Product yields for the Photofission of 232Th, 234,238U, 237Np and 239,240,242Pu actinides at various incident photon energies

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

Product yields for the Photofission of ²³²Th, ^234,238U, ²³⁷Np and ^239,240,242Pu actinides at various incident photon energies

M. R. Pahlavani，

P. Mehdipour

Nuclear Science and Techniques

Vol.29, No.10

Article number 146

Published in print 01 Oct 2018

Available online 31 Aug 2018

DOI：10.1007/s41365-018-0482-1

50200

Photofission fragments mass yield for ²³²Th, ^234,238U, ²³⁷Np, and ^239,240,242Pu isotopes are investigated. The calculations are done using a developed approach based on Gorodisskiy’s phenomenological formalism. The Gorodisskiy’s method is developed to be applied for neutron induced fission. Here we revised it for application to photofission. The effect of emitted neutron prior to fission on fission fragment mass yields has also been studied. The peak to valley ratio is extracted for the ²⁴⁰Pu isotope as a function of energy. Obtained results of present formalism are compared with the available experimental data. Satisfactory agreement is achieved between results of present approach and experimental data.

Fission fragmentsFragment mass yieldPhotofissionNeutron fissionHeavy nucleiPeak to vally

1 Introduction

In spite of peaceful application in energy production, different features of various kinds of fission are not known yet. Photofission is one way to induce fission on actinides (U, Pu, Np, etc.). Although photofission was discovered more than fifty years ago [1], its theoretical modeling was produced with many challenges. Various theoretical approaches [2-7] have been presented to evaluate fragment mass yields for neutron/proton induced fission, but there is no comprehensive method to study photofission. The distribution of fragment mass yield plays a significant role in examining the validity of every theoretical method of fission. If the calculated fragment mass yields are well fitted with measured data, this means that the selected model is suitable to be applied for the fission process. In the statistical model of Wilkins [8, 9] and its developed version by Moreau [10], the probability of fission for neutron-induced fission is presented as follows:

P \sim e^{(- E_{total} / T)} = e^{(- (E_{mac} + E_{mic}) / T)},

(1)

where P is the probability of fission and E_total includes all kinds of energies presented in the evaluation of compound nucleus from saddle to scission point. E_mac and E_mic are the macroscopic and microscopic energies, respectively. The relation between fission fragments mass yield and fragment mass number indicates that the probability appears in the Gaussian form. Therefore, the theoretical neutron induced fission models are developed in the form of a multi-Gaussian shape and are formulated as a sum of several Gaussian type function [11- 13].

The old formalism presented by Wang Fu-cheng [5] was applicable for reproducing low-energy fission and contains five Gaussian terms. The formalism of Robert Mills[14], in which the dependence on energy was neglected, includes four Gaussian terms in the quadratic structure. The GEF computer code [15] is developed based on this method with various assumptions and many adjustable parameters. The complicated Langevin-Brownian method has been employed to study neutron induced fission and lunched some success in recent years [16-20]. The Gorodisskiy phenomenological method [21] has been recently successful in reproducing the experimental data for neutron induced fission of actinides. It is clear that the photofission process is similar to neutron induced fission except in multi-polarity absorptions and time scale of the fission process. These absorptions happen in the beginning of the process, therefore, after formation of the compound nucleus, the fission process of the neutron induced fission will become similar to photofission. According to the Bohr’s hypothesis, the compound nucleus rapidly loses its total formation memory except conserved degrees of freedom. Therefore the mode of decay of the compound nucleus does not depend on the way the compound nucleus is formed. Thus for the same excited configuration of the compound nucleus, the photofission fragment mass yield is expected to be approximately similar to the neutron induced fission. In the present study, we examined an extended version of the Gorodisskiy’s method to generate the photofission fragment mass yield. We modified the relative contributions of the symmetric and the asymmetric yields, (Y_s and Y_a) of the Gorodisskiy’s method and the variance of asymmetric fission (σ_a) as presented in the next section. The paper is constructed as follows: Theoretical model used to calculate fission fragments mass distribution is presented in Sect. 2. In Sect. 3, theoretical results are compared with experimental data for different excitation energies and different numbers of emitted neutron prior to fission for ²³²Th, ^234,238U, ²³⁷Np, and ^239,240,242Pu actinides. Finally, a short conclusion is presented in Sect. 4.

2 Description of theoretical Model

Studies of fission fragments mass distribution of most fissioning nuclei indicate that the yield of fission fragments as a function of fragment mass number (A) is a Gaussian function with 4 [23] or 3 [22] modes. These modes are symmetric (SL) or asymmetric (S1, S2, and S3) related to the type of the compound nucleus and its excitation energy.

