Microscopic analysis of dipole electric and magnetic strengths in $^{156}$Gd

Abstract

The dipole electric (E1) and magnetic (M1) strengths in strongly deformed $^{156}$Gd are investigated within a fully self-consistent Quasiparticle Random Phase Approximation (QRPA) with Skyrme forces SVbas, SLy6 and SG2. We inspect, on the same theoretical footing, low-lying dipole states and the isovector giant dipole resonance in E1 channel and the orbital scissors resonance as well as the spin-flip giant resonance (SFGR) in M1 channel. Besides, E1 toroidal mode and low-energy spin-flip M1 excitations are considered. The deformation splitting and dipole-octupole coupling of electric excitations are analyzed. The origin of SFGR gross structure, impact of the residual interaction and interference of orbital and spin contributions to SFGR are discussed. The effect of the central exchange ${\textbf {J}}^2$-term from the Skyrme functional is demonstrated. The calculations show a satisfactory agreement with available experimental data, except for the recent NRF measurements of M. Tamkas et al for M1 strength at 4–6 MeV, where, in contradiction with our calculations and previous $(p,p')$ data, almost no M1 strength was observed.

Appendix A: Skyrme functional

The Skyrme functional has the form [12, 48, 49]

$$\begin{aligned} \mathcal {H}_\textrm{Sk}= & {} \frac{b_0}{2} \rho ^2- \frac{b'_0}{2} \sum \rho _{q}^2 + \frac{b_3}{3} \rho ^{\alpha +2} - \frac{b'_3}{3} \rho ^{\alpha } \sum \rho ^2_q \nonumber \\{} & {} +b_1 (\rho \tau - {\textbf {j}}^{\;2}) - b'_1 \sum (\rho _q \tau _q - {\textbf {j}}^{\;2}_q) \nonumber \\{} & {} - \frac{b_2}{2} \rho \Delta \rho + \frac{b'_2}{2} \sum \rho _q \Delta \rho _q \nonumber \\{} & {} - b_4 (\rho \nabla \cdot {\textbf {J}}\!+\!(\nabla \!\times \!{\textbf {j}}) \!\cdot \! {\textbf {s}}) \nonumber \\{} & {} - b'_4 \sum (\rho _q \nabla \cdot {\textbf {J}}_q\! +\!(\nabla \!\times \!{\textbf {j}}_q) \!\cdot \! {\textbf {s}}_q) \nonumber \\{} & {} + \frac{\tilde{b}_0}{2} {\textbf {s}}^{\;2} - \frac{\tilde{b}'_0}{2} \sum {\textbf {s}}_{q}^{\;2} + \frac{\tilde{b}_3}{3} \rho ^{\alpha } {\textbf {s}}^{\;2} - \frac{\tilde{b}'_3}{3} \rho ^{\alpha } \sum {\textbf {s}}^{\;2}_q \nonumber \\{} & {} -\frac{\tilde{b}_2}{2} {\textbf {s}} \!\cdot \! \Delta {\textbf {s}} + \frac{\tilde{b}'_2}{2} \sum {\textbf {s}}_q \!\cdot \!\Delta {\textbf {s}}_q \nonumber \\{} & {} + \tilde{b}_1 ({\textbf {s}}\!\cdot \!{\textbf {T}}\!-\!{\textbf {J}}^{2}) + \tilde{b}'_1 \sum ({\textbf {s}}_q\!\cdot \!{\textbf {T}}_q \!-\!{\textbf {J}}_q^{2}). \end{aligned}$$

(11)

