Monte Carlo simulation of prompt neutron emission during acceleration in fission

Abstract

The possibility of prompt neutron emission during acceleration of fission fragments (FFs) was examined by means of Monte Carlo method and statistical neutron emission model. Multimodal random neck-rapture model was used to describe the initial distribution of mass, charge, and total kinetic energy of the primary fragments. Statistical model was used to simulate the de-excitation process of the fragments from the moment of scission until full acceleration. By random number sampling, the fission process was simulated in order to obtain the basic physical quantities, and their correlations were analyzed to verify the adequacy of the model. It was found that, on the average, ∼10% and ∼16% of prompt neutrons for ²³⁵U(n _th,f) and ²⁵²Cf(sf), respectively, were emitted before reaching 90% of the final fragment kinetic energy.

Keywords:

• neutron emission during acceleration
• prompt neutron spectrum
• multimodal random-neck rapture model
• statistical model
• Monte Carlo simulation

1. Introduction

It has been considered that most of the prompt fission neutrons are emitted after full acceleration of fission fragments (FFs) due to strong Coulomb repulsion working between them. On the other hand, however, there has also been a discussion on a possibility of neutron emission during acceleration (NEDA). This phenomenon is interesting from physics point of view, as it provides knowledge on the timescale of de-excitation of excited nuclei and on possible competition with Coulomb acceleration of FFs. This phenomenon is also interesting from application point of view, since neutron emission from FFs before full acceleration implies an enhancement of low-energy component of the prompt fission neutron spectrum (PFNS), because the NEDA neutron receives smaller linear momentum from the FF.

One of the present authors [1] examined the effect of NEDA upon the PFNS on the basis of deterministic model of Eismont [2], treating the fraction and timing of NEDA as parameters. In the present work, we examine the possibility of NEDA using the Monte Carlo (MC) method and statistical model for neutron emission from FFs and their effect on the PFNS. We believe that MC method is adequate to examine the problem, because fission process, including division of mass, charge, kinetic energy, and neutron emission from FFs, is a stochastic process in its nature. However, it should be emphasized here that the aim of the present work is not to produce the PFNS by means of the MC method but to examine the effect of NEDA on the PFNS. The PFNS was calculated here with the theory of Madland and Nix [3] with consideration of the multimodal fission [4], because, at present, the full MC calculation of the PFNS still involves some problems that remains to be examined.

Section 2 describes the model and assumptions of the present work, and Section 3 discusses the results and comparison with experiments. Section 4 summarizes the conclusion obtained in this work.

2. Methods

2.1. Multimodal model of fission

In performing MC simulation of neutron emission from FFs, it is necessary to define the primary fragment distribution as a function of mass, charge, and total kinetic energy (TKE). In the present study, the multimodal random neck-rapture (MM-RNR) model [4] was used as a framework of representation of the primary distribution, since it provides best account of the two-dimensional distributions of mass and TKE of the FFs for many actinides [5,6]. In this model, the fission process is considered to proceed along several definite deformation paths, symbolically named Standard-1 (S1), Standard-2 (S2), Standard-X (SX), and Superlong (SL). In terms of scission point model of Wilkins et al. [7], the path S1 is considered to pass through, or close to, the dip on the deformation energy surface caused by the double magic number of 50 protons and 82 neutrons. The mode S2 corresponds to a path proceeding through a weaker dip of deformed shell of 86 neutrons and SL a mass-symmetric deformation path. The mode SX was introduced to explain wider mass distributions of FFs observed in trans-neptunium nuclides. The branching ratios for each mode depend on nuclides and the excitation energy of the fissioning nuclei. The multimodal parameters were determined from experimental data [5,6,8–10] or taken from systematics [11,12] when adequate measured data were not available. The mass, charge, and TKE distributions of the primary FFs for each fission mode were represented with the Gaussian functions:

where indices L and H stand for light and heavy fragments, respectively, for each mode. The most probable charge ⟨Z _L,H⟩ for fragment with mass A _L,H was determined by the corrected unchanged charge distribution (UCD) assumption, which states ⟨Z _L,H⟩ = A _L,H(Z _C/A _C) ± 0.5, where index C stands for the compound (fissioning) nucleus and + and − signs refers to light and heavy fragments, respectively. For symmetric fission, the correction term ±0.5 was omitted. The charge distribution width σ_Z was assumed to be 0.56 [13] for all cases. The results in this work were found not to be sensitive to this quantity.

