.. _magnetism:

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.

.. figure::
    mag_img/mag_vector.png

The magnetic scattering length density is then

.. math::
    \beta_M = \dfrac{\gamma r_0}{2\mu_B}\sigma \cdot
    \mathbf{M_\perp} = D_M\sigma \cdot \mathbf{M_\perp}

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:

.. math::
    -- &= ((1-u_i)(1-u_f))^{1/4} \\
    -+ &= ((1-u_i)(u_f))^{1/4} \\
    +- &= ((u_i)(1-u_f))^{1/4} \\
    ++ &= ((u_i)(u_f))^{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 $u_f$.

.. figure::
    mag_img/M_angles_pic.png

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

.. math::
    \beta_{\pm\pm} =  \beta_N \mp D_M M_{\perp x'}
    \text{ for non spin-flip states}

and

.. math::
    \beta_{\pm\mp} =  -D_M (M_{\perp y'} \pm iM_{\perp z'})
    \text{ for spin-flip states}

where

.. math::
    M_{\perp x'} &= M_{0q_x}\cos(\theta_{up})+M_{0q_y}\sin(\theta_{up}) \\
    M_{\perp y'} &= M_{0q_y}\cos(\theta_{up})-M_{0q_x}\sin(\theta_{up}) \\
    M_{\perp z'} &= M_{0z} \\
    M_{0q_x} &= (M_{0x}\cos\phi - M_{0y}\sin\phi)\cos\phi \\
    M_{0q_y} &= (M_{0y}\sin\phi - M_{0x}\cos\phi)\sin\phi

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

.. math::
    M_{0x} &= M_0\cos\theta_M\cos\phi_M \\
    M_{0y} &= M_0\sin\theta_M \\
    M_{0z} &= -M_0\cos\theta_M\sin\phi_M

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