Matrix elements of the quadratic Stark effect on atoms with hyperfine structure

The perturbation of atomic hyperfine levels by a weak uniform static electric field is discussed. For nearly all atoms, the effects linear in the field are identically zero, and the levels suffer small shifts proportional to the square of the field. The calculation of these shifts using ordinary second-order time-independent perturbation theory is presented. The advantages of separating the interaction into kinematical and dynamical parts and techniques for performing the separation are discussed. The algebra of irreducible spherical tensors is used to separate the interaction into parts having monopolar and quadrupolar symmetry. Some numerical values of the matrix elements of the quadrupole part are given (the monopole elements are trivial), and the shifts of the hyperfine levels due to this interaction are plotted as a function of field for some typical atoms.