Magnetic scattering of neutrons by a relativistic atom

We examined the magnetic scattering of neutrons by an atom described by a relativistic Hamiltonian. It is shown that the magnetic-scattering amplitude can be expressed in terms of relativistic magnetic and electric multipole operators. An effective magnetic-scattering amplitude is then defined so that a relativistic result is obtained by taking matrix elements of this effective operator between nonrelativistic states of the atom. It is shown that the effective magnetic-scattering amplitude can be expressed in terms of relativistic radial integrals and the generalized Racah tensors $W^{\mathrm{}(0,k)k}(nl,n^{\prime }{l}^{\prime })$ and $W^{(1,k^{\prime })k}(nl,n^{\prime }{l}^{\prime })(k^{\prime }=k,k\pm 1)$. The relativistic radial integrals can be evaluated using one-electron radial wave functions, obtained by Hartree-Fock-Dirac calculations. The matrix elements of the Racah tensors can be obtained by standard techniques and their evaluation is considerably simplified by selection rules based on the symmetry properties of the atomic states. The formalism has been applied to the magnetic scattering of neutrons by an atom in the $l^N$ electronic configuration. In this case, explicit expressions for the relativistic radial integrals have been derived, and the nonrelativistic limit of the effective magnetic-scattering amplitude has been examined.