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Implementation of algorithms
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Triclinic crystal cell parameters
Triclinic crystal cell in 3-d is described by cell edge lengthes, a, b, c, and associated angles between edges, alpha, beta, and gamma, as shown in the plot.
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*----------*
/ \ / \
/ \ / \
/ \ gamma \
/ *----------*
/ / / /
/alpha / / /
*-------/--* c
\ / \beta /
a / \ /
\ / \ /
*-----b----* |
As a starting point in this analysis we use parameters from previous study
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a = 18.36 A b = 26.65 A c = 4.81 A alpha = 90.00 deg beta = 90.00 deg gamma = 102.83 deg |
See: Crystal structure, Bravais lattice, Crystal system, Primitive cell, Lattice constant
3-d space primitive vectors
Crystal cell parameters can be transformed to 3-d primitive vectors
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a1 = (18.36, 0.0, 0.0) a2 = (5.917873795354449, 25.984635262829016, 0.0) a3 = (0.0, 0.0, 4.81) |
Reciprocal space primitive vectors
See: Reciprocal lattice, Surface diffraction, Miller index3-d primitive vectors can be converted in reciprocal space primitive vectors
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b1 = [ 0.05446623 0.01240442 0. ] b2 = [ 0. 0.03848428 0. ] b3 = [ 0. 0. 0.20790021] |
See: Reciprocal lattice, Surface diffraction, Miller index
Table of lattice nodes sorted by radius
Lattice nodes in reciprocal space can be evaluated from primitive vectors and associated Muller indexes h,k,l. Lattice nodes ordered by ascending R(h,k,l) - distance between (h,k,l) and (0,0,0) points, is presented in table
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2-d case 3-d case ( h, k) R(h,k)[1/A] ( h, k, l) R(h,k,l)[1/A] ___________________ ________________________ ( 0, 0) 0.0000 ( 0, 0, 0) 0.0000 ( 0, 1) 0.0385 ( 0, 1, 0) 0.0385 ( 1, 0) 0.0559 ( 1, 0, 0) 0.0559 (-1, 1) 0.0604 (-1, 1, 0) 0.0604 ( 1, 1) 0.0745 ( 1, 1, 0) 0.0745 ( 0, 2) 0.0770 ( 0, 2, 0) 0.0770 (-1, 2) 0.0845 (-1, 2, 0) 0.0845 ( 1, 2) 0.1047 ( 1, 2, 0) 0.1047 (-2, 1) 0.1098 (-2, 1, 0) 0.1098 ( 2, 0) 0.1117 ( 2, 0, 0) 0.1117 ( 0, 3) 0.1155 ( 0, 3, 0) 0.1155 (-1, 3) 0.1166 (-1, 3, 0) 0.1166 (-2, 2) 0.1208 (-2, 2, 0) 0.1208 ( 2, 1) 0.1260 ( 2, 1, 0) 0.1260 ( 1, 3) 0.1390 ( 1, 3, 0) 0.1390 (-2, 3) 0.1417 (-2, 3, 0) 0.1417 ( 2, 2) 0.1491 ( 2, 2, 0) 0.1491 (-1, 4) 0.1517 (-1, 4, 0) 0.1517 ( 0, 4) 0.1539 ( 0, 4, 0) 0.1539 (-3, 1) 0.1634 (-3, 1, 0) 0.1634 ( 3, 0) 0.1676 ( 3, 0, 0) 0.1676 (-3, 2) 0.1682 (-3, 2, 0) 0.1682 (-2, 4) 0.1689 (-2, 4, 0) 0.1689 ( 1, 4) 0.1750 ( 1, 4, 0) 0.1750 ( 2, 3) 0.1776 ( 2, 3, 0) 0.1776 ( 3, 1) 0.1801 ( 3, 1, 0) 0.1801 (-3, 3) 0.1812 (-3, 3, 0) 0.1812 (-1, 5) 0.1881 (-1, 5, 0) 0.1881 ( 0, 5) 0.1924 ( 0, 5, 0) 0.1924 ( 3, 2) 0.1993 ( 3, 2, 0) 0.1993 (-2, 5) 0.1999 (-2, 5, 0) 0.1999 (-3, 4) 0.2008 (-3, 4, 0) 0.2008 ( 2, 4) 0.2093 ( 0, 0, 1) 0.2079 1st contributing index with l=1 ( 1, 5) 0.2119 ( 2, 4, 0) 0.2093 (-4, 1) 0.2181 ( 0, 1, 1) 0.2114 (-4, 2) 0.2196 ( 1, 5, 0) 0.2119 ( 4, 0) 0.2234 ( 1, 0, 1) 0.2153 ( 3, 3) 0.2236 (-1, 1, 1) 0.2165 (-3, 5) 