PHYSICS

M. K. VOLKOV, T. VERESH

ALLOWANCE FOR SUPERFLUID EFFECTS IN $\alpha$-DECAY TO VIBRATIONAL LEVELS OF DAUGHTER NUCLEI

(Presented by Academician N. N. Bogolyubov on 6 VIII 1952)

In a recent paper by V. G. Solov’ev ($^1$), a theory of $\alpha$-decay was formulated within the framework of the superfluid model of the nucleus, and it was shown that allowance for pair correlations leads to improved agreement between the theory and the corresponding experimental data. Simultaneously with the indicated work, in an article by Mang and Rasmussen ($^2$), pair correlations were taken into account for transitions to the ground band. In the present article a further extension is carried out of the method proposed in ($^1$) to the case of $\alpha$-decay to collective levels. In the calculations, the Hamiltonian of the model with quadrupole–quadrupole interaction in the region of deformed nuclei was used ($^3$). Neutron–proton correlations were taken into account by the method proposed in the work of B. L. Birbrair ($^4$).

The matrix element of $\alpha$-decay of the parent nucleus from the ground state $\psi_0(N,Z)$ to the vibrational state $\psi'_k(N-2,Z-2)$ may be written as ($^1$)

\[ M=\psi'_k(N-2,Z-2)A\psi_0(N,Z), \tag{1} \]

where

\[ A=\frac{1}{4}\sum_{(\nu,\mu,r,\sigma)} W_{r_1r_2\sigma_1\sigma_2}(p\nu_1\nu_2/n\mu_1\mu_2) a_{p\nu_1r_1}a_{p\nu_2r_2}a_{n\mu_1\sigma_1}a_{n\mu_2\sigma_2}; \tag{2} \]

$a$ are nucleon annihilation operators; $(r,\sigma)=\pm1$. The summation over $(\nu,\mu)$ is carried out over all one-particle levels of the mean field.

We shall be interested only in even-even nuclei. The function of the ground state of the parent nucleus is

\[ \psi_0=\prod_{s,t}(u_{t,s}+v_{t,s}a^+_{ts+}a^+_{ts-})\psi_0; \tag{3} \]

$t$ is the isotopic index, $\psi_0$ is the vacuum function.

We shall seek the function of the vibrational state of the daughter nucleus $\psi'_k$ by the Tamm–Dancoff method, similarly to how this was done in Yoshida’s work ($^5$).

Let us write the Hamiltonian for nucleons moving in an axially symmetric deformed potential, with allowance for pairing and quadrupole–quadrupole interaction:

\[ H=H_0+H_Q; \tag{4} \]

\[ H_0=\sum_{t,\nu}\varepsilon_{t\nu}(a^+_{t\nu+}a_{t\nu+}+a^+_{t\nu-}a_{t\nu-}) -G\sum_{t\nu\nu'}a^+_{t\nu+}a^+_{t\nu-}a_{t\nu'-}a_{t\nu'+}; \tag{4a} \]

\[ H_Q=-\frac{\chi}{2} \sum_{\mu,t,t',\lambda,\lambda',\lambda_1,\lambda'_1} (q_{2\mu})_{t\lambda\lambda'}(q^*_{2\mu})_{t'\lambda'_1\lambda_1} a^+_{t\lambda}a^+_{t'\lambda_1}a_{t\lambda'}a_{t'\lambda'_1}; \tag{4b} \]

here $\varepsilon_{t\nu}$ is the one-particle energy measured from the Fermi sphere, $q_{2\mu}=r^2Y_{2\mu}$.

