CHARGE DISTRIBUTION IN NUCLEAR FISSION

It is known that, upon neutron capture, nuclei undergo fission into fragments that are asymmetric in mass. The mass of the light fission fragments of \( \mathrm{U}^{235} \) is concentrated around values \( A = 95 \), and the mass of the heavy fragments around \( A = 140 \). The charge of the initial nucleus is distributed between the fragments, the fragments are radioactive, and by means of \(\beta\)-decays are transformed into stable nuclei. To explain the charge distribution of the nucleus in fission, the following assumptions were previously put forward:

1. In each of the fragments the ratio of charge to mass remains the same as it was in the initial nucleus; i.e., the charge-distribution densities in the fragment nuclei and in the initial nucleus are identical.

Fig. 1

1. The “minimum radioactive decay energy” postulate of Wigner and Way. According to these authors’ assumptions, the charge of the initial nucleus is distributed among the fragments in such a way that the total energy of the \(\beta\)-, \(\gamma\)-, and neutrino radiation of all subsequent decays of the fragments down to the stable state is minimal.

2. The postulate of Glendenin and others on the “equal length” of the decay chains of the fragments, i.e., that the differences \( Z_A - Z_p \) are the same for both fragments, where \( Z_A \) is the stable charge of a nucleus of mass \( A \), and \( Z_p \) is the initial charge of the fragment of mass \( A \). This postulate and the consequences following from it were regarded in a number of previous works as the most reliable.

In nuclear fission only part (up to 20) of the electrons in the atomic shells of the fragments is stripped off. As the fragments slow down in the medium, they acquire the missing electrons. For these reasons, direct determination of the initial charge of the fragments is impossible.

The study of fragments is usually carried out by methods of radiochemical analysis; however, because of their radioactivity, the products investigated are separated from the primary fission fragments by a precisely known number of \(\beta\)-decays, and therefore the determination from them of the initial charge of the fragments and the finding of the fission probability with a given charge distribution are unreliable.

In fission, a large spread is observed in the magnitude of the kinetic energy of the fragments. Fig. 1 shows the curve of the kinetic energy of fission fragments of \( \mathrm{U}^{235} \) with masses \( A_H = 141 \), \( A_L = 95 \).

According to this curve, the maximum kinetic energy is almost 170 MeV, while deviations in the value of the kinetic energy reach 40 MeV. The author of the paper under review¹ uses this large spread in the kinetic energy of the fragments to calculate the initial charge of the fragments. He assumes that the spread in the kinetic energy of the fragments is a consequence of variations in the charge distribution, since variations in the forces of Coulomb repulsion of the fragments will then occur, and the fragments acquire kinetic energy at the expense of the forces of Coulomb repulsion.

The energy-balance equation in fission has the form:

\[ E_{\mathrm{p}} = E_{\mathrm{k}} + E_{\mathrm{r}} + E_{0}, \]

where \(E_{\mathrm{p}}\) is the total fission energy, \(E_{\mathrm{k}}\) is the kinetic energy; \(E_{\mathrm{r}}\) is the energy of the neutrons and \(\gamma\)-rays emitted in fission; \(E_{0}\) is the energy of \(\beta\)-, \(\gamma\)- and neutrino radiation in the subsequent decays of the fragments.

For the observed maximum value \(\Delta E_{\mathrm{k}} = 40\) MeV we obtain:

\[ \Delta E_{\mathrm{p}} = 40\ \text{MeV} + \Delta E_{\mathrm{r}} + \Delta E_{0}. \tag{1} \]

According to Wigner, \(\Delta E_{\mathrm{k}}\) is caused mainly by a variation in the number of neutrons emitted in fission. The emission of each neutron reduces the left-hand side of (1) by the binding energy of the neutron in one of the fragments, i.e. by 8 MeV. When a neutron is emitted, the fragment becomes closer to the stable state, which reduces \(\Delta E_{0}\) in the right-hand side of (1) by 2–3 MeV. The resulting effect \(\Delta E_{\mathrm{k}}\) will amount to 5–6 MeV, and to explain the observed difference of 40 MeV a difference in the number of neutrons by 6–8 is required, which is too large in comparison with the known mean number of neutrons emitted in fission.

