Statistical properties and broken symmetries within the shell model

Shell-model calculations in the $s-d$ shell have been utilized to examine how the statistical behavior of eigenvalues and reduced transition probabilities are affected by broken isospin symmetry. Calculations have been performed for the nuclides ${}^{22}\mathrm{Na},{}^{26}\mathrm{Al}$, and ${}^{34}\mathrm{Cl}$ and have been compared to existing experimental data for ${}^{26}\mathrm{Al}$ and ${}^{30}\mathrm{P}$. The eigenvalue statistics depend on the magnitude of the Coulomb matrix element, and this is reflected in a sensitivity to the choice of single-particle energies. The distributions of reduced transition probabilities are not universal but depend upon the particular transition mode chosen; this behavior is qualitatively similar to experimental results.

Figure 1

Nearest-neighbor spacing distributions for shell-model eigenvalues and experimental eigenvalues in ${}^{26}\mathrm{Al}$. The labels on each graph indicate which quantum numbers identify the sequences and which Hamiltonian was used in the calculation. Solid curves show GOE distributions, and dashed curves show the corresponding Poisson distributions.

Figure 2

Nearest-neighbor spacing distributions for shell-model eigenvalues in ${}^{34}\mathrm{Cl}$. The labels on each graph indicate which quantum numbers identify the sequences and which Hamiltonian was used in the calculation. Solid curves show GOE distributions, and dashed curves show the corresponding Poisson distributions.

Figure 3

${\Delta }_3$ for shell-model eigenvalues and experimental eigenvalues in ${}^{26}\mathrm{Al}$. The labels on each graph indicate which quantum numbers identify the sequences and which Hamiltonian was used in the calculation. Solid curves show GOE distributions, and dashed curves show the corresponding Poisson distributions.

Figure 4

${\Delta }_3$ for shell-model eigenvalues in ${}^{22}\mathrm{Na}$. The labels on each graph indicate which quantum numbers identify the sequences and which Hamiltonian was used in the calculation. Solid curves show GOE distributions, and dashed curves show the corresponding Poisson distributions.

Figure 5

(Color) distributions of reduced transition probabilities for the IC (in red) and ${\mathrm{INC}}_B$ (in blue) calculations. The smooth black curves show the Porter-Thomas distribution.

Figure 6

Distributions of reduced transition probabilities for ${}^{30}\mathrm{P}$ experimental transition sequences. The dashed curves show the Porter-Thomas distribution.

Figure 7

Combined $M1$ and $E2$ distributions of reduced transition probabilities for ${}^{26}\mathrm{Al}$ and ${}^{30}\mathrm{P}$ experimental transition sequences. The dashed curves show the Porter-Thomas distributions.

Figure 8

Combined distributions of reduced transition probabilities for the four shell-model INC calculations. The dashed curves show the Porter-Thomas distributions.