# source:sasmodels/doc/guide/magnetism/magnetism.rst@4f5afc9

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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:

• Non spin-flip \$(+ +)\$ and \$(- -)\$
• Spin-flip \$(+ -)\$ and \$(- +)\$

Each measurement is an incoherent mixture of these spin states based on the fraction of \$+\$ neutrons before (\$u_i\$) and after (\$u_f\$) the sample, with weighting:

−  −   = ((1 − ui)(1 − uf))1 ⁄ 4  −  +   = ((1 − ui)(uf))1 ⁄ 4  +  −   = ((ui)(1 − uf))1 ⁄ 4  +  +   = ((ui)(uf))1 ⁄ 4

Ideally the experiment would measure the pure spin states independently and perform a simultaneous analysis of the four states, tying all the model parameters together except \$u_i\$ and \$u_f\$. 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 for non spin-flip states

and

β±∓ =  − DM(My±iMz) for spin-flip states

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\$ mtheta:sld \$theta_M\$ mphi:sld \$phi_M\$ up:angle \$theta_mathrm{up}\$ up:frac_i \$u_i\$ = (spin up)/(spin up + spin down) before the sample up:frac_f \$u_f\$ = (spin up)/(spin up + spin down) 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
2017-11-15 Paul Kienzle

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