Introduction

Experiments with K and B mesons indicate that time-reversal invariance (T) is violated through phases in the Cabibbo-Kobayashi-Maskawa matrix that affect weak interactions [Naboulsi()]. The suspicion that extra-Standard-Model physics, e.g. supersymmetry, also violates T has motivated a different kind of experiment: measuring the electric dipole moments (EDMs) of the neutron and of atoms. Because any such dipole moment must be proportional to the expectation value of the T-odd spin operator, it can only exist when T (and parity) is violated [Sachs(1987),Khriplovich and Lamoreaux(1997)]. So far the experiments have seen no dipole moments, but they continue to improve and even null results are useful, since they seriously constrain new physics. Whatever the experimental situation in the future, therefore, it is important to determine theoretically what the presence or absence of EDMs at a given level implies about T-violating interactions at elementary-particle scales. Our focus here is atoms, which for some sources of T violation currently provide limits as good or better than the neutron [Romalis et al.(2001)Romalis, Griffith, Jacobs, and Fortson].

One way an atom can develop an EDM is through T and P violation in its nucleus. Let us assume that given a fundamental source of the broken symmetry one can use effective-field theory and QCD to calculate the strength of the resulting T-violating nucleon-pion interaction. One then needs to connect the strength of that interaction to the resulting nuclear ``Schiff moment'', which, because the nuclear EDM is screened [Schiff(1963)], is the quantity responsible for inducing an EDM in electrons orbiting the nucleus. The Schiff moment is defined classically as a kind of radially weighted dipole moment:

$$\bm{S} = {\textstyle{\frac{1}{10}}} \int {\rm d}^3r \rho_{\r... ...xtstyle{\frac{5}{3}}} \overline{r_{\rm ch}^2} \right) \bm{r} ,$$ (1)

where $\rho_{\rm ch}$ is the nuclear charge density and $\overline{r_{\rm ch}^2}$ is the mean-square charge radius. Recent papers [Spevak et al.(1997)Spevak, Auerbach, and Flambaum,Auerbach et al.(1996)Auerbach, Flambaum, and Spevak] have argued that because of their asymmetric shapes, octupole-deformed nuclei in the light-actinide region should have collective Schiff moments that are 100 to 1000 times larger than the Schiff moment in $^{199}$Hg, the system with the best experimental limit on its atomic EDM [Romalis et al.(2001)Romalis, Griffith, Jacobs, and Fortson]. Ref. [Engel et al.(1999)Engel, Friar, and Hayes] suggested that certain many-body effects may make the enhancement a bit less than that. The degree of enhancement is important because several experiments in the light actinides are contemplated, planned, or underway [Chupp(),Holt()]. They may see nonzero EDMs, and even if they don't we will need to be able to compare their limits on fundamental sources of T violation to those of experiments in other isotopes.

Perhaps the most attractive octupole-deformed nucleus for an experiment is $^{225}$Ra. Though radioactive, it has a ground-state angular momentum $J=1/2$, which minimizes the effect of stray quadrupole electric fields in an experiment to measure a dipole moment. In addition, the associated atom has close-lying electronic levels of opposite parity and is relatively easy to trap and manipulate. As a result, at least one group is at work on a measurement in $^{225}$Ra [Holt()]. Here we calculate its Schiff moment, attempting to incorporate the effects discussed in Ref. [Engel et al.(1999)Engel, Friar, and Hayes] through a symmetry-unrestricted mean-field calculation. We begin in the next section by describing the physics of the Schiff moment in octupole-deformed nuclei, briefly reviewing prior work in the process. In Section III we test our mean-field approach by calculating properties of even Ra isotopes. In Section IV we discuss issues peculiar to mean-field calculations in odd nuclei and then present our results for the Schiff moment of $^{225}$Ra, focusing particularly on the degree of enhancement. Section V is a brief conclusion.