The total excitation energy (TXE) of the FFs was calculated by the energy conservation relation as follows:

where the total energy release of fission E _R(A _L, A _H, Z _L , Z _H) for each mode was calculated using mass formulas [14,15]. The average quantities ⟨TXE⟩ and ⟨TKE⟩ were calculated from an equation similar to Equation (4) averaged over distribution. The charge center distance l of the two FFs for each mode was obtained from the relation as follows:

which is also a stochastic quantity. The initial distributions of the FFs for each mode were determined using the MC sampling for mass, charge numbers, and TKE according to Equations (1)–(3).

2.2. Excitation energy partition between the fragments

Partition of the TXE between the two fragments at scission point is an important but not yet fully solved problem. In this work, three assumptions were used regarding the energy partition, in accordance with Talou [16,17]:

Hypothesis 0 (H0): Equipartition of energy to particle degrees of freedom is assumed, neglecting collective degrees of freedom at the scission point:

Hypothesis 1 (H1): Thermal equilibrium (equal temperature) of the two fragments is assumed, which presumes that scission process is slow enough compared with particle motion within the nucleus. This assumption implies that the TXE is shared according to the level density parameters (LDPs) for the two fragments, i.e.:

where the LDPs a were obtained by solving the transcendent equation of Ignatyuk [18]. Iteration method was used to solve the transcendent Equation (7).

Hypothesis 2 (H2): The average prompt neutron multiplicity ⟨ν(A)⟩ is considered to be approximately proportional to the fragment excitation energy and shows a saw-toothed structure as a function of FF mass A. This in turn suggests that the excitation energy is shared in proportion to ν(A):

where ⟨ν_i (A _L)⟩ and ⟨ν_i (A _H)⟩ stand for averages over LF and HF for mode i, respectively.

The neutron multiplicity ⟨ν_i (A_i )⟩ as a function of mass A for mode i was calculated by dividing E*(A) obtained by the following formulas by the neutron separation energy and averaging over LF and HF regions. The deformation energy of the spheroidally deformed fragment just after scission is given [4] as a function of eccentricity ϵ and fissility parameter x:

Here the prescission shape parameters were taken from Fan et al. [12] and Büyükmumcu and Kildir [19]. The excitation energy of a fragment of mass A for each mode is expressed as the sum of deformation and internal excitation energies as E* = E _def +(E _s */A _C)A, where E _s * stands for the internal excitation energy of the prescission nucleus [4]. Equation (9) is based on the liquid model, and thus naturally has a limited validity; however, it would be worthwhile to attempt to estimate ⟨ν_i (A)⟩ with Equation (9) and compare the results with empirical data.

2.3. Statistical model of neutron emission

The neutron emission from hot nuclei was treated with the statistical neutron decay model of the nucleus. When considering sequential emission of neutrons from FF, the dependence of average lifetime of neutron emission on fragment and the cascade neutron emission were considered in the analysis. The average lifetime τ_k for the k-th emitted neutron was calculated with the Ericson formula [20] as follows:

where m _n is the neutron mass, and ρ_C(E_k *) and ρ_R(E_k  − S_{n ,k}  − ϵ) are the level densities of the compound and residual nucleus, respectively. The inverse reaction cross-section σ_C(ϵ) was calculated using the spherical optical model potential of Becchetti and Greenlees [21] and computer code ELIESE-3 [22] for each FF. From Equation (10), we see that the average lifetime varies according to the excitation energy, neutron separation energy, and level density, which are significantly influenced by the shell and pairing effects.

The energy distribution of evaporated neutrons was calculated with the Weisskopf-Ewing distribution:

where T(A – 1,Z, ⟨TKE⟩) is the nuclear temperature for the residual nucleus after neutron emission and S _n the neutron binding energy. The LDP a(A – 1,Z, ⟨TKE⟩) was obtained from Ignatyuk's model [18].

The neutron energy ϵ in the center-of-mass (CM) system of the fragment was determined by using random number sampling. The excitation energy of the residual nucleus after emission of the k-th neutron is given by E_k * = E _k−1*−S_n,k (A_k,Z)−ϵ _k . The neutron emission is assumed to continue as long as E _k *> S_n,k (A_k,Z) + Δ _k , where Δ _k is the pairing energy; the rest of the excitation energy was assumed to be emitted as gamma rays. The neutron energy in the CM frame (ϵ) is transformed to the laboratory frame (E) by

where E _f ^L(H) is the kinetic energy of the initial light (heavy) fragment, given by

The angle θ was determined by random number sampling assuming isotropic emission in the CM system.

