On the optimisation of the spectral resolution in spectrographs for cold neutrons based on refraction at grazing incidence

Recently the wavelength dispersion of cold neutrons in the refraction process at inclined interfaces was identified as an efficient tool for neutron spectrographs, in which a larger wavelength band can be registered simultaneously. This registration mode reduces the data acquisition time significantly as no need to monochromatise the incident neutron beam by use of inefficient choppers exists. In the related studies the spectrograph performance is treated with rather complex equations. This study instead provides a theoretical treatment of the dispersion properties with simpler analytical equations, which were previously used in connection with X-rays. It can be shown, that the spectral resolution in the original spectrographs is mostly limited by the finite size of the refracted beam, which is inconveniently increasing upon refraction at grazing internal incidence onto an inclined refracting interface. The blurring of the beam size of a monochromatic beam at the detector due to the angular spread of the incident beam is mostly negligible. It is thus proposed that a significant improvement in the spectral resolution of such a spectrograph can be achieved, when the beam size at the detector is reduced by introducing focusing in the refraction process. It is shown, that the spectral resolution can then ultimately be limited by the smaller size of the blurred image caused by the angular spread of the beam. Then the improvement in this beam divergence limit can be by an order of magnitude and it is achieved by refraction upon internal incidence onto a concave interface. It is found that such a configuration will focus appropriately in a larger wavelength interval. By this means for wavelengths between 5 Å and 12 Å spectral resolutions of below 1% are feasible, which are not yet reported for such prism spectrographs.