Here $b_i$, $b'_i$, $\tilde{b}_i$, $\tilde{b}'_i$ are the force parameters. The relation of these parameters with standard ones can be found in Refs. [12, 48, 49, 73]. Functional (11) includes time-even (nucleon $\rho _q$, kinetic-energy $\tau _q$, spin-orbit ${\textbf {J}}_q$) and time-odd (current ${\textbf {j}}_{ q}$, spin ${\textbf {s}}_q$, spin kinetic-energy ${\textbf {T}}_q$) densities. The label $q\in \{p,n\}$ q denotes protons and neutrons. Densities without index q, like $\rho = \rho _p + \rho _n$, denote total densities. The contributions with $b_i$ (i = 0, 1, 2, 3, 4) and $b'_i$ (i = 0, 1, 2, 3) are the standard terms responsible for ground state properties and electric excitations of even-even nuclei [48]. The isovector spin-orbit interaction is usually linked to the isoscalar one by $b'_4=b_4$. The terms with $\tilde{b}_i,\tilde{b}'_i$ (i = 0, 1, 2, 3) represent the spin–isospin channel important for odd nuclei and magnetic modes in even-even nuclei. The last line of (11) includes tensor spin-orbit terms $\propto \tilde{b}_1,\tilde{b}'_1$ which can affect both mean-field ground-state properties and magnetic modes. In general, these terms embrace contributions from both central-exchange interaction and non-central tensor interaction [54,55,56]. In our calculations, ${\textbf {J}}^2$ is taken into account only for SG2 and only with central exchange contribution. In this case, $\tilde{b}_i$ and $\tilde{b}'_i$ are fully expressed through the standard Skyrme parameters $t_1, t_2, x_1, x_2$ (see Eqs. (23)–(24) below) and have the values ${\tilde{b}}_1$= $-$ 40.1, ${\tilde{b}}'_1$ = $-$ 47.8.

The parameters $b_i$, $b'_i$, $\tilde{b}_i$, $\tilde{b}'_i$ are related with standard Skyrme parameters as [12, 48, 49, 73].

$$\begin{aligned} b_0= & {} t_0\left( 1 + \frac{1}{2} x_0\right) , \end{aligned}$$

(12)

$$\begin{aligned} b'_0= & {} t_0\left( \frac{1}{2} + x_0\right) , \end{aligned}$$

(13)

$$\begin{aligned} b_1= & {} \frac{1}{4}\left[ t_1\left( 1 + \frac{1}{2}x_1\right) + t_2\left( 1 + \frac{1}{2}x_2\right) \right] , \end{aligned}$$

(14)

$$\begin{aligned} b'_1= & {} \frac{1}{4}\left[ t_1\left( \frac{1}{2} + x_1\right) - t_2\left( \frac{1}{2} + x_2\right) \right] , \end{aligned}$$

(15)

$$\begin{aligned} b_2= & {} \frac{1}{8}\left[ 3t_1\left( 1 + \frac{1}{2}x_1\right) - t_2\left( 1 + \frac{1}{2}x_2\right) \right] , \end{aligned}$$

(16)

$$\begin{aligned} b'_2= & {} \frac{1}{8}\left[ 3t_1\left( \frac{1}{2} + x_1\right) + t_2\left( \frac{1}{2} + x_2\right) \right] , \end{aligned}$$

(17)

$$\begin{aligned} b_3= & {} \frac{1}{4}t_3\left( 1 + \frac{1}{2} x_3\right) , \end{aligned}$$

(18)

$$\begin{aligned} b'_3= & {} \frac{1}{4}t_3\left( \frac{1}{2} + x_3 \right) , \end{aligned}$$

(19)

$$\begin{aligned} b_4= & {} b'_4=\frac{1}{2}t_4, \end{aligned}$$

(20)

$$\begin{aligned} {\tilde{b}}_0= & {} \frac{1}{2} t_0x_0, \end{aligned}$$

(21)

$$\begin{aligned} {\tilde{b}}'_0= & {} \frac{1}{2} t_0 \end{aligned}$$

(22)

$$\begin{aligned} {\tilde{b}}_1= & {} \frac{1}{8}(t_1x_1 + t_2x_2), \end{aligned}$$

(23)

$$\begin{aligned} \tilde{b}'_1= & {} -\frac{1}{8}(t_1 - t_2), \end{aligned}$$

(24)

$$\begin{aligned} {\tilde{b}}_2= & {} \frac{1}{16}(3t_1x_1 - t_2x_2), \end{aligned}$$

(25)

$$\begin{aligned} \tilde{b}'_2= & {} \frac{1}{16}(3t_1 + t_2), \end{aligned}$$

(26)

$$\begin{aligned} {\tilde{b}}_3= & {} \frac{1}{8} t_3x_3, \end{aligned}$$

(27)

$$\begin{aligned} {\tilde{b}}'_3= & {} \frac{1}{8} t_3. \end{aligned}$$

(28)

As seen from Table 8, Skyrme parameters $t_1, t_2, x_1, x_2$, entering ${\textbf {J}}^2$ parameters ${\tilde{b}}_1$ and ${\tilde{b}}'_1$, are rather different for SVbas, SLy6 and SG2.