2.4. Multimodal Madland–Nix model [22,23]

The PFNS was calculated with the Madland–Nix model [3]. Here, the multimodal model in fission was considered [23,24]. The maximum temperature of the triangular distribution is calculated with

where

W(A_i ,Z_i ) being the contribution of mode i for FF of (A_i ,Z_i ). The average number of neutrons emitted per fission for each mode was given by

where

The relation in Equation (17) was originally proposed by Fréhaut [25]; the coefficients p and q readjusted by Vladuca et al. [26] were used here:

The average quantities ⟨E _R⟩ and ⟨S _n⟩ were calculated with formulas similar to Equations (15b) and (15c). Attention should be paid to the fact that the average neutron separation energy was calculated with , in order to smooth out even–odd effects in the emission cascade. The modal PFNS was given by weighted average of spectra from LF and HF:

The total PFNS was calculated by averaging the modal PFNS:

where w_i is the mode branching ratio and ν_i the average neutron multiplicity for mode i.

2.5. Angular anisotropy of neutron emission in the fragment CM system

It is considered from experimental evidences that FFs have high angular momentum of ∼7 ħ perpendicular to the fission axis, due to strong Coulomb torque at the scission point. This leads to anisotropic neutron emission in the fragment CM system. The angle-dependent neutron spectrum in the CM system is expressed as follows [27]:

where the anisotropy parameter b is defined as b = W(θ)/W(90°) – 1, W(θ) being the neutron emission probability at the CM angle θ. Then the laboratory system spectrum of neutrons is calculated, in the framework of Madland–Nix model, by [28]

One of the present authors [29] has shown that consideration of angular anisotropy has the effect of enhancing the low-energy (<0.6 MeV) and high-energy (>4 MeV) components of the neutron spectrum.

2.6. NEDA effect

It has been known [2] from point-charge model that the relation

holds between the acceleration time t _acc after scission and the relative acceleration χ = KE/KE_final of fragments at the time t _acc, where v _final is the final velocity, l the charge-center distance at scission, given by l = Z _L Z _H e ²/TKE with consideration of fluctuation in TKE. It is to be noted here that the quantity l here is a stochastic quantity randomly sampled in accordance with Equations (3) and (5). The probability of emission of the first neutron at time t after scission is expressed by P(t) = 1 − exp(−t/τ). Using Equation (23), we can describe the distribution P(χ) as a function of relative acceleration χ:

The PFNS with consideration of NEDA effect was obtained by integrating the total PFNS over distribution of P(χ):

3. Results

3.1. Basic input data

Data of mass and TKE distributions for FFs were taken from measurements [5,6,8–10] for ²³⁵U(n _th,f), ²³⁹Pu(n _th,f), and ²⁵²Cf(sf). For ²³³U(n _th,f), systematics [11,12] were used to estimate the distributions for lack of adequate measured data. Mass excess data [14] and mass formula [15] were used to evaluate the total energy release and neutron binding energies; this choice was made to make the condition uniform and make comparison easier with the work of Lemaire et al. [30,31]. The LDPs were obtained from the Ignatyuk formula [18] together with parameters of Kawano et al. [32] and Koura et al. [15]. The optical potential parameters of Becchetti and Greenlees [21] were adopted to calculate the inverse reaction cross-sections to be used to calculate the average lifetime Equation (10) of neutron emission and Weisskopf evaporation spectra, Equation (11). The average neutron multiplicity ⟨ν(A _L)⟩ used in H2 were calculated according to the method described below Equation (8) with prescission shape parameters of Fan et al. [12] and Büyükmumcu and Kildir [19].

3.2. Results and discussion

3.2.1. Neutron multiplicity

The neutron multiplicity distribution for ²³⁵U(n _th,f) is shown in Figure 1. We can see that present calculations under different hypotheses H0, H1, and H2 give results approximately consistent with experimental data, also in view of the average neutron multiplicity shown in Table 1. This implies that the simulated TXE and its distribution are fairly consistent with reality.

Figure 1. Frequency distribution of total number of prompt neutrons for ²³³U(n _th,f), ²³⁵U(n _th,f), ²³⁹Pu(n _th,f), and ²⁵²Cf(sf). The measured data (points) were taken from Holden and Zucker [40].

