Definition of observables

Since pairing is neglected in this work, the charge quadrupole moments $Q_{20}$ and $Q_{22}$ are defined microscopically as sums of expectation values of the s.p. quadrupole moment operators and $\hat{q}_{22}$ of the occupied proton states, i.e.,

$$Q_{20} = \sum_\mu \left< \mu \vert \hat{q}_{20} \vert \mu \right>,$$ (1)

$$Q_{22} = \sum_\mu \left< \mu \vert \hat{q}_{22} \vert \mu \right>,$$ (2)

where and $\hat{q}_{22}$ are defined in three-dimensional Cartesian coordinates as [22] (conserved signature symmetry is assumed)

$$\hat{q}_{20} = 2z^2-x^2-y^2,$$ (3)

$$\hat{q}_{22} = \sqrt{3}(x^2-y^2).$$ (4)

The factor of $\sqrt{3}$ is included in the definition of $\hat{q}_{22}$ in order to have the following expressions for the total quadrupole moment $Q_2$ and the associated Bohr angle $\gamma$:

$$Q_2 = \sqrt{Q_{20}^2+Q_{22}^2},$$ (5)

$$\tan(\gamma) = Q_{22}/Q_{20}.$$ (6)

Note that the sums in Eqs. (1-2) run only over proton states. The neutrons, having zero electric charge, do not appear in the sums explicitly, but they influence the charge quadrupole moments indirectly via the quadrupole polarization (deformation changes) induced by occupying/emptying single-neutron states.

It should be noted that with the definitions (1-6), the spherical components of the quadrupole tensor are $Q_{20}$ and $Q_{22}/\sqrt{2}$. This fact is important for the definition of the so-called transition quadrupole moment $Q_t$ [23,24]. This moment gives the measure of the transition strength of the $\Delta I$=2 (stretched) $E2$ radiation in the limit of large deformation and angular momentum, and it is proportional to the component $Q^{\omega}_{22}/\sqrt{2}$ of the spherical quadrupole tensor when the quantization axis coincides with the vector of rotational velocity $\omega$, i.e.,

$$Q^{\omega}_{20} = D^2_{0, 0}(\psi^{\omega},\theta^{\omega},\phi^{\omega})Q_{20} + \... ...}(\psi^{\omega},\theta^{\omega},\phi^{\omega})\right] \frac{Q_{22}}{\sqrt{2}} ,$$ (7)

$$\frac{Q^{\omega}_{22}}{\sqrt{2}} = D^2_{2, 0}(\psi^{\omega},\theta^{\omega},\phi^{\omega})Q_{20} + \... ...2}(\psi^{\omega},\theta^{\omega},\phi^{\omega})\right] \frac{Q_{22}}{\sqrt{2}}.$$ (8)

Here, symbols $D^\lambda_{\mu\nu}$ denote the Wigner functions [25], with their arguments $\psi^{\omega},\theta^{\omega},\phi^{\omega}$ being the Euler angles that rotate the $z$ axis (the standard quantization axis for spherical tensors) onto the direction of the angular velocity.

For the cranking axis coinciding with the $y$-axis of the intrinsic system, as is the case for the code HFODD [26,27] used in the present study, the Euler angles are $\psi$=0, $\theta$=$\pi/2$, and $\phi$=$\pi/2$, which gives:

$$Q^{\omega\parallel{y}}_{20} = -{1\over 2}Q_{20} - \sqrt{{3\over 2}}\frac{Q_{22}}{\sqrt{2}},$$ (9)

$$\frac{Q^{\omega\parallel{y}}_{22}}{\sqrt{2}} = \sqrt{3\over 8}Q_{20} -{1\over 2}\frac{Q_{22}}{\sqrt{2}}.$$ (10)

The second of these equations gives the definition of the transition quadrupole moment used in this work:

$$Q^{\omega\parallel{y}}_{t} = \sqrt{8\over 3}\frac{Q^{\omega... ...sqrt{2}} = Q_{20} - \sqrt{{2\over3}} \frac{Q_{22}}{\sqrt{2}}.$$ (11)

In order to provide a link to studies that employ the $x$-axis cranking, like, e.g., Refs. [23,24] and our earlier papers [21,20], we repeat derivations for the Euler angles $\psi$=$\pi/2$, $\theta$=$\pi/2$, $\phi$=$\pi$, which rotate the $z$ axis onto the $x$ axis:

$$Q^{\omega\parallel{x}}_{20} = -{1\over 2}Q_{20} + \sqrt{{3\over 2}}\frac{Q_{22}}{\sqrt{2}},$$ (12)

$$\frac{Q^{\omega\parallel{x}}_{22}}{\sqrt{2}} = -\sqrt{3\over 8}Q_{20} -{1\over 2}\frac{Q_{22}}{\sqrt{2}},$$ (13)

hence

$$Q^{\omega\parallel{x}}_{t} = -\sqrt{8\over 3}\frac{Q^{\omeg... ...sqrt{2}} = Q_{20} + \sqrt{{2\over3}} \frac{Q_{22}}{\sqrt{2}}.$$ (14)

Although definitions (11) and (14) differ by signs of the second terms, values of $Q^{\omega\parallel{y}}_{t}$ and $Q^{\omega\parallel{x}}_{t}$ obtained in self-consistent calculations must be identical because they cannot depend on the direction of the cranking axis. It means that values of $Q_{22}$ obtained in cranking calculations along the $y$ and $x$ axes have opposite signs. In what follows, we employ definition (11) of the transition moment and drop the superscripts that denote the direction of the cranking axis, e.g.,

$$Q_{t} = Q_{20} - \sqrt{{1\over3}}\, Q_{22},$$ (15)

$$\hat{q}_{t} = \hat{q}_{20} - \sqrt{{1\over3}}\, \hat{q}_{22}.$$ (16)

Finally, the expectation value of the total angular momentum $J$ (its projection on the cranking axis) is defined as a sum of the expectation values of the s.p. angular momentum operators $\hat{j}_y$ of the occupied states

$$J\equiv \langle \hat{J}_y\rangle = \sum_\mu \left< \mu \vert \hat{j}_y \vert \mu \right> .$$ (17)

The value of $J$ can be expressed in terms of the total spin $I$ via the cranking relation [28]

$$J=\sqrt {I(I+1)} \approx I+\frac 12.$$ (18)