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12       Indexing

The following algorithm is based on the iterative method of Coelho (2003). Unlike lp_serach it requires the extraction of d-spacings. The INDEXING directory contains example INP files, for example:

index_zero_error

try_space_groups “2 75”

load index_d {

   8.912

   7.126

   4.296

   …

}

Individual space groups can be tried or for simplicity all of the Bravais lattices can be tried by placing them in the INP file using the standard macros as follows:

Bravais_Cubic_sgs

Bravais_Trigonal_Hexagonal_sgs

Bravais_Tetragonal_sgs

Bravais_Orthorhombic_sgs

Bravais_Monoclinic_sgs

Bravais_Triclinic_sgs

On termination of Indexing a *.NDX file is created with a name corresponding to the name of the INP file and placed in the same directory as the INP file. The *.NDX file contains solutions found as well as a detailed summary of the best 20 solutions. Here’s an example of an NDX file:

' Indexing method - Alan Coelho (2003), J. Appl. Cryst. 36, 86-95

' Time: 2.015 seconds

 

     'Sg     Status UNI      Vol       Gof     Zero      Lps…

 

Indexing_Solutions_With_Zero_Error_2 {

 

   0) P42/nmc    3   0    1187.321    38.82   0.0000    11.1924  …

   1) P42/nmc    3   0    1187.057    38.64   0.0000    11.1896  …  

   2) P42/nmc    3   0    1187.458    38.61   0.0000    11.1914  …  

}

/*

======================================================================

   0) P-1         0     985.652    30.80   0.0111     7.0877  …  

 

   h   k   l       dc       do    do-dc     2Thc     2Tho   2Tho-2Thc

   0   0   1   15.857   15.830   -0.027    5.569    5.578    0.009

   0   1   0    8.765    8.750   -0.015   10.084   10.101    0.017

   0   0   2    7.928    7.910   -0.018   11.151   11.177    0.026

   0   1   1    7.788    7.780   -0.008   11.352   11.364    0.012

   0  -1   1    7.559    7.560    0.001   11.698   11.696   -0.002

*/

12.1  Reprocessing solutions - DET files

Details of solutions can be obtained at a later stage by including solution lines found in the NDX file into the INP file. For example, supposing details of solutions 50 and 51 were sought then the following (see example INDEXING\EX10.INP) could be used:

index_lam 1.540596

index_zero_error     

try_space_groups 2

 

Indexing_Solutions_With_Zero_Error_2 {

  50) P-1        1   0    2064.788     9.74   0.0000   …

  51) P-1        3   0    3128.349     9.61   0.0115   …

}

load index_d {

   15.83 good

    8.75

    7.91

    …

}

After running this INP file a *.DET file is created containing details of the supplied solutions.

12.2  Keywords and data structures

The data structures for indexing are as follows:

Tindexing

[]]//[[#i1|index_lam]]//[[#i1| ]][[#i1| !E1.540596]

[]]//[[#i2|index_min_lp]]//[[#i2| !E2] ]][[#i2| ]][[#i2|[]]//[[#i2|index_max_lp]]//[[#i2| !E]

[]]//[[#i3|index_max_Nc_on_No]]//[[#i3| !E5]

[]]//[[#i4|index_max_number_of_solutions]]//[[#i4| #3000]

[]]//[[#i5|index_max_th2_error]]//[[#i5| !E0.05]

[]]//[[#i6|index_max_zero_error]]//[[#i6| #0.2]

[]]//[[#i7|index_th2]]//[[#i7| ]][[#i7| !E | ]]//[[#i7|index_d]]//[[#i7| !E]…]] [//index_I//  E1 [good]] [[#i8|[]]//[[#i8|index_x0]]//[[#i8| !E]

[]]//[[#i9|index_zero_error]]//[[#i9|]

[]]//[[#i11|seed]]//[[#i11|]

[try_space_groups $]...