Table 1. Comparison of average number of prompt neutrons.

Reaction		⟨v ^L⟩	⟨v ^H⟩	⟨v ^tot⟩
^233U(n th, f)
Present	H0	0.901	1.523	2.424
	H1	1.126	1.320	2.446
	H2	1.080	1.361	2.441
Exp	Holden and Zucker [40]			2.484 ± 0.0080
	Boldeman and Hines [54]			2.48
	Nefedov et al. [56]			2.578
^235U(n th, f)
Present	H0	0.888	1.548	2.436
	H1	1.116	1.337	2.453
	H2	1.093	1.359	2.452
Other MCS	Lemaire et al. [30] – H1*	1.15	1.58	2.73
	Lemaire et al. [30] – H2*	1.58	1.09	2.67
	Talou [17] − R T  = 1.0	1.11	1.36	2.47
	Talou [17] − R T = 1.2	1.40	1.07	2.47
Exp	Nishio et al. [42]	1.42	1.01	2.47
	Müller et al. [55]*	1.44	1.02	2.46
	Holden and Zucker [40]			2.43 ± 0.0070
	Boldeman and Hines [54]			2.4056
	Nefedov et al. [56]			2.383
^239Pu(n th, f)
Present	H0	1.083	1.725	2.808
	H1	1.329	1.489	2.818
	H2	1.289	1.526	2.814
Other MCS	Talou [17] – R T  = 1.0	1.42	1.47	2.89
	Talou [17] – R T = 1.2	1.72	1.17	2.89
Exp	Holden and Zucker [40]			2.881 ± 0.0090
	Boldeman and Hines [54]			2.8794
	Nefedov et al. [56]			2.89
^252Cf(sf)
Present	H0	1.582	2.446	4.028
	H1	1.755	2.272	4.026
	H2	1.833	2.156	3.989
Other MCS	Lemaire et al. [30] – H1	1.74	2.44	4.18
	Lemaire et al. [30] – H2	2.18	1.91	4.09
	Büyükmumcu and Kildir [19]			3.87
	Talou [17] – R T = 1.0	1.67	2.11	3.78
	Talou [17] – R T = 1.2	2.06	1.73	3.79
	Litaize and Serot [39]	2.06	1.70	3.76
Exp	Vorobiev et al. [45]	2.051	1.698	3.756 ± 0.031
	Holden and Zucker [40]			3.766 ± 0.01
	Boldeman and Hines [54]			3.757
Note: *E n = 0.5 MeV.

The average neutron multiplicity as a function of fragment mass A calculated with different hypothesis is shown in Figure 2. Although Equation (9) is a formula based on the classical liquid drop model, the saw-tooth structure is approximately reproduced by hypothesis H2. We thus adopted H2 as a working hypothesis, and the analyses below were done with H2.

Figure 2. The average number of prompt neutrons emitted from FF of mass A. The measured data were taken from Nishio et al. [41], Tsuchiya et al. [43], Budtz-Jørgensen and Knitter [44], Vorobiev et al. [45], Batenkov et al. [46].

3.2.2. Prompt neutron multiplicity vs. TKE

Correlation between total prompt neutron multiplicity ⟨ν⟩ and TKE is shown in Figure 3. The present results are in good agreement with MC calculation of Lemaire et al. [30,31] and in accordance with their result that the calculations do not depend on hypotheses H0–H2. However, it is apparent that the calculations deviate from experiments by overestimating ⟨ν⟩ for low ⟨TKE⟩, and the calculated slopes are rather steeper for ²³⁵U(n _th,f). The reason for this discrepancy should be examined both from theoretical and experimental points of view.

Figure 3. Correlation between ⟨ν(A)⟩ and TKE. The measured data were taken from Nishio et al. [42], Tsuchiya et al. [43], and Budtz-Jørgensen and Knitter [44].

3.2.3. Average neutron multiplicity

The average number of total prompt neutrons ⟨ν _tot ⟩ calculated under different hypotheses are compared with experimental data in Table 1. It can be seen that (i) ⟨ν _tot ⟩ is not sensitive to the choice of hypothesis H0–H2 and (ii) the agreement with experimental data and previous MC calculations is fairly good. Considering that ⟨ν _tot ⟩ is determined by TXE, we conclude that partition of the total energy release between TXE and TKE in the present calculation is essentially sound. On another note, comparing the experimental and calculated values for ⟨ν _L ⟩ and ⟨ν _H ⟩ for ²³⁵U(n _th,f) and ²⁵²Cf(sf) in Table 1, we observe a discrepancy. The reason might be due to the liquid-drop approximation Equation (9) with prescission shape parameters [12,19], while the actual energy partition is governed by quantum effects such as shell and superfluid effects on the nucleon exchange around the scission point. This implies that further examination should be made on the energy partition between the fragments.

