Symmetries in Scattering of Slow Neutrons

Reciprocity, detailed balance, and other symmetry laws for slow-neutron scattering are discussed for the case of anisotropic media. While Friedel's law, stating that $\Sigma (\mathbf{v}\to {\mathbf{v}}^{\prime })=\Sigma (-\mathbf{v}\to -{\mathbf{v}}^{\prime })$, may fail for polar absorbing media, reciprocity as a consequence of time-reversal invariance, and of thermal equilibrium of the medium, remains valid. The optical theorem helps to derive a reciprocity relation for the total cross section $\Sigma \left(\mathbf{v}\right)=\Sigma (-\mathbf{v})$. Similar conclusions are derived for the matrices used to describe scattering of polarized neutrons. The two reciprocity relations lead to a macroscopic reciprocity theorem, expressing a symmetry property of Green's functions of the linearized Boltzmann equation.