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# Polarisation/Magnetic Scattering

Models which define a scattering length density parameter can be evaluated
as magnetic models. In general, the scattering length density (SLD = \$beta\$) in each region where the SLD is uniform, is a combination of the nuclear and magnetic SLDs and, for polarised neutrons, also depends on the spin states of the neutrons.

For magnetic scattering, only the magnetization component \$mathbf{M_perp}\$ perpendicular to the scattering vector \$mathbf{Q}\$ contributes to the magnetic scattering length. The magnetic scattering length density is then

βM = (γr0)/(2μB)σM = DMσM

where \$gamma = -1.913\$ is the gyromagnetic ratio, \$mu_B\$ is the Bohr magneton, \$r_0\$ is the classical radius of electron, and \$sigma\$ is the Pauli spin.

Assuming that incident neutrons are polarized parallel (+) and anti-parallel (-) to the \$x'\$ axis, the possible spin states after the sample are then

No spin-flips (+ +) and (- -)

Spin-flips (+ -) and (- +) If the angles of the \$Q\$ vector and the spin-axis \$x'\$ to the \$x\$ - axis are \$phi\$ and \$theta_{up}\$, respectively, then, depending on the spin state of the neutrons, the scattering length densities, including the nuclear scattering length density (\$beta{_N}\$) are

β±± = βNDMMx when there are no spin-flips

and

β±∓ =  − DM(My±iMz) when there are

where

Mx = M0qxcos(θup) + M0qysin(θup) My = M0qycos(θup) − M0qxsin(θup) Mz = M0z M0qx = (M0xcosφ − M0ysinφ)cosφ M0qy = (M0ysinφ − M0xcosφ)sinφ

Here, \$M_{0x}\$, \$M_{0x}\$, \$M_{0z}\$ are the x, y and z components of the magnetization vector given in the laboratory xyz frame given by

M0x = M0cosθMcosφM M0y = M0sinθM M0z =  − M0cosθMsinφM

and the magnetization angles \$theta_M\$ and \$phi_M\$ are defined in the figure above.

The user input parameters are:

 M0_sld = \$D_M M_0\$ Up_theta = \$theta_mathrm{up}\$ M_theta = \$theta_M\$ M_phi = \$phi_M\$ Up_frac_i = (spin up)/(spin up + spin down) neutrons before the sample Up_frac_f = (spin up)/(spin up + spin down) neutrons after the sample

Note

The values of the 'Up_frac_i' and 'Up_frac_f' must be in the range 0 to 1.

Document History

2015-05-02 Steve King
2017-05-08 Paul Kienzle

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