  The dependence of the nuclear level density on intrinsic deformation is an
important input to dynamical nuclear processes such as fission. Auxiliary-field
Monte Carlo (AFMC) method is a powerful method for computing nuclear level
densities. However, the statistical distribution of intrinsic shapes is not
readily accessible due to the formulation of AFMC in a spherical
configuration-interaction shell-model approach. Instead, theory of deformation
up to now 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 ($^{148}$Sm,
$^{150}$Sm, $^{152}$Sm, $^{154}$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.