The author considers the instantaneous fission energy \(E_{\mathrm{m}}\), i.e. the energy liberated in the act of fission. It is equal to the sum of the kinetic energy of the fragments \(E_{\mathrm{k}}\) and the excitation energy of the fragments \(E_{\mathrm{v}}\):

\[ E_{\mathrm{m}} = E_{\mathrm{k}} + E_{\mathrm{v}} = M(235,92) + n - [M(A_L, Z_L) + M(A_H, Z_H)]. \tag{2} \]

Using Finberg’s semi-empirical formula for \(M(A,Z)\):

\[ M(A,Z) = M(A,Z_A) + \alpha[(Z - Z_A)^2/A], \]

where \(\alpha\) and \(Z_A\) are constants, the author calculates \(E_{\mathrm{m}}\) as a function of the chosen values of the fragment charges \(Z_H\) and \(Z_L\). For the calculation it is necessary to know the possible limits of the quantities \(Z_H\) and \(Z_L\). Positron activity of the fragments has not been observed; therefore the upper limit for the charge of the fragments may be taken so that \(Z_H\) and \(Z_L\) correspond to the values of the charges of stable nuclei with masses \(A_H\) and \(A_L\). The lower limit is subject to the condition of charge conservation: \(Z_H + Z_L = 92\). As a result of calculations, using formula (2), one obtains the dependence of \(E_{\mathrm{m}}\) on \(Z_H\) and \(Z_L\), which is shown in Fig. 2. Further: \(E_{\mathrm{k}} = E_{\mathrm{m}} - E_{\mathrm{v}}\), and in Fig. 2 the curves \(E_{\mathrm{k}}\) are plotted, obtained under different assumptions about the magnitude \(E_{\mathrm{v}}\), which may vary within small limits. \(E_{\mathrm{k}}\) corresponds to \(E_{\mathrm{v}} = \mathrm{const}\), and \(E_{\mathrm{k}_2}\) and \(E_{\mathrm{k}_3}\) to the extreme limiting values of \(E_{\mathrm{v}}\). Fig. 1 gives the dependence of the relative probability of fission with a given kinetic energy of the fragments on the value of the energy, while Fig. 2 gives the magnitude of the kinetic energy (\(E_{\mathrm{k}_1}\), or \(E_{\mathrm{k}_2}\), \(E_{\mathrm{k}_3}\)) as a function of the charge distribution. Consequently, a joint consideration of the curves in Fig. 1 and Fig. 2 makes it possible to obtain the curve of the relative-

probability curve for the charge distribution in fission as a function of the charge of fragments of given masses. This curve is shown in Fig. 3. The relative probability of the charge distribution was calculated for

Fig. 2.

all possible values of the excitation energy of the fragments \(E_{\mathrm{B}}\), which corresponds to the values \(E_{\mathrm{k}1}\), \(E_{\mathrm{k}2}\), and \(E_{\mathrm{k}3}\). The hatched area in Fig. 3 gives the relative probability of charge distribution for all possible values of \(E_{\mathrm{k}}\). In precisely the opposite direction

Fig. 3.

is the curve obtained in Glendenin’s work, shown in Fig. 3 by the dotted line. The author’s work proves the close coincidence of variations in the instantaneous fission energy and, as a consequence, variations in the kinetic energy of the fragments with the distribution of the charge of the initial nucleus between the fragments, and it is shown that variations in the energy of the fragments are mainly due to variations in the charge distribution of the nucleus.

K. T.

References

1. D. Brunton, Phys. Rev. 76, 1798 (1949).
2. K. Way and E. Wigner, Phys. Rev. 73, 1318 (1948).
3. Cotyell and Glendenin, Phys. Rev. 75, 337 (1949).