Isospin nonconservation in $sd$-shell nuclei

The question of isospin-symmetry breaking in nuclei of the $sd$ shell is addressed. We propose a new global parametrization of the isospin-nonconserving shell-model Hamiltonian which accurately describes experimentally known isobaric mass splittings. The isospin-symmetry violating part of the Hamiltonian consists of the Coulomb interaction and effective charge-dependent forces of nuclear origin. Particular attention has been paid to the effect of the short-range correlations. The behavior of $b$ and $c$ coefficients of the isobaric-mass-multiplet equation (IMME) is explored in detail. In particular, a high-precision numerical description of the staggering effect is proposed and contribution of the charge-dependent forces to the nuclear pairing is discussed. The Hamiltonian is applied to the study of the IMME beyond a quadratic form in the $A=32$ quintet, as well as to calculation of nuclear structure corrections to superallowed $0^{+}\to 0^{+}$ Fermi $\beta$ decay and to amplitudes of Fermi transitions to nonanalog states in $sd$-shell nuclei.

Figure 1

Harmonic oscillator energy spacing, $\hslash \omega$. (i) Blomqvist-Molinari formula [54] (solid black line), (ii) Blomqvist-Molinari formula with an additional scaling factor given by Eq. (3.7) of Ref. [21] (double-dot-dashed blue line); (iii) Blomqvist-Molinari formula refitted by Kirson [55] (dashed red line); (iv) $\hslash \omega$ deduced from the experimental nuclear charge radii [56] following the procedure indicated in Ref. [55] with ${\hslash }^2/m=41.458$ MeV fm${}^2$ (black dots). Only $A=2,...,60$ is indicated.

Figure 2

$V_{\text{coul}}\left(r\right)$ without SRCs (dotted black line), $V_{\text{coul}}\left(r\right)$ with proposed parameters for Jastrow-type SRC function on the basis of coupled-cluster calculation with CD-Bonn (double-dashed red line) or with Argonne V18 [48] (solid blue line), $V_{\text{coul}}\left(r\right)$ with Miller-Spencer parametrized Jastrow-type SRC function [57] (dashed purple line); $V_{\text{coul}}\left(R_{+}^I\left(r\right)\right)$ from UCOM (dot-dashed green line), $V_{\text{coul}}\left(R_{+}^{II}\left(r\right)\right)$ from UCOM (double-dot-dashed brow line). The inset enlarges the left-hand part of this figure.

Figure 3

Average Coulomb strength parameter, ${\overline{\lambda }}_{\text{coul}}$, as obtained from the fit with the USD interaction to the Range I data selection (see Sec. 3a for details). Down (blue) triangles correspond to the fit with Coulomb force alone. The ${\overline{\lambda }}_{\text{coul}}$ obtained from $V_{\text{coul}}$ and $V_{\pi }$ are depicted by (purple) dots, and up (black) triangles are ${\overline{\lambda }}_{\text{coul}}$ obtained from $V_{\text{coul}}$ and $V_{\rho }$ combination, whereas (red) squares represent ${\overline{\lambda }}_{\text{coul}}$ from a fit with the $V_{\text{coul}}$ and $V_0$ combination of the two-body charge-dependent forces. $y$-axis tic labels: 1, “without SRC”; 2, Miller-Spencer; 3, CD-Bonn; 4, AV-18; 5, UCOM.

Figure 4

Experimental and theoretical $\left|b\right|$ coefficients in $sd$-shell nuclei plotted as a function of $A$. Minor $x$-axis tics are $J$ states, which are arranged in an increment of 0.05 for every $\frac{1}{2}$ step.

Figure 5

Experimental and theoretical $\left|b\right|$ coefficients in $sd$-shell nuclei plotted as a function of $A^{2/3}$. Dot-dashed line is $\left|b\right|=\frac{3e^2(A-1)}{5r_0{A}^{1/3}}$, double-dot-dashed line is $\left|b\right|=\frac{3e^2}{5r_0}{A}^{2/3}$, the dashed line represent $b$ values from Möller and Nix model Eq. (29).

Figure 6

Experimental and theoretical $c$ coefficients plotted as a function of $A$. Minor $x$-axis tics are $J$ states, which are arranged in an increment of 0.05 for every $\frac{1}{2}$ step.