0.2254 (-4, 1, 0) 0.2181 (-4, 3) 0.2276 (-4, 2, 0) 0.2196 ( 4, 1) 0.2350 ( 1, 1, 1) 0.2209 (-4, 4) 0.2416 ( 0, 2, 1) 0.2217 ( 2, 5) 0.2430 ( 4, 0, 0) 0.2234 ( 3, 4) 0.2515 ( 3, 3, 0) 0.2236 ( 4, 2) 0.2520 (-1, 2, 1) 0.2244 (-4, 5) 0.2605 (-3, 5, 0) 0.2254 (-5, 2) 0.2727 (-4, 3, 0) 0.2276 ( 4, 3) 0.2733 ( 1, 2, 1) 0.2328 (-5, 1) 0.2733 ( 4, 1, 0) 0.2350 (-5, 3) 0.2775 (-2, 1, 1) 0.2351 ( 5, 0) 0.2793 ( 2, 0, 1) 0.2360 ( 3, 5) 0.2818 ( 0, 3, 1) 0.2378 (-5, 4) 0.2874 (-1, 3, 1) 0.2383 ( 5, 1) 0.2903 (-2, 2, 1) 0.2404 ( 4, 4) 0.2982 (-4, 4, 0) 0.2416 (-5, 5) 0.3019 ( 2, 5, 0) 0.2430 ( 5, 2) 0.3057 ( 2, 1, 1) 0.2431 ( 5, 3) 0.3251 ( 1, 3, 1) 0.2501 ( 4, 5) 0.3257 ( 3, 4, 0) 0.2515 ( 5, 4) 0.3476 (-2, 3, 1) 0.2516 ( 5, 5) 0.3727 ( 4, 2, 0) 0.2520 |
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and generates summary tables/images with results as listed below.
Crystal in h
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-k space
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k : -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 h=-4: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 h=-3: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0222 0.0618 0.0000 1.0000 0.0075 0.0680 0.0000 0.0000 h=-2: 0.0000 0.0000 0.0156 0.0158 0.0013 0.0000 0.2334 0.6190 0.0272 0.0000 0.0078 0.3472 0.0000 h=-1: 0.0000 0.0070 0.0395 0.0000 0.0023 0.0000 0.1424 0.2158 0.3511 0.0036 0.0273 0.0019 0.0000 h= 0: 0.0000 0.0015 0.0131 0.0135 0.0031 0.0000 0.4409 0.0000 0.0290 0.0600 0.0389 0.0015 0.0000 h= 1: 0.0000 0.0000 0.0319 0.0000 0.3228 0.1522 0.1187 0.0038 0.0000 0.0000 0.0173 0.0089 0.0000 h= 2: 0.0000 0.3036 0.0131 0.0011 0.0043 0.5837 0.2267 0.0000 0.0039 0.0000 0.0000 0.0000 0.0000 h= 3: 0.0000 0.0000 0.0708 0.0049 0.7698 0.0000 0.0699 0.0172 0.0014 0.0035 0.0000 0.0000 0.0000 h= 4: 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 |
Crystal in reciprocal space
Try peak_finder_v3
We try to use peak_finder_v3 with parameters (see Peak finders for more detail. ).
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alg_arc = PyAlgos(windows=winds_arc, mask=mask_arc, pbits=0)
alg_arc.set_peak_selection_pars(npix_min=5, npix_max=100, amax_thr=0, atot_thr=2000, son_min=6)
alg_equ = PyAlgos(windows=winds_equ, mask=mask_equ, pbits=0)
alg_equ.set_peak_selection_pars(npix_min=5, npix_max=100, amax_thr=0, atot_thr=2000, son_min=6)
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peaks_arc = alg_arc.peak_finder_v3(nda, rank=5, r0=5, dr=0.05)
peaks_equ = alg_equ.peak_finder_v3(nda, rank=5, r0=5, dr=0.05)
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It finds approximately 3× more (seed) peaks with higher atot_thr than peak_finder_v2.
This makes event selection more complicated and in result the same matching algorithm finds 4× less peaks. 
References
- Crystal structure, Bravais lattice, Crystal system, Primitive cell, Lattice constant - 3-d lattice
- Reciprocal lattice, Ewald's sphere, Surface diffraction, Miller index - reciprocal space
- PDG-2014
- X-ray crystallography, Bragg's law, Fiber diffraction
- my not on units for q and reciprocal space
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