We apply the method of the canonical transformation of N. N. Bogolyubov ($^6$) to diagonalize $H_0$ and find the function of the vibrational state.

of the daughter nucleus, following Yoshida [5], in the form

\[ \psi_k'=\sum_{t\lambda\lambda'} f_{t\lambda\lambda'}^{(k)} A_{t\lambda\lambda'}^{+}\psi_0', \tag{5} \]

where \(A_{t\lambda\lambda'}^{+}=\alpha_{t\lambda}^{+}\beta_{t\lambda'}^{+}\); \(\beta_{t\lambda}^{+}\) \((\alpha_{t\lambda}^{+})\) is the operator corresponding to the creation of a quasiparticle with negative (positive) projection of the angular momentum on the symmetry axis of the nucleus; \(\psi_0'\) is the quasiparticle vacuum function of the daughter nucleus. As a result, with the assumption of the usual approximations [7, 5], we arrive at the following equation for the eigenvalues of the vibrational energy of the nucleus

\[ f_{t\lambda\lambda'}^{(k)}(E_{t\lambda}+E_{t\lambda'}-\hbar\omega_k) -\chi(q_{2k}^{*})_{t\lambda\lambda'}\chi_{t\lambda\lambda'} \sum_{t_1\lambda_1\lambda_1'}(q_{2k})_{t_1\lambda_1\lambda_1'}\chi_{t_1\lambda_1\lambda_1'}f_{t_1\lambda_1\lambda_1'}^{(k)}=0, \]

where

\[ \chi_{t\lambda\lambda'}=u_{t\lambda}v_{t\lambda'}+u_{t\lambda'}v_{t\lambda},\qquad E_{t\lambda}=\sqrt{\varepsilon_{t\lambda}^{2}+\Delta_t^{2}}. \]

From (6) we obtain for \(f_{t\lambda\lambda'}^{(k)}\)

\[ f_{t\lambda\lambda'}^{(k)} = c_k \frac{(q_{2k}^{*})_{t\lambda\lambda'}\chi_{t\lambda\lambda'}} {E_{t\lambda}+E_{t\lambda'}-\hbar\omega_k}, \tag{7} \]

where

\[ c_k= \left\{ \sum_{t\lambda\lambda'} \frac{\left|q_{2k}\right|_{t\lambda\lambda'}^{2}\chi_{t\lambda\lambda'}^{2}} {(E_{t\lambda}+E_{t\lambda'}-\hbar\omega_k)^2} \right\}^{-1/2} \tag{8} \]

is the normalization factor.

We can now proceed to the calculation of the matrix element (1). Using (2), (3), and (5), one may write (1) in the form

\[ \begin{aligned} M_k &=\frac14\,\psi_k' \sum_{\nu_1\nu_2\mu_1\mu_2 r_1 r_2\sigma_1\sigma_2} W_{r_1r_2\sigma_1\sigma_2}(p\nu_1\nu_2\mid n\mu_1\mu_2) a_{p\nu_1r_1}a_{p\nu_2r_2}a_{n\mu_1\sigma_1}a_{n\mu_2\sigma_2}\psi_0 = \\ &=c_k\Bigg[ \sum_\mu v_{n\mu}u_{n\mu}'\prod_{i\ne\mu}(u_{n_i}'u_{n_i}+v_{n_i}'v_{n_i}) \sum_{\lambda\lambda'\nu_1\nu_2} W_{-+}(p\nu_1\nu_2\mid n) \frac{(q_{2k})_{p\lambda\lambda'}\chi_{p\lambda\lambda'}'} {E_{p\lambda}'+E_{p\lambda'}'-\hbar\omega_k} \times \\ &\quad\times \prod_{j\ne\lambda\lambda'\nu_1\nu_2} (u_{p_j}'u_{p_j}+v_{p_j}'v_{p_j}) \Big\{ \delta_{\lambda\lambda'}\delta_{\nu_1\nu_2} \big[(1-\delta_{\lambda\nu_1})u_{p\nu_1}'v_{p\nu_1}(u_{p\lambda}'v_{p\lambda}-v_{p\lambda}'u_{p\lambda}) -\delta_{\lambda\nu_1}v_{p\lambda}'v_{p\lambda}\big] \\ &\quad\quad -\delta_{\lambda\nu_1}\delta_{\lambda'\nu_2}(1-\delta_{\lambda\lambda'})v_{p\lambda}'v_{p\lambda'} \Big\} +\sum_\nu v_{p\nu}u_{p\nu}' \prod_{i\ne\nu}(u_{p_i}'u_{p_i}+v_{p_i}'v_{p_i}) \times \\ &\quad\times \sum_{\lambda\lambda'\mu_1\mu_2} W_{-+}(p\mid n\mu_1\mu_2) \frac{(q_{2k})_{n\lambda\lambda'}\chi_{n\lambda\lambda'}'} {E_{n\lambda}'+E_{n\lambda'}'-\hbar\omega_k} \prod_{j\ne\lambda\lambda'\mu_1\mu_2} (u_{n_j}'u_{n_j}+v_{n_j}'v_{n_j}) \times \\ &\quad\times \Big\{ \delta_{\lambda\lambda'}\delta_{\mu_1\mu_2} \big[(1-\delta_{\lambda\mu_1})u_{n\mu_1}'v_{n\mu_1}(u_{n\lambda}'v_{n\lambda}-v_{n\lambda}'u_{n\lambda}) -\delta_{\lambda\mu_1}v_{n\lambda}'v_{n\lambda}\big] \\ &\quad\quad -\delta_{\lambda\mu_1}\delta_{\lambda'\mu_2}(1-\delta_{\lambda\lambda'})v_{n\lambda}'v_{n\lambda'} \Big\} \Bigg]. \tag{9} \end{aligned} \]

