Mean-field calculations

Self-consistent mean-field theory is widely used for describing bulk properties of nuclei [Bender et al.(2003)Bender, Heenen, and Reinhard]. In the guise of density-functional theory, it is also used throughout atomic and molecular physics. The approach is more ``microscopic" -- nucleons are the only degrees of freedom -- and far less phenomenological than the collective particle-rotor model. Self-consistency connects the single-particle states and the actual density distribution. The variational principle that determines the single-particle wave functions thus optimizes all multipole moments not fixed by global symmetries. The density distributions of neutrons and protons are not proportional to each other; they have slightly different deformations and radial profiles. In odd-$A$ nuclei, self-consistent calculations include rearrangement due to the unpaired particle. Rearrangement causes polarization of the even-$A$ core through orbital-current and spin-density terms in the effective interaction. Core polarization is one of the effects on the Schiff moment of $^{225}$Ra that we investigate below.

Our approach is nonrelativistic and employs Skyrme interactions. To get an idea of the range of results this kind of calculation can produce, we use four different parameterizations of the Skyrme energy functional, i.e., four different Skyrme forces. The four give similar results for many observables near stability, but still have significant differences. Our favorite interaction, for reasons explained below, is SkO' [Bender et al.(2002)Bender, Dobaczewski, Engel, and Nazarewicz,Reinhard et al.(1999)Reinhard, Dean, Nazarewicz, Dobaczewski, Maruhn, and Strayer], but we also show results for the commonly used forces SIII [Beiner et al.(1975)Beiner, Flocard, Giai, and Quentin], SkM$^*$ [Bartel et al.(1982)Bartel, Quentin, Brack, Guet, and Håkansson], and SLy4 [Chabanat et al.(1998)Chabanat, Bonche, Haensel, Meyer, and Schaeffer].