source: sasmodels/doc/guide/magnetism/magnetism.rst @ 6b86bee

.. _magnetism:

Polarisation/Magnetic Scattering
================================

(Version 2: Spherical Polarimetry in SANS: Allow for freedom in field/polarisation axis away from the detector plane,
i.e. allow in-beam direction or oscillatory/rotational fields...)

For magnetic systems, the scattering length density (SLD = $\beta$) is a combination
of the nuclear and magnetic SLD. For polarised neutrons, the resulting effective SLD
depends on the spin state of the neutron before and after being scattered in the sample.

Models in Sasview, which define a SLD parameter, can be evaluated also as magnetic models introducing
the magnetisation (vector) $\mathbf{M}=M (\sin\theta_M \cos \phi_M, \sin \theta_M \sin \phi_M,\cos\theta_M )$ and the associated magnetic SLD given by
the simple relation $\beta_M= b_H M$, where $b_H=\dfrac{\gamma r_0}{2\mu_B}=2.7$ fm
denotes the magnetic scattering length and $M=\lvert \mathbf{M} \rvert$ the magnetisation
magnitude, where $\gamma = -1.913$ is the gyromagnetic ratio, $\mu_B$ is the
Bohr magneton, $r_0$ is the classical radius of electron.

It is assumed that the magnetic SLD in each region of the model is uniformly for
nuclear scattering and has one effective magnetisation orientation

The external field $\mathbf{H}=H \mathbf{P}$coincides with the polarisation axis
$\mathbf{P}=(\sin\theta_P \cos \phi_P, \sin \theta_P \sin \phi_P,\cos\theta_P )$ for the neutrons, which is the quantisation axis
for the Pauli spin operator.

.. figure::
    mag_img/M_angles_pic.png

.. note::
    The polarisation axis at the sample position is the determining factor and determines
    the scattering geometry. Before and after the field at the sample position,
    the polarisation turns adiabatically to the guide field of the instrument.
    This operation does not change the observed spin-resolved scattering at the detector.
    Anyway the magnetic field is the vector defining a symmetry axis of the
    system and the magnetisation vector will orient with respect to the field.


.. note::
    For AC oscillating/rotation field varying in space with time, you can coupling the magnetisation
    with the field axis via a constrained fit. This will allow to easily parametrise
    a phase shift of the magnetisation lagging behind a magnetic field varying from time frame to time frame.
    Anyway the magnetic field is the vector defining a symmetry axis of the
    system and the magnetisation vector will most often orient symmetrically with respect to the field.


The neutrons are polarised parallel (+) or antiparallel (-) to $\mathbf{P}$. One can
distinguish 4 spin-resolved cross sections:

 * Non-spin-flip (NSF) $(+ +)$ and $(- -)$

 * Spin-flip (SF) $(+ -)$ and $(- +)$

The spin-dependent magnetic scattering length densities are defined as (see Moon, Riste, Koehler)

.. math::
    \beta_{M, s_{in} s_{out}}  = b_H\sigma \cdot \mathbf{M_\perp}

where  $\sigma$ is the Pauli spin, and $s_{in/out}$ describes the spin state of the neutron before and
after the sample.

For magnetic neutron scattering, only the magnetisation component or Halpern-Johnson vector 
$\mathbf{M_\perp}$ perpendicular to the scattering vector
$\mathbf{Q}=q \mathbf{n}=q (\cos\theta, \sin \theta,0)$ contributes to the magnetic scattering:

.. math::
    \mathbf{M_\perp} = \mathbf{n} [\mathbf{n} \cdot \mathbf{M}] -\mathbf{M}

with $\mathbf{n}$ the unit scattering vector and $\theta$ denotes the angle
between $\mathbf{Q}$ and the x-axis.

.. figure::
    mag_img/mag_vector.png

The two NSF cross sections are given by

.. math::
    I^{\pm\pm} = N^2 \mp \mathbf{P}\cdot(N^{\ast}\mathbf{M_\perp} +N\mathbf{M_\perp}^{\ast}) 
        + (\mathbf{P}\cdot \mathbf{M_\perp})^2

and the two SF channels:

.. math::
    I^{\pm\mp} = \mathbf{M_\perp}\cdot \mathbf{M_\perp} - (\mathbf{P}\cdot \mathbf{M_\perp})^2
        \mp i \mathbf{P}\cdot \mathbf{M_\perp} \times \mathbf{M_\perp}^{\ast}

with $i=\sqrt{-1}$, and $^{\ast}$ denoting the complex conjugate quantity, and
$\times$ and $\cdot$  the vector and scalar product, respectively.

