Indexing

Content

Implementation of algorithms

pyimgalgos/src/FiberIndexing.py self-documentation

Script

cxif5315/make-index-table.py

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.

                *----------*
               / \        / \
              /   \      /   \
             /     \ gamma    \
            /       *----------*
           /       /  /       /
          /alpha  /  /       /
         *-------/--*       c
          \     /    \beta /
           a   /      \   /
            \ /        \ /
             *-----b----*

As a starting point in this analysis we use parameters from previous study

  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

  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

3-d primitive vectors can be converted in reciprocal space primitive vectors

  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

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

Run 169 photon energy

See: PDG-2014

photon energy = 6003.1936 eV
wavelength = 2.0653 A
wave number/Evald radius k = 1/lambda = 0.484187 1/A

Lookup table generator

def make_lookup_table(b1 = (1.,0.,0.), b2 = (0.,1.,0.), b3 = (0.,0.,1.),\
                      hmax=3, kmax=2, lmax=1, cdtype=np.float32,\
                      evald_rad=3, sigma_q=0.001, fout=None, bpq=None, bpomega=None, bpbeta=None) :
    """Makes lookup table - peak information as a function of angle beta and omega, where
       beta  [deg] - fiber axis tilt,  
       omega [deg] - fiber rotation around axis,  
       For each crysal orientation (beta, gamma) lookup table contains info about lattice nodes
       closest to the Evald's sphere: 
       # beta 20.00  omega 178.50 degree
       # index   beta     omega   h  k  l     dr [1/A]   R(h,k,l)   qv [1/A]   qh [1/A]   P(omega)
         1078    20.00   178.50   1 -5  0     0.000262   0.211944  -0.016779   0.211221   0.964192
         1078    20.00   178.50   0 -1  0     0.002470   0.038484   0.000343   0.038306   0.038686
         1078    20.00   178.50   0  1  0     0.000582   0.038484  -0.000344  -0.038455   0.834544

       where:
       index - orientation index (just an unique integer number)
       beta, omega [deg] - crystal orientation angles,
       h, k, l - Miller indeces
       dr [1/A] - distance between lattice node and Evald's sphere
       R(h,k,l) [1/A] - distance between nodes (h,k,l) and (0,0,0)
       qv, qh [1/A] - vertical and horizontal components of scattering vector q
       P(omega) - un-normalized probability (<1) evaluated for dr(omega) using sigma_q.

       File name is generated automatically with current time stamp like
       lut-cxif5315-r0169-2015-10-23T14:58:36.txt

       Input parameters:
       b1, b2, b3 - reciprocal lattice primitive vectors,
       hmax, kmax, lmax - lattice node indeces
       cdtype - data type for lattice node coordinates,
       evald_rad - Evald's sphere radius,
       sigma_q - expected q resolution,
       fout - open output file object,
       bpq, bpomega, bpbeta - binning parameters for q, omega, and beta
       NOTE: Units of b1, b2, b3, evald_rad, and sigma_q should be the same, for example [1/A].

       Returns 2-d numpy array for image; summed for all beta probobility(omega vs. q_horizontal).
    """

Lookup table content

Metadata

# file name: lut-cxif5315-r0169-2015-10-23T16:03:06.txt

# Triclinic crystal cell parameters:
#   a = 18.36 A
#   b = 26.65 A
#   c = 4.81 A
#   alpha = 90.00 deg
#   beta  = 90.00 deg
#   gamma = 102.83 deg

# 3-d space primitive vectors:
#   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:
#   b1 = [ 0.05446623 -0.01240442  0.        ]
#   b2 = [ 0.          0.03848428  0.        ]
#   b3 = [ 0.          0.          0.20790021]

# photon energy = 6003.1936 eV
# wavelength = 2.0653 A
# wave number/Evald radius k = 1/lambda = 0.484187 1/A
# sigma_q   = 0.000484 1/A (approximately pixel size/sample-to-detector distance = 100um/100mm)
# 3*sigma_q = 0.001453 1/A

Data

# beta 20.00  omega 170.50 degree
# index   beta     omega   h  k  l     dr [1/A]   R(h,k,l)   qv [1/A]   qh [1/A]   P(omega)
  1062    20.00   170.50   0  4  0     0.000595   0.153937  -0.008679  -0.153596   0.469484

# beta 20.00  omega 171.00 degree EMPTY
# beta 20.00  omega 171.50 degree EMPTY
# beta 20.00  omega 172.00 degree
# index   beta     omega   h  k  l     dr [1/A]   R(h,k,l)   qv [1/A]   qh [1/A]   P(omega)
  1065    20.00   172.00   0  3  0    -0.001336   0.115453  -0.005511  -0.115473   0.022208

