source: sasmodels/doc/guide/magnetism/magnetism.rst @ 6e0c1df

magnetic_model
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[9f60c06]1.. _magnetism:
2
3Polarisation/Magnetic Scattering
[990d8df]4================================
[9f60c06]5
[6e0c1df]6Spherical Polarimetry in SANS: This description allows for freedom in
7field/polarisation axis away from the detector plane
8i.e. allows in-beam direction or AC or rotational fields.
[9f60c06]9
[6b86bee]10For magnetic systems, the scattering length density (SLD = $\beta$) is a combination
11of the nuclear and magnetic SLD. For polarised neutrons, the resulting effective SLD
12depends on the spin state of the neutron before and after being scattered in the sample.
13
14Models in Sasview, which define a SLD parameter, can be evaluated also as magnetic models introducing
15the 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
16the simple relation $\beta_M= b_H M$, where $b_H=\dfrac{\gamma r_0}{2\mu_B}=2.7$ fm
17denotes the magnetic scattering length and $M=\lvert \mathbf{M} \rvert$ the magnetisation
18magnitude, where $\gamma = -1.913$ is the gyromagnetic ratio, $\mu_B$ is the
19Bohr magneton, $r_0$ is the classical radius of electron.
20
21It is assumed that the magnetic SLD in each region of the model is uniformly for
22nuclear scattering and has one effective magnetisation orientation
23
[6e0c1df]24The external field $\mathbf{H}=H \mathbf{P}$ coincides with the polarisation axis
[6b86bee]25$\mathbf{P}=(\sin\theta_P \cos \phi_P, \sin \theta_P \sin \phi_P,\cos\theta_P )$ for the neutrons, which is the quantisation axis
26for the Pauli spin operator.
27
28.. figure::
29    mag_img/M_angles_pic.png
30
31.. note::
32    The polarisation axis at the sample position is the determining factor and determines
33    the scattering geometry. Before and after the field at the sample position,
34    the polarisation turns adiabatically to the guide field of the instrument.
35    This operation does not change the observed spin-resolved scattering at the detector.
36    Anyway the magnetic field is the vector defining a symmetry axis of the
37    system and the magnetisation vector will orient with respect to the field.
38
39
40.. note::
41    For AC oscillating/rotation field varying in space with time, you can coupling the magnetisation
42    with the field axis via a constrained fit. This will allow to easily parametrise
43    a phase shift of the magnetisation lagging behind a magnetic field varying from time frame to time frame.
44    Anyway the magnetic field is the vector defining a symmetry axis of the
45    system and the magnetisation vector will most often orient symmetrically with respect to the field.
46
47
48The neutrons are polarised parallel (+) or antiparallel (-) to $\mathbf{P}$. One can
49distinguish 4 spin-resolved cross sections:
50
51 * Non-spin-flip (NSF) $(+ +)$ and $(- -)$
52
53 * Spin-flip (SF) $(+ -)$ and $(- +)$
54
55The spin-dependent magnetic scattering length densities are defined as (see Moon, Riste, Koehler)
56
57.. math::
58    \beta_{M, s_{in} s_{out}}  = b_H\sigma \cdot \mathbf{M_\perp}
59
60where  $\sigma$ is the Pauli spin, and $s_{in/out}$ describes the spin state of the neutron before and
61after the sample.
62
[6e0c1df]63For magnetic neutron scattering, only the magnetisation component or Halpern-Johnson vector
[6b86bee]64$\mathbf{M_\perp}$ perpendicular to the scattering vector
65$\mathbf{Q}=q \mathbf{n}=q (\cos\theta, \sin \theta,0)$ contributes to the magnetic scattering:
66
67.. math::
68    \mathbf{M_\perp} = \mathbf{n} [\mathbf{n} \cdot \mathbf{M}] -\mathbf{M}
69
70with $\mathbf{n}$ the unit scattering vector and $\theta$ denotes the angle
71between $\mathbf{Q}$ and the x-axis.
[9f60c06]72
73.. figure::
[0cd9158]74    mag_img/mag_vector.png
[9f60c06]75
[6b86bee]76The two NSF cross sections are given by
[9f60c06]77
78.. math::
[6e0c1df]79    I^{\pm\pm} = N^2 \mp \mathbf{P}\cdot(N^{\ast}\mathbf{M_\perp} +N\mathbf{M_\perp}^{\ast})
[6b86bee]80        + (\mathbf{P}\cdot \mathbf{M_\perp})^2
81
82and the two SF channels:
[9f60c06]83
[6b86bee]84.. math::
85    I^{\pm\mp} = \mathbf{M_\perp}\cdot \mathbf{M_\perp} - (\mathbf{P}\cdot \mathbf{M_\perp})^2
86        \mp i \mathbf{P}\cdot \mathbf{M_\perp} \times \mathbf{M_\perp}^{\ast}
[9f60c06]87
[6b86bee]88with $i=\sqrt{-1}$, and $^{\ast}$ denoting the complex conjugate quantity, and
89$\times$ and $\cdot$  the vector and scalar product, respectively.