[x_angle_scaler #0.1]

[x_scaler #]

Values for most keywords are automatically determined or have default values (appearing as numbers above) adequate for difficult indexing problems. In the following example from UPPW (service provided by Armel Le Bail to the SDPD mailing list at http://sdpd.univ-lemans.fr/uppw/) only a few keywords are necessary. Also note the use of the dummy keyword; this allows for the exclusion of 2q and I values without having to edit the columns of data.

 

seed

index_lam  0.79776

index_zero_error   

index_max_Nc_on_No 6

try_space_groups 3

load index_th2 dummy dummy index_I dummy  {

    ' d (A)  2Theta     Height      Area     FWHM

     1.724  26.50645    2758.3   23303.7    0.0450

     2.646  17.27733  150393.8  747063.6    0.0250

     3.235  14.13204   98668.8  493153.7    0.0250

     3.417  13.37776   11102.6   53185.0    0.0250

     5.190   8.80955     782.7    3910.9    0.0250

     …

}

12.3  Keywords in detail

[index_lam  !E1.540596]

Defines the wavelength in Å.

[index_min_lp !E2.5] [index_max_lp !E]

Defines the minimum and maximum allowed lattice parameters. Typically the maximum is determined automatically.

[index_max_Nc_on_No !E5]

Determines the maximum ratio of the number of calculated to observed lines. The value of 6 allows for up to 83% of missing lines.

[index_max_number_of_solutions #1000]

The number of best solutions to keep.

[index_max_th2_error !E0.05]

Used for determining impurity lines (un-indexed lines UNI in *.NDX). Large values, 1 for example, forces the consideration of more observed input lines. For example if it is know that there are none or maybe just one impurity line then a large value for index_max_th2_error will speed up the indexing procedure.

[index_max_zero_error !E0.2]

Excludes solutions with zero errors greater than index_max_zero_error.

[index_th2  !E | index_d !E]…

[index_I  E1 [good]]

index_th2 or index_d defines a reflection entry in 2q degrees or d-spacing in Å.

index_I is typically set to the area under the peak; it is used to weight the reflection.

good signals that the corresponding d-spacing is not an impurity line. A single use of good on a large d-spacing decreases the number of possible solutions and hence speeds up the indexing process (see examples INDEXING\EX10.INP).

[index_x0 !E]

Defines Xhh in the reciprocal lattice equation:

In a triclinic lattice the highest d-spacing can probably be indexed as 100 or 200 etc. Thus

index_x0 = 1/(dmax)^2;

speeds up the indexing process (if, in this case, the first line can be indexed as 100) and additionally the chances of finding the correct solution is enhanced. Example EX13.INP demonstrates this. Note that if the data is in 2Th degrees then the following can be used:

index_x0 = (2 Sin(2Thmin Pi/360) / wavelength))^2;

The two macros Index_x0_from_d and Index_x0_from_th2 simplify the use of index_x0.

[index_zero_error]

Includes a zero error.

[seed]

Seeds the random number generator.

[try_space_groups $]…

[x_angle_scaler #0.1]

[x_scaler #]

Defines the space groups to be searched. The macros Bravais_Cubic_sgs etc… (see TOPAS.INC) defines lowest symmetry Bravais space groups. It is almost always sufficient to use only these. Higher symmetry space groups for the Bravais lattices corresponding to the 10 best solutions is subsequently searched. Here are some examples of using try_space_groups.

Search Use
Primitive monoclinic try_space_groups 3
The two monoclinic Bravais lattices of lowest symmetry. Bravais_Monoclinic_sgs
C-centered monoclinic of lowest symmetry. try_space_groups 5
All orthorhombic space groups individually. Unique_Orthorhombic_sgs

Below is a list showing which space groups have identical hkls in regards to powder data.

x_scaler is a scaling factor used for determining the number of steps to search in parameter space. x_scaler needs to be less than 1. Increasing x_scaler searches parameter space in finer detail. Default values are as follows:

Cubic                            0.99

Hexagonal/Trigonal         0.95

Tetragonal                     0.95

Orthorhombic                 0.89

Monoclinic                     0.85

Triclinic                         0.72

x_angle_scaler is a scaling factor for determining the number of angular steps for monoclinic and triclinic space groups. Small values, 0.05 for example, increases the number of angular steps. The dult value of 0.1 is usually sufficient.