3.2.4. Neutron emission lifetime and NEDA probability

The neutron emission lifetime calculated with Equation (10) for FF from ²³⁵U(n _th,f) is shown as a function of fragment mass in Figure 4. It can readily be observed that (i) the lifetime varies over six orders of magnitude from fragment to fragment, (ii) it fluctuates greatly between neighboring FF due to even–odd effect, and (iii) the gross structure is determined by shell effect on the LDP and S _n and do not depend strongly on the hypotheses H0–H2. It is to be noted that the lifetime shown in Figure 4 represents the weighted averages for four fragments close to the most probable charge Z _p with respective weight of occurrence. The very long lifetime at masses around 75 is due to the fact that the level density in the denominator of Equation (10) is remarkably small owing to low calculated excitation energy of FFs around mass 75 under H2, as can be seen from Figure 2. However, this does not affect the following results, because the fission yield in this region is very small.

Figure 4. The neutron emission lifetime for FF as a function of fragment mass for ²³⁵U(n _th,f).

The NEDA probability Equation (24) calculated with MC method for ²³⁵U(n _th,f) is plotted in Figure 5. We confirm that, although neutron emission probability is high in the final stage of acceleration (χ ≈ 1), a certain fraction of neutrons are certainly emitted before full acceleration. The apparently higher probability near χ = 0 is due to the fact that it takes time to accelerate the FFs, which are assumed to be at rest just after scission.

Figure 5. NEDA probability, averaged over all fission modes, as a function of relative acceleration χ for ²³⁵U(n _th,f).

Dependence on other assumptions was examined, and our findings were as follows:

Figure 6. NEDA probability as a function of relative acceleration χ for different fission modes calculated with energy-dependent [21] and constant inverse reaction cross-sections for ²³⁵U(n _th,f).

1.	Fission modes: NEDA probability as a function of relative acceleration χ for different fission modes calculated with energy-dependent and constant inverse reaction cross-sections for 235U(n th,f) in Figure 6. We found that NEDA probability strongly depends on fission modes. This is natural, since the average lifetime ⟨τ⟩ given by Equation (10) is dependent on the excitation energy, neutron binding energy, and the LDP, which are different for FFs produced in different modes, as can be seen in Figure 4.
2.	Excitation energy partition: Comparison was made for different hypotheses H0–H2, but the relative acceleration χ was not sensitive to these choices.
3.	Fissioning nuclei: Calculations were carried out for 233U(n th,f), 233U(n th,f), 239Pu(n th,f), and 252Cf(sf). It was found that the tendency is more or less similar to each other; only for 252Cf(sf), higher values of P(χ) were obtained, which were explained by higher TXE for this case, compared with other nuclides, because of higher total fission energy release E R for 252Cf(sf).
4.	Multiple neutron emission: A possibility of multiple NEDA was also examined. It was found that emission of a second and third neutron, if possible, occurred at later stage of acceleration, thus contributing little to the total NEDA fraction.
5.	LDP: Comparison was made for different LDPs; Gilbert and Cameron [33] and those used in Myers and Swiatecki [34] and Möller et al. [35], taken from RIPL-2 [36]. It was found that the absolute value of P(χ) was rather sensitive to the choice of LDP, although the functional shape remained similar.
6.	Inverse reaction cross-section: As was shown in Figure 6, consideration of energy dependence of σC(ϵ) instead of constant cross-section assumption significantly shortens the emission time due to higher cross-section at lower energies, thus enhances the NEDA probability.

The integral values of P(χ) from χ = 0 up toχ = 0.9, 0.95, and 0.99 for LF and HF for possible fission modes are compared in Table 2. It is seen that the probability of NEDA integrated up to 90% of E _final is 2.2–8.6% for S1, 10.0–11.9% for S2, and 18.1% for SL for ²³⁵U(n _th,f) under the hypothesis H2. This shows that the higher the excitation energy is, the greater is the NEDA probability, as anticipated. This tendency is confirmed by comparing the P(χ) values for LF and HF and also by comparing the ²³⁵U(n _th,f) and ²⁵²Cf(sf) cases, the fission Q values for the last being greater than the former.