In the matrix element corresponding to \(\gamma\)-vibrations, essentially \(\lambda\ne\lambda'\). This introduces certain simplifications into expression (9):

\[ \begin{aligned} M_\gamma &=c_\gamma\Bigg\{ \sum_{\mu\lambda\lambda'} v_{n\mu}u_{n\mu}' \prod_{i\ne\mu}(u_{n_i}'u_{n_i}+v_{n_i}'v_{n_i}) W_{-+}(p\lambda\lambda'\mid n) \frac{(q_{22})_{p\lambda\lambda'}\chi_{p\lambda\lambda'}'} {\hbar\omega_\gamma-E_{p\lambda}'-E_{p\lambda'}'} \times \\ &\quad\times v_{p\lambda}v_{p\lambda'}' \prod_{i\ne\lambda\lambda'} (u_{p_i}'u_{p_i}+v_{p_i}'v_{p_i}) +\sum_{\mu\lambda\lambda'} v_{p\mu}u_{p\mu}' \prod_{i\ne\mu}(u_{p_i}'u_{p_i}+v_{p_i}'v_{p_i}) \times \\ &\quad\times W_{-+}(p\mid n\lambda\lambda') \frac{(q_{22})_{n\lambda\lambda'}\chi_{n\lambda\lambda'}'} {\hbar\omega_\gamma-E_{n\lambda}'-E_{n\lambda'}'} v_{n\lambda}v_{n\lambda'}' \prod_{j\ne\lambda\lambda'} (u_{n_j}'u_{n_j}+v_{n_j}'v_{n_j}) \Bigg\}. \tag{10} \end{aligned} \]

Still greater simplifications can be achieved in the matrix element corresponding to \(\beta\)-vibrations. In calculating the quadrupole matrix elements \((q_{20})_{\lambda\lambda'}\) according to Nilsson’s scheme \((^{8})\), one may neglect terms corresponding to the transitions \(N\to N\pm2\), which make a substantially smaller contribution in comparison with the terms corresponding to the transitions \(N\to N\). In this approximation we have \(\lambda=\lambda'\) and

