  In any context in which color superconductivity arises in nature, it is
likely to involve pairing between species of quarks with differing chemical
potentials. For suitable values of the differences between chemical potentials,
Cooper pairs with nonzero total momentum are favored, as was first realized by
Larkin, Ovchinnikov, Fulde and Ferrell (LOFF). Condensates of this sort
spontaneously break translational and rotational invariance, leading to gaps
which vary periodically in a crystalline pattern. Unlike the original LOFF
state, these crystalline quark matter condensates include both spin zero and
spin one Cooper pairs. We explore the range of parameters for which crystalline
color superconductivity arises in the QCD phase diagram. If in some shell
within the quark matter core of a neutron star (or within a strange quark star)
the quark number densities are such that crystalline color superconductivity
arises, rotational vortices may be pinned in this shell, making it a locus for
glitch phenomena.
