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sasmodels/doc/guide/magnetism/magnetism.rst
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Polarisation/Magnetic Scattering
(Version 2: Spherical Polarimetry in SANS: Allow for freedom in field/polarisation axis away from the detector plane, i.e. allow inbeam 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) $mathbf{M}=M (sintheta_M cos phi_M, sin theta_M sin phi_M,costheta_M )$ and the associated magnetic SLD given by the simple relation $beta_M= b_H M$, where $b_H=dfrac{gamma r_0}{2mu_B}=2.7$ 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 $mathbf{P}=(sintheta_P cos phi_P, sin theta_P sin phi_P,costheta_P )$ for the neutrons, which is the quantisation axis for the Pauli spin operator.
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 spinresolved 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 spinresolved cross sections:
 Nonspinflip (NSF) $(+ +)$ and $( )$
 Spinflip (SF) $(+ )$ and $( +)$
The spindependent magnetic scattering length densities are defined as (see Moon, Riste, Koehler)
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 HalpernJohnson vector $mathbf{M_perp}$ perpendicular to the scattering vector $mathbf{Q}=q mathbf{n}=q (costheta, sin theta,0)$ contributes to the magnetic scattering:
with $mathbf{n}$ the unit scattering vector and $theta$ denotes the angle between $mathbf{Q}$ and the xaxis.
The two NSF cross sections are given by
and the two SF channels:
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 HalpernJohnson vector parallel to $P$ contributes
The component perpendicular to the polarisation gives rise to SF scattering. The perpendicular plane is constructed with the two vectors
and a third vector perpendicular to both $mathbf{P}$ and $mathbf{M}_{perp,perp P } $ :
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
and
with $lvertmathbf{M}_{perp,perp P }rvert= (mathbf{M}_{perp,perp P } cdot mathbf{M}_{perp,perp P })^{1/2} =(M_{perp,x}^2+M_{perp,y}^2+M_{perp,z}^2(M_{perp,x} P_x+ M_{perp,y} P_y + M_{perp,z} P_z )^2 )^{1/2}$.
Every magnetic scattering cross section can be constructed from an incoherent mixture of the 4 spinresolved spin states depending on the efficiency parameters before ($u_i$) and after ($u_f$) the sample. For a halfpolarised 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 fielddependence 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
[1]
 Moon and T. Riste and W. C. Koehler, Phys. Rev., 181 (1969) 920.
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