Separability of the TAC rotation

To complete the presentation of our results, we here formulate and discuss the separability rule, by which our planar (2D) and chiral (3D) HF TAC solutions turn out to be simple superpositions of independent HF PAC (1D) rotations about 2 or 3 principal axes, respectively.

Figure: Comparison of the full TAC (points) total angular-momentum alignments (lower part) and Routhians (upper part) with analogous quantities obtained within the PAC approach, as discussed in the text (lines). The alignments $I_i$ on the principal axes $i=s,m,l$ are shown as functions of the corresponding components, $\omega _i$, of the angular frequency. The increments, $\Delta H'$ (27) and $\Delta\tilde{H}'$ (26), in the full TAC and equivalent PAC Routhians with respect to $\omega =0$ are plotted in function of the total frequency $\omega$. In the latter plots, a quadratic function of $\omega$ is added to stretch the scale. The HF results for the planar (left) and chiral (right) solutions in $^{132}$La are shown for the $T$ variant of the calculations (see Sec. 3.2) with the SkM* force.

Take, for instance, the total alignments on the principal axes, $I_i(\omega_i)$, where $i=s,m,l$, as functions of the corresponding components of the angular frequency. Such quantities were examined in Section 3.4 for the PAC calculations, but they can be equally well extracted for the planar and chiral solutions. A comparison of the PAC and TAC results in $^{132}$La is shown in the lower part of Fig. 11 for the $T$ variant of calculations with the SkM* force. The PAC values of $I_s$, $I_m$, and $I_l$ are plotted with lines, respectively: dotted, solid, and dashed. The TAC results are marked with symbols, respectively: down-triangles, diamonds, and up-triangles. The left and right parts of the Figure show the TAC values of $I_i(\omega_i)$ for the planar and chiral solutions, respectively. It is obvious that the results obtained for the rotation about tilted axes are almost identical to those obtained from the independent PAC calculations.

Unlike the angular momentum, the average value of the total TAC Routhian, $\langle\hat{H}'\rangle$ of Eq. (1), cannot be trivially decomposed into contributions from rotations about the three principal axes. Yet, for each TAC solution, characterized by an angular frequency $\omega$, we have its three intrinsic components, $\omega _i$, and we can consider a sum, $\tilde{H}'$, of the corresponding values of the PAC Routhians, $\langle\hat{H}'_i\rangle(\omega_i)$, i.e.,

$$\tilde{H}'(\omega) = \sum_{i=s,m,l}\langle\hat{H}'_i\rangle(\omega_i)~,$$ (25)

which we call equivalent Routhian. We then compare the difference relative to its value at $\omega$=0,

$$\Delta\tilde{H}'(\omega)=\tilde{H}'(\omega)-\tilde{H}'(0),$$ (26)

with the analogous difference,

$$\Delta H'(\omega)=\langle\hat{H}'\rangle(\omega)-\langle\hat{H}'\rangle(0),$$ (27)

computed for the full TAC Routhian. In the upper part of Fig. 11, the differences (26) and (27) are plotted in function of $\omega$ as lines and points, respectively.

For the chiral solution, as $\langle\hat{H}'\rangle(0)$ we take the same value as for the planar case, because the chiral band can be regarded as a continuation of the planar one, as discussed in Section 3.7. The equivalent Routhian can be plotted only in such a frequency range in which the PAC solutions are obtained for the corresponding components $\omega _i$. In that range, $\Delta\tilde{H}'$ deviates from $\Delta H'$ by not more than 30keV, as it can be seen from the Figure, whereas the total Routhian itself drops by about 8MeV between $\omega =0$ and 0.6MeV/$\hbar$. Therefore, we conclude that the equivalent Routhian, constructed out of the PAC solutions, reproduces the full TAC Routhian to a very high accuracy.

Another piece of information that is contained in the TAC results, and not directly in the PAC results, is how the total angular frequency, $\omega$, should be distributed over the three principal axes for each given point of a band representing the rotation about a tilted axis. That, however, can also be determined from the PAC solutions by means of the Kerman-Onishi requirement [37] (see Section 2) that the angular-momentum vector must be parallel to the angular-frequency vector. Indeed, this condition is equivalent to the system of three non-linear equations,

$$\begin{array}{rcl} I_s(\omega_s)/\omega_s &=& \mu , \\ I_m(\o... ...ega_m &=& \mu , \\ I_l(\omega_l)/\omega_l &=& \mu . \end{array}$$ (28)

By plotting the three PAC functions $I_i(\omega_i)/\omega_i$ on the same plot as functions of $\omega _i$ we obtain a planar or chiral solution whenever two or three, respectively, of these functions cross a horizontal line. Then, values of $\omega _i$ corresponding to the crossing points define the requested distribution of the components within the total angular frequency $\omega$. The agreement of the PAC and TAC alignments shown in Fig. 11 guarantees that the above procedure gives a correct result.

We have thus demonstrated that within our HF results, the general 2D or 3D rotation separates into two or three, in a sense independent, 1D rotations about the principal axes. This is possible mainly because our solutions are very stiff against deformation changes with rotational frequency. Had they been soft, rotations about different (principal or tilted) axes could cause different shape polarizations that would prevent such a simple superposition of motions. Yet, whenever the motions are separable, the PAC calculations supplemented with the Kerman-Onishi condition are actually sufficient to describe the 2D or 3D rotations. Although this rule has to be, in principle, confirmed numerically in each particular case, it is plausible that it will hold in all analogous cases of stiff alignments, by which the difficult TAC calculations can be replaced by much more easily performed PAC calculations.