M1 dipole strength from projected generator coordinate method calculations in the sd-shell valence space

TL;DR

This work addresses the challenge of describing low-energy $M1$ strength and the associated low-energy enhancement (LEE) in sd-shell nuclei, where standard QRPA approaches face limitations. It adopts the projected generator coordinate method (PGCM) to restore symmetries and incorporate collective dynamics, benchmarking it against exact shell-model results in $^{24}$Mg using the USDB interaction. Two generator-coordinate sets, Set A and Set B, are explored to break time-reversal symmetry, and the PGCM reproduces $1^+$ energies, magnetic dipole moments, and $B(M1)$ strengths, including the LEE, with remarkable accuracy and efficient convergence (roughly 200 constrained states suffice for dominant contributions). The method is extended to other sd-shell nuclei, showing good agreement in overall $B(M1)$ distributions and cumulated strength, highlighting PGCM as a viable route for systematic, large-scale calculations in nuclear structure and reaction modeling.

Abstract

The low-energy enhancement observed in the deexcitation $γ$-ray strength functions, attributed to magnetic dipole (M1) radiations, has spurred theoretical efforts to improve on its description. Among the most widely used approaches are the quasiparticle random-phase approximation (QRPA) and its extensions. However, these methods often struggle to reproduce the correct behavior of the M1 strength at the lowest $γ$ energies. An alternative framework, the projected generator coordinate method (PGCM), offers significant advantages over QRPA by restoring broken symmetries and incorporating both vibrational and rotational dynamics within a unified description. Due to these features, PGCM has been proposed as a promising tool to study the low-energy M1 strength function in atomic nuclei. However, comprehensive investigations employing this method are lacking. The PGCM is presently used within the frame of sd-shell valence space calculations based on the USDB shell-model interaction to benchmark its performance against the solutions obtained via exact diagonalization. The reliability of two different sets of generator coordinates in the PGCM calculations is gauged using ${}^{24}$Mg as a test case. The ability of the PGCM to reproduce results from exact diagonalization in the sd valence space is demonstrated for $1^{+}$ states and M1 transitions. Future work will need to assess whether the proposed method can be applied systematically and extended to large-scale calculations while maintaining a reasonable computational cost.

Figures (9)

• Figure 1: Projected energy and norm surfaces in $^{24}$Mg as a function of the isovector and isoscalar pairing amplitudes $\delta^{T=1,0}_{pn}$ based on constrained HFB states associated with Set B. Fixed constraints $\langle J^{\mathrm{is}}_{x}\rangle = 0$ (left panels) and $\langle J^{\mathrm{is}}_{x}\rangle = 5$ (right panels) are employed. Panels: (a)-(b) VAPNP energy surfaces, (c)-(d) $J^\pi_\sigma=0^{+}_1$ projected norm overlap, (e)-(f) $0^{+}_1$ projected energy, (g)-(h) $1^{+}_1$ projected norm overlap and (i)-(j) $1^{+}_1$ projected energy.
• Figure 2: PGCM energies of the $0^{+}_{1}$ ground state and $1_{\sigma}^{+}$ excited states in $^{24}$Mg as a function of the dimension of the natural basis, relative to the exact ground-state energy. PGCM energies from Set A are represented by full lines, exact shell-model ones are indicated by blue diamonds.
• Figure 3: The same as in Fig. \ref{['fig:pgcm_ener_nat_setA']} but for Set B.
• Figure 4: Magnetic dipole transition probabilities, $B(M1; 1_{\sigma}^{+}\rightarrow 0^{+}_{1})$, as a function of the excitation energy $E(1^+_\sigma)$ in $^{24}$Mg. Exact diagonalization (blue bars) and PGCM calculations with Set A (red bars) using different maximum values for the cranking constraint are shown. The comparison of the excitation energies of the lowest one hundred $1_{\sigma}^{+}$ states is shown in the insets.
• Figure 5: The same as Fig. \ref{['fig:pgcm_bm1_setA']} but for Set B (green bars).