Nuclear Matrix Elements for Tensor Interactions that Violate Local Lorentz Invariance

The principle of local Lorentz invariance can be tested by measurements using a $^{21}$Ne-Rb-K comagnetometer [M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin and M. V. Romalis, Phys. Rev. Lett. {\bf 107}, 171604 (2011)]. This experiment puts strong limits on the possible anisotropy in the maximum attainable velocity for a massive particle. In order to convert from magnetic field units one needs the nuclear matrix element of the quadrupole momentum tensor operator proportional to $2p_z^2-p_x^2-p_y^2$ for the $3/2^+$ ground state of $^{21}$Ne. In the paper above the simple $d_{3/2}$ single-particle model was used. We use the full sd-shell model wavefunctions. We also consider the effect of core polarization. Results for $^{131}$Xe and $^{201}$Hg are also considered.