Energy minima in the $N=75$ isotones

Table: Quadrupole $\beta$ and $\gamma$ deformation parameters, parameters of the classical model, $\mathcal{J}_{s,m,l}$ $[\hbar^2/$MeV$]$, $s_{s,l}$ $[\hbar]$, classical estimates for the critical frequencies and spins, $\omega _{\text{crit}}^{\text{clas}}$ $[\mathrm{MeV}/\hbar]$ and $I_{\text{crit}}^{\text{clas}}$ $[\hbar]$, and the full HF TAC results for those quantities, $\omega _{\text{crit}}^{\text{HF}}$ $[\mathrm{MeV}/\hbar]$ and $I_{\text{crit}}^{\text{HF}}$ $[\hbar]$, for the $N=75$ isotones. The HF results with the SLy4 and SkM* forces are shown for the $N$, $G$, and $T$ variants of calculation defined in Sec. 3.2.

nucleus	force		$\beta$	$\gamma$	$\mathcal{J}_s$	$\mathcal{J}_m$	$\mathcal{J}_l$	$s_s$	$s_l$	$\omega^{\text{clas}}_{\text{crit}}$	$I^{\text{clas}}_{\text{crit}}$	$\omega^{\text{HF}}_{\text{crit}}$	$I^{\text{HF}}_{\text{crit}}$
$^{130}$Cs	SLy4	$N$	0.24	49	4.81	29.2	17.0	5.41	4.86	0.46	12.8
		$G$	0.24	49	5.50	37.1	21.3	5.45	5.16	0.37	13.2
		$T$	0.24	49	4.16	29.4	19.9	5.49	5.20	0.59	16.8
	SkM*	$N$	0.23	47	5.86	31.3	17.9	5.43	5.01	0.43	13.0
		$G$	0.23	47	6.55	36.7	21.1	5.47	4.97	0.37	13.0
		$T$	0.23	47	5.69	33.4	20.0	5.49	5.14	0.43	13.9
$^{132}$La	SLy4	$N$	0.26	46	7.18	28.7	19.1	5.44	4.90	0.57	15.9	0.68	18.8
		$G$	0.26	46	8.45	36.0	23.7	5.60	5.21	0.47	16.4	0.60	20.3
		$T$	0.26	46	7.12	31.7	22.2	5.64	5.26	0.60	18.5
	SkM*	$N$	0.25	45	8.19	30.8	20.3	5.47	5.03	0.54	16.0	0.62	17.8
		$G$	0.25	45	8.81	35.9	23.5	5.60	5.06	0.46	15.9	0.54	17.8
		$T$	0.25	45	8.37	34.0	22.4	5.63	5.21	0.50	16.5	0.58	18.5
$^{134}$Pr	SLy4	$N$	0.26	58	6.11		25.4	5.00	4.36
	oblate	$G$	0.26	58	7.21		31.3	4.92	4.84
		$T$	0.26	56	3.68		29.8	5.26	4.60
	SkM*	$N$	0.23	22	18.5	28.1	20.7	5.38	5.28	0.91	25.0
	triaxial	$G$	0.23	22	21.4	32.6	24.3	5.46	5.44	0.82	26.1
		$T$	0.23	22	20.7	30.8	24.8	5.52	5.57	1.08	32.7
$^{136}$Pm	SLy4	$N$	0.25	53			24.7		4.62
	oblate	$G$	0.25	53			30.3		5.38
		$T$	0.25	52			29.0		5.05
	SkM*	$N$	0.22	19	15.0	27.5	12.4	5.32	5.38	0.56	14.8
	triaxial	$G$	0.22	19	17.9	33.2	13.0	5.50	5.42	0.45	14.4
		$T$	0.22	19	17.4	29.8	13.4	5.51	5.53	0.56	16.1

As the first step, the HF calculations without cranking were performed to find the $\pi{}h_{11/2}^1~\nu{}h_{11/2}^{-1}$ bandheads. Obviously, the energetically most favored state of this configuration is obtained if the valence proton particle and neutron hole occupy the lowest and highest levels of the $h_{11/2}$ multiplets, respectively. Table 1 gives the obtained $\beta$ and $\gamma$ deformations for each isotone and Skyrme parameter set. They practically do not depend on the included time-odd terms, and remain almost constant when cranking is applied later on in our calculations. Note that the present values of $\beta$ are up to 1.5 times larger than those found in the PQTAC/SCTAC calculations quoted in Section 3.1. Also the values of $\gamma$ are more distant from the maximum triaxiality of $30^\circ$ as compared to the earlier results by other authors.

In $^{134}$Pr and $^{136}$Pm, two minima with the same $\pi h_{11/2}^1~\nu h_{11/2}^{-1}$ configuration were found, which differ by the occupation of positive-parity states. The energetically lower minima have similar positive-parity s.p. structure as in $^{130}$Cs and $^{132}$La, but they correspond to almost oblate shapes of $\gamma=53^{\circ}$-to- $58^{\circ}$. The other minima have $\gamma=19^{\circ}$-to- $22^{\circ}$, thus corresponding to triaxial shapes. In the following, those two kinds of minima in $^{134}$Pr and $^{136}$Pm are conventionally referred to as oblate and triaxial. Such a structure of minima and configurations appears for both interactions studied here, SkM* and SLy4, and the corresponding sets of results are very similar to one another. Therefore, to save space, below only the SkM* results are shown for the triaxial minima, and only the SLy4 results for the oblate ones.

In Section 2, we introduced the intrinsic frame of a nucleus as formed by the principal axes of the quadrupole tensor. Below, by the short ($s$), medium ($m$), and long ($l$) axes of our triaxial solutions we understand the intrinsic axes indexed so that $\langle x_s^2\rangle<\langle x_m^2\rangle<\langle x_l^2\rangle$, where $x_i$ is the Cartesian coordinate for axis $i=s$, $m$, or $l$.