Electromagnetic transition strengths in soft deformed nuclei

Spectroscopic observables such as electromagnetic transition strengths can be related to the properties of the intrinsic mean-field wave function when the latter are strongly deformed, but the standard rotational formulas break down when the deformation decreases. Nevertheless there is a well-defined, nonzero, spherical limit that can be evaluated in terms of overlaps of mean-field intrinsic deformed wave functions. We examine the transition between the spherical limit and the strongly deformed limit for a range of nuclei, comparing the two limiting formulas with exact projection results. We find a simple criterion for the validity of the rotational formula depending on $\langle \Delta {\vec{J}}^2\rangle$, the mean square fluctuation in the angular momentum of the intrinsic state. We also propose an interpolation formula which describes the transition strengths over the entire range of deformations, reducing to the two simple expressions in the appropriate limits.

Figure 1

The ratio $B{\left(E2\right)}_{\mathrm{ROT}}/B{\left(E2\right)}_{\mathrm{PROJ}}$ is plotted as a function of the deformation parameter ${\beta }_2$ for a range of nuclei. The solid line connects calculated values. The dashed line is calculated from the interpolating formula, Eqs. (26) and (27).

Figure 2

Lowest $\langle \Delta {\vec{J}}^2\rangle$ values of intrinsic wave functions that meet our criterion for using the rotational formula (see text).

Figure 3

The spherical-limit transition strengths of Eq. (18) are displayed for several isotopic chains as a function of neutron number $N$ in panel (a) and as a function of Z in panel (b). The isotopic chains correspond to $Z$ values between 12 and 94 in steps of six units. Strengths are given in Weisskopf units, 1 W.u. $=$ $5.94\times {10}^{-6}{A}^{4/3}{e}^2{\text{b}}^2$.

Figure 4

Accuracy of Eq. (23) to calculate $B\left(E2\right)$ values in the spherical limit. Plotted is the ratio of the $B\left(E2\right)$ from Eq. (23) to the value obtained by a full projection of the wave functions at ${\beta }_2=0.005$. The horizontal scale gives the ${\beta }_2$ values used in Eq. (23). The ratios are offset for clarity, with the dotted lines indicating equal values.

Figure 5

The ratio $B{(E3,3^{-}\to 0^{+})}_{|\mathrm{PROJ}}/B{(E3,3^{-}\to 0^{+})}_{|\mathrm{ROT}}$ is plotted as a function of the ${\beta }_2(+)$ quadrupole deformation parameter of the positive-parity intrinsic state.