Anisotropic muonium with random hyperfine distortions: A new static relaxation theory

The spin dynamics associated with random hyperfine distortions of an ensemble of muons, which have thermalized in a solid as static muonium atoms, is approximated with the motion generated by averaging the spin evolution of a single isolated anisotropic muonium atom over a product of continuous hyperfine frequency distributions. That is, since each muonium atom in the ensemble is assumed to have a given set of nonunique anisotropic hyperfine frequencies, then the frequency distributions represent the ensemble average of many such muonium atoms. The hyperfine tensor includes an isotropic Fermi contact term and a symmetric traceless dipole-dipole term. There is no antisymmetric contribution. In this work the hyperfine tensor is expanded in terms of second-rank spherical tensors while the expansion coefficients are used to parametrize the distortion. Thus there are six frequencies associated with an anisotropic muonium atom, one associated with the isotropic part of the hyperfine tensor and five corresponding to the anisotropic components. Relaxation functions are calculated for muonium in both the zero- and high-field limits. The different anisotropic components of the hyperfine tensor lead to different (anisotropic) observable motion functions. For an amorphous or powder sample, only the isotropic motion function is observable, and the different anisotropic frequencies lead to motion functions with differing shapes. These functions have been applied to the case of fused quartz where the spin relaxation is well known to be entirely due to random hyperfine distortions.