# Changeset 6b86bee in sasmodels

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Timestamp:
Mar 31, 2019 12:42:57 PM (5 months ago)
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magnetic_model
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6e0c1df
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5e1875c
git-author:
Dirk Honecker <honecker@…> (03/31/19 12:23:01)
git-committer:
Dirk Honecker <honecker@…> (03/31/19 12:42:57)
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Documentation for generalised magnetic SANS with orientation of magnetisation and polarisation in line with particle orientation notation. Addresses ticket SasView?/sasview#993 and Sasview/sasmodels#113 .

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 rdf87acf ================================ 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. (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 scattering, only the magnetization component $\mathbf{M_\perp}$ perpendicular to the scattering vector $\mathbf{Q}$ contributes to the magnetic scattering length. 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 magnetic scattering length density is then The two NSF cross sections are given by .. math:: \beta_M = \dfrac{\gamma r_0}{2\mu_B}\sigma \cdot \mathbf{M_\perp} = D_M\sigma \cdot \mathbf{M_\perp} 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 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: and the two SF channels: .. math:: -- &= (1-u_i)(1-u_f) \\ -+ &= (1-u_i)(u_f) \\ +- &= (u_i)(1-u_f) \\ ++ &= (u_i)(u_f) 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} 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 $u_f$. with $i=\sqrt{-1}$, and $^{\ast}$ denoting the complex conjugate quantity, and $\times$ and $\cdot$  the vector and scalar product, respectively. .. figure:: mag_img/M_angles_pic.png 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 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 .. 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 D_M M_{\perp x'} \text{ for non spin-flip states} \beta_{\pm\pm} =  \beta_N \mp b_H math{P}\cdot M_{\perp } \text{ for non-spin-flip states} and .. math:: \beta_{\pm\mp} =  -D_M (M_{\perp y'} \pm iM_{\perp z'}) \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} 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 with $\lvert\mathbf{M}_{\perp,\perp P }\rvert= (\mathbf{M}_{\perp,\perp P } \cdot \mathbf{M}_{\perp,\perp P })^{1/2} =(M_{\perp,x}^2+M_{\perp,y}^2+M_{\perp,z}^2-(M_{\perp,x} P_x+ M_{\perp,y} P_y + M_{\perp,z} P_z )^2 )^{1/2}$. 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. 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       $D_M M_0$ sld_M0       $b_H M_0$ sld_mtheta   $\theta_M$ sld_mphi     $\phi_M$ 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 up_angle     $\theta_\mathrm{up}$ 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:: The values of the 'up_frac_i' and 'up_frac_f' must be in the range 0 to 1. 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* | 2017-11-15 Paul Kienzle | 2018-06-02 Adam Washington | 2019-03-29 Dirk Honecker