Radiative strength functions from the energy-localized Brink-Axel hypothesis

Abstract

Radiative strength functions (RSFs) model the bulk electromagnetic response of highly-excited nuclei and are critical inputs for statistical reaction codes. In this paper, we present a definition of the RSF that is consistent with Hauser-Feshbach reaction codes and that can be efficiently computed with the shell model using the Lanczos strength-function (LSF) method. We introduce a variant of the shell model LSF method that exploits the energy-localized Brink-Axel hypothesis, which makes it possible to compute both electric and magnetic RSFs across all energies relevant to capture reactions. We verify agreement with the conventional definition of RSFs with benchmark calculations of $^{24}$Mg, then present novel results for $^{56}$Fe. For $^{56}$Fe we find that: (i) the M1 RSF shape evolves smoothly with excitation energy, consistent with the energy-localized Brinkl-Axel hypothesis, (ii) both M1 and E1 transitions contribute significantly to the radiative strength below the photo-absorption threshold, and (iii) within the sdpf model space, the strength below 3 MeV observed in Oslo-type experiments cannot be fully reproduced. These results pave the way for a coherent microscopic description of the RSFs and further motivate the use of energy-dependent RSFs in modern reaction codes.

Figures (8)

• Figure 1: Depiction of the radiative strength function, Eq. \ref{['eq:defgsf']}. The red arrows indicate partial decay widths from levels $c \to d$; the box indicates an energy-average over partial decay widths from within $E_c \pm \Delta E/2$.
• Figure 2: Agreement of the Bartholomew formula (dashed) and the present level-density-free (LDF) formula (solid), demonstrated for the M1 RSF of $^{24}$Mg. Panel a) was computed using the first 1000 levels and panel b) with the first 5000 levels (see text for details). All calculations used a bin size $\Delta E= 200$ keV. The gray band shows the standard deviation of the RSFs $f_d$ within each bin, which are averaged to obtain $f_\text{(SA)}^{\mathrm{X}L}$.
• Figure 3: Contribution of two individual strength functions $f_d^\mathrm{M1}(E_\gamma)$ to the overall reduced strength $f_\mathrm{(SA)}^{\mathrm{X}L}(E_\gamma)$, showing that the ground state has a unique shape, while a random excited state has a partial radiative strength function with a shape compatible with the total reduced radiative strength function. The ground state ($d=1$) has no strength below about 7 MeV due to a physical lack of states close enough in energy. The $d=239$ strength, on the other hand, has significant strength approaching $E_\gamma=0$.
• Figure 4: The radiative strength function for a single state can be computed accurately and to higher energy with the LSF method. We demonstrate with a level with $E_d=17.05$ near the neutron separation energy $S_n(^{24}\mathrm{Mg})=16.531$ MeV. We compare the result from the complete list of transitions against that from the ILSF method (see text). The state was selected to have $j_d=0$ so that Eq. \ref{['eq:bjf0']} applies, simplifying the calculation.
• Figure 5: The Lanczos strength-function method with interior eigenvalues (ILSF) can be further improved by averaging the partial strength functions of several approximate excited levels. Here we use five $j_d=0$ interior eigenvectors with energies between 15 and 18 MeV as pivots. The fluctuations of individual strengths averages out to closely follow the average RSF $f_\mathrm{(SA)}^\mathrm{M1}$ computed with 5000 levels.