Changeset 6e0c1df in sasmodels


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

correct typos.

File:
1 edited

Legend:

Unmodified
Added
Removed
  • doc/guide/magnetism/magnetism.rst

    r6b86bee r6e0c1df  
    44================================ 
    55 
    6 (Version 2: Spherical Polarimetry in SANS: Allow for freedom in field/polarisation axis away from the detector plane, 
    7 i.e. allow in-beam direction or oscillatory/rotational fields...) 
     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. 
    89 
    910For magnetic systems, the scattering length density (SLD = $\beta$) is a combination 
     
    2122nuclear scattering and has one effective magnetisation orientation 
    2223 
    23 The external field $\mathbf{H}=H \mathbf{P}$coincides with the polarisation axis 
     24The external field $\mathbf{H}=H \mathbf{P}$ coincides with the polarisation axis 
    2425$\mathbf{P}=(\sin\theta_P \cos \phi_P, \sin \theta_P \sin \phi_P,\cos\theta_P )$ for the neutrons, which is the quantisation axis 
    2526for the Pauli spin operator. 
     
    6061after the sample. 
    6162 
    62 For magnetic neutron scattering, only the magnetisation component or Halpern-Johnson vector  
     63For magnetic neutron scattering, only the magnetisation component or Halpern-Johnson vector 
    6364$\mathbf{M_\perp}$ perpendicular to the scattering vector 
    6465$\mathbf{Q}=q \mathbf{n}=q (\cos\theta, \sin \theta,0)$ contributes to the magnetic scattering: 
     
    7677 
    7778.. 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}) 
    7980        + (\mathbf{P}\cdot \mathbf{M_\perp})^2 
    8081 
     
    9394 
    9495.. 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 
    9798 
    9899The component perpendicular to the polarisation gives rise to SF scattering. The perpendicular 
     
    101102.. math:: 
    102103    \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 
    104105 
    105106and a third vector perpendicular to both $\mathbf{P}$ and $\mathbf{M}_{\perp,\perp P } $ : 
    106107 
    107108.. 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 Halpern-Johnson 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})] 
    110111 
    111 For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\matbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \matbf{O}^{\ast}=0$ 
     112 
     113For symmetric, collinear spin structures ($\mathbf{M}_{\perp}^{\ast}=\mathbf{M}_{\perp}^{\ast}$), $\mathbf{O}\cdot \mathbf{O}^{\ast}=0$ 
    112114since  $\mathbf{M}_{\perp} \times \mathbf{P} \cdot \mathbf{M}_{\perp} \times \mathbf{P} = \mathbf{M}_{\perp,\perp P }$. 
    113115 
     
    115117Depending on the spin state of the 
    116118neutrons, the scattering length densities, including the nuclear scattering 
    117 length density $(\beta{_N})$ are 
     119length density $(\beta{_N})$ are for the non-spin-flip states 
    118120 
    119121.. math:: 
    120     \beta_{\pm\pm} =  \beta_N \mp b_H math{P}\cdot M_{\perp } 
    121     \text{ for non-spin-flip states} 
     122    \beta_{\pm\pm} =  \beta_N \mp b_H \mathbf{P}\cdot \mathbf{M}_{\perp } 
    122123 
    123 and 
     124 
     125and for spin-flip states 
    124126 
    125127.. 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 spin-flip 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})) 
    128129 
    129130 
    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 
     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}. 
    132137 
    133138 
Note: See TracChangeset for help on using the changeset viewer.