Spherical Neutron Polarimetry

Spherical neutron polarimetry is a neutron scattering technique where, prior to scattering, the initial state polarization vector of the incident neutron beam is precisely oriented and, after scattering, the final state neutron beam polarization vector is precisely measured. The initial and final state neutron beam polarizations are assumed to be constant only changing upon scattering from some scattering medium, such as a single crystal. The transformation from initial state polarization to final state polarization through scattering is used to charaterize the magnetic structure of the scattering medium. Through spherical neutron polarimetry, this transformation can be measured as a polarization property tensor of that scattering medium.

Polarization Mechanics

The neutron is a two dimension quantum object. The neutron has a spin quantum number that can be +-1/2 and can therefore be described in a two-dimensional Hilbert space. A neutron beam is an ensemble of neutrons and has a three-dimensional polarization with a real space direction. When a neutron beam is subjected to a static magnetic field the beam polarization will precess around an axis parallel to the field direction. See below.

The frequency of the precession is known as the Larmor frequency.

When the neutron beam is subjected to a magnetic field gradient the polarization will begin to wobble. In the frame of a single neutron a magnetic field gradient will look like a rotating magnetic field. If the rotation rate is slow enough when compared to the Larmor frequency, the adiabatic limit is reached and the polarization will then follow the magnetic field direction to within the extent of the wobble.

Both spin precession and adiabatic rotation can be used to precisely orient the polarization vector of the neutron beam.

Measurement of the neutron beam polarization is different. Measurement invloves the decomposition of the neutron beam polarization vector into its constituent components. Since the polarization is three-dimensional, measurement of the polarization requires measurement of three orthogonal components. Measurement of a single component is recoded as the fractional change between the number of +1/2 spin states and -1/2 spins states. If U is the number of +1/2 spin states and D is the number of -1/2 spin states then the ${j}^{th}$ component of the polarization ${P}_{j}$ can be written as,

${P}_{j}=\genfrac{}{}{0.1ex}{}{U-D}{U+D}$

This means that for an unpolarized neutron beam, where the {${P}_{j}$} are all zero, 50% of the beam will be in the +1/2 state and 50% of the beam will be in the -1/2 state.

A neutron beam may be polarized by passing it through a polarized 3He filter. Filtering removes intensity of the unwanted spin state leaving intensity of the desired spin state. Since filtering is not perfect, there will always be residual intensity in the unwated spin state. This is why the polarization is defined as the factional difference between the two possible spin states. In the figure above, the intensity composition relative to the unpolarized beam before and after filtering is depicted.

Now consider that the spin state of the neurton is only defined in the presence of a magnetic field, the component of the polarization vector measured will then depend on the orientation of that field. By orienting the maganetic field along three mutually orthogonal directions within some frame of reference, the three vector components of the polarization may be independently measured.

In the figure above, the neutron beam polarization is oriented at 60 degrees above the horizon. To measure the vertical z-component and horizontal x-compenent of polarization, the magnetic field is applied separately to the beam along those two directions. For either direction the intensity of +1/2 sates and -1/2 states is measured. The pie-charts depict the intensity compositions, relative to the polarized beam, that one would expect to measure in this geometry.

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