Low energy magnetic radiation enhancement in the $f_{7/2}$ shell

Studies of the $\gamma$-ray strength functions can reveal useful information concerning underlying nuclear structure. Accumulated experimental data on the strength functions show an enhancement in the low $\gamma$ energy region. We have calculated the $M1$ strength functions for the ${}^{49,50}\mathrm{Cr}$ and ${}^{48}\mathrm{V}$ nuclei in the $f_{7/2}$ shell-model basis. We find a low energy enhancement for $\gamma$ decay similar to that obtained for other nuclei in previous studies, but for the first time we are also able to study the complete distribution related to $M1$ emission and absorption. We find that $M1$ strength distribution peaks at zero transition energy and falls off exponentially. The height of the peak and the slope of the exponential are approximately independent of the nuclei studied in this model space and the range of initial angular momenta. We show that the slope of the exponential fall off is proportional to the energy of the $T=1$ pairing gap.

Figure 1

Summed $B\left(M1\right)$ strength for a range of initial states in ${}^{50}\mathrm{Cr}$.

Figure 2

The average $M1$ strength distribution for states between 6 and 8 MeV in ${}^{50}\mathrm{Cr}$.

Figure 3

Average $B\left(M1\right)$ values as a function of $\gamma$-ray energy $E_{\gamma }$ for ${}^{50}\mathrm{Cr}$ and initial energies, $E_i$, in various 2-MeV ranges. The lowest panel is for 0–2 MeV and the highest for 10–12 MeV. Each $M1$ distribution is compared to the same exponential, red line, with parameters $B_0=0.75{\mu }_N^2$ and $T_B=1.33$ MeV.

Figure 4

Average $B\left(M1\right)$ values as a function of $\gamma$-ray energy $E_{\gamma }$ for ${}^{49}\mathrm{Cr}$, ${}^{50}\mathrm{Cr}$, and ${}^{48}\mathrm{V}$ for initial energies, $E_i$, in the interval 6–8 MeV. Each $M1$ distribution is compared to the same exponential, red line, with parameters $B_0=0.75{\mu }_N^2$ and $T_B=1.33$ MeV.

Figure 5

Average $B\left(M1\right)$ values for ${}^{50}\mathrm{Cr}$ as a function of $\gamma$-ray energy, $E_{\gamma }$, for different initial spin ranges. Each $M1$ distribution is compared to the same exponential, red line, with parameters $B_0=0.75{\mu }_N^2$ and $T_B=1.33$ MeV.

Figure 6

Average $B\left(M1\right)$ values as a function of $\gamma$-ray energy $E_{\gamma }$ (black line) for ${}^{50}\mathrm{Cr}$ for initial energy, $E_i$, in the interval 6–8 MeV, compared with the average $B\left(M1\right)$ values derived using (a) $k_0=0.0$, $k_1=1.0$, (b) $k_0=0.5$, $k_1=1.0$, (c) $k_0=1.0$, $k_1=0.0$, (d) $k_0=1.0$, and $k_1=0.5$, red line. The green line is the exponential fit with $B_0=0.75{\mu }_N^2$, $T_B=1.33$ MeV.

Figure 7

Calculated $\gamma \mathrm{SF}$ values for ${}^{48}\mathrm{V}$ using the $f_{7/2}$ model space (black dashed stair line) and the GX1A interaction in the $pf$ model space allowing first the $f_{7/2}$ orbital to be occupied (red dotted stair line), then the $f_{7/2},f_{5/2}$ (blue heavy stair line) and the $f_{7/2},p_{3/2},f_{5/2}$ (green double dot-dashed stair line) orbitals, and last all the $pf$ shell (orange stair line), as a function of $\gamma$-ray energy, $E_{\gamma }$, for initial energies, $E_i$, in the interval 6–8 MeV.

Figure 8

Calculated $\gamma \mathrm{SF}$ values for ${}^{48}\mathrm{V}$ using the $f_{7/2}$ model space (black dashed stair line) and the GX1A interaction in the $pf$ model space allowing first the $f_{7/2}$ orbital to be occupied (red dotted stair line), then the $f_{7/2},p_{3/2}$ (violet heavy stair line) and the $f_{7/2},p_{3/2},p_{1/2}$ (purple double dot-dashed stair line) orbitals, and last all the $pf$ shell (orange stair line), as a function of $\gamma$-ray energy, $E_{\gamma }$, for initial energies, $E_i$, in the interval 6–8 MeV.

Figure 9

Calculated $\gamma \mathrm{SF}$ values for ${}^{48}\mathrm{V}$ in the $f_{7/2}$ (black dashed stair line) and the $pf$ (orange stair line) model spaces compared to the $\gamma \mathrm{SF}$ values for ${}^{56}\mathrm{Fe}$ in the $pf$ (red dot-dashed stair line) model space, as a function of $\gamma$-ray energy, $E_{\gamma }$, for initial energies, $E_i$, in the interval 6–8 MeV.

Figure 10

Calculated $\gamma \mathrm{SF}$ values for ${}^{48}\mathrm{V}$ in the $f_{7/2}$ (black dashed stair line) and the $pf$ (orange stair line) model spaces compared to the experimental low limit (blue downward-pointing triangles) and high limit (red circles) data of ${}^{50}\mathrm{V}$ ([35]), as a function of $\gamma$-ray energy, $E_{\gamma }$, for initial energies, $E_i$, in the interval 6–8 MeV.