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doc/source/manual/appendix.rst

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-Appendix=============Diffraction simulation----------------------The simulation of a white-beam diffraction pattern from a known materialstarts from the calculation of monochromatic beam diffraction patternsfor the specific detector location, *I(θ,E)*. Then these mono-beamdiffraction patterns are integrated over the entire energy range withweighting factor being the flux of photons with different energy, F(E).where E\ :sub:`1` and E\ :sub:`2` are typically 1 keV and 60 keV,respectively. To improve the calculation speed, discrete diffractionpeaks are considered, meaning that *I(θ,E)* is normally replaced by aseries of I\ :sub:`hkl`\ (θ,E). Equation above then becomesIn our simulations, the diffraction peak shape is described using thepseudo-Voigt function.Essentially, the white-beam diffraction intensity at a given scatteringangle is the convolution of the input diffraction intensity of differentatomic planes with the energy spectrum of the x-rays.|image24|Figure 23: Simulation of white-beam diffractionHow to obtain the sample absorption file----------------------------------------    **Option 1**: Ask the beamline scientists for it. If users are not    confident to calculate the attenuation curve, asking the beamline    scientists is always the best choice. Provide sample parameters    including chemical formula, mass density, and thickness. For    example, sample Al\ :sub:`2`\ O\ :sub:`3`, density of 3.95    g/cm\ :sup:`3`, thickness 500 µm.    **Option 2**: Use the web tool of The Center of X-Ray Optics    (http://henke.lbl.gov/optical_constants/). Calculate the x-ray    transmission for a solid with procedure shown below and in Fig. 24.-  Fill in sample information-  Select energy range-  Click “Submit Request” to do the calculation. Another window will   then pop up with the plot-  Click “data file here” to show the values-  Copy the data (without the sample info lines), and save it as a   \*.txt file    Note that this online tool can only calculate the transmission for    energy ranging from 0-30 keV, which often not suffice for white-beam    experiments.|image25|Figure 24: Calculate x-ray transmission at the website of CXRO    **Option 3**: Self calculate based on the x-ray mass attenuation    coefficients from NIST. At NIST webpage    (http://physics.nist.gov/PhysRefData/XrayMassCoef/tab3.html), the    attenuation coefficients (*µ/ρ*) are provided for elements from    Hydrogen to Uranium. The x-ray attenuation *T* is calculated as:    where *µ/ρ* is value directly from the webpage, *ρ* is the mass    density of the sample, and *t* is the sample thickness.    For compounds, the attenuation coefficients will simply be the    weighted-sum of each element.    where *w\ :sub:`i`* is the weight (or mass) fraction of element *i.*    For example of Al\ :sub:`2`\ O\ :sub:`3`, density 3.95    g/cm\ :sup:`3`, thickness 500 µm, and for x-ray energy 10 keV. The    mass coefficient of the sample will be:    cm\ :sup:`2`/g    The attenuation will then be:Some comparisons to guide the usage-----------------------------------1) Scaling factor: 4 vs 2 vs 1   Always, the larger the scaling factor, the faster the processing   time, yet the smaller the resolution. For the example shown in Fig.   25, the simulation time for scaling factor 4, 2, and 1 are 11 min,   1.5 min, and 14 sec, respectively. (Note the CPU is Intel i7, 3.4   GHz) Clearly, the small peak at 48° can only be visualized in the   patterns with scaling factor 2 and 1. The effect of scaling factor on   the analysis of experiment data is not as obvious, as the   signal-to-noise ratio of single-pulse diffraction patterns is rather   low and radially averaging is more related to the parameters “Points”   and “Q res”. For image with dimension of about 1k x 1k, the scaling   factor of 2 may be a good option.