Skyrme random-phase-approximation description of spin-flip $M1$ giant resonance

The spin-flip $M1$ giant resonance is explored in the framework of the random-phase-approximation (RPA) on the basis of the Skyrme energy functional. A representative set of eight Skyrme parametrizations ($\mathrm{SkT}6$, $\mathrm{SkM}*$, $\mathrm{SLy}6$, $\mathrm{SG}2$, $\mathrm{SkO}$, $\mathrm{SkO}’$, $\mathrm{SkI}4$, and $\mathrm{SV}$-bas) is used. Light and heavy, spherical and deformed nuclei (${}^{48}\mathrm{Ca}$, ${}^{158}\mathrm{Gd}$, ${}^{208}\mathrm{Pb}$, and ${}^{238}\mathrm{U}$) are considered. The calculations show that spin densities play a crucial role in forming the collective shift in the spectrum. The interplay of the collective shift and spin-orbit splitting determines the quality of the description. None of the considered Skyrme parametrizations is able to describe simultaneously the $M1$ strength distribution in closed-shell and open-shell nuclei. It is found that the problem lies in the relative positions of proton and neutron spin-orbit splitting. This calls for a better modeling of the tensor and isovector spin-orbit interaction.

Figure 1

Collective shifts of the $M1$ peak for different Skyrme parametrizations (as indicated along the $x$ axis) in ${}^{48}\mathrm{Ca}$, ${}^{208}\mathrm{Pb}$, ${}^{158}\mathrm{Gd}$, and ${}^{238}\mathrm{U}$. The plots exhibit the total (black boxes) shifts as well as the partial ones with ${\tilde{b}}_0$ and ${\tilde{b}}_3$ (open circles), ${\tilde{b}}_2$ (open triangles), and ${\tilde{b}}_1$ (stars). The ${\tilde{b}}_1$ contribution exists only for SkT6, SG2, and SkO’. For better viewing, the symbols are connected by lines.

Figure 2

(a) Spin-flip $M1$ resonance in ${}^{238}\mathrm{U}$ calculated with SG2 force: total (bold solid line), proton (dotted line), and neutron (solid line) strengths. The experimental data [15] are given by black boxes with bars. (b)–(d) Proton and neutron unperturbed spin-orbit splittings for different Skyrme forces in ${}^{158}\mathrm{Gd}$, ${}^{238}\mathrm{U}$, and ${}^{208}\mathrm{Pb}$.

Figure 3

Spin-flip $M1$ resonance in ${}^{48}\mathrm{Ca}$ (left) and ${}^{208}\mathrm{Pb}$ (right) calculated within SRPA for eight Skyrme forces as indicated. For better distinction, the strength in ${}^{208}\mathrm{Pb}$ for SG2, SLy6, SkT6 and SkM* is depicted by the bold line. The experimental energies in ${}^{48}\mathrm{Ca}$ [17] and ${}^{208}\mathrm{Pb}$ [18] are marked by vertical arrows.

Figure 4

Spin-flip $M1$ resonance in ${}^{158}\mathrm{Gd}$ described with eight Skyrme parametrizations as indicated. The experimental data are from Refs. [15, 16].

Figure 5

Same as Fig. 4, but for ${}^{238}\mathrm{U}$.

Figure 6

Dependence of unperturbed spin-orbit splitting for proton $\pi (h_{11/2}^{-1},h_{9/2})$ and neutron $\nu (i_{13/2}^{-1},i_{11/2})$ configurations in ${}^{208}\mathrm{Pb}$ on (a) the attenuation $0⩽{\gamma }_T⩽1$ for tensor interaction and (b) the parameter $b_4^{\text{'}}$ of isovector spin-orbit interaction. Skyrme parametrizations with varied ${\gamma }_T$ or $b_4^{\text{'}}$ are used [5]. The proton (neutron) splittings are marked by circles (boxes). Filled symbols mark results for parametrizations refitted for given ${\gamma }_T$ or $b_4^{\text{'}}$. Open symbols stand for SV-bas, where only ${\gamma }_T$ or $b_4^{\text{'}}$ are varied, i.e., without refitting.