Statistical theory of deformation distributions in nuclear spectra

The dependence of the nuclear level density on intrinsic deformation is an important input to dynamical nuclear processes such as fission. The auxiliary-field Monte Carlo (AFMC) method is a powerful method for computing state densities. However, the statistical distribution of intrinsic shapes is not readily accessible due to the formulation of the AFMC method in a spherical configuration-interaction shell-model approach. Instead, the theory of deformation has largely relied on a mean-field approximation which breaks rotational symmetry. We show here how the distributions of the intrinsic quadrupole deformation parameters can be calculated within the AFMC method and present results for a chain of even-mass samarium nuclei (${}^{148}\mathrm{Sm}, {}^{150}\mathrm{Sm}, {}^{152}\mathrm{Sm}, {}^{154}\mathrm{Sm}$) which includes spherical, transitional, and strongly deformed isotopes. The method relies on a Landau-like expansion of the Helmholtz free energy in invariant polynomials of the quadrupole tensor. We find that an expansion to fourth order provides an excellent description of the AFMC results.

Figure 1

The laboratory-frame axial quadrupole distribution $P\left(q_{20}\right)$ for ${}^{154}\mathrm{Sm}$ at three distinct temperatures: (a) a high temperature, $T=4$ MeV; (b) an intermediate temperature, $T=1.1$ MeV, near the shape transition; and (c) a low temperature, $T=0.07$ MeV. Solid lines are the marginal distributions $P\left(q_{20}\right)$ obtained from the Landau-like expansion of the intrinsic shape distribution [Eq. (18)], with parameters $a, b$, and $c$ determined from the AFMC moments of $q_{20}$. Open circles are the direct AFMC calculation of $P\left(q_{20}\right)$ using Eqs. (10) and (11). For clarity, only every fifth AFMC point is included in the plot. The uncertainties in the AFMC results are smaller than the size of the symbols.

Figure 2

The (a) second, (b) third, and (c) fourth moments of ${\widehat{Q}}_{20}$, evaluated from the AFMC distributions $P\left(q_{20}\right)$ as a function of temperature $T$ for the even-mass samarium isotopes ${}^{148-154}\mathrm{Sm}$. The error bars are included but are typically smaller than the size of the symbols.

Figure 3

The expansion parameters (a) $a$, (b) $b$, and (c) $c$ versus temperature $T$ for the even-mass samarium isotopes ${}^{148-154}\mathrm{Sm}$, as determined from the moments in Fig. 2 by solving Eqs. (22) (symbols). The solid lines describe the smoothing spline interpolation (see text).

Figure 4

Distributions $P(T,\beta ,\gamma )$ (shown in a logarithmic scale) in the $\beta -\gamma$ plane for the even-mass samarium isotopes at different temperatures: (a)–(d) a high temperature, $T=4$ MeV; (e)–(h) an intermediate temperature, $T=0.8$ MeV; and (i)–(l) a low temperature, $T=0.07$ MeV. A thermal shape transition from prolate to spherical shape is evident for all but the spherical nucleus ${}^{148}\mathrm{Sm}$ as the temperature increases. A quantum shape transition from a spherical to a prolate shape is also observed near the ground state ($T=0.07$ MeV) as the neutron number increases.

Figure 5

The distribution $P(T,\beta ,\gamma =0)$ (shown on a logarithmic scale) as a function of the axial deformation parameter $\beta$ for the even-mass samarium isotopes (a) ${}^{148}\mathrm{Sm}$, (b) ${}^{150}\mathrm{Sm}$, (c) ${}^{152}\mathrm{Sm}$, and (d) ${}^{154}\mathrm{Sm}$. The solid, dashed, and dotted lines correspond, respectively, to temperatures of $T=0.07$ MeV, $T=0.8$ MeV, and $T=4$ MeV.

Figure 6

The parameter $\tau =ac/b^2$ as a function of temperature $T$ for the even-mass samarium isotopes ${}^{148-154}\mathrm{Sm}$. The values computed from the spline interpolation of $a, b$, and $c$ are described by the solid lines. $\tau <0$ describes deformed shapes while $\tau >9/32$ describes spherical shapes. The interval $0<\tau <9/32$ is a mixed regime with a first-order shape transition at $\tau =1/4$.

Figure 7

Partition of the $(\beta ,\gamma )$ plane into spherical, prolate, and oblate regions.

Figure 8

The probabilities of spherical (open circles), prolate (solid circles), and oblate (pluses) shapes (defined as in Fig. 7) as a function of temperature $T$ for the even-mass samarium isotopes (a) ${}^{148}\mathrm{Sm}$, (b) ${}^{150}\mathrm{Sm}$, (c) ${}^{152}\mathrm{Sm}$, and (d) ${}^{154}\mathrm{Sm}$.

Figure 9

The first derivatives (a)–(d) $da/dT$, (e)–(h) $db/dT$, and (i)–(l) $dc/dT$ of the Landau-like expansion parameters calculated in the AFMC approach (open circles with error bars) and the derivatives of the smoothing spline interpolation of $a, b$, and $c$ (dashed lines with uncertainties shown as shaded bands).

Figure 10

The total state densities $\rho \left(E_x\right)$ computed directly from the thermal energy $E\left(T\right)$ for the even-mass samarium isotopes ${}^{148-154}\mathrm{Sm}$.

Figure 11

The shape probabilities ${\rho }_{\mathrm{shape}}\left(E_x\right)/\rho \left(E_x\right)$ as a function of excitation energy $E_x$ for each of the three regions in Fig. 7: spherical (open circles), prolate (solid circles), and oblate (pluses) for (a) ${}^{148}\mathrm{Sm}$, (b) ${}^{150}\mathrm{Sm}$, (c) ${}^{152}\mathrm{Sm}$, and (d) ${}^{154}\mathrm{Sm}$. The solid lines are the sum of these three probabilities with statistical errors shown by the shaded bands. The sum rule (35) is satisfied within the statistical errors.