The polarisation axis at the sample position is the determining factor and determines
the scattering geometry. For the NSF scattering the component of the Halpern-Johnson
vector parallel to $P$ contributes

.. math::
    \mathbf{M}_{\perp,\parallel P } = ( mathbf{P}\cdot \mathbf{M}_{\perp }) mathbf{P}
    \text{ magnetisation component parallel to polarisation for NSF scattering}

The component perpendicular to the polarisation gives rise to SF scattering. The perpendicular
plane is constructed with the two vectors

.. math::
    \mathbf{M}_{\perp,\perp P } = \mathbf{M}_{\perp } - (\mathbf{P}\cdot \mathbf{M}_{\perp }) \mathbf{P}
    \text{ magnetisation component perpendicular to polarisation for SF scattering}

and a third vector perpendicular to both $\mathbf{P}$ and $\mathbf{M}_{\perp,\perp P } $ :

.. math::
    \mathbf{O} = \mathbf{M}_{\perp} \times \mathbf{P} - \mathbf{M}_{\perp,\perp P } = [\mathbf{q}\cdot(\mathbf{M}\times\mathbf{P})(\mathbf{q}-\mathbf{P}\times\mathbf{q})]
    \text{ vector perpendicular to polarisation and Halpern-Johnson vector for SF scattering}

For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\matbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \matbf{O}^{\ast}=0$
since  $\mathbf{M}_{\perp} \times \mathbf{P} \cdot \mathbf{M}_{\perp} \times \mathbf{P} = \mathbf{M}_{\perp,\perp P }$.


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 b_H math{P}\cdot M_{\perp }
    \text{ for non-spin-flip states}

and

.. math::
    \beta_{\pm\mp} =  -b_H (\lvert\mathbf{M}_{\perp,\perp P }\rvert \pm i \mathbf{q}\cdot (\mathbf{M}\times \mathbf{P}  (1-\mathbf{P}\cdot\mathbf{q}))
    \text{ for spin-flip states}


with $\lvert\mathbf{M}_{\perp,\perp P }\rvert= (\mathbf{M}_{\perp,\perp P } \cdot \mathbf{M}_{\perp,\perp P })^{1/2}
=(M_{\perp,x}^2+M_{\perp,y}^2+M_{\perp,z}^2-(M_{\perp,x} P_x+ M_{\perp,y} P_y + M_{\perp,z} P_z   )^2 )^{1/2}$.




Every magnetic scattering cross section can be constructed from an incoherent mixture
of the 4 spin-resolved spin states depending on the efficiency parameters before
($u_i$) and after ($u_f$) the sample. For a half-polarised experiment(SANSPOL with $u_f=0.5$) or
full (longitudinal) polarisation analysis, the accessible spin states are measured
independently and a simultaneous analysis of the measured states is performed,
tying all the model parameters together except $u_i$ and $u_f$, which are set based
on the (known) polarisation efficiencies of the instrument.

.. note::
    The values of the 'up_frac_i' ($u_i$) and 'up_frac_f' ($u_f$) must be in the range 0 to 1.
    The parameters 'up_frac_i' and 'up_frac_f' can be easily associated to
    polarisation efficiencies 'e_in/out' (of the instrument). Efficiency values range from 0.5
    (unpolarised beam)  to 1 (perfect optics). For 'up_frac_i/f'  <0.5
    a cross section is constructed with the spin reversed/flipped with respect
    to the initial supermirror polariser. The actual polarisation efficiency
    in this case is however  'e_in/out' = 1-'up_frac_i/f'.




The user input parameters are:

===========   ================================================================
 sld_M0       $b_H M_0$
 sld_mtheta   $\theta_M$
 sld_mphi     $\phi_M$
 up_frac_i    $u_i$ polarisation efficiency *before* the sample
 up_frac_f    $u_f$ = polarisation efficiency *after* the sample
 p_theta      $\theta_P$
 p_phi        $\phi_P$
===========   ================================================================


.. note::
    P.S. of Dirk:
    This is the most general description of magnetic SANS ever written and will supersede prior art!
    Works for fully magnetically saturated systems. If you figure out how to
    implement an isotropic ensemble of particle magnetisation ( similar for orientations).
    This is needed to generate two populations with spin pointing in opposite directions in order to describe
    field-dependence correctly, i.e. the different variation of mean magnetisation vs
    square mean quantities.
    With proper generalised orientation distribution, you cover all "normal" use cases
   (except of the fancy stuff one have to simulate).


References
----------

    .. [#] R. M. Moon and T. Riste and W. C. Koehler, *Phys. Rev.*, 181 (1969) 920.

*Document History*

| 2015-05-02 Steve King
| 2017-11-15 Paul Kienzle
| 2018-06-02 Adam Washington
| 2019-03-29 Dirk Honecker