The definition of the shape parameters

Last modification: 1996/3/25

For certain purposes, the shape or deformation parameters are more useful than multipole moments, although the former are model-dependent while the latter are directly related to the experimental observables. In a widely-used method of the Strutinsky shell correction, the deformation parameters are built in the theory in order to specify the single-particle potential. On the other hand, for mean-field solutions, one has to define the deformation parameters from the density distributions of nucleons.

In this paper, we define the deformation parameters as those of a sharp-surface uniform-density liquid drop which has the same mass moments as the HF+BCS solution has. More specifically, the mass density of the liquid drop is expressed as

rho(r) = rho₀ theta ( R(r)-r),
R(r) = R₀ ( 1 + sum_l,m a_lm Y_lm (r)).

The necessary and sufficient conditions on a_lm to fulfill the reality of R(r) and the D_2h symmetry are that l and m are even numbers and a_lm =a_lm^* =a_l-m. We set a_lm=0 for l >= 6 and determine the remaining seven parameters rho₀, R₀, a₂₀, a₂₂, a₄₀, a₄₂, and a₄₄ such that the liquid drop has the same particle number, mean-square mass radius, and mass quadrupole (r² Y_2m) and hexadecapole (r⁴ Y_4m) moments as the HF+BCS solution has.

Let us check the feasibility of thus-defined shape parameters by comparing them with model-independent quantities.

A figure ( GIF file of3.7KB or PS file of 75KB ) shows that the liquid drop radius R₀ agrees well with the r.m.s. radius r_rms multiplied by (5/3)^1/2 (the diagonal line). Owing to deformation and surface diffuseness, the former tends to be slightly smaller than the latter. For 5361 points plotted in the figure (mass, proton, and neutron moments of 1029 ground and 758 first-excited solutions), the maximum and the r.m.s. deviations of R₀ from (5/3)^1/2 r_rms are 0.3 fm and 0.06 fm, respectively.

As for the quadrupole deformation parameter, a figure ( GIF file of3.6KB or PS file of 76KB ) shows the relation between a₂₀ and delta defined as 3 < Q > / 4 < r² > . The solid line represents a₂₀ = (16/45 pi)^1/2 delta, which is the leading-order expression for thin-surface density distributions. If we fit a polynomial through order three with the leading order coefficient fixed to the above value, the resulting function is a₂₀ = (16/45 pi)^1/2 delta -0.47 delta² +0.78 delta³. The maximum and the r.m.s. deviations from this function are 0.05 and 0.007, respectively.