[4f5afc9]90
[6b86bee]91The polarisation axis at the sample position is the determining factor and determines
92the scattering geometry. For the NSF scattering the component of the Halpern-Johnson
93vector parallel to $P$ contributes
[4f5afc9]94
[6b86bee]95.. math::
[6e0c1df]96    \mathbf{M}_{\perp,\parallel P } = ( \mathbf{P}\cdot \mathbf{M}_{\perp }) \mathbf{P}
97
[9f60c06]98
[6b86bee]99The component perpendicular to the polarisation gives rise to SF scattering. The perpendicular
100plane is constructed with the two vectors
[4f5afc9]101
102.. math::
[6b86bee]103    \mathbf{M}_{\perp,\perp P } = \mathbf{M}_{\perp } - (\mathbf{P}\cdot \mathbf{M}_{\perp }) \mathbf{P}
[6e0c1df]104
[9f60c06]105
[6b86bee]106and a third vector perpendicular to both $\mathbf{P}$ and $\mathbf{M}_{\perp,\perp P } $ :
107
108.. math::
[6e0c1df]109    \mathbf{O} = \mathbf{M}_{\perp} \times \mathbf{P} - \mathbf{M}_{\perp,\perp P }
110        = [\mathbf{n}\cdot(\mathbf{M}\times\mathbf{P})(\mathbf{n}-\mathbf{P}\times\mathbf{n})]
[6b86bee]111
[6e0c1df]112
113For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\mathbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \mathbf{O}^{\ast}=0$
[6b86bee]114since  $\mathbf{M}_{\perp} \times \mathbf{P} \cdot \mathbf{M}_{\perp} \times \mathbf{P} = \mathbf{M}_{\perp,\perp P }$.
[9f60c06]115
116
[6b86bee]117Depending on the spin state of the
[9f60c06]118neutrons, the scattering length densities, including the nuclear scattering
[6e0c1df]119length density $(\beta{_N})$ are for the non-spin-flip states
[9f60c06]120
121.. math::
[6e0c1df]122    \beta_{\pm\pm} =  \beta_N \mp b_H \mathbf{P}\cdot \mathbf{M}_{\perp }
123
[9f60c06]124
[6e0c1df]125and for spin-flip states
[9f60c06]126
127.. math::
[6e0c1df]128    \beta_{\pm\mp} =  -b_H (\lvert\mathbf{M}_{\perp,\perp P }\rvert \pm i \mathbf{n}\cdot (\mathbf{M}\times \mathbf{P}  (1-\mathbf{P}\cdot\mathbf{n}))
[9f60c06]129
130
[6e0c1df]131
132with
133
134.. math::
135    \lvert\mathbf{M}_{\perp,\perp P }\rvert= (\mathbf{M}_{\perp,\perp P } \cdot \mathbf{M}_{\perp,\perp P })^{1/2}
136        =(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}.
[6b86bee]137
138
139
140
141Every magnetic scattering cross section can be constructed from an incoherent mixture
142of the 4 spin-resolved spin states depending on the efficiency parameters before
143($u_i$) and after ($u_f$) the sample. For a half-polarised experiment(SANSPOL with $u_f=0.5$) or
144full (longitudinal) polarisation analysis, the accessible spin states are measured
145independently and a simultaneous analysis of the measured states is performed,
146tying all the model parameters together except $u_i$ and $u_f$, which are set based
147on the (known) polarisation efficiencies of the instrument.
148
149.. note::
150    The values of the 'up_frac_i' ($u_i$) and 'up_frac_f' ($u_f$) must be in the range 0 to 1.
151    The parameters 'up_frac_i' and 'up_frac_f' can be easily associated to
152    polarisation efficiencies 'e_in/out' (of the instrument). Efficiency values range from 0.5
153    (unpolarised beam)  to 1 (perfect optics). For 'up_frac_i/f'  <0.5
154    a cross section is constructed with the spin reversed/flipped with respect
155    to the initial supermirror polariser. The actual polarisation efficiency
156    in this case is however  'e_in/out' = 1-'up_frac_i/f'.
[9f60c06]157
158
159
160
161The user input parameters are:
162
163===========   ================================================================
[6b86bee]164 sld_M0       $b_H M_0$
[df87acf]165 sld_mtheta   $\theta_M$
166 sld_mphi     $\phi_M$
[6b86bee]167 up_frac_i    $u_i$ polarisation efficiency *before* the sample
168 up_frac_f    $u_f$ = polarisation efficiency *after* the sample
169 p_theta      $\theta_P$
170 p_phi        $\phi_P$
[9f60c06]171===========   ================================================================
172
[6b86bee]173
[9f60c06]174.. note::
[6b86bee]175    P.S. of Dirk:
176    This is the most general description of magnetic SANS ever written and will supersede prior art!
177    Works for fully magnetically saturated systems. If you figure out how to
178    implement an isotropic ensemble of particle magnetisation ( similar for orientations).
179    This is needed to generate two populations with spin pointing in opposite directions in order to describe
180    field-dependence correctly, i.e. the different variation of mean magnetisation vs
181    square mean quantities.
182    With proper generalised orientation distribution, you cover all "normal" use cases
183   (except of the fancy stuff one have to simulate).
184
185
186References
187----------
188
189    .. [#] R. M. Moon and T. Riste and W. C. Koehler, *Phys. Rev.*, 181 (1969) 920.
[9f60c06]190
[59485a4]191*Document History*
[990d8df]192
[59485a4]193| 2015-05-02 Steve King
[befe905]194| 2017-11-15 Paul Kienzle
195| 2018-06-02 Adam Washington
[6b86bee]196| 2019-03-29 Dirk Honecker
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