12.4  Identifying dominant zones

Here are two example output lines from an NDX file.

0) P42/nmc  3   0    1187.124    38.82   0.0000    11.1904    11.1904     9.4799     90.000     90.000     90.000 ' ===  24  19

6) P-421c   3   0    1187.124    35.67   0.0000    11.1904    11.1904     9.4799     90.000     90.000     90.000 ' ===  24  19

Ø       The 1st column corresponds to the rank of the solution.

Ø       The 2nd corresponds to the space group.

Ø       The 3rd corresponds to the Status of the solution with meaning of the number as follows:

  Status 1: Weighting applied as defined in Coelho (2003)
  Status 2: Zero error attempt applied
  Status 3: Zero error attempt successful and impurity lines removal attempt successful
  Status 4: Impurity line(s) removed

Ø       The 4th  column corresponds to the number of un-indexed lines.

Ø       The 5th column corresponds to the volume of the lattice.

Ø       The 6th corresponds to the goodness of fit value.

Ø       The 7th corresponds to the zero error if index_zero_error is included.

Ø       Columns 8 to 13 contains the lattice parameters.

The last 2 columns contain the number of non-zero h2 + k2 + h k and l2 values used in the indexed lines. These represent the hkl coefficient for X0 and X1 respectively for Trigonal/Hexagonal systems. When one of these numbers are zero then the corresponding lattice parameters is not represented and the number is therefore displayed as the negative number of –999. This facility is particularly useful for identifying dominant zones. For example, if the smallest lattice parameter is 3Å and the smallest d-spacings is 4Å then it is impossible to determine the small lattice parameter. In these cases values of –999 will be obtained.

The following table gives the hkl coefficients corresponding to the Xnn reciprocal lattice parameters for the 7 crystal systems.

  X0 X1 X2 X3 X4 X5
Cubic h2+k2+l2          
Hexagonal Trigonal h2+k2+h k l2        
Tetragonal h2+k2 l2        
Orhtorhombic h2 k2 l2      
Monoclinic h2 k2 l2 h l    
Triclinic h2 k2 l2 h k h l k l

12.5  %%***%% Probable causes of Failure %%***%%

The most probable cause of failure is the inclusion of too many d-spacings. If it is assumed that the smallest lattice parameter is greater than 3Å then it is problematic to include d-spacings with values less than about 2.5Å when there are already 30 to 40 reflections with d values greater than 2.5Å. Some of the problems caused by very low d-spacings are:

Ø       The number of calculated lines increases dramatically and thus index_max_Nc_on_No will need to be increased.

Ø       The low d-spacings are probably inaccurate due to peak overlap at the high angles they are observed at.

A situation where it is necessary to include low d-spacings is when there are only a few d-spacings available as in higher symmetry lattices.

