UDC 539.142.3+539.144.3

PHYSICS

V. G. SOLOV’EV, P. FOGEL’

STATES CLOSE TO ONE-QUASIPARTICLE STATES IN ODD DEFORMED NUCLEI

(Presented by Academician N. N. Bogolyubov, January 27, 1966)

In even-even nuclei, interactions between quasiparticles lead to the formation of collective nonrotational states, described as one-phonon excitations. In an odd nucleus there is one quasiparticle in addition to the phonons and quasiparticles of the even-even nucleus. In \((^1)\) the interaction of quasiparticles with phonons of an even-even nucleus was considered in odd deformed nuclei. In \((^2)\) investigations of the secular equation were carried out, and an analysis was made of the structural features of excited nonrotational states in odd nuclei. In the present work we give the results of calculations of the energies and wave functions of states close to one-quasiparticle states in odd deformed nuclei in the region \(153 \leq A \leq 183\), and investigate the influence of admixtures to one-quasiparticle states on the values of the decoupling parameters \(a\) and spectroscopic factors in direct nuclear reactions.

The secular equation determining the energies \(\eta_j\) of the ground and excited states in odd deformed nuclei has the form \((^1)\)

\[ \varepsilon(\rho)-\eta_j-\frac{1}{4}\sum_{\lambda\mu i}\sum_\nu \frac{V_{\rho\nu}^{2}}{Y^{i}(\lambda\mu)} \frac{f^{\lambda\mu 2}(\rho\nu)+\bar f^{\lambda\mu 2}(\rho\nu)} {\varepsilon(\nu)+\omega_i^{\lambda\mu}-\eta_j}=0, \tag{1} \]

where the energies of the collective states \(\omega_i^{\lambda\mu}\) and the quantity \(Y^i(\lambda\mu)\) were calculated in \((^3,\,^4)\); \(\varepsilon(\nu)=\sqrt{C^2+(E(\nu)-\lambda)^2}\) (\(C\) is the correlation function, \(\lambda\) is the chemical potential for the odd nucleus); \(V_{\rho\nu}=\mu_s\mu_\nu-v_\rho v_\nu\), \(f^{\lambda\mu}(\rho\nu)\), \(\bar f^{\lambda\mu}(\rho\nu)\) are matrix elements of the multipole-moment operator \((\lambda\mu)\). The summation over \(\lambda\mu i\) is due to the fact that interactions of quasiparticles with quadrupole \(\lambda=2,\ \mu=0,\ \mu=2\) and octupole \(\lambda=3,\ \mu=0,1,2\)-phonons are taken into account for the first two roots \(i=1,2\) of the secular equations for even-even nuclei. The wave function describing a state with a given value of \(K\pi\) has the form

\[ \Psi(K\pi)=\Omega^{+}(K\pi)\Psi_0,\qquad \Omega^{+}(K\pi)=\frac{1}{\sqrt{2}}\,C_\rho \left\{ \sum_\sigma a_{\rho\sigma}^{+} + \sum_{\lambda\mu i}\sum_{\nu\sigma} D_{\rho\nu\sigma}^{\lambda\mu i}a_{\nu\sigma}^{+}Q_i^{+}(\lambda\mu) \right\}; \tag{2} \]

\[ Q_i(\lambda\mu)\Psi_0=0, \tag{3} \]

where \(Q_i(\lambda\mu)\) is the phonon operator of multipolarity \((\lambda\mu)\); \(a_{\nu\sigma}\) is the quasiparticle operator; \(\sigma=\pm1\); \(\rho\) denotes mean-field levels with the given values \(K\pi\); \(\nu\) denotes the remaining ones.

\[ C_\rho^{-2}=1+\frac{1}{4}\sum_{\lambda\mu i}\sum_\nu \frac{V_{\rho\nu}^{2}}{Y^{i}(\lambda\mu)} \frac{f^{\lambda\mu 2}(\rho\nu)+\bar f^{\lambda\mu 2}(\rho\nu)} {\left(\varepsilon(\nu)+\omega_i^{\lambda\mu}-\eta_j\right)^2}, \tag{4} \]

\[ D_{\rho\nu\sigma}^{\lambda\mu i} =-\frac{1}{2}\, \frac{V_{\rho\nu}}{\sqrt{Y^{i}(\lambda\mu)}} \frac{f^{\lambda\mu}(\rho\nu)-\sigma\bar f^{\lambda\mu}(\rho\nu)} {\varepsilon(\nu)+\omega_i^{\lambda\mu}-\eta_j}. \tag{5} \]

The quantity \(C_\rho^2\) determines the contribution of the one-quasiparticle state with the given \(\rho\), while \(C_\rho^2(D_{\rho\nu}^{\lambda\mu i})^2\) determines the contribution of the component with quasiparticle \(\nu\) and phonon \(\lambda\mu i\) to the state under consideration, described by \(\Psi(K\pi)\).

