) Ypre ( A, Z ) = Ypre ( A) Ppre ( Z A) . The primary isobaric chain yields and their charge dis tributions are presented in the form of the products of a smoothed distribution and an even–odd factor,
Y pre ( A ) = Ypre ( A ) FAoe, (8) Ppre ( Z A ) = Ppre ( Z A ) FZoe. The smoothed charge distribution of the isobaric chain of primary fragments is approximated by the normal distribution ⎡ ( Z − Z ( A)) 2 ⎤ 1 exp ⎢− Ppre ( Z A) = ⎥, 2 σ Z ( A) 2π 2σ Z ( A) ⎦⎥ ⎣⎢ (9) Z Z ( A) = A sc + δZ ( A) . Asc The parameters of this distribution are determined by the conditions at the scission point, which represents
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the configuration of two weakly overlapping axially deformed fragments with a common axis of symmetry. Weak nucleon exchange takes place through an over lapping region with radius rneck ≈ 2 fm. The typical period of isovector density oscillations along the axis of symmetry for the compound nucleus at the scission point is much shorter than the periods of other collec tive oscillations, so the charge distribution variance σ Z ( A ) can be calculated for fixed mass asymmetry and distance between the fragment centers. To describe small charge oscillations, we must determine the harmonic potential parameters 2 (10) V ( Z ) = V ( Z ) + 1 C ZZ ( Z − Z ) . 2 Stiffness parameter C ZZ and the average charge of the isobaric chain of primary fragments Z ( A ) are calcu lated using the scission point method, in which the fragment deformation energy is calculated using the Strutinsky method [21]. Singleparticle neutron and proton spectra are calculated using the deformed Woods–Saxon potential with universal nuclear poten tial parameters [24]. The potential energy at the scis sion point is minimized with respect to the deforma tion parameters of both fragments for fixed mass split ting upon different charge splitting, thereby obtaining the dependence Vmin ( Z A ) approximated by parabola (10). The charge distribution variance σ Z ( A ) is determined by zero point oscillations in the harmonic potential:
(11) σ Z ( A) = 1 . 2 M ZZ C ZZ The parameterization obtained in [25] is used to cal culate the mass parameter for isovector oscillations along the symmetry axis at the scission point: ACN l neck + 2rneck M ZZ = 16 r03m , 2 Z CN N CN 9 rneck
(12)
where r0 is the nucleus radius constant, m is the nucleon mass, and lneck is the neck length (lneck = 2–4 fm). The smoothed fragment mass distribution is described using the multimodal approach in the nuclear fission [26]. Five fission modes are used to approximate the primary mass distribution [23, 27]: 5
Ypre ( A) =
∑C m=1
m
(A ,Z sc
)
* Ym ( A )
sc, E sc
= CSY YSY ( A) + CSIYSI ( A) + CSIIYSII ( A ) + + CSAIYSAI ( A) + CSAIIYSAII ( A) .
(13)
The symmetric component Y SY is connected with the properties of a homogeneously charged nuclear liquid drop on the potential surface. The asymmetric stan dardI mode is generated by the influence of nuclear shells Z = 50 and N = 82 (the 132Sn mode); the other
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yield in the region of symmetric fission as the excita tion energy of the compound nucleus increases. The neutron multiplicity from separate fragments F M n ( A, Z ) and the spectra in the fragment center of mass system are calculated using different versions of statistical theory. Most consistent are calculations of prompt fission neutron emission using Hauser–Fesh bach theory [1, 4, 12].
Mn 264Fm
5 4 3 2
NUMERICAL RESULTS
1
0
50
100
150 200 τsddsc/10–22 s
Fig. 1. Fission neutron multiplicity as a function of the time needed to descend to the saddle point for spontane ous 264Fm fission: (dashed line) neutrons emitted in descending; (dashedanddotted line) neutrons emitted from fragments; (solid line) total number of neutrons.
