Commit 472f13dc by Peterson, Peter Committed by GitHub

Merge pull request #30688 from mantidproject/30679_doc_latex_error

D7AbsoluteCrossSections doc latex errors
parents a69d4f7e f243bf9f
 ... ... @@ -68,9 +68,11 @@ This method is an expansion of the Uniaxial method, that requires measurements o M = 2 \cdot \left(2 \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} - \left(\frac{d\sigma_{x}}{\text{d}\Omega}\right)_{\text{nsf}} - \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{nsf}} \right) N = \frac{1}{6} \cdot \left(2 \cdot \left( \left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{d\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} \right) .. math:: N = \frac{1}{6} \cdot \left(2 \cdot \left( \left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{d\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} \right) - \left( \left( \frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{sf}} + \left( \frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{sf}} + \left( \frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{sf}} \right) \right) - \left( \left( \frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{sf}} + \left( \frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{sf}} + \left( \frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{sf}} \right) \right) .. math:: I = \frac{1}{2} \cdot \left(\left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{sf}} - M \right) ... ... @@ -84,19 +86,27 @@ The 10-point method is an expansion of the XYZ method, that requires measurement where :math:\alpha is the Sharpf angle, which for elastic scattering is equal to half of the (signed) in-plane scattering angle and :math:\theta_{0} is an experimentally fixed offset (see more in Ref. [3]). .. math:: M_{1} = (2 c_{0} - 4) \cdot \left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{nsf}} + (2c_{0} + 2) \cdot \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{nsf}} + (2-4c_{0}) \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} .. math:: M_{1} = (2 c_{0} - 4) \cdot \left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{nsf}} + (2c_{0} + 2) \cdot \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{nsf}} + (2-4c_{0}) \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} .. math:: M_{2} = (2 c_{4} - 4) \cdot \left(\frac{\text{d}\sigma_{x+y}}{\text{d}\Omega}\right)_{\text{nsf}} + (2c_{4} + 2) \cdot \left(\frac{\text{d}\sigma_{x-y}}{\text{d}\Omega}\right)_{\text{nsf}} + (2-4c_{4}) \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} .. math:: M_{2} = (2 c_{4} - 4) \cdot \left(\frac{\text{d}\sigma_{x+y}}{\text{d}\Omega}\right)_{\text{nsf}} + (2c_{4} + 2) \cdot \left(\frac{\text{d}\sigma_{x-y}}{\text{d}\Omega}\right)_{\text{nsf}} + (2-4c_{4}) \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} .. math:: .. math:: M = M_{1} \cdot \text{cos}(2\alpha) + M_{2} \cdot \text{sin}(2\alpha), M = M_{1} \cdot \text{cos}(2\alpha) + M_{2} \cdot \text{sin}(2\alpha), where :math:c_{0} = \text{cos}^{2} \alpha and :math:c_{4} = \text{cos}^{2} (\alpha - \frac{\pi}{4}) .. math:: N = \frac{1}{12} \cdot \left(2 \cdot \left( \left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{nsf}} + 2 \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{\text{d}\sigma_{x+y}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{\text{d}\sigma_{x-y}}{\text{d}\Omega}\right)_{\text{nsf}} \right) .. math:: - \left( \left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{x+y}}{\text{d}d\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{x-y}}{\text{d}\Omega}\right)_{\text{sf}} \right) \right) N = \frac{1}{12} \cdot \left(2 \cdot \left( \left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{nsf}} + 2 \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{\text{d}\sigma_{x+y}}{\text{d}\Omega}\right)_{\text{nsf}} + \left(\frac{\text{d}\sigma_{x-y}}{\text{d}\Omega}\right)_{\text{nsf}} \right) - \left( \left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{x+y}}{\text{d}d\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{x-y}}{\text{d}\Omega}\right)_{\text{sf}} \right) \right) .. math:: I = \frac{1}{4} \cdot \left(\left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}d\Omega}\right)_{\text{sf}} + 2 \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{x+y}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{x-y}}{\text{d}\Omega}\right)_{\text{sf}} - M \right) .. math:: I = \frac{1}{4} \cdot \left(\left(\frac{\text{d}\sigma_{x}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{y}}{\text{d}d\Omega}\right)_{\text{sf}} + 2 \cdot \left(\frac{\text{d}\sigma_{z}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{x+y}}{\text{d}\Omega}\right)_{\text{sf}} + \left(\frac{\text{d}\sigma_{x-y}}{\text{d}\Omega}\right)_{\text{sf}} - M \right) Sample data normalisation ... ... @@ -106,7 +116,9 @@ The sample data normalisation is the final step of data reduction of D7 sample, There are three options for the normalisation; it uses either the input from a reference sample with a well-known cross-section, namely vanadium, or the output from the cross-section separation, either magnetic or spin-incoherent cross-sections. A relative normalisation of the sample workspace to the detector with the highest counts is always performed. .. math:: S \text{(#,} \pm \text{, chn/t/meV)} = I \text{(#,} \pm \text{, chn/t/meV)} \cdot D \text{(#,} \pm \text{, chn/t/meV)}, .. math:: S (\#, \pm \text{, chn/t/meV)} = I (\#, \pm \text{, chn/t/meV)} \cdot D (\#, \pm \text{, chn/t/meV)}, where I is the sample intensity distribution corrected for all effects, and D is the normalisation factor. ... ... @@ -114,17 +126,17 @@ where I is the sample intensity distribution corrected for all effects, and D If the data is to be expressed in absolute units, the normalisation factor is the reduced vanadium data, normalised by the number of formula units in the sample :math:N_{S}: .. math:: D \text{(#,} \pm \text{, chn/t/meV)} = \frac{1}{N_{S}} \cdot V \text{(#,} \pm \text{, chn/t/meV)} .. math:: D (\#, \pm \text{, chn/t/meV)} = \frac{1}{N_{S}} \cdot V (\#, \pm \text{, chn/t/meV)} If data is not to be expressed in absolute units, the normalisation factor depends only on the vanadium input: .. math:: D \text{(#,} \pm \text{, chn/t/meV)} = \frac{V \text{(#,} \pm \text{, chn/t/meV)}}{\text{max}(V \text{(#,} \pm \text{, chn/t/meV)})} .. math:: D (\#, \pm \text{, chn/t/meV)} = \frac{V (\#, \pm \text{, chn/t/meV)}}{\text{max}(V (\#, \pm \text{, chn/t/meV)})} 2. Paramagnetic This normalisation is not valid for TOF data, and requires input from XYZ or 10-point cross-section separation. The paramagnetic measurement does not need to have background subtracted, as the background is self-subtracted in an XYZ measurement. .. math:: D \text{(#)} = \frac{2}{3} \frac{(\gamma r_{0})^{2} S(S+1)}{P \text{(#)}}, .. math:: D (\#) = \frac{2}{3} \frac{(\gamma r_{0})^{2} S(S+1)}{P (\#)}, where :math:\gamma is the neutron gyromagnetic ratio, :math:r_{0}` is the electron's classical radius, and S is the spin of the sample. ... ... @@ -134,11 +146,11 @@ Similarly to the paramagnetic normalisation, it is also not valid for TOF data, The data can be put on absolute scale if the nuclear-spin-incoherent (NSI) cross-section for the sample is known, then: .. math:: D \text{(#)} = \frac{\text{d} \sigma_{NSI}}{\text{d} \Omega} \frac{1}{NSI \text{(#)}}. .. math:: D (\#) = \frac{\text{d} \sigma_{NSI}}{\text{d} \Omega} \frac{1}{NSI (\#)}. If only the detector efficiency is to be corrected, then it is sufficient to use only the nuclear-spin-incoherent cross-section input: .. math:: D \text{(#)} = \frac{\text{max}(NSI \text{(#)})}{NSI \text{(#)}}. .. math:: D (\#) = \frac{\text{max}(NSI (\#))}{NSI (\#)}. Usage ... ...
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