\[ \begin{aligned} M_\beta ={}& c_\beta W'(p\mid n) \Biggl\{ \sum_{\lambda\nu\mu} v_{n\mu}u'_{n\mu} \prod_{i\ne\mu}(u'_{n_i}u_{n_i}+v'_{n_i}v_{n_i}) \frac{(q_{20})_{p\lambda\lambda}\Delta'_p}{E'_{p\lambda}(2E'_{p\lambda}-\hbar\omega_\beta)} \times \\ &\times \left[(1-\delta_{\lambda\nu})u'_{p\nu}v_{p\nu} (u'_{p\lambda}v_{p\lambda}-v'_{p\lambda}u_{p\lambda}) -\delta_{\lambda\nu}v'_{p\lambda}v_{p\lambda}\right] \prod_{j\ne\nu\lambda}(u'_{p_j}u_{p_j}+v'_{p_j}v_{p_j}) \\ &\quad +\sum_{\lambda\nu\mu} v_{p\nu}u'_{p\nu} \prod_{i\ne\nu}(u'_{p_i}u_{p_i}+v'_{p_i}v_{p_i}) \frac{(q_{20})_{n\lambda\lambda}\Delta'_n}{E'_{n\lambda}(2E'_{n\lambda}-\hbar\omega_\beta)} \prod_{j\ne\lambda\mu}(u'_{n_j}u_{n_j}+v'_{n_j}v_{n_j}) \times \\ &\times \left[(1-\delta_{\mu\lambda})u'_{n\mu}v_{n\mu} (u'_{n\lambda}v_{n\lambda}-v'_{n\lambda}u_{n\lambda}) -\delta_{\lambda\mu}v'_{n\lambda}v_{n\lambda}\right] \Biggr\}. \end{aligned} \tag{11} \]

Comparing (11) with the matrix element for the case when the daughter nucleus is in the ground state,

\[ M^0 = W'(p\mid n) \sum_{\mu,\nu} v_{n\mu}u'_{n\mu}v_{p\nu}u'_{p\nu} \prod_{i\ne\mu}(u'_{n_i}u_{n_i}+v'_{n_i}v_{n_i}) \prod_{j\ne\nu}(u'_{p_j}u_{p_j}+v'_{p_j}v_{p_j}), \tag{12} \]

we see that in the expression for the hindrance factor of \(\alpha\)-decay to a \(\beta\)-vibrational level

\[ F=\frac{M^{0\,2}}{M_\beta^2} \tag{13} \]

there will remain only the “superfluid” parts of the matrix elements

\[ \begin{aligned} F={}& \sum_\lambda \left[ \frac{\lvert q_{20}\rvert_{p\lambda\lambda}^{2}\Delta_p'^{\,2}} {E_{p\lambda}'^{\,2}(2E'_{p\lambda}-\hbar\omega_\beta)^2} + \frac{\lvert q_{20}\rvert_{n\lambda\lambda}^{2}\Delta_n'^{\,2}} {E_{n\lambda}'^{\,2}(2E'_{n\lambda}-\hbar\omega_\beta)^2} \right] \times \\ &\times \Biggl\{ \sum_{\lambda,\nu} \frac{(q_{20})_{p\lambda\lambda}\Delta'_p} {E'_{p\lambda}(2E'_{p\lambda}-\hbar\omega_\beta)} \frac{ \left[(1-\delta_{\lambda\nu})u'_{p\nu}v_{p\nu} (u'_{p\lambda}v_{p\lambda}-v'_{p\lambda}u_{p\lambda}) -\delta_{\lambda\nu}v'_{p\lambda}v_{p\lambda}\right] }{ \left[\sum_\nu v_{p\nu}u'_{p\nu} \prod_{i\ne\nu}(u'_{p_i}u_{p_i}+v'_{p_i}v_{p_i})\right] } \times \\ &\times \prod_{j\ne\lambda\nu}(u'_{p_j}u_{p_j}+v'_{p_j}v_{p_j}) + \sum_{\lambda,\nu} \frac{(q_{20})_{n\lambda\lambda}\Delta'_n} {E'_{n\lambda}(2E'_{n\lambda}-\hbar\omega_\beta)} \times \\ &\times \frac{ \left[(1-\delta_{\lambda\nu})u'_{n\nu}v_{n\nu} (u'_{n\lambda}v_{n\lambda}-v'_{n\lambda}u_{n\lambda}) -\delta_{\lambda\nu}v'_{n\lambda}v_{n\lambda}\right] }{ \left[\sum_\nu v_{n\nu}u'_{n\nu} \prod_{i\ne\nu}(u'_{n_i}u_{n_i}+v'_{n_i}v_{n_i})\right] } \prod_{j\ne\lambda\nu}(u'_{n_j}u_{n_j}+v'_{n_j}v_{n_j}) \Biggr\}^{-2}. \end{aligned} \tag{14} \]

