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

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


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


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


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
2018-06-02 Adam Washington

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