Figure 7

Experimental and theoretical $c$ coefficients in $sd$-shell nuclei plotted as a function of $A^{-1/3}$. The black dashed line represents $c$ coefficients from a charged sphere model, while the double-dot-dashed line shows the prediction according to Möller and Nix model [Eq. (30)].

Figure 8

Staggering effect of the $b$ coefficients of the ground-state doublets and of the lowest-lying quartets in $sd$-shell nuclei. Plot (a): $\left|b\right|$ values of the ground-state doublets (left $y$ axis) and the lowest-lying quartets (right $y$ axis). For $T=1/2$, the solid (blue) line $b=0.7447A^{2/3}-1.2551$ (MeV), the dashed (black) line $0.7442A^{2/3}-1.1987$, and the dot-dashed (black) line $b=0.7463A^{2/3}-1.3329$ (MeV), represent best fits to experimental values for all the $A$, $A=4n+3$ and $A=4n+1$ isobars, respectively. For $T=3/2$, the double-dot-dashed line $b=0.7441A^{2/3}-1.2547$ (MeV) is obtained by a fit to all experimental $b$ coefficients. Plots (b)–(d): deviations of $b$ coefficients from the respective fitted to all data points lines. The experimental $b$ values are represented by (blue) triangles and $b$ obtained in a shell-model fit are shown by (red) squares.

Figure 9

Contributions of the various charge-dependent forces to doublet ($T=1/2$) $b$ coefficients. The $\left|b\right|$ values are plotted as a function of $A^{2/3}$. The total $\left|b\right|$ values refer to the left $y$ axis, while contributions from $V_{\text{coul}}^{\left(1\right)}$, from $V_0^{\left(1\right)}$, and from the summed ISPE ${\sum }_i{\epsilon }_i^{\left(1\right)}$, refer to the right $y$ axis. The contribution from $V_0^{\left(1\right)}$ is multiplied by 20.

Figure 10

Contributions of the various charge-dependent forces to the lowest-lying quartet ($T=3/2$) $b$ coefficients. Refer to Fig. 9 for further description.

Figure 11

Staggering effects of $c$ coefficients of the lowest-lying triplets [plot (a)], and quartets and quintets [plot (b)]. See text for discussion. In the bottom panel, the dot-dashed (black) line, $c=651.6A^{-1/3}+2.6$ (keV), is an unweighted fit to the experimental $c$ coefficients. The solid (blue) line, $c=520.7A^{-1/3}+44.1$ (keV), is an unweighted fit to the $c$ coefficients obtained in a shell-model fit.

Figure 12

Contributions of the various charge-dependent forces to the lowest-lying triplet $c$ coefficients. All $c$ values refer to the left $y$ axis, while the contributions from $V_{\text{coul}}^{\left(2\right)}$ and $V_0^{\left(2\right)}$ refer to the right $y$ axis.

Figure 13

Contributions of the various charge-dependent forces to the lowest-lying quintet $c$ coefficients. All $c$ values and the contribution from $V_{\text{coul}}^{\left(2\right)}$ refer to the left $y$ axis, while the contribution from $V_0^{\left(2\right)}$ refers to the right $y$ axis.

Figure 14

Deviations of experimental or theoretical masses of the $A=32$ quintet from the corresponding quadratic IMME fit. (Purple) squares are quoted from Ref. [82]. (Purple) solid squares are quoted from Ref. [84]. (Green) circles are quoted from Ref. [85], Table , set A; (green) solid circles are from set B; up (blue) triangles are from set C; up (blue) solid triangles are from set D; down (black) triangles are from set E; down (black) solid triangles are from set F. The recent theoretical work of Signoracci and Brown [95] is presented as (red) pentagons, whereas (red) solid pentagons are the present calculation.

Figure 15

Deviations of experimental or theoretical masses of the $A=32$ quintet from the corresponding cubic IMME fit. Refer to Fig. 14 for description.

Figure 16

Comparison of $\mathcal{F}t$ values in $sd$-shell space. Plot (a) depicts experimental $ft$ values. The $ft$ value of ${}^{38}{K}^m$ is not shown; it is 3051.9 (10) s. Plot (b) shows $\mathcal{F}t$ values. The horizontal (gray) strip is 1 standard deviation according to Ref. [5].