DEFORMATION POTENTIAL IN THE REGION OF HEAVY NUCLEI

V. E. Marshalkin, L. V. Zakrevskaya

(Presented by the scientific seminar of the Scientific Research Institute of Nuclear Physics)

It is known that heavy nuclei \((A > 220)\) possess a stable deformation in the ground state. This is explained by the deforming action of the nucleons of an unfilled shell, owing to which the system has the least internal energy (deformation potential) in a form different from spherical. The substantial influence of an odd nucleon on the behavior of the internal energy of a system [1, 2] at large deformations is also known. These two facts play an important role in constructing the potential barrier for the fission of atomic nuclei. In the present work the influence of the internal structure of the system on its deformability is investigated, and the influence of an odd nucleon on the fission barrier is estimated as a manifestation of the quantum-mechanical properties of the system.

It is assumed [3] that the change in the internal energy of the nucleons (the energy of the nuclear interaction) in the process of deformation is determined sufficiently well by the change in the sum of the single-particle energies. Without taking the Coulomb interaction into account, this change has the form:

\[ U(z_0)=\sum_{i=1}^{A} E_i(z_0), \tag{1} \]

where \(z_0\) is the deformation parameter.

The calculation of the single-particle spectrum for deformations up to the separation of fragments was carried out earlier [4]. However, the accuracy of the computations was insufficient for quantitative calculations of the influence of the internal structure on the characteristics of the fissioning system. Therefore the calculation of the energy spectrum was refined on the “Minsk-1” machine. In the notation of the cited work, the quantum numbers \(n_z\), characterizing the motion of nucleons along the \(z\)-axis, are given in Table 1. The eigenenergies of the nucleons were calculated by the formula:

\[ E_{n_z,n_\perp,\Omega,\sum}(z_0) = \hbar \omega (z_0) \left[ n_z+n_\perp+\frac{3}{2} +(-0.1)\Lambda \sum + \right. \]

\[ \left. +(-0.05)\,\mu\, \left(\Lambda^2+2n_z n_\perp+2n_z+n_\perp\right) \right], \tag{2} \]

where \(n_z, n_\perp\) are the oscillator quantum numbers for oscillations along the \(z\)-axis and in the plane perpendicular to it, respectively.

Fig. 1. Dependence of the internal energy of nucleons on nuclear deformation

Fig. 2. Behavior of the specific energy of the fissioning system

\(\Omega, \Lambda, \Sigma\) are the projections of the total angular momentum, orbital angular momentum, and spin of the particle onto the symmetry axis of the nucleus.

Table 1

According to the scheme described, a calculation of \(U(z_0)\) was carried out for the nucleus \({}_{92}U^{238}\) at \(z_0 = 0;\ 0.2;\ 0.3;\ 0.4\). At each deformation, an even number of nucleons of the same type, because of nonradiative transitions [5], occupied the lowest levels in accordance with the Pauli principle. In Fig. 1 the minimum of the internal energy for \({}_{92}U^{238}\) at \(0.2 < z_0 < 0.3\) is clearly manifested.

It should be noted that the calculation was performed for \(\hbar \omega_0 = \dfrac{54}{A^{1/3}}\) MeV. It is also clear that, as a result of taking the Coulomb interaction into account, the minimum will shift somewhat to the right. The relation between the parameter of ellipsoidal deformation and the parameter \(z_0\) used by us is found from the relation:

\[ \alpha_2 = \frac{2}{R_0} \sqrt{\frac{5}{2}} \int_0^{\frac{\pi}{2}} \left[ |z_0 \cdot \cos \Theta| + \sqrt{z_0^2 \cos^2 \Theta + (R_1^2 - z_0^2)} \right] \cdot \frac{1}{4}(3\cos 2\Theta + 1)\sin \Theta \, d\Theta, \tag{3} \]

where \(R_1\) is the radius of the separating fragment.

In the calculations it is necessary to take into account the influence of the change

\[ \hbar \omega_0 = \frac{C}{A^{1/3}} \]

(due to the choice of the constant \(C\) or of the mass number) on the behavior of the internal energy and, correspondingly, of the stable deformation. If \(\dfrac{\omega_0}{\omega_0^1} = \gamma\), then \(E' = \gamma E\), and \(z'_0 = \sqrt{\gamma}\, z_0\).

The results of calculations of the internal energy of other nuclei \((U^{233}, U^{235}, Am^{241}, Pu^{239})\) indicate a shift of the stable deformation that is in qualitative agreement with the experimental values.

The situation becomes substantially more complicated if we have a nucleus with an odd mass number. In this case, the transition of the last odd nucleon to lower levels in the process of deformation is hindered owing to the conservation law for the projection of the angular momentum of the system. Therefore, in the case of an odd nucleus the deformation potential includes a “specific” energy [1], determined by the difference between the energy of the fixed state \(\Omega\) and the lowest state \(\Omega'\).

at different deformations. This quantity may be regarded as a measure of the extent to which the fission barrier of odd nuclei raises the corresponding barrier in neighboring even-even nuclei. Figure 2 shows the behavior of the specific energy for the nuclei \(U^{235}\), \(U^{233}\), \(P_u^{239}\), and \(A_m^{241}\). By definition, the specific energy at the stable deformation becomes zero. In Fig. 2 the specific energy of each of the nuclei is indicated to within a term independent of the deformation (the Coulomb interaction is not taken into account, and the minimum of the internal energy is determined with a comparatively large error).

The calculations make it possible only to say that the stable deformation of the nucleus \(U^{235}\) is greater than that of \(U^{233}\), and that of \(P_u^{239}\) greater than that of \(A_m^{241}\). Consequently, the specific energy of \(U^{235}\) is somewhat less than the specific energy of \(U^{233}\), but the specific energy of \(P_u^{239}\) is greater than that of \(A_m^{241}\). These conclusions are in complete agreement with the half-lives for spontaneous fission [6]. From Fig. 2 it is seen that the specific energy increases the height of the barrier only slightly, but substantially changes its width.

The analysis carried out allows us to say that investigation of the internal structure of the fissioning system makes it possible to understand essential features of the process of fission of atomic nuclei.

In conclusion, we express our deep gratitude to P. A. Cherdantsev for useful advice and discussions.

REFERENCES

1. J. A. Wheeler. Fission and the stability of nuclei. In the collection “Niels Bohr and the Development of Physics,” edited by W. Pauli, IL, 1958.

2. S. A. E. Johansson, Nucl. Phys., 12, 449, North-Holland Publishing Co., Amsterdam, 1959.

3. Steven Moszkowski. Nuclear models. In the collection Structure of the Atomic Nucleus, edited by A. S. Davydov, IL, 1958.

4. P. A. Cherdantsev, V. E. Marshalkin. Izv. Vuzov SSSR, Fizika, No. 3, Publishing House of TSU, 1966.

5. P. A. Cherdantsev, V. E. Marshalkin. Report “Nonradiative transitions of nucleons in the process of fission of atomic nuclei” at the XVI Annual Conference on Nuclear Spectroscopy and Nuclear Structure in Moscow, 1966.

6. A. Kraut. Advances in the physics of nuclear fission. In the collection Physics of Nuclear Fission, edited by V. S. Stavinskii. Gosatomizdat, 1963.