The investigations carried out in (2) showed that the lowering of the energies \(\eta_j\) relative to \(\varepsilon(\rho)\) and relative to the first pole \(\varepsilon(\nu)+\omega_i^{\lambda\mu}\) is determined mainly by the terms (1) with \(\lambda=2,\ \mu=2,\ i=1\) and \(\lambda=3,\ \mu=0,\ i=1\); in a number of cases an important role is played by the terms in (1) with \(\lambda=2,\ \mu=0,\ i=1\) and \(\lambda=2,\ \mu=2,\ i=2\).

Let us consider states in odd deformed nuclei which, in their structure, are close to one-quasiparticle states. In the wave functions the contribution of the state \(\rho\) is predominant, and the quantity \(C_\rho^2\) is somewhat less than unity. A state has a structure close to one-quasiparticle if the condition

\[ \varepsilon(\rho)\ll \min\{\varepsilon(\nu)+\omega_i^{\lambda\mu}\} \tag{6} \]

is fulfilled (and in a number of other cases).

Calculations performed for the levels of odd deformed nuclei in the region \(153\le A\le 183\) showed that the interaction of quasiparticles with phonons leads to a nonuniform decrease, relative to \(\varepsilon(\rho)\), of the energies of states close to one-quasiparticle states. Therefore, in a number of nuclei the calculated sequence of excited states differs from the sequence of levels in the Nilsson scheme.

Thus, in the isotopes of lutetium and tantalum the \(K\pi=3/2-\) state is very close to the one-particle \(514\uparrow\) state (the contribution of \(\rho\) is more than 90%), and the lowering \(\varepsilon(514\uparrow)-\eta_1=(10\div20)\) keV. At the same time, in these nuclei, in \(K\pi=7/2+\) states close to \(404\downarrow\), admixtures play a somewhat larger role, since the contribution of \(\rho\) is of order 97%, and the lowering \(\varepsilon(404\downarrow)-\eta_1=(50\text{--}100)\) keV. Therefore—

Table 1

States close to one-particle states in odd nuclei

therefore, in all lutetium isotopes and in the tantalum isotopes with \(A=177,\ 179\), and 181, the state \(^{7}/_{2}+404\downarrow\) is the ground state, while the state \(^{9}/_{2}-514\uparrow\) is in all cases excited. Another example: in nuclei with \(N=91\), according to the Nilsson scheme, the state \(^{11}/_{2}-505\uparrow\) should be the ground state, whereas according to the calculations carried out, in \(^{153}\mathrm{Sm}\) the ground state is \(^{3}/_{2}+651\uparrow\), and in \(^{155}\mathrm{Gd}\) the ground state is \(^{3}/_{2}-502\downarrow\), which agrees with experiment. Thus, the state \(^{11}/_{2}-505\uparrow\) is not the ground state in nuclei with \(N=91\text{--}93\), although \(\varepsilon(505\uparrow)-\eta_1 \sim 100\) keV.

Fig. 1. Decoupling parameter \(a\) for the state \(^{1}/_{2}-524\downarrow\) at
\(\delta=0.3\) \((a^N=0.89)\), \(1\)—experiment, \(2\)—calculation

Thus, the interaction of quasiparticles with phonons has a weak influence on states with \(K=^{11}/_{2},\ ^{9}/_{2}\), which are close to one-quasiparticle states, and a somewhat stronger influence on states with smaller \(K\). As a result, states with \(K=^{11}/_{2},\ ^{9}/_{2}\) are not the ground states in odd deformed nuclei if, in the mean-field level scheme, levels with smaller values of \(K\) lie close to these states.

