Polarization and Magnetic Scattering


1. Theory
In general, a particle with a volume V can be described by an ensemble containing N 3-dimensional rectangular pixels where each pixels are much smaller than V. Assuming that all the pixel sizes are same, the elastic scattering intensity by the particle


where βj and rj are the scattering length density and the position of the j'th pixel respectively. And the total volume


for βj ≠ 0 where vj is the volume of the j'th pixel.
The scattering length density (SLD) of each pixel where the SLD is uniform, is a combination of the nuclear and magnetic SLDs and depends on the spin states of the neutrons as follows:

For magnetic scattering, only the magnetization component, Mperp, perpendicular to the scattering vector Q contributes to the the magnetic scattering length. (Figure below).


The magnetic scattering length density is then


where γ = -1.913 the gyromagnetic ratio, μB is the Bohr magneton, r0 is the classical radius of electron, and σ is the Pauli spin.
For polarized neutron, the magnetic scattering is depending on the spin states. Let's consider that the incident neutrons are polarized parallel (+)/anti-parallel (–) to the x' axis (See both Figures above). The possible out-coming states then are + and - states for both incident states.
- Non-spin-flips: (+ +) and (- -)
- Spin-flips: (+ -) and (- +)



Now, let's assume that the angles of the Q vector and the spin-axis (x') from x-axis are φ and θup, respectively (See Figure above). Then, depending upon the polarization (spin) state of neutrons, the scattering length densities , including the nuclear scattering length density (β N) are given as, for non-spin-flips,



for spin-flips,



where



Here, the M0x, M0y and M0z are the x, y and z components of the magnetization vector given in the xyz lab frame.

2. GUI


After the computation, the result will be listed in the 'Theory' box in the data explorer panel on the main window.
The 'Up_frac_i' and 'Up_frac_f' are the ratio, (spin up) /(spin up + spin down) neutrons before the sample and at the analyzer, respectively.


*Note: The values of 'Up_frac_i' and 'Up_frac_f' must be in the range between 0 and 1.