Changeset 6e0c1df in sasmodels
 Timestamp:
 Apr 1, 2019 7:02:50 AM (4 weeks ago)
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 magnetic_model
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 6b86bee
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doc/guide/magnetism/magnetism.rst
r6b86bee r6e0c1df 4 4 ================================ 5 5 6 (Version 2: Spherical Polarimetry in SANS: Allow for freedom in field/polarisation axis away from the detector plane, 7 i.e. allow inbeam direction or oscillatory/rotational fields...) 6 Spherical Polarimetry in SANS: This description allows for freedom in 7 field/polarisation axis away from the detector plane 8 i.e. allows inbeam direction or AC or rotational fields. 8 9 9 10 For magnetic systems, the scattering length density (SLD = $\beta$) is a combination … … 21 22 nuclear scattering and has one effective magnetisation orientation 22 23 23 The external field $\mathbf{H}=H \mathbf{P}$ coincides with the polarisation axis24 The external field $\mathbf{H}=H \mathbf{P}$ coincides with the polarisation axis 24 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 25 26 for the Pauli spin operator. … … 60 61 after the sample. 61 62 62 For magnetic neutron scattering, only the magnetisation component or HalpernJohnson vector 63 For magnetic neutron scattering, only the magnetisation component or HalpernJohnson vector 63 64 $\mathbf{M_\perp}$ perpendicular to the scattering vector 64 65 $\mathbf{Q}=q \mathbf{n}=q (\cos\theta, \sin \theta,0)$ contributes to the magnetic scattering: … … 76 77 77 78 .. math:: 78 I^{\pm\pm} = N^2 \mp \mathbf{P}\cdot(N^{\ast}\mathbf{M_\perp} +N\mathbf{M_\perp}^{\ast}) 79 I^{\pm\pm} = N^2 \mp \mathbf{P}\cdot(N^{\ast}\mathbf{M_\perp} +N\mathbf{M_\perp}^{\ast}) 79 80 + (\mathbf{P}\cdot \mathbf{M_\perp})^2 80 81 … … 93 94 94 95 .. math:: 95 \mathbf{M}_{\perp,\parallel P } = ( mathbf{P}\cdot \mathbf{M}_{\perp })mathbf{P}96 \text{ magnetisation component parallel to polarisation for NSF scattering} 96 \mathbf{M}_{\perp,\parallel P } = ( \mathbf{P}\cdot \mathbf{M}_{\perp }) \mathbf{P} 97 97 98 98 99 The component perpendicular to the polarisation gives rise to SF scattering. The perpendicular … … 101 102 .. math:: 102 103 \mathbf{M}_{\perp,\perp P } = \mathbf{M}_{\perp }  (\mathbf{P}\cdot \mathbf{M}_{\perp }) \mathbf{P} 103 \text{ magnetisation component perpendicular to polarisation for SF scattering} 104 104 105 105 106 and a third vector perpendicular to both $\mathbf{P}$ and $\mathbf{M}_{\perp,\perp P } $ : 106 107 107 108 .. math:: 108 \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})]109 \text{ vector perpendicular to polarisation and HalpernJohnson vector for SF scattering}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})] 110 111 111 For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\matbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \matbf{O}^{\ast}=0$ 112 113 For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\mathbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \mathbf{O}^{\ast}=0$ 112 114 since $\mathbf{M}_{\perp} \times \mathbf{P} \cdot \mathbf{M}_{\perp} \times \mathbf{P} = \mathbf{M}_{\perp,\perp P }$. 113 115 … … 115 117 Depending on the spin state of the 116 118 neutrons, the scattering length densities, including the nuclear scattering 117 length density $(\beta{_N})$ are 119 length density $(\beta{_N})$ are for the nonspinflip states 118 120 119 121 .. math:: 120 \beta_{\pm\pm} = \beta_N \mp b_H math{P}\cdot M_{\perp } 121 \text{ for nonspinflip states} 122 \beta_{\pm\pm} = \beta_N \mp b_H \mathbf{P}\cdot \mathbf{M}_{\perp } 122 123 123 and 124 125 and for spinflip states 124 126 125 127 .. math:: 126 \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})) 127 \text{ for spinflip states} 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})) 128 129 129 130 130 with $\lvert\mathbf{M}_{\perp,\perp P }\rvert= (\mathbf{M}_{\perp,\perp P } \cdot \mathbf{M}_{\perp,\perp P })^{1/2} 131 =(M_{\perp,x}^2+M_{\perp,y}^2+M_{\perp,z}^2(M_{\perp,x} P_x+ M_{\perp,y} P_y + M_{\perp,z} P_z )^2 )^{1/2}$. 131 132 with 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}. 132 137 133 138
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