Tilted-axis cranking

The tilted-axis cranking is realized by using in the energy functional the extended cranking term, cf. Eq. (I-23), which now reads

$${\cal E}^{\mbox{\scriptsize {cran}}} = \sum_{a=x,y,z}\left[... ...box{{\boldmath {$J$}}}}_{0}\rangle} {\omega_{J0}}\right)^2 .$$ (1)

The first two terms under the sum are simple generalizations of the standard linear and quadratic spin constraints to three dimensions. Different stiffness constants $C_{Ja}$ are allowed in three Cartesian directions $a$=$x,y,z$. These standard constraints act on the standard isoscalar (total) average angular momentum vector, $\langle\hat J_{0a}\rangle$= $\langle\hat J_{na}\rangle$+ $\langle\hat J_{pa}\rangle$, which is a sum of the neutron ($n$) and proton ($p$) component. The third term under the sum constitutes a linear constraint on the isovector average angular momentum vector, $\langle\hat J_{1a}\rangle$= $\langle\hat J_{na}\rangle$$-$ $\langle\hat J_{pa}\rangle$. It is introduced to facilitate the fixing of separate neutron and proton angular momentum vectors, which can be essential when trying to localize, e.g., the shears configurations. Of course, after a proper configuration is found, the final constraints should involve only the isoscalar component of the angular momentum vector. The last term in Eq. (1) constraints to zero the angle between the angular frequency and angular momentum vectors. Since in the self-consistent solutions this angle must be equal to zero [7], the last constraint helps to reach the self-consistent solution faster. However, since it is build upon the angular frequency vector pertaining to the linear isoscalar constraint, it cannot be used neither in conjunction with the quadratic nor with the isovector constraint.

In practical calculations, it turns out that the angle between the angular frequency and angular momentum converges to zero extremely slowly. This is so because for an angular frequency vector fixed in space, the whole nucleus must turn in space in order to align its angular momentum with the angular frequency. Therefore, a much faster procedure is to proceed in an opposite way, i.e., in each iteration to force the angular frequency to be aligned or anti-aligned with the current angular momentum vector (see switch IMOVAX in Sec. 3.5). This is a purely heuristic procedure, because it does not correspond to a minimization of any given Routhian. However, once the self-consistent solution is found (the angular frequency and angular momentum vectors become parallel or antiparallel) it is the Routhian for the final angular frequency which has taken the minimum value.