Perturbative cranking approximation

The perturbative cranking approximation (ATDHFB-C$^{\rm p}$) has been widely used for the evaluation of the collective mass tensor. In this approximation, apart from neglecting the time-odd interaction terms in the ATDHFB equation and off-diagonal matrix elements of the HFB energy matrix (40), the derivatives are not evaluated explicitly but are obtained using a perturbative approach. A complete description of the perturbative cranking model as applied to the nuclear fission process can be found in Refs. [17,18,19,20,21].

The perturbative cranking expression for the mass tensor is obtained by approximating the mean-field derivatives in Eq. (36) by canonical-basis expressions. For instance, the matrix element of $h^i$ can be approximated by

$$\langle\nu\vert h^i\vert\mu\rangle\approx (\breve{h}_{\mu}-\breve{h}_{\nu})\langle\nu\vert\partial_i\mu\rangle,$$ (54)

for $\mu\neq\nu$ which, together with

$$\langle\mu\vert{h}^i\vert\mu\rangle \approx \partial_i \breve{h}_{\mu} = \breve{h}^i_\mu,$$ (55)

$$\partial_i \breve{\Delta}_{\mu} = \breve{\Delta}^i_\mu,$$ (56)

for $\breve{h}_{\mu}\equiv\breve{h}_{\mu\mu}$ and $\breve{\Delta}_{\mu}\equiv-\breve{\Delta}_{\mu\bar{\mu}}s_{\bar\mu}^*$, leads to the following expression for the cranking mass tensor

$${\cal M}^{C^{\rm p}}_{ij} \approx \sum_{\mu \neq \nu} \frac... ...-_{\mu\nu})^2 + \sum_\mu \frac{F_\mu^i F_\mu^j}{2\breve E_\mu}$$ (57)

where

$$F_\mu^i \equiv F_{\mu\bar\mu}^i = -\frac{1}{2\breve{E}_\mu... ...ambda^i) - (\breve{h}_{\mu}-\lambda)\breve{\Delta}_{\mu}^i].$$ (58)

Assuming a weak state dependence of $\breve{\Delta}_\mu$ [22], neglecting the derivatives of $\Delta$ and $\lambda$ [21], and using the identity

$$\frac{\eta^-_{\mu\nu}}{\eta^+_{\mu\nu}}= \frac{(\breve{h}_{\... ...E}_\mu\breve{\Delta}_{\nu}+\breve{E}_\nu\breve{\Delta}_{\mu}},$$ (59)

one arrives at the following perturbative cranking mass tensor:

$${\cal M}^{C^{\rm p}}_{ij} \approx \sum_{\mu\nu} \frac{\langl... ...rangle} {(\breve{E}_\mu+\breve{E}_\nu)^3} (\eta^+_{\mu\nu})^2,$$ (60)

where the sums run over the whole set of canonical states. This expression resembles the standard cranking expression for the collective mass tensor [17,18,20,21] originally derived for a phenomenological mean field $h$.