# beta 20.00  omega 172.50 degree
# index   beta     omega   h  k  l     dr [1/A]   R(h,k,l)   qv [1/A]   qh [1/A]   P(omega)
  1066    20.00   172.50   1 -4  0     0.001294   0.175032  -0.011014   0.174446   0.028162
  1066    20.00   172.50   0  3  0    -0.000396   0.115453  -0.005158  -0.115384   0.715354

Plots

Constant fiber tilt angle β = 15°

Superposition of tilt angles β = 5, 25°

Comparison of these images show that tiny difference (doubled peaks for β = 5, 25°) is observed for large values of |qh|.

Comparison of data with triclinic lattice prediction

Re-tune geometry

Left- and right-hand side aro not quite symmetric. Re-tune geometry Geometry re-alignment

The same plots for 2015-11-06 geometry:

Now data peaks look symmetric relative to the center of image, but predicted peaks are misaligned relative to data.

Re-rune lattice parameters

Measurement of peak radial parameters. Two dash-curve circles have radial parameters 100 and 300 pixels, respectively.

Radial parameters of the brightest peaks/rings

R (pix)	h,k	R-Ratio	qH (1/Å)
71.4	0,1	1	0.0375
104.7	1,0	1.466	0.0550
114.3	-1,1	1.601	0.0601
144.6	1,1/0,2	2.025	0.0725/0.0751
159.9	-1,2	2.239	0.0837
210.1	-2,1/2,0	2.942	0.1089/0.1100
330.1		4.622
400.4		5.608
Ceramic	rings
438.7		6.144
626.1		8.769
784.6		10.989

The three peaks with smallest radial parameters gives gamma = 101.53°.

Evaluated lattice parameters:

# Triclinic crystal cell parameters:
#   a = 18.55 A
#   b = 27.19 A
#   c = 4.86 A
#   gamma = 78.47 deg   or 101.53

# 3-d space primitive vectors:
#   a1 = (18.55, 0.0, 0.0)
#   a2 = (-5.435623727109098, 26.645523961582906, 0.0)
#   a3 = (0.0, 0.0, 4.8601)

# reciprocal space primitive vectors:
#   b1 = [ 0.05390836  0.01099718 -0.        ]
#   b2 = [ 0.          0.03752976  0.        ]
#   b3 = [ 0.         -0.          0.20575708]

Table of lattice node parameters sorted by radius

( h, k) R(h,k)[1/A]
( 0, 0) 0.0000
( 0, 1) 0.0375 cut
( 1, 0) 0.0550 *
(-1, 1) 0.0601 *
( 1, 1) 0.0725 
( 0, 2) 0.0751
(-1, 2) 0.0837 *
( 1, 2) 0.1015
(-2, 1) 0.1089 +
( 2, 0) 0.1100 +
( 0, 3) 0.1126
(-1, 3) 0.1150
(-2, 2) 0.1202
( 2, 1) 0.1232
( 1, 3) 0.1348
(-2, 3) 0.1408
( 2, 2) 0.1451
(-1, 4) 0.1492
( 0, 4) 0.1501
(-3, 1) 0.1618
( 3, 0) 0.1651
(-3, 2) 0.1671
(-2, 4) 0.1675
( 1, 4) 0.1699
( 2, 3) 0.1724
( 3, 1) 0.1764
(-3, 3) 0.1803
(-1, 5) 0.1847
( 0, 5) 0.1876
( 3, 2) 0.1945
(-2, 5) 0.1977
(-3, 4) 0.1997
( 2, 4) 0.2031
( 1, 5) 0.2058
(-4, 1) 0.2157
( 3, 3) 0.2176
(-4, 2) 0.2179
( 4, 0) 0.2201
(-1, 6) 0.2209
(-3, 5) 0.2238
( 0, 6) 0.2252
(-4, 3) 0.2263

Indexing optimization

Optimization parameters

sigma_q - in lookup table, works for omega; fraction of the wave number from 0.001 to 0.003
sigma_q - in record processing, works for dq = r(node) - r(Evald); 1/A
range of omega angle, ex: 0-180, 360 - stands for 360 points in the range from 0 to 180°
range(s) of beta angle, usually used for two bands, small angles and symmetric with +180° offset.

beta ranges from 0 to 30° and from 0 to -30° produces the same results.

