Physics

A. N. Kushnirenko

QUADRUPOLE MOMENTS OF NUCLEI

(Presented by Academician N. N. Bogolyubov on 20 VI 1957)

The electric quadrupole moments of nuclei have been studied in works \((^{1-8})\). Since within the framework of the shell model this question has not received a definitive solution, the author decided to consider it once again. We shall represent the wave function of the nucleus in the form of a linear combination of Slater functions composed of one-nucleon wave functions. In the case of strong spin-orbit coupling, the latter have the form \((^9)\):

\[ \psi_{\alpha_i}(\overline r_i, s_i, t_i) = R_{\beta_i}(r_i) \left| \begin{array}{c} a_i Y_{l_i,m_i}(\theta_i,\varphi_i)\\ b_i Y_{l_i,m_i+1}(\theta_i,\varphi_i) \end{array} \right| T_{\tau_i}(t_i), \tag{1} \]

where \(\overline r_i, s_i, t_i\) are, respectively, the spatial, spin, and isotopic-spin coordinates of the \(i\)-th nucleon; \(\alpha_i=(n_i,j_i,j_{zi},\tau_{zi})\), \(\beta_i=(n_i,l_i,j_i)\) are sets of quantum numbers characterizing the state of the nucleon; \(n_i\) is the principal quantum number; \(l_i\) is the orbital quantum number; \(j_{zi}\) is the quantum number of the projection of the total angular momentum on an arbitrary \(z\)-axis; \(m_i\) is the magnetic quantum number; \(\tau_{zi}\) is the quantum number of the isotopic spin of the \(i\)-th nucleon; \(R_{\beta_i}(r_i)\) is the radial wave function; \(Y_{l,m}(\theta,\varphi)\) is a spherical function; \(T_{\tau_i}(t_i)\) is the wave function of isotopic spin; \(a_i\) and \(b_i\) are known coefficients \((^9)\).

The coefficients in the expansion of the nuclear wave function in Slater functions are found from the conservation laws for the projection of angular momentum on the \(z\)-axis, the square of angular momentum, and the square of the isotopic spin of the nucleus. Using such a wave function and the general formula for the quadrupole moment \((^{10})\), we find the formula for the quadrupole moment of the nucleus:

\[ Q_{zz} = \sum_{i,k} |c_k|^2 \frac{A_i^{(k)}}{2j_i^{(k)}\bigl(j_i^{(k)}+1\bigr)} \left[ j_i^{(k)}\bigl(j_i^{(k)}+1\bigr)-3j_{zi}^{(k)2} \right] + \]

\[ + \sum_{\substack{i,k,j\\(k\ne j)}} c_k^{*}c_j \delta_{\tau_j^i,\tau_k^i} \prod_{\substack{n\\(n\ne i)}} \delta_{\alpha_n^k,\alpha_n^j} A_i^{(k,j)}B_i^{(k,j)}, \tag{2} \]

where \(i\) is the number of the proton; \(k\) and \(j\) are the numbers of the Slater functions; \(c_k\) is the amplitude of the \(k\)-th Slater function; the number \(n\) runs over all integers from 1 to \(A\) (except \(n=i\)); \(A\) is the number of particles in the nucleus;

\[ A_i^{(k)} = \int R_{\beta_i^{(k)}}^{2}(r)\,r^{4}\,dr; \qquad A_i^{(k,j)} = \int R_{\beta_i^{(k)}}(r)R_{\beta_i^{(j)}}(r)\,r^{4}\,dr; \tag{3} \]

\[ B_i^{(k,j)} = \int (3\cos^{2}\theta-1) \left\{ a_i^{(k)}a_i^{(j)*} Y_{l_i^{(k)},m_i^{(k)}}^{*} Y_{l_i^{(j)},m_i^{(j)}} + \right. \]

\[ \left. + b_i^{(k)}b_i^{(j)*} Y_{l_i^{(k)},m_i^{(k)}+1}^{*} Y_{l_i^{(j)},m_i^{(j)}+1} \right\} \,d\Omega_i. \tag{4} \]

It can be shown that the value of the coefficient \(A_i^{(k)}\) lies in the interval \((R^2/3, R^2)\), and its mean value is approximately equal to \(0.6\,R^2\), where \(R\) is the nuclear radius.

Table 1 gives the electric quadrupole moments of certain light nuclei calculated by formula (2), under the assumption that \(A_i^{(k)} \approx 0.6\,R^2\) (the calculation was carried out without taking configuration admixtures into account).

Table 1

The last (triple) sum in formula (2) shows that, in the presence of mixed shell configurations, an additional electric quadrupole moment arises. Thus, for example, analysis of shell structures indicates a close location of the proton levels \(4d_{5/2}\), \(6g_{7/2}\), and \(6h_{11/2}\). Such levels begin to be filled by protons almost simultaneously. Therefore the protons filling these levels enter into complex states that are linear combinations of the states \(4d_{5/2}\), \(5g_{7/2}\), and \(6h_{11/2}\). It may be assumed that such complex shell configurations are responsible for at least part of the anomalously large positive electric quadrupole moment of certain nuclei with \(Z>50\).

In conclusion, I express my gratitude to D. D. Ivanenko, S. I. Larin, and Ya. A. Smorodinsky for valuable comments and discussion.

Kyiv State University
named after T. G. Shevchenko

Received
30 V 1957

CITED LITERATURE

1. B. H. Flowers, Phil. Mag., 43, 1330 (1952).
2. A. Bohr, B. R. Mottelson, Kgl. Danske Videnskab. Mat.-fys. Medd., 27, No. 16 (1953).
3. M. A. Perks, Proc. Phys. Soc., A 68, 1083 (1955).
4. H. Horie, A. Arima, Phys. Rev., 99, 778 (1955).
5. J. Rainwater, Phys. Rev., 79, 432 (1950).
6. R. Van Wageningen, J. de Boer, Physica, 18, 369 (1952).
7. S. A. Moszkowski, C. H. Townes, Phys. Rev., 93, 306 (1953).
8. F. I. Kliman, ZhETF, 18, No. 4, 346 (1948).
9. D. I. Blokhintsev, Principles of Quantum Mechanics, 1949.
10. J. Blatt, V. Weisskopf, Theoretical Nuclear Physics, Moscow, 1954.