# Changeset 6e0c1df in sasmodels for doc/guide/magnetism/magnetism.rst

Ignore:
Timestamp:
Apr 1, 2019 7:02:50 AM (4 years ago)
Branches:
magnetic_model
Parents:
6b86bee
Message:

correct typos.

File:
1 edited

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Unmodified
 r6b86bee ================================ (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...) Spherical Polarimetry in SANS: This description allows for freedom in field/polarisation axis away from the detector plane i.e. allows in-beam direction or AC or rotational fields. For magnetic systems, the scattering length density (SLD = $\beta$) is a combination nuclear scattering and has one effective magnetisation orientation The external field $\mathbf{H}=H \mathbf{P}$coincides with the polarisation axis 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. after the sample. For magnetic neutron scattering, only the magnetisation component or Halpern-Johnson vector 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:: I^{\pm\pm} = N^2 \mp \mathbf{P}\cdot(N^{\ast}\mathbf{M_\perp} +N\mathbf{M_\perp}^{\ast}) 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 .. math:: \mathbf{M}_{\perp,\parallel P } = ( mathbf{P}\cdot \mathbf{M}_{\perp }) mathbf{P} \text{ magnetisation component parallel to polarisation for NSF scattering} \mathbf{M}_{\perp,\parallel P } = ( \mathbf{P}\cdot \mathbf{M}_{\perp }) \mathbf{P} The component perpendicular to the polarisation gives rise to SF scattering. The perpendicular .. 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} \mathbf{O} = \mathbf{M}_{\perp} \times \mathbf{P} - \mathbf{M}_{\perp,\perp P } = [\mathbf{n}\cdot(\mathbf{M}\times\mathbf{P})(\mathbf{n}-\mathbf{P}\times\mathbf{n})] For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\matbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \matbf{O}^{\ast}=0$ For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\mathbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \mathbf{O}^{\ast}=0$ 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 length density $(\beta{_N})$ are for the non-spin-flip states .. math:: \beta_{\pm\pm} =  \beta_N \mp b_H math{P}\cdot M_{\perp } \text{ for non-spin-flip states} \beta_{\pm\pm} =  \beta_N \mp b_H \mathbf{P}\cdot \mathbf{M}_{\perp } and and for spin-flip states .. 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} \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})) with $\lvert\mathbf{M}_{\perp,\perp P }\rvert= (\mathbf{M}_{\perp,\perp P } \cdot \mathbf{M}_{\perp,\perp P })^{1/2} =(M_{\perp,x}^2+M_{\perp,y}^2+M_{\perp,z}^2-(M_{\perp,x} P_x+ M_{\perp,y} P_y + M_{\perp,z} P_z )^2 )^{1/2}$. with .. math:: \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}.