File:Shape deformations 5.jpg

Summary

Single-particle spectrum for an axially-symmetric harmonic-oscillator potential. The energy eigenvalue of the one-particle orbit (\(n_{x},n_{y},n_{z}\)) is written as \[\varepsilon(N,n_{z})=\hbar\bar{\omega}\left(N+\frac{3}{2}-\frac{1}{3}\delta_{osc}(3n_{z}-N)\right)\tag{1} \] where the principal quantum number \(N=n_{\perp}(\equiv n_{x}+n_{y})+n_{z}\) and \(\bar{\omega}=(2\omega_{\perp}+\omega_{z})/3\), while the deformation parameter \(\delta_{osc}\), which is approximately equal to \(\beta\approx\delta\), is defined by \[\delta_{osc}=3\,\frac{\omega_{\perp}-\omega_{z}}{2\omega_{\perp}+\omega_{z}}\approx\frac{R_{z}-R_{\perp}}{R_{av}}\] where \(R_{av}\) denotes the mean radius. At \(\delta_{osc}=0\) (spherical shape) the spectrum is regularly bunched with the equal energy spacings \(\hbar\omega_{0}\). Each major shell with the energy \((N+\frac{3}{2})\hbar\omega_{0}\) has the degeneracy \((N+1)(N+2)\) including the nucleon spin degree of freedom. The particle numbers, 2, 8, 20, 40, 70, 112, ..., would be the magic numbers for this potential and associated with an especially stable spherical shape. Single-particle levels belonging to a given major shell have the same parity \(\pi=(-1)^{N}\) with the maximum orbital angular momentum \(\ell_{max}=N\). For \(\delta_{osc}\neq0\) the levels split into \((N+1)\) levels with eigenvalues \(\varepsilon(N,n_{z})\) in (1), and each level has a degeneracy of 2(\(n_{\perp}+1\)), where a factor 2 comes from the nucleon spin \(\frac{1}{2}\) while (\(n_{\perp}+1\)) from possible values of \(n_{x}\)(\(=0,1,2,...,n_{\perp}\)). For prolate (oblate) deformation, \(\omega_{z}<\omega_{\perp}\) or \(\delta_{osc}>0\) (\(\omega_{z}>\omega_{\perp}\) or \(\delta_{osc}<0\)), the levels with larger (smaller) \(n_{z}\) become energetically lower. Eigenvalues expressed in units of the mean frequency \(\bar{\omega}\) have a linear dependence on \(\delta_{osc}\), and the slope of the lowest level with a given \(N\) for prolate shape (\(n_{z}=N\)) is twice that for oblate shape (\(n_{z}\) = 0) with an opposite sign. The arrows mark the deformations corresponding to the indicated rational ratios of frequencies \(\omega_{\perp}:\omega_{z}\). The figure gives the total particle number corresponding to completed shells in the potentials with \(\omega_{\perp}:\omega_{z}\) = \(1:1\) (spherical), \(2:1\) (prolate) and \(1:2\) (oblate). The figure is taken from Ref. #BM75.

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