Figure of merit

Total number of selected events with ≥2 peaks 1915

Number of indexed events with ≥2 peaks, and distribution of events indexed with 1, 2, 3 peaks

Optimal parameters

sigma_q= 0.002 in both cases
range omega (0-180,360)
ranges beta (0-50,11) and (180-230,11)

In run 169 this look-up table yields in 1171 events indexed for ≥2 peaks

Plots

Image of the look-up table for low range of beta angles, superposition of low and high ranges, projection of the lookup table on q_h

Comparison of the look-up table and data (geometry and lattice parameters are re-tuned)

Indexing

Script

proc-cxif5315-r0169-peaks-from-file-v2.py - loop over peaks in event and try to associate event peaks with a list of predicted for each crystal orientation. Most probable combination is saved in the file.

File record for event

There are three states for matching peaks in event indexing:

MATCHED - peak is matched to predicted crystal node,
PEAK-NM - peak in data is not matched with any of predicted nodes,
NODE-NM - predicted crystal node is on Evald sphere, but peak in data is missing.

Event indexing table contains records for all peaks from data and for all predicted nodes from crystal orientation. Each record consists of three parts; status word, peak information, and orientation information. In case if orientation or peak information is missing it is replaced by zeros, like shown in example below.

  STATE phi-fit beta-fit   qh-fit    qv-fit  dqh[1/A]   Exp     Run  time(sec)   time(nsec) fiduc    Evnum  Reg  Seg  Row  Col  Npix      Amax      Atot   rcent   ccent rsigma  csigma rmin rmax cmin cmax    bkgd     rms     son  imrow   imcol     x[um]     y[um]     r[um]  phi[deg]   index    beta   omega   h   k   l   dr[1/A]  R(h,k,l)   qv[1/A]   qh[1/A]  P(omega)
NODE-NM   -7.98  -12.45  0.000000  0.000000  0.000000 cxif5315  169  1424601179  232228024  105684   85184  N/A    0    0    0     0       0.0       0.0     0.0     0.0   0.00    0.00    0    0    0    0    0.00    0.00    0.00      0       0         0         0         0      0.00    230   180.00  114.50   3  -4   0  0.001658  0.199684  0.000000  0.199336  0.230902
PEAK-NM   -7.98  -12.45  0.083921  0.000014  0.000000 cxif5315  169  1424601179  232228024  105684   85184  EQU    1   72   44    31     220.9    1976.6    73.4    42.7   2.49    1.45   70   80   39   46   16.23   19.99   10.23    559     730     17471      2112     17598      6.89      0     0.00    0.00   0   0   0  0.000000  0.000000  0.000000  0.000000  0.000000
MATCHED   -7.98  -12.45  0.059485 -0.000022  0.000448 cxif5315  169  1424601179  232228024  105684   85184  EQU    1  119   39    59    1293.5   11106.4   119.0    39.5   1.69    1.50  112  125   36   45   17.81   31.87   40.03    564     683     12306      1551     12403      7.18    230   180.00  114.50   1  -1   0  0.001936  0.060084  0.000000  0.059933  0.135565
MATCHED   -7.98  -12.45 -0.166810 -0.000001  0.000288 cxif5315  169  1424601179  232228024  105684   85184  EQU   16  121  101    88     276.3    6888.4   120.0   103.1   2.52    2.93  115  127   98  110   39.18   36.32    6.53    637     247    -35645     -6412     36217   -169.80    230   180.00  114.50  -3   2   0  0.000051  0.167107  0.000000 -0.167098  0.998636

This information for all events goes in the file with name like peak-idx-cxif5315-r0169-2015-11-13T17:04:37.txt.

Result presenter

cxif5315/proc-cxif5315-r0169-idx-file.py

reads file obtained in previous section using

from pyimgalgos.TDFileContainer import TDFileContainer
from pyimgalgos.TDMatchRecord   import TDMatchRecord

and generates summary tables/images with results as listed below.

Crystal in h-k space

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. ).

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)
...
        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)

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

Page tree

Indexing

Content

Implementation of algorithms

Script

Triclinic crystal cell parameters

3-d space primitive vectors

Reciprocal space primitive vectors

Table of lattice nodes sorted by radius

Run 169 photon energy

Lookup table generator

Lookup table content

Metadata

Data

Plots

Re-tune geometry

Re-rune lattice parameters

Indexing optimization

Optimization parameters

Figure of merit

Optimal parameters

Plots

Indexing

Script

File record for event

Result presenter

Crystal in h-k space

Crystal in reciprocal space

Try peak_finder_v3

References