1. Introduction

The average number of prompt neutrons is one of two (along with the fission cross section) major nuclear-physical parameters required for practical calculations. This value is determined in detail and accurately for neutron-induced reactions for the most nuclides. At the same time, the experimental data and evaluation of the average number of prompt neutrons in the case of photofission are much scarcer, old-fashioned or obtained by indirect mode [1, 2]. With the increasing interest in the methods of nuclear fuel burning and long-lived actinide decontamination the need in precise values of nuclear constants is evident during last years. This is especially true for photonuclear constants. Therefore, the search was focused on a formula that can be used to estimate the value of for photofission of arbitrary actinide.

2. General Approach

Let’s consider a general approach to modeling the average number of the prompt neutrons in the case of neutron activated fission. This value is a function of three variables: the mass of the compound nucleus A, its charge Z and the incident neutron energy

In previous studies [3-5] it has been shown that expanding the charge, mass and energy dependence of in the form of truncated Taylor series apparently yields a reasonable representation for if the zero-, first- and one second-order cross term are kept in truncation for all isotopes, i. e. it is sufficient to use a linear approximation over all three variables, at least up to the threshold (γ,2nf), taking into account the contribution of the even-odd effect in a common form:

(1)

where the intercept

(2)

and slope

(3)

(4)

Z₀, A₀ and E₀ are the values, about which the expansion is to be made [3].

The coefficients C_i in equations (1) – (3) were evaluated by the least-square method [5]. A wide range of existing data for (A,Z,E_n) for actinides from ²³²Th to ²⁴⁵Cm was analyzed and the best simplified parameterization of the slope and intercept was recently found in [6]:

(5)

(6)

(7)

These calculations were performed in the case of neutron-induced actinide fission without taking into account other reactions.

Generally speaking, all mentioned approaches have to be versatile and suitable for all other cases of fission, including photofission. Therefore, we applied the model (1-4) in the case of photofission, where (E_n – E₀) is replaced by (Е_γ – E_s), Е_γ – photon energy, E_s – nucleon separation energy.

To do this, we take into account the wide experimental data set of photofission (A,Z,Eγ) in the energy range ~ 8 – 20 MeV for actinides ²³²Th [7, 8], and ~ 5 – 20 MeV for ²³³U [9, 10], ²³⁴U [9, 10], ²³⁵U [7, 9], ²³⁶U [7, 9], ²³⁸U [7, 9], ²³⁷Np [9, 10] and ²³⁹Pu [9, 10]. We used all these 219 experimental points for our calculation.

The values of the nucleon separation energy for each actinide were taken from [11, 12] and are presented in Table 1.

To evaluate the quality of linear approximation (1) – (4), modified for photofission, we calculate for each actinide separately. The best possible energy approximation for have to be lie within the linear approach [7, 10]:

(8)

Table 1. Nucleon separation energy

i, j – actinide number, n=i - j, N=8, θ(n) - step function

for

where and are the lower and upper bound of photon energy interval for respective i-th actinide, Eγ–E_s>0.

The common quality of the description of the data set for each actinide by (8) is evaluated by [13]:

(9)

where each i-th interval out of total of N intervals contains n data points and two parameters give a resulting combination to in fitting eq. (8) with result (dot line in Fig. 1, 2). Consequently it is naturally that we have to build the most general formula describing the average prompt neutrons for photofission depending on the photon energy of all actinides, which is close to the best results of local linear calculation. As the initial formula, we chose that similar to (1):

(10)

where the slope and the intercept а(A,Z) are chosen as (2) – (3).

The results are given in Fig. 1, 2 (solid line). Coefficients C_i were calculated by least-square method [14]. The final formula for calculating the average number of prompt neutrons for photofission of actinides is:

(11)

(12)

Figure 1. The results of fitting experimental data for average neutron multiplicities using (10)-(12) (solid lines) for 232Th ‒ a), 233U ‒ b), 234U ‒ c), 235U ‒ d), 236U ‒ e): open circles (○) from [7], down triangles (s) from [8], up triangles (Δ) from [9]. Squares (□) – experimental data from [10]; dash lines – ENDF/B-VII.1 [14]. Dot line – formula (8)

Figure 2. The results of fitting experimental data for average neutron multiplicities using (10)-(12) (solid lines) for 237Np ‒ a) and 239Pu ‒ b): open circles (○) from [7], down triangles (s) from [8], up triangles (Δ) from [9]. Squares (□) – experimental data from [10]; dash lines – ENDF/B-VII.1 [14]. Dot line – formula (8)

The quality of the description of the experimental data in this case: χ²/dof = 2.5.

This value is in accord with the best linear approximation for each actinide.

We also checked the quality of the recommended values from ENDF/B-VII-1 [15] (dash lines in Fig. 1, 2.) for the same data set. All the approaches are in accord with each other at least up to the threshold (γ,2nf).

Also using formulas (10) – (12) we have calculated the photon energy dependence of in the case of the ²⁴⁰Pu and ²⁴¹Am [16] isotopes (see Fig. 3).

Figure 3. The results of the prediction of average neutron multiplicities for 240Pu – a) and 241Am –b) (solid lines), dash lines – ENDF/B-VII.1 [14], circles (●) – experimental data from [15]

3. Conclusions

The empirical formula for estimation of the average number of prompt neutrons as a linear function of the (Eγ – E_s(A,Z)) of actinides photofission was obtained. The formula contains 7 free parameters taking into account the even-odd effect. The quality of description of by formulae (10) – (12) ( χ²/dof = 2.5) is comparable (see Fig.1.) with that of a local description for each actinide separately (χ²/dof = 2.1). All this gives us enough reason to use the resulting formula to calculate the average number of prompt neutrons for any actinide photofission.