Optimal Larmor precession magnetic field shapes: application to neutron spin echo three-axis spectrometry

Abstract

Resolution characteristics of neutron spectrometers using Larmor precession of the neutron spin are limited by magnetic field homogeneities of a special type. The line integral of the modulus of the magnetic induction along a neutron trajectory of length L is a measure of the amount of precession performed by the neutron. Hence it should be precisely the same for all neutron trajectories in a diverging beam. We present an analytical solution to the variational problem , for the case of cylindrical magnets (better than any lower symmetry geometry) coaxial to the beam axis. This solution describes the best irrotational (rot B = 0) field shape along the beam axis z. It can be obtained in practice by superposing a number of solenoids of different lengths. The optimal homogeneity is significantly better than for a simple solenoid of comparable dimensions, the only magnets used until now. For realistic lengths L and beam radii r, it is however not good enough for very high-resolution measurements. We therefore introduce a technique to correct both for the residual inhomogeneities of optimized cylinder magnets and the line integral variations due to path length differences resulting from finite angular beam divergence. Such corrections can only be done by introducing current distributions in the beam. Their optimal distributions can also be calculated analytically. Until now only the residual inhomogeneities have been corrected by in beam currents. With the two concepts of optimal field shape (OFS) and path length corrections described here, the resolution properties of Larmor precession techniques can be pushed to their intrinsic limits. As a further result of the correction technique introduced here, wider angular divergences can be used, for example using multi-detectors, resulting in substantially improved neutron economy. Several neutron spin echo (NSE) spectrometers based on above ideas have in the meantime been constructed by different working parties. The experimental results confirm the calculations reported here; no significant polarization drop is observed at maximum field. For the investigation of dispersive elementary excitations in solids where the neutron energy change depends on the momentum transfer, a special type of gradient coils is needed. We describe their design for OFS precession magnets.