Enhancement of Schiff Moments in Octupole-Deformed Nuclei - Previous Work

where is the member of the ground-state multiplet with , the sum is over excited states, and is the operator

(3) |

where arrows denote isovector operators, is +1 for neutrons, is the nucleon mass, and we are using the convention . The 's are the unknown isoscalar, isovector, and isotensor T-violating pion-nucleon couplings, and is the usual strong coupling.

In a nucleus such as Hg, with no intrinsic octupole deformation, many
intermediate states contribute to the sum in Eq. (2). By
contrast, the asymmetric shape of Ra implies the existence of a very
low-energy state, in this case 55 keV above the ground state
, that dominates the sum because of the
corresponding small denominator. To very good approximation,
then,

where is the ground-state angular momentum, equal to 1/2 for Ra, and the brackets indicate expectation values in the intrinsic state. The intrinsic-state expectation value is generated by the collective quadrupole and octupole deformation of the entire nucleus; it is much larger than a typical matrix element in a spherical or symmetrically deformed nucleus. Together with the small energy denominator, this large matrix element is responsible for the enhancement of laboratory-frame Schiff moments in nuclei such as Ra.

The amount of the enhancement is not easy to calculate accurately, however.
The
reason is that the matrix element of the two-body spin-dependent operator
in Eq. (5) depends sensitively on the behavior
of a few valence particles,
which carry most of the spin. In the approximation that particles (or
quasiparticles) move in independent orbits generated by a mean field, the
potential can be written as an effective density-dependent one-body operator
that we will denote , defined implicitly by

The authors of Refs. [Spevak et al.(1997)Spevak, Auerbach,
and
Flambaum,Auerbach et al.(1996)Auerbach,
Flambaum,
and Spevak] used a version of the
particle-rotor model [Leander and Sheline(1984)] to represent the odd- nucleus. In
this
model, all but one of the nucleons are treated as a rigid core, and the last
valence nucleon occupies a deformed single-particle orbit, obtained by
solving
a Schrödinger equation for a Nilsson or deformed Wood-Saxon potential. The
model implies that the core carries no intrinsic spin whatever, that the
neutron and proton densities are proportional, and that the exchange terms on
the right-hand side of Eq. (7) are negligible. Under these
assumptions, , which now acts only on the single valence
nucleon, reduces to [Sushkov et al.(1984)Sushkov, Flambaum,
and Khriplovich]

Ref. [Engel et al.(1999)Engel, Friar, and Hayes] confirmed the collectivity of the intrinsic Schiff moments obtained in Refs. [Spevak et al.(1997)Spevak, Auerbach, and Flambaum,Auerbach et al.(1996)Auerbach, Flambaum, and Spevak], but questioned the accuracy of some of the assumptions used to evaluate the matrix element of , suggesting that either core-spin polarization or self-consistency in the nuclear wave function might reduce laboratory Schiff moments. The zero-range approximation and the neglect of exchange in are also open to question. As a result, it is not clear whether the Schiff moment of Ra is 1000 times that of Hg or 100 times, or even less. In what follows, we provide a (tentative) answer by moving beyond the particle-rotor model. Our calculation is not the final word on Schiff moments in octupole-deformed nuclei -- we only do mean-field theory, neglecting in particular to project onto states with good parity, and do not fully account for the pion's nonzero range -- but is a major step forward.