Some results of the calculations are presented in Table 1, where the experimental and calculated energy values are given, as well as \(\varepsilon(\rho)-\varepsilon(K)\) (\(\varepsilon(K)\) refers to the ground state)—the excitation energies in the independent-quasiparticle model—and the structure of these states. The experimental data were taken from \((^{5-12})\), and also from private communications by M. Bunker, Ch. Reich, D. Burke, and B. Elbek. It is seen from the table that the admixtures increase with increasing excitation energy for a given state. In individual cases, even in comparatively strongly excited states, the role of admixtures is small; for example, in the state with \(K\pi=^{1}/_{2}+\) and energy 612 keV in \(^{181}\mathrm{Ta}\), the contribution of \(411\downarrow\) is 95%; in the state with \(K\pi=^{7}/_{2}+\) and energy 995 keV in \(^{175}\mathrm{Yb}\), the contribution of \(633\uparrow\) is 98%, etc.

The calculated energy values of levels close to one-quasiparticle states agree somewhat better (especially for highly excited levels) with the experimental data than do calculations according to the independent-quasiparticle model with allowance for the blocking effect \((^{13,\ 14})\). However, this agreement is not sufficiently good, since the determining role is played by the position of the mean-field levels. In a number of cases, an important role is played by the Coriolis interaction, which we do not take into account.

The circumstance that states in odd nuclei are not one-quasiparticle states, but contain admixtures to them, manifests itself in the probabilities of \(\beta\)-decays, in the magnitudes of spectroscopic factors in direct nuclear reactions, in the values of the decoupling parameters \(a\) for states with \(K=^{1}/_{2}\), etc.

Let us investigate the influence of the interaction of quasiparticles with phonons on the magnitude of the decoupling parameter \(a\). Taking the wave function \(\Psi(K\pi)\) in the form

(2), we obtain

\[ a=C_\rho^2\left\{a_{\rho\rho}^{N}+\sum_{\nu'\nu}\left(a_{\nu\nu'}^{N}D_{\rho\nu}^{20i}D_{\rho\nu'}^{20i}-a_{\nu\nu'}^{N}D_{\rho\nu}^{30i}D_{\rho\nu'}^{30i}\right)\right\}, \tag{7} \]

where \(a_{\rho\rho}^{N}\) are the decoupling parameters calculated with Nilsson wave functions \((5)\). Figure 1 shows the experimental \((^{15})\) and calculated values of \(a\) for states close to the one-quasiparticle state \(521\downarrow\). It is seen from the figure that when the level \(521\downarrow\) lies near the Fermi surface, as in \(^{169}\mathrm{Er}\), \(^{171}\mathrm{Yb}\), \(^{173}\mathrm{Hf}\), the admixtures are small and the values of \(a\) are close to \(a^{N}\), and also that allowance for the interaction of quasiparticles with phonons has made it possible to explain the changes in the behavior of \(a\) for states close to \(521\downarrow\) in different nuclei. However, in some cases, for example for states close to \(510\uparrow\), allowance for the interaction of quasiparticles with phonons does not lead to the removal of the discrepancies between the calculated and experimental values of \(a\).

Table 2

Spectroscopic factor and amplitude of the one-particle state for states \(1/2\), close to \(1/2-510\uparrow\)

Let us consider the influence of admixtures on the values of spectroscopic factors in direct nuclear reactions. Thus, upon excitation of a one-quasiparticle state \(\rho\) in a \((dp)\)-reaction on an even-\(A\) target, the spectroscopic factor is equal to \(\mu_\rho^2\). If, however, admixtures are taken into account, i.e., if it is assumed that the wave function has the form (2), then the spectroscopic factor is equal to \(C_\rho^2u_\rho^2\). Table 2 gives the calculated values of the spectroscopic factors for excited states with \(K\pi=1/2-\), close to \(510\uparrow\), in ytterbium isotopes, the values of \(C_\rho^2\), and also the energies of these states. The calculated values of the spectroscopic factors \(C_\rho^2u_\rho^2\) correctly reproduce the behavior of the reaction cross sections*, and the decrease in the values of \(C_\rho^2u_\rho^2\) in the light Yb isotopes is connected with a decrease in the magnitude of \(C_\rho^2\).

Thus, the interaction of quasiparticles with phonons in a number of cases has a substantial influence on states close to one-quasiparticle states in odd deformed nuclei, and it should be taken into account in studying the structure of excited states.

In conclusion we express our deep gratitude to Prof. D. Berg, B. Elbek, C. Ritchie, and M. Bunker for interesting discussions, and to A. A. Korneichuk and G. Jungclaussen for carrying out the numerical calculations.

Joint Institute
for Nuclear Research

Received
10 I 1966

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* D. Berg and B. Elbek—private communication.