Polarization and Magnetic Scattering

The magnetic scattering is implemented in five (2D) models, SphereModel, CoreShellModel, CoreMultiShellModel, CylinderModel, and ParallelepipedModel. In general, the scattering length density (SLD) in each regions where the SLD (=β) 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, M_perp, 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, μ_B 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') against x-axis are φ and θ_up, 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 M_0x, M_0y and M_0z are the x, y and z components of the magnetization vector given in the xyz lab frame. The angles of the magnetization, θ_M and φ_M as defined in the Figure (above),

The user input parameters are M0_sld = D_MM₀, Up_theta = θ_up, M_theta = θ_M, and M_phi = φ_M. 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 the 'Up_frac_i' and 'Up_frac_f' must be in the range between 0 and 1.