Formula (14) makes it possible to carry out numerical calculations and comparison with experiment, which has been done for the \(\alpha\)-decay of the nuclei \(U^{234}\), \(Pu^{238}\), \(Cm^{242}\).

The matrix elements \(\lvert q_{20}\rvert_{\lambda\lambda}\) are calculated according to Nilsson’s scheme \((^{8})\). The values \(\Delta'_t\), \(E'_{t\lambda}\), \(u_{t\lambda}\), \(v_{t\lambda}\) are taken from (9). The quantities of the frequencies \(\omega_\beta\) are calculated from the position of the \(\beta\)-excitation levels of the corresponding nuclei \((^{10})\).

The theoretical and experimental \(^{(11,12)}\) values of the hindrance factors for \(\alpha\)-decay to \(\beta\)-vibrational levels of daughter nuclei are given in Table 1. The calculated data, in the approximation in which they were obtained, can claim only a qualitative comparison with experiment.

Table 1

It should also be noted that in the calculations the blocking effect for the \(\beta\)-vibrational state of the daughter nucleus was not taken into account.

However, despite the very substantial approximations indicated, the theoretically calculated values of the hindrance factors for the \(\alpha\)-decays of the nuclei listed above are in satisfactory agreement with the experimental data. It should especially be noted that the clearly expressed increase in the theoretical values of the \(\alpha\)-decay hindrance factor from \(\mathrm{Cm}^{242}\) to \(\mathrm{U}^{234}\) is in complete agreement with the experimentally obtained values.

Thus, the calculations carried out make it possible to conclude that the hindrance factors for \(\alpha\)-decays of nuclei to \(\beta\)-vibrational levels of daughter nuclei are explained mainly by taking into account pairing-correlation effects of the superconducting type and by introducing a quadrupole–quadrupole interaction into the Hamiltonian.

In conclusion, the authors consider it a pleasant duty to express their gratitude to V. G. Solov’ev for proposing the topic and for very fruitful comments in the course of the work.

Joint Institute for Nuclear Research

Received
19 VI 1962

REFERENCES

1. V. G. Solov’ev, DAN, 144, No. 6 (1962).
2. H. J. Mang, J. O. Rasmussen, Shell Model Calculation of Alpha Decay Rates..., Preprint.
3. D. F. Zaretskii, M. G. Urin, ZhETF, 41, 898 (1961).
4. B. L. Birbrair, ZhETF, 42, 479 (1962).
5. S. Yoshida, Phys. Rev., 123, 2122 (1961).
6. N. N. Bogolyubov, V. V. Tolmachev, D. V. Shirkov, A New Method in the Theory of Superconductivity, Acad. Sci. USSR Publ., 1958.
7. M. Baranger, Phys. Rev., 120, 957 (1960).
8. S. Nilsson, Mat. Fus. Medd. Dan. Vid. Selsk., 29, 16 (1955).
9. T. Veber, V. G. Solov’ev, T. Shiklosh, Izv. AN SSSR, Ser. Fiz., 96, No. 8 (1962).
10. H. Londoldt-R. Börnstein, Zahlenwerte und Funktionen, N. S. 1, Energie-Niveaus der Kerne, Berlin—Göttingen—Heidelberg, 1961.
11. J. O. Rasmussen, Phys. Rev., 113, 1593 (1959).
12. F. Asaro, I. Perlman, Proc. Int. Conf. on Nuclear Structure, Kingston, Canada, Amsterdam, 1960.