In Gorodisskiy original approach that has developed for neutron induced fission, two main parts are considered to generate the mass distribution of fission fragments. The yield of fragment with mass number higher than the half of compound nucleus is indicated by Y_H and its conjugate with the mass number lower than the half of compound nucleus is shown by Y_L. These modes are presented as an exponential function with a small deviation from Gaussian shape. Therefore, it is convenient to use the Charlier’s distribution [21, 24, 25] to construct the fission fragment mass yield

f (u) = 1 - γ_{1} (1 / 2 u - 1 / 6 u^{3}) + γ_{2} (1 / 24 u^{4} - 1 / 4 u^{2} + 1 / 8),

(2)

where $u = \frac{(M - < M >)}{σ}$ and coefficients γ₁ and γ₂ are respectively called dissymmetry and excess and are given by

\begin{array}{l} γ_{1} = \frac{< (M - < M {>)}^{3} >}{σ^{3}} \\ γ_{2} = \frac{< (M - < M {>)}^{4} >}{σ^{4}} - 3. \end{array}

(3)

When γ₁=γ₂=0, the value of Charlier’s distribution is equal to one. Values of these parameters for different actinides with their given atomic number are presented in Table 1 [21].

Values of γ₁ and γ₂ for various actinides with definite Z_cn.

Z_cn	90	91	92	93	94	95
γ₁	0	-0.08	0.36	0.23	0.3	0.38
γ₂	-0.36	-0.07	-0.34	-0.27	-0.3	-0.34

In the original approach of Gorodisskiy, the yield of heavy fragment is defined by the following relation

Y_{H} = \frac{1}{\sqrt{2 π}} (\frac{Y_{s} e^{- 1 / 2 u_{s}^{2}}}{σ_{s}} + \frac{Y_{a} e^{- 1 / 2 u_{a}^{2}} f (u)}{σ_{a}}),

(4)

where Y_s and Y_a respectively correspond to the relative contributions of the symmetric and asymmetric modes of fission in fragment yield, respectively. Parameters γ₁ and γ₂ are adjusted with the experimental data for each compound nucleus formed after absorbtion of photon [21]. Also, u_a, u_s, σ_a, and σ_s are defined as follows:

σ_{a} = \frac{(A_{cn} - υ_{pre}) (Z_{cn} - 73) (0.074 + 0.0296 \sqrt[4]{E})}{Z_{cn}},

(5)

σ_{s} = \frac{0.031 (A_{cn} - υ_{pre}) \sqrt[4]{E}}{\sqrt{90.54 - 1.9 \frac{Z_{cn}^{2}}{A_{cn} - υ_{pre}}}} + 9.64,

(6)

u_{s} = \frac{A - (A_{cn} - υ_{pre}) / 2}{σ_{s}}

(7)

and

u_{a} = \frac{A - α}{σ_{a}} .

(8)

where $α = 54 \frac{A_{cn} - υ_{pre}}{Z_{cn}}$ for Z_cn=90-91 and $α = 28.6 \frac{A_{c n} - υ_{p r e}}{Z_{cn}} + 0.708 Z_{cn}$ for Z_cn≥ 92. A_cn and Z_cn are respectively the mass and charge numbers of compound nucleus. υ_pre is the average pre-scission neutron multiplicity and E is the gamma-rays energy. Also, A is the mass number of the nascent fragment. In order to calculate the values of Y_a and Y_s in Eq. (3), the ratio $\frac{Y_{a}}{Y_{s}}$ is obtained through the fitting method with experimental data as follows:

\frac{Y_{a}}{Y_{s}} = 1.244 (1 - e^{- 0.0027 {(| E - 5.7 |)}^{3 / 2}}) (\sqrt[4]{E} + 100 (\frac{Z_{cn}}{A_{cn}} - 0.4)),

(9)

and the values of Y_a and Y_s are obtained by replacing it in the above equations

Y_{a} = 200 {(| \frac{Y_{a}}{Y_{s}} | + 2)}^{- 1}

(10)

and

Y_{s} = 200 - 2 Y_{a} .

(11)

In a similar way, the yield of light conjugate fragment(Y_L)is also obtained by replacing M_L with A_cn- A. Finally, the fission fragment mass yield is evaluated using the following equation

Y = Y_{H} + Y_{L} .