Table 2. Fraction of prompt neutrons emitted before full acceleration of fission fragments for each fission mode in ²³⁵U(n _th, f) and ²⁵²Cf(sf). Monte Carlo calculations were made under hypothesis 2 (H2).

	S1 HF	S1 LF	S2 HF	S2 LF	SL
^235U(n th,f)
∼90%	8.6%	2.2%	10.0%	11.9%	18.1%
∼95%	14.9%	3.9%	16.7%	19.1%	27.5%
∼99%	42.6%	13.4%	42.9%	43.6%	53.2%
	S1 HF	S1 LF	S2 HF	S2 LF	SX HF	SX LF	SL
^252Cf(sf)
∼90%	16.0%	15.3%	11.3%	15.2%	15.7%	15.4%	18.0%
∼95%	25.9%	23.3%	18.5%	23.0%	24.7%	23.7%	27.9%
∼99%	59.1%	45.9%	44.8%	46.3%	52.2%	48.9%	55.8%

3.3. Prompt fission neutron spectra

Results of calculation of PFNS with and without consideration of NEDA effects are compared in Figure 7. Results are also shown for anisotropic emission of neutrons. Note that the spectra are represented as the ratio to the Maxwellian distribution with temperature parameters T _M = 1.324 MeV, 1.324 MeV, 1.38 MeV, and 1.42 MeV for ²³³U(n _th,f), ²³⁵U(n _th,f), ²³⁹Pu(n _th,f), and ²⁵²Cf(sf), respectively.

Figure 7. Prompt fission neutron spectra calculated with and without NEDA effect. Calculations with consideration of angular anisotropy of neutron emission are also shown. The measured data were taken from Lajtai et al. [47], Starostov et al. [48], Boycov et al. [49], Lajtai et al. and D'yachenko et al. [50,51], and Mannhart [52,53]. The spectra are represented as the ratio to the Maxwellian distribution with temperature parameters T _M  = 1.324 MeV, 1.324 MeV, 1.38 MeV, and 1.42 MeV for ²³³U(n _th,f), ²³⁵U(n _th,f), ²³⁹Pu(n _th,f), and ²⁵²Cf(sf), respectively.

Consideration of NEDA effect was found to enhance the low-energy (<1 MeV) part and reduce the high-energy (>3 MeV) part of the spectrum, because of less boosting to NEDA neutrons. Consideration of CM angular anisotropy (with b = 0.05 [36] and b = 0.1 [37]) of neutron emission has the effect of enhancing both the low-energy (<0.6 MeV) and high-energy (>4 MeV) parts of the spectrum. One of our interests was to examine what would be the result if both effects are considered simultaneously. The present result shows that the agreement between calculation and experiments is improved at the lowest energies. On the other hand, the agreement seems to be worsened a bit at energies over 3 MeV, except for the case of ²⁵²Cf(sf). One of the reasons would be that, since the overall integral of the spectrum is normalized to unity, an increase at lower energies leads to a decrease at higher energies. A remedy for this discrepancy remains to be studied.

4. Conclusion

By applying the MC method to simulate the NEDA phenomena in the fission process, we obtained the results that (a) the NEDA probability before acceleration up to 90% of final TKE of the fragments varies according to the fission mode, due to different excitation energy pertinent to the fission mode, and (b) the NEDA probability ranges from 2.2% at minimum (LF in the S1 mode) to 18.1% at maximum (SL mode), the average being around 10% for²³⁵U(n _th,f) under hypothesis H2. The NEDA probability for ²⁵²Cf(sf) is higher, ranging from 15.3 (LF in the S1-mode) to 18.0% (SL-mode), the average being 16% under hypothesis H2. This is due to the fact that the average emission time τ, as given by Ericson [20], depends strongly to the excitation energy of the FF.

The NEDA phenomenon has an effect of enhancing the low-energy part and reducing the high-energy part of the PFNS. Thus, taking into account the NEDA effect, together with accounting for CM anisotropy of neutron emission in the multimodal Madland–Nix model significantly enhances the low-energy part and improves the agreement with experimental data in the region less than 0.6 MeV. The apparent discrepancy observed at energies over 3 MeV for the cases except for ²⁵²Cf(sf) remains to be remedied in the future.