Comprehensive Test of the Brink-Axel Hypothesis in the Energy Region of the Pygmy Dipole Resonance

The validity of the Brink-Axel hypothesis, which is especially important for numerous astrophysical calculations, is addressed for ${}^{116,120,124}\mathrm{Sn}$ below the neutron separation energy by means of three independent experimental methods. The $\gamma$-ray strength functions (GSFs) extracted from primary $\gamma$-decay spectra following charged-particle reactions with the Oslo method and with the shape method demonstrate excellent agreement with those deduced from forward-angle inelastic proton scattering at relativistic beam energies. In addition, the GSFs are shown to be independent of excitation energies and spins of the initial and final states. The results provide a critical test of the generalized Brink-Axel hypothesis in heavy nuclei, demonstrating its applicability in the energy region of the pygmy dipole resonance.

Figure 1

Experimental primary $\gamma$-ray matrix $P(E_{\gamma },E_x)$ for ${}^{120}\mathrm{Sn}$. The yellow dashed line indicates the neutron threshold $S_n$, while the dashed red and blue lines (regions 1 and 2) confine transitions to the ground state and the first excited $J^{\pi }=2^{+}$ state at $E_x=1.171\mathrm{MeV}$. The solid blue lines (region 3) mark the region $4.5\le E_x\le 9.1\mathrm{MeV}$, $E_{\gamma }\ge 1.3\mathrm{MeV}$ used for the Oslo method analysis.

Figure 2

Comparison of the GSFs for ${}^{116,120,124}\mathrm{Sn}$ obtained from the Oslo method (blue) and from the $(p,p^{\prime })$ experiments [47] (orange). The total error bands for the Oslo method (light blue) are asymmetric and include all uncertainties. The dark blue band represents statistical and systematic uncertainties from the unfolding and the extraction of primary $\gamma$ rays. The $E_{\gamma }$ bin widths are 128 and 200 keV for the Oslo data and the $(p,p^{\prime })$ measurements respectively.

Figure 3

GSF of ${}^{120}\mathrm{Sn}$ for narrow initial excitation-energy bins at (a) 5.6 MeV, (b) 6.3 MeV, and (c) 7.4 MeV with a width of 256 keV (red data points) compared to the Oslo method result for the full excitation-energy range $4.5\le E_x\le 9.1\mathrm{MeV}$ (blue). For the Oslo method, the total error band is shown. For both approaches, an $E_{\gamma }$ bin width of 128 keV is used.

Figure 4

Comparison of the GSFs for ${}^{120}\mathrm{Sn}$ extracted with the Oslo method (blue band) from selective decay to the ground state and the first excited $2^{+}$ state utilizing the shape method (red and green triangles) and from the $(p,p^{\prime })$ data [47] (orange band). The $E_{\gamma }$ bin widths are 128 keV for the Oslo and shape-method data and 200 keV for the $(p,p^{\prime })$ data.