Thermal shape and orientation fluctuation corrections for the hot giant dipole resonance within the static path approximation

The $\gamma$-absorption cross section [i.e., giant dipole resonance (GDR)] of ${}^{106}\mathrm{Sn}$ is calculated at temperature $T=2\mathrm{}$ MeV and angular momenta $J=40\mathrm{}$ and 55 following the linear response theory at finite temperature and incorporating the thermal shape as well as orientation fluctuation corrections within the static path approximation approach. The orientation fluctuation corrections have been included for the first time in such a calculation, and are found to be rather unimportant. Then without the inclusion of these effects the GDR is calculated at several more temperature and spin values. We find that at $T=4\mathrm{}$ MeV the GDR width $\Gamma$ shows an increase of only about 1.5 MeV compared to its value at $T=0.5\mathrm{}$ MeV. However, at a fixed temperature $T=2\mathrm{}$ MeV, the increase in the width at $J=69\mathrm{}$ is by about 3 MeV compared to its value at $J=0.\mathrm{}$ Finally, we have also compared the increase of $\Gamma$ at $T=2\mathrm{}$ MeV as a function of J for ${}^{106}\mathrm{Sn}$ and ${}^{120}\mathrm{Sn}.$ The increase is steeper for the lighter isotope, as expected.