asymmetric standardII mode is associated with the influence of the deformed nuclear shell N = 86–90 of the heavy fragment. The superasymmetric SA I and superasymmetric SA II modes are determined by the influence of the nuclear shells N = 50 and Z = 28 (the 78Ni mode). The mass distribution of each mode is normalized, ∑ Y m ( A ) = 2, and approximated by the A sum of two normal distributions,
Ym ( A) =
1 σm
{exp ⎡⎣− ( A − A ) 2π m
2
2σ 2m ⎤ ⎦
}
2 + exp ⎡− ( A + Am − Asc ) 2σ 2m ⎤ . ⎣ ⎦
(14)
The parameters of distributions (14) and the weights of separate modes C m were determined by comparing the calculated and experimental mass distributions and independent yields of heavy nuclei fission products [23] over a wide range of excitation energies for com pound nuclei. The positions of primary mass distribu tion modes A m depend weakly on the nucleon compo sition and excitation energy of compound nuclei at the scission point. The distribution variance of separate modes σ m is proportional to the temperature at the scission point Tsc. The contributions from separate fis sion modes C m depend strongly on the nucleon com position of compound nuclei and the excitation energy below E sc* ≈ 40 MeV (see Fig. 5 in [23]). The competition between the symmetrical mode and the sum of contributions from asymmetric modes is most strongly evident in the fast growth of the fragment
The theoretical model described above were used to develop the FIPRODY (Fission PRODuct Yields) software package for calculating the characteristics of spontaneous fission and fission by thermal neutrons and light particles. The model parameters were deter mined by comparing theoretical calculations of the basic characteristics and the experimental values obtained for the fissioning of heavy nuclei from Th to Cf by neutrons and protons. An important prediction of the proposed model for prompt fission neutron (PFN) emission is neutron evaporation at the stage of descending from the saddle point to the scission point even for the spontaneous fission of heavy nuclei. The properties of this PFN component differ in two ways from those PFNs that evaporate from accelerated fission fragments. First, their angular distribution in the lab system is isotropic even for a particular direction of fragment emission. Second, their spectrum is shifted toward low energies because of the limited available average excitation energy at the descent stage. For a more accurate description of neutron evaporation upon descending from the saddle point, we must perform dynamic cal culations for all fission modes, a very complicated problem at present. Here we use an approximate approach in which evaporation from the compound nucleus occurs in fixed time τ sdsc and the average exci tation energy in descending is determined by the potential energy at the scission point. The average neutron multiplicity at the descent stage M nsdsc as a function of time interval τ sdsc calculated for spontane ous 264Fm fission is shown in Fig. 1. The total calcu lated PFN multiplicity equal to the sum M nsdsc + M npost is virtually independent of the descent time. Figure 2 shows the calculated prefission neutron spectrum ( M nsdsc = 0.254) emitted in descending for τ sdsc = 10–20 s and the total PFN spectrum for sponta neous 252Cf fission. The possible existence of socalled scission neutrons not emitted from accelerated excited fragments was found more than 50 years ago in corre lation measurements of fragments and neutrons in the spontaneous 252Cf fission [28]. This problem of scis sion neutrons has yet to be solved satisfactorily, and the considered prescission mechanism of PFN emission can be used to explain this phenomenon.
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A CONSISTENT MODEL FOR DESCRIBING PROMPT FISSION NEUTRONS dMn/dEn, MeV–1
887
Ratio 1.1
1
10
1.0 100
0.9
252Cf
10–1
0.8 0.7
10–2
252
Cf
0.6
10–3
0.5
10–4
0.4
235
U(nth, f )
0.3 10–5
0.2 1
0.1
10–6 0
5
10
15
20 En, MeV
Fig. 2. Calculated neutron spectrum for (solid line) spon taneous 252Cf fission and (dashed line) neutrons emitted upon descending to the scission point.
The integral PFN spectrum is comprised of multi ple sources with different parameters, and can be approximated by a Maxwellian distribution. Figure 3 shows the calculated spectra for spontaneous 252Cf fis sion and 235U fission by thermal neutrons as ratios to the Maxwellian distribution with the corresponding temperature parameter. The temperature parameters Т = 1.42 MeV for spontaneous 252Cf fission and Т = 1.32 MeV for 235U fission by thermal neutrons nor mally used to estimate PFN spectra were applied. The character of the theoretical spectra’s deviation from Maxwellian is typical for corresponding esti mates [29]. Neutron emission in the spontaneous fission of Fm isotope over a wide range of mass numbers is of great interest, as these isotopes lie in the region of transition to superheavy nuclei, the identification of which is possible through neutron emission [30]. The calcu lated average neutron multiplicity and average kinetic energy of spontaneous fission fragments for Fm iso topes with А = 244–257 are compared to experimental data [31, 32] in Fig. 4. Good agreement with the experimental data was obtained for Fm isotopes with А < 257, for which the properties of fission modes are typical for the actinide region. The calculated mass distribution of primary frag ments and PFN multiplicity as a function of the frag ment mass in the case of spontaneous 246Fm fission is shown in Fig. 5. Symmetric bimodal fission occurs for 258Fm and heavier isotopes; there is in this case a com pact symmetric mode with anomalously high fragment kinetic energy and a strongly stretched symmetric mode with low kinetic energy [32].
