Cranking approximation

In most of the studies, the time-odd interaction matrix $E_1$ appearing in Eq. (27) is neglected. In the following, this approximation will be referred to as the cranking approximation (ATDHFB-C). In the absence of the term involving $E_1$, the $Z$-matrix can be easily obtained in the quasiparticle basis from the equation:

$$-i F^i_{\mu \nu} = (E_\mu + E_\nu) {Z}^i_{\mu \nu}\,\,\,$$ (33)

and the collective cranking mass tensor is given by:

$${\cal M}^C_{ij} = \frac {1} {2 {\dot q_i} {\dot q_j}} \sum... ...{i}_{\mu \nu} F^{j\ast}_{\mu \nu} \right) } {E_\mu + E_\nu}.$$ (34)

It should be noted that Eq. (33) is diagonal in the quasiparticle basis and not in the canonical basis. The essential input to the ATDHFB-C mass tensor (34) is the matrix $F$. In the following, $F$ is evaluated in both canonical and quasiparticle basis.