(12)

The Eq. (9) is revised by considering two adjustable parameters β and δ, instead of their fixed values in the original formalism of Gorodisskiy as,

\frac{Y_{a}}{Y_{s}} = 1.244 (1 - e^{- 0.0027 {(| E - 5.7 |)}^{δ}}) (\sqrt[β]{E} + 100 (\frac{Z_{cn}}{A_{cn}} - 0.4)) .

(13)

The parameters β and δ are obtained using the fitting method with experimental data. The developed approach is used to calculate the photofission fragment mass yield of various actinides at different energies.

3 Results and discussion

The revised formalism presented in this paper is applied to calculate the fragments mass distribution in photofission reactions of ²³²Th at 16 MeV, ²³⁸U at 17.2 MeV, ²³⁷Np at 24 MeV, and ²⁴²Pu at 20 MeV and compared with results of Gorodisskiy’s formalism as well as experimental data in Figs. 1, 2, 3, 4. As it is clear from Figs. 1, 2, 3, 4, the fission fragment mass yields of the present formalism are in better agreement with experimental data than Gorodisskiy’s original formalism. However there is some inconsistency between theoretical and experimental data especially in most probable fragmentation at maxima of fragment yields.

Figure 1:

(Color online) Photofission fragment mass yield at 16 MeV for ²³²Th: comparison between experimental data [27], Gorodisskiy’s formalism and our formalism.

Figure 2:

(Color online) Photofission fragment mass yield at 17.2 MeV for ²³⁸U: comparison between experimental data [30], Gorodisskiy’s formalism, and our formalism.

Figure 3:

(Color online) Photofission fragment mass yield at 24 MeV for ²³⁷Np: comparison between experimental data [33], Gorodisskiy’s formalism, and our formalism.

Figure 4:

(Color online) Photofission fragment mass yield at 20 MeV for ²⁴²Pu: comparison between experimental data [37], Gorodisskiy’s formalism, and our formalism.

In Figs. 5, 6, 7, and 8 the calculated results of fission fragment yields considering 1, 2, 3 and 4 pre-scission neutron multiplicity prior to fission are compared with experimental data for photofission reactions of ²³²Th at 14 MeV, ²³⁸U at 17.2 MeV, ²³⁷Np at 24 MeV, and ²⁴²Pu at 20 MeV. These figures indicate that the emission of neutrons prior to fission does not considerably affect the photofission fragment mass yield. This is one of the Gorodisskiy formalism drawbacks that is not able to properly indicate the role played by neutron emission in photofission. In order to calculate the fission fragment mass yield as a function of fragment atomic number, (Z), following the semi-empirical formula between fragments mass, (A) and its atomic number, (Z) is employed

Figure 5:

(Color online) Photofission fragments mass yield at 14 MeV for ²³²Th: comparison between experimental data [27] and our formalism considering 1, 2, and 3 neutrons emitted prior to fission.

Figure 6:

(Color online) Photofission fragment mass yield at 17.2 MeV for ²³⁸U: comparison between experimental data [30] and our formalism considering 1, 2, and 3 neutrons emitted prior to fission.

Figure 7:

(Color online) Photofission fragment mass yield at 24 MeV for ²³⁷Np: comparison between experimental data [33] and our formalism considering 1, 2, and 3 neutrons emitted prior to fission.

Figure 8:

(Color online) Photofission fragment mass yield at 20 MeV for ²⁴²Pu: comparison between experimental data [37] and our formalism considering 1, 2 and 3 neutrons emitted prior to fission.

A = \frac{A_{cn}}{Z_{cn}} (Z - 2.5) .

(14)

The fragment mass yield of photofission for ²³⁸U at 11.39 MeV and 13.39 MeV are calculated and indicated as a function of fragments atomic number in Figs. 9(a) and 9(b), respectively. This figure shows that the fragment mass yield of the present formalism for ²³⁸U is better fitted with experimental data [30].

Figure 9:

(Color online) Photofission fragment mass yield at 11.39 MeV(Figure 9a) and 13.39 MeV (Fig. 9(b)) for ²³⁸U: comparison between experimental data [30] and our formalism.

Generally, high-energy photons used in photofission studies, are generated through the Bremsstrahlung effect of accelerated electron beam crossing dense conversion targets. The maximum electron energy is called end-point energy and the average compound nucleus excitation energy corresponding to such end-point energy is an important factor to reproduce experimental data. Naik and his co-authors [29] determined the photofission fragment mass yield for the ²⁴⁰Pu compound nucleus at 10 MeV photon end-point energy, which is equivalent to the average excitation energy equal to 7.61 MeV. In Fig. 10, the calculated results of fragment mass yield using the present formalism are compared with the experimental data. As it is clear from this figure, by considering the end-point energy instead of average excitation energy, the calculated fragment mass yield is better fitted with experimental data.