10 En, MeV
Fig. 3. Ratio of theoretical PFN spectra and Maxwellian distribution for spontaneous 252Cf (Т = 1.42 Mev) and 235U fission by thermal neutrons (Т = 1.32 MeV).
As example of the particle induced fission consider the 14.7 MeV neutron induced fission of U235 is considered. The calculated spectra are presented in Fig. 6 where the total spectum is shown by solid line. The preequilibrium spectrum (marked by stars in Fig. 6) contributes remarkably to the total spectrum at
4.5
(a)
Mn
Fm
4.0 3.5 3.0 2.5 Ekin, MeV 205
(b)
200 195 190 244
246
248
250
252
254
256 258 Mass number
Fig. 4. PFN multiplicities and average fragment kinetic energies for spontaneous fission of Fm isotopes: (white dots) calculated and (black dots) experimental.
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RUBCHENYA dMn/dEn, MeV–1 101
Fissioning 0.06 246
Fm
235U(n(14.7),
100
f)
10–1
0.04
10–2 10–3 0.02 10–4 10–5 0
5
10
15
0 4
Mn
Fig. 6. Calculated PFN spectra for 235U fission by 14.7 MeV neutrons: (solid line) total spectrum; (stars) pre equilibrium neutrons; (circles) prefission neutrons; and (triangles) postfission neutrons.
3 2 1 0 80
20 En, MeV
90
100 110 120 130 140 150 160 170 Mass number
Fig. 5. Calculated primary fragment yield and neutron multiplicity as functions of fragment mass for spontaneous 246Fm fission.
an energy of around 10 MeV. This effect was observed in [33] for somewhat lower energies. The neutron spectrum before scission is the sum of the evaporation spectra for the second and third fission chances, and at the descent stage from saddle to the scission point, and the multiplicity of these neutrons strongly exceeds that of the preequilibrium component. The prescission neutron emission results in decreasing the total excita tion energy of the compound nucleus at the scission point being much lower than the excitation energy in the input channel, so the multiplicity of postscission neutrons is close to that of thermal neutron fission. CONCLUSIONS In conclusion, a consistent model for calculating PFN characteristics for spontaneous fission, fission by thermal neutrons, and nucleoninduced fission over a wide range of energies was developed. The fission pro cess up to the scission point was simulated using the Monte Carlo method including the preequilibrium stage the preequilibrium stage at incident particle energy above 10 MeV. Dynamic corrections due to
nuclear friction were considered in calculating the fis sion width. The effect of nuclear friction results in the possibility of neutron emission at the descent from the saddle point even for spontaneous fission. For exam ple, in the spontaneous fission of the heavy Fm iso tope, this component can be remarkale. Postfission neutrons emitted by accelerated fragments are calcu lated by averaging over the primary mass and charge distributions, and the primary fragment kinetic energy distributions. The excitation energy and kinetic energy of primary fragments are determined using a model of nascent fragments with inclusion of the shell and pair ing corrections. The mass distribution of primary frag ments for certain compound nuclei at the scission point are determined in a multimodal approximation using five fission modes. The PFN multiplicity spectra were calculated for the spontaneous fissioning of 252Cf and Fm isotopes (with masses А = 244–257), and for 235U fission by thermal and 14.7 MeV neutrons. The main character istic feature of this model, as compared to earlier approximations [4–12], is consistent calculations of the distributions necessary for PFN calculations. This model can be used for calculation and evaluation of the neutron data in the fission of heavy nuclei involved in the nuclear fuel cycle. The model is planned to extend to the mass region А ≥ 258 and superheavy nuclei where the bimodal fis sion emerges and the fission mode parameters may be changed remarkably. REFERENCES 1. Browne, J.C. and Dietrich, F.S., Phys. Rev. C, 1974, vol. 10, p. 2545. 2. Madland, D.G. and Nix, J.R., Nucl. Sci. Eng., 1982, vol. 81, p. 213.
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