Examples of previous cranking approaches

As argued in Sec. 3.1.7, there are good reasons to use in cranking calculations the subgroup $\{\hat{\cal{R}}_{l},\hat{\cal{S}}_{m}^T,\hat{\cal{P}}\}$(Table 1) of conserved D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ symmetries. This generic three-generator subgroup appears in three space orientations, i.e., for l=x, y, or z, and each of these possibilities was employed in one of the HF(B) or phenomenological-mean-field cranking analyses to date.

In particular, traditionally the x axis was chosen as the direction of the cranking angular momentum, see e.g. Ref. [11], and therefore, the standard Goodman basis [10] corresponds to the l=x subgroup, with phases of single-particle states (and quasiparticle states, for that matter) fixed by using the $\hat{\cal{R}}_{z}^T$ operator. Then, by dropping the parity operator from the symmetry group $\{\hat{\cal{R}}_{x},\hat{\cal{S}}_{z}^T,\hat{\cal{P}}\}$, most octupole-cranking calculations were performed within the 2-III_A subgroup $\{\hat{\cal{S}}_{x},\hat{\cal{S}}_{y}^T\}$ of Table 1.

Another choice was made in the HO-basis [9] and coordinate-space [12,13] HF(B) calculations, where the z axis was used as the cranking axis. Such choice was motivated by the standard representation of spinors, that are eigenstates of $\hat{\sigma}_z$, and hence the l=zsubgroup $\{\hat{\cal{R}}_{z},\hat{\cal{S}}_{y}^T,\hat{\cal{P}}\}$ was employed. In these approaches, phases of single-particle states were fixed by using the $\hat{\cal{S}}_{y}^T$operator, and the parity-broken calculations were done within the $\{\hat{\cal{S}}_{z},\hat{\cal{S}}_{y}^T\}$ subgroup.

Finally, in the recent Cartesian HO-basis HF approach of Ref. [14], the code HFODD was constructed for the conserved l=y subgroup $\{\hat{\cal{R}}_{y},\hat{\cal{S}}_{x}^T,\hat{\cal{P}}\}$, and the y direction was used for the cranking axis. The choice of this symmetry, and the resulting choice of the y cranking axis, was motivated by the fact that it allows for using real electric multipole moments, cf. Ref.[6]. Phases of single-particle states were in Ref. [14] fixed by using the $\hat{\cal{K}}_z$ operator (37), and calculations were performed within the basis of the $\hat{\cal{S}}_{y}$ eigenstates, Table 4. The HFODD code allows for calculations with one symmetry plane, and this is done within the $\{\hat{\cal{S}}_{y}\}$ conserved symmetry group of Table 1. The code can also optionally perform the two-symmetry-plane cranking calculations for the 2-III_Asubgroups $\{\hat{\cal{S}}_{y},\hat{\cal{S}}_{x}^T\}$ and $\{\hat{\cal{S}}_{y},\hat{\cal{S}}_{z}^T\}$.