Analysis of multibeam data for neutron reflectivity

We offer mathematical proof that multiple-beam neutron reflectivity, corresponding to simultaneous collection of data at multiple angles (wavevector transfers) does not perform better, errorwise for counting noise, than single-beam data collection for the same total number of reflected neutrons-and may perform much worse, depending on the beam modulation strategy used. The basic idea is that the nominal statistical benefit of summing data at, say, N different wavevector transfers is undone by needing to collect N differently modulated (i.e., weighted) sums in order to extract the reflectivities. To our knowledge, a general proof of this behavior for arbitrary strategies has been lacking. The formal result can be summarized by saying that the best nondiagonal matrix modulation strategies are orthogonal (unitary) matrices, or constant multiples thereof, and that these can do no better than diagonal--i.e., single-beam--strategies.