|image26|Figure 25: Effect of scaling factor2) Direct beam position: wrong vs rightFigure 26 show the overlap of diffraction rings with the referencepeaks in cases of wrong and correct direct beam position input. Thecorrect parameters should satisfy all other data, either same samplewith different undulator gap (i.e. energy spectrum) or other samples.For this specific example in Fig. 26, all the diffraction rings aregenerated by the 1\ :sup:`st` harmonic energy, so they are all labeled,but this is not always the case. Figure 27 show the data from anotherundulator gap, in which some diffraction rings are generated by2\ :sup:`nd` harmonic energy. Users may simply input the photon energyused for labeling in the “E1 (keV)” space.|image27|Figure 26: Effect of direction beam locating on calculation of q map|image28|Figure 27: Reference peaks corresponds to 1st and 2nd harmonic energies3) Pointes and Q res: rough vs smooth plotsSmooth plot looks good, but may lose some structure information due tothe over averaging.|image29|Figure 28: Rough and smooth (yet low-resolution) 1D plots4) Region of interest: without ROI vs with ROIThis is not necessary if all the pixels on the detector functionproperly.|image30|Figure 29: Effect of ROI on the 1D plot
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-.. |image24| image:: figures/image25.png.. |image25| image:: figures/image26.png.. |image26| image:: figures/image27.png.. |image27| image:: figures/image28.png.. |image28| image:: figures/image29.png.. |image29| image:: figures/image30.png.. |image30| image:: figures/image31.png
+Appendix=============Diffraction simulation----------------------The simulation of a white-beam diffraction pattern from a known materialstarts from the calculation of monochromatic beam diffraction patternsfor the specific detector location, *I(θ,E)*. Then these mono-beamdiffraction patterns are integrated over the entire energy range withweighting factor being the flux of photons with different energy, F(E)... math::  I_{white} = \int_{E_{1}}^{E_{2}} I(\theta,E) \cdot F(E)\,dE.where E\ :sub:`1` and E\ :sub:`2` are typically 1 keV and 60 keV,respectively. To improve the calculation speed, discrete diffractionpeaks are considered, meaning that *I(θ,E)* is normally replaced by aseries of I\ :sub:`hkl`\ (θ,E). Equation above then becomes:.. math::  I_{white} = \int_{E_{1}}^{E_{2}} \left(\displaystyle\sum_{i=1}^{n} I_{hkl_{i}}(\theta,E)\right) \cdot F(E)\,dE.In our simulations, the diffraction peak shape is described using thepseudo-Voigt function:.. math::  V(x) = \left(\frac{1-\eta}{\sigma\sqrt{2\pi}}\right) \cdot exp\left({\frac{x^2}{2\sigma^2}}\right) + \eta \cdot \frac{\sigma}{2\pi} \cdot \frac{1}{x^2 + \left(\frac{\sigma}{2}\right)^2}Essentially, the white-beam diffraction intensity at a given scatteringangle is the convolution of the input diffraction intensity of differentatomic planes with the energy spectrum of the x-rays.|image24|Figure 23: Simulation of white-beam diffractionHow to obtain the sample absorption file----------------------------------------    **Option 1**: Ask the beamline scientists for it. If users are not    confident to calculate the attenuation curve, asking the beamline    scientists is always the best choice. Provide sample parameters    including chemical formula, mass density, and thickness. For    example, sample Al\ :sub:`2`\ O\ :sub:`3`, density of 3.95    g/cm\ :sup:`3`, thickness 500 µm.    **Option 2**: Use the web tool of     `The Center of X-Ray Optics <http://henke.lbl.gov/optical_constants/>`_.    Calculate the x-ray transmission for a solid with procedure shown below     and in Fig. 24.    -  Fill in sample information    -  Select energy range    -  Click “Submit Request” to do the calculation. Another window will then pop up with the plot    -  Click “data file here” to show the values    -  Copy the data (without the sample info lines), and save it as a \*.txt file    Note that this online tool can only calculate the transmission for    energy ranging from 0-30 keV, which often not suffice for white-beam    experiments.|image25|Figure 24: Calculate x-ray transmission at the website of CXRO    **Option 3**: Self calculate based on the x-ray mass attenuation    coefficients from the    `NIST <http://physics.nist.gov/PhysRefData/XrayMassCoef/tab3.html>`_    web page. There the attenuation coefficients (*µ/ρ*) are provided     for elements from Hydrogen to Uranium. The x-ray attenuation *T*      is calculated as:.. math::  T = exp\left({-\frac{\mu}{\rho}}\cdot \rho \cdot t \right)    where *µ/ρ* is value directly from the `web page <http://physics.nist.gov/PhysRefData/XrayMassCoef/tab3.html>`_,*ρ* is the mass density of the sample, and *t* is the sample thickness.For compounds, the attenuation coefficients will simply be the weighted-sum of each element:.. math::  \frac{\mu}{\rho} = \displaystyle\sum_{i} w_i \cdot \left(\frac{\mu}{\rho}\right)_iwhere w\ :sub:`i` is the weight (or mass) fraction of element *i*.For example of Al\ :sub:`2`\ O\ :sub:`3`, density 3.95g/cm\ :sup:`3`, thickness 500 µm, and for x-ray energy 10 keV, themass coefficient of the sample will be:.. math::  \frac{\mu}{\rho} = \frac{27 \times 2}{27 \times 2 + 16 \times 3} \times 25.43 + \frac{16 \times 3}{27 \times 2 + 16 \times 3} \times 5.565 = 16.08  cm^2/g.The attenuation will then be:.. math::  T = exp(-16.08 \times 3.95 \times 500 \times 10^{-4}) = 0.042Some comparisons to guide the usage-----------------------------------1) Scaling factor: 4 vs 2 vs 1   Always, the larger the scaling factor, the faster the processing   time, yet the smaller the resolution. For the example shown in Fig.   25, the simulation time for scaling factor 4, 2, and 1 are 11 min,   1.5 min, and 14 sec, respectively. (Note the CPU is Intel i7, 3.4   GHz) Clearly, the small peak at 48° can only be visualized in the   patterns with scaling factor 2 and 1. The effect of scaling factor on   the analysis of experiment data is not as obvious, as the   signal-to-noise ratio of single-pulse diffraction patterns is rather   low and radially averaging is more related to the parameters “Points”   and “Q res”. For image with dimension of about 1k x 1k, the scaling   factor of 2 may be a good option.|image26|Figure 25: Effect of scaling factor2) Direct beam position: wrong vs rightFigure 26 show the overlap of diffraction rings with the referencepeaks in cases of wrong and correct direct beam position input. Thecorrect parameters should satisfy all other data, either same samplewith different undulator gap (i.e. energy spectrum) or other samples.For this specific example in Fig. 26, all the diffraction rings aregenerated by the 1\ :sup:`st` harmonic energy, so they are all labeled,but this is not always the case. Figure 27 show the data from anotherundulator gap, in which some diffraction rings are generated by2\ :sup:`nd` harmonic energy. Users may simply input the photon energyused for labeling in the “E1 (keV)” space.|image27|Figure 26: Effect of direction beam locating on calculation of q map|image28|Figure 27: Reference peaks corresponds to 1st and 2nd harmonic energies3) Pointes and Q res: rough vs smooth plotsSmooth plot looks good, but may lose some structure information due tothe over averaging.|image29|Figure 28: Rough and smooth (yet low-resolution) 1D plots4) Region of interest: without ROI vs with ROIThis is not necessary if all the pixels on the detector functionproperly.|image30|Figure 29: Effect of ROI on the 1D plot.. |image24| image:: figures/image25.png.. |image25| image:: figures/image26.png.. |image26| image:: figures/image27.png.. |image27| image:: figures/image28.png.. |image28| image:: figures/image29.png.. |image29| image:: figures/image30.png.. |image30| image:: figures/image31.png