On the theory of coherent neutrino scattering from crystals

Recently, Weber has analyzed the scattering of (anti)neutrinos from a crystal, predicting that the neutrino-nuclear cross-section is multiplied by a coherence factor O(N2), where N is the number of nuclei in the crystal. This leads to a macroscopic force on the crystal, for which he reports finding preliminary evidence using a fission reactor as a source of electron antineutrinos. Butler has since argued that a factor O(N2) can enter only when the neutrino wave-length, is comparable with the crystal dimension, indicating that coherence occurs only in this ultra-low-energy regime. The scattering is here considered from an ideal crystal of (anti)neutrinos with much smaller wave-lengths, comparable to the lattice spacing or considerably less, leading to coherent Bragg-laue scattering, analogous to that of X-rays or thermal neutrons. An explicit formula is obtained for the coherent force exerted on the crystal expressed in terms of energy moments of the neutrino source. The coherent force produced by neutrinos of such wave-lengths is linear in N. Hence, in agreement with Butler, it is found that a coherence factor O(N2) can occur only for neutrinos with ultra-long wave-lengths