Finite-amplitude method for charge-changing transitions in axially deformed nuclei

We describe and apply a version of the finite amplitude method for obtaining the charge-changing nuclear response in the quasiparticle random-phase approximation. The method is suitable for calculating strength functions and beta-decay rates, both allowed and forbidden, in axially-deformed open-shell nuclei. We demonstrate the speed and versatility of the code through a preliminary examination of the effects of tensor terms in Skyrme functionals on beta decay in a set of spherical and deformed open-shell nuclei. Like the isoscalar pairing interaction, the tensor terms systematically increase allowed beta-decay rates. This finding generalizes previous work in semimagic nuclei and points to the need for a comprehensive study of time-odd terms in nuclear density functionals.

Figure 1

Schematic representation of the integration contour $C$ used to evaluate beta-decay rates. Only the poles of the strength function below the endpoint energy ${\omega }_{\max }$ contribute to the decay rate.

Figure 2

A 14th-order polynomial approximation to the phase space integral $f\left(E_0\right)$ [Eq. (23)] for the beta decay of ${}^{148}\mathrm{Ba}$. The solid line is calculated with the exact Fermi function $F_0$ [Eq. (24)] and the points correspond to the polynomial approximation (30).

Figure 3

The imaginary part of the integrand in Eq. (31) that determines the $K=0$ allowed decay rate of ${}^{142}\mathrm{Ba}$, with SkO and the tensor interaction. The integrand behaves well enough to allow simple quadrature. The origin in the complex plane corresponds to $t=\pi$ (see text).

Figure 4

Comparison of the pnFAM Gamow-Teller strength function (points) in the deformed nucleus ${}^{22}\mathrm{Ne}$ with the same function from the matrix QRPA (lines), smeared with a Lorentzian. We use the Skyrme functional SkM* without including ${\mathbb{J}}^2$ terms or pairing in the QRPA.

Figure 5

Partial Gamow-Teller half-lives in several nuclei as a function of rotational energy correction, normalized to the uncorrected values. The marker on each curve indicates the correction obtained from Eq. (21).

Figure 6

The ratio of computed and experimental partial Gamow-Teller half-lives. Two functionals are used: the bare SkO functional and the SkO functional with an added tensor piece. Isoscalar pairing is absent here.

Figure 7

Gamow-Teller strength functions in several isotopes. The solid (blue) lines represent the strength without tensor terms and the dashed (red) lines the strength with those terms.

Figure 8

(a) Ratio of calculated to experimental half-lives with the functional ${\mathrm{SkO}}^{\prime }\mathrm{and}$ first-forbidden contributions included. The bare ${\mathrm{SkO}}^{\prime }$ half-lives (solid blue line) are systematically longer than experiment and can be reduced by introducing either isoscalar pairing (dashed green line), a full tensor interaction (dashed red line), or only the time-odd components associated with that interaction (dotted line). (b) Allowed and first-forbidden contributions to the decay rates associated with panel a). Left (blue) bars are rates computed without tensor terms, and right (red) ones the rates with tensor terms in both time-odd and time-even channels. The darker upper parts correspond to first-forbidden contributions and the lower lighter parts to the allowed ones. Values are normalized to the total rate without tensor terms in each isotope.

Figure 9

The effects of a tensor force from a microscopic $G$ matrix.