Figure 10:

(Color online) Photofission fragment mass yield at 10 MeV for ²⁴⁰Pu: comparison between experimental data [29] and our formalism.

The average mass of the light (〈A_L〉) and heavy (〈A_H〉) photofission fragments for ²⁴⁰Pu and ²³²Th actinides are also obtained using the following equation

〈 A_{L} 〉 = \frac{\sum (A \times Y_{L})}{\sum Y_{L}}, 〈 A_{H} 〉 = \frac{\sum (A \times Y_{H})}{\sum Y_{H}},

(15)

and the calculated results are compared respectively with experimental data of Thierens et al. [26] and Naik et al.[30] in Table 2. As it can be seen from Table 2, there is a fair agreement between results of our formalism and experimental data for ²⁴⁰Pu isotope. A small inconsistency between results of the present formalism and experimental data [27] for ²³²Th isotope is produced due to neutron emission through the fission process at high excitation energies. Photofission fragment mass yields for the ²³⁸U isotope at 300 and 500 MeV are presented in Figs. 11(a) and 11(b). It is clear from these figures that, the calculated results of the present formalism are in good agreement with experimental data [28]. It should be noted that the original approach of Gorodisskiy was not successful in reproducing experimental data of high incident photons.

The average mass of the light and heavy photofission fragments for ²⁴⁰Pu at 30 MeV and ²³²Th at 80 MeV photon energies are compared with experimental data.

	Our formalism	Thierens [26]	Our formalism	Naik [27]
	²⁴⁰Pu (30 MeV)		²³²Th (80 MeV)
〈A_L〉	100.57	100.29	92.93	91.74
〈A_H〉	139.42	139.71	139.14	136.75

Figure 11:

(Color online) Photofission fragment mass yield at two different high energies, 300 MeV (Figure 11a ) and 500 MeV (Figure 11b), for ²³⁸U: comparison between experimental data [28] and our formalism.

The peak to valley ratio of fission fragment yield for the ²⁴⁰Pu compound nucleus is calculated and compared with experimental data of Naik [29, 31] in Fig. 12. It is clearly indicated in this figure that the peak to valley ratio decreases as the Bremsstrahlung end-point photon energy grows.

Figure 12:

(Color online) Peak to valley ratio as a function of photon end-point energies (10→30 MeV) for ²⁴⁰Pu: comparison between experimental data [29] and our formalism.

The photofission fragment mass yields for the ²³⁹Pu isotope at 25 MeV incident photon energy and ²³⁴U isotope at 5.77 and 6.11 MeV energies are compared with experimental data in Figs. 13 and 14. These Figures indicate satisfactory agreement between our results and experimental data except around maximum fragment mass yields where the results of the present formalism does not match with experimental data properly.

Figure 13:

(Color online) Photofission fragment mass yield at 25 MeV for ²³⁹Pu: comparison between experimental data [35] and our formalism.

Figure 14:

(Color online) Photofission fragment mass yield at 5.77 MeV (Figure 13a) and 6.11 MeV (Figure 13b) for ²³⁴U: comparison between experimental data [38] and our formalism.

4 Conclusion

A phenomenological formula with some adjustable parameters is developed to calculate fission fragment mass yield for ²³²Th, ^234,238U, ²³⁷Np, and ^239,240,242Pu actinides at various energies. Calculated results are compared with results of Gorodisskiy’s formalism as well as experimental data. Satisfactory agreement has been achieved between results of the present formalism and experimental data especially at low and intermediate photon energies. In this research, the role played by neutron emission prior to fission has also been investigated. It has been stated earlier that, the present approach is not able to consider the effect of neutron multiplicity in photofission. The peak to valley ratio for photofission of the ²⁴⁰Pu isotope is also obtained and indicated in Fig. 12. Figure 12 shows that, the values of peak to valley ratio is decreased with the increase of the Bremsstrahlung end-point energies. Generally, it has been shown that the calculated results of the present study are in good agreement with experimental data as compared with data reported in Ref. [29].

The average mass of light (〈A_L〉) and heavy (〈A_H〉) fragments for ²³²Th and ²⁴⁰Pu actinides are obtained and compared with experimental data [26, 27]. Good agreement between the results of the present approach with experimental data has also been achieved.

References

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Photo-Fission of Uranium and Thorium

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