12.6  Unique space group hkls in Powder diffraction

Space group numbers with identical hkls Space group symbols with identical hkls
Triclinic
1 2 P1 P-1
Monoclinic
9 15 Cc C2/c
5 8 12 C2 Cm C2/m
14 P21/c
7 13 Pc P2/c
4 11 P21 P21/m
3 6 10 P2 Pm P2/m
Orthorhombic
70 Fddd
43 Fdd2
22 42 69 F222 Fmm2 Fmmm
68 Ccca
73 Ibca
37 66 Ccc2 Cccm
45 72 Iba2 Ibam
41 64 Aba2 Cmca
46 74 Ima2 Imma
36 40 63 Cmc21 Ama2 Cmcm
39 67 Abm2 Cmma
20 C2221
23 24 44 71 I222 I212121 Imm2 Immm
21 35 38 65 C222 Cmm2 Amm2 Cmmm
52 Pnna
56 Pccn
60 Pbcn
61 Pbca
48 Pnnn
54 Pcca
50 Pban
33 62 Pna21 Pnma
34 58 Pnn2 Pnnm
32 55 Pba2 Pbam
30 53 Pnc2 Pmna
29 57 Pca21 Pbcm
27 49 Pcc2 Pccm
31 59 Pmn21 Pmmn
26 28 51 Pmc21 Pma2 Pmma
19 P212121
18 P21212
17 P2221
16 25 47  P222 Pmm2 Pmmm
Tetragonal
142 I41/acd
110 I41cd
141 I41/amd
109 122 I41md I-42d
108 120 140 I4cm I-4c2 I4/mcm
88 I41/a
80 98 I41 I4122
79 82 87 97 107 119 121 139  I4 I-4 I4/m I422 I4mm I-4m2 I-42m I4/mmm
130 P4/ncc
126 P4/nnc
133 P42/nbc
103 124 P4cc P 4/mcc
104 128 P4nc P 4/mnc
106 135 P42bc P 42/mbc
137 P42/nmc
138 P42/ncm
134 P42/nnm
125 P4/nbm
114 P-421c
105 112 131 P42mc P-42c P42/mmc
102 118 136 P42nm P-4n2 P42/mnm
101 116 132 P42cm P-4c2 P42/mcm
100 117 127 P4bm P-4b2 P4/mbm
86 P42/n
85 129 P4/n P4/nmm
92 96 P41212 P43212
94 P42212
76 78 91 95 P41 P43 P4122 P4322
77 84 93 P42 P 42/m P4222
90 113 P4212 P-421m
75 81 83 89 99 111 115 123   P4 P-4 P4/m P422 P4mm P-42m P-4m2 P4/mmm
Trigonal & Hexagonal
161 167 R3c R-3c
146 148 155 160 166 R3 R-3 R32 R3m R-3m
184 192 P6cc P6/mcc
159 163 186 190 194 P31c P-31c P63mc P-62c P63/mmc
158 165 185 188 193 P3c1 P-3c1 P63cm P-6c2 P63/mcm
169 170 178 179 P61 P65 P6122 P6522
144 145 151 152 153 154 171 172 180 181   P31 P32 P3112 P3121 P3212 P3221 P62 P64 P6222 P6422
173 176 182 P63 P63/m P6322
 143 147 149 150 156 157 162 164 168 174 175 177 183 187 189 191 P3 P-3 P312 P321 P3m1 P31m P-31m P-3m1 P6 P-6 P6/m P622 P6mm P-6m2 P-62m P6/mmm
Cubic
228 Fd-3c
219 226 F-43c Fm-3c
203 227 Fd-3 Fd-3m
210 F4132
196 202 209 216 225    F23 Fm-3 F432 F-43m Fm-3m
230 Ia-3d
220 I-43d
206 Ia-3
214 I4132
197 199 204 211 217 229 I23 I213 Im-3 I432 I-43m Im-3m
222 Pn-3n
218 223 P-43n Pm-3n
201 224 Pn-3 Pn-3m
205 Pa-3
212 213 P4332 P4132
198 208 P213 P4232
195 200 207 215 221    P23 Pm-3 P432 P-43m Pm-3m

12.7  Equations in Indexing - Background

a, b and c lattice vectors can be converted to Cartesian coordinates with a collinear with the Cartesian x axis and b coplanar with the Cartesian x-y plane as follows:

a = ax i b = bx i + by j c = cx i + cy j  + cz k (12‑1)

where

ax = a

bx = b cos(g),   by = b sin(g)

cx = c cos(b),   cy =  c (cos(a) – cos(b) cos(g)) / sin(g),   cz2 = c2 - (cx)2– (cy)2

a, b, c are the lattice parameters and a, b, g the lattice angles. The reciprocal lattice vectors A, B, and C calculated from the lattice vectors of Eq. (12‑1) become:

A= Ax i + Ay j  + Azk

B = By j + Bz k

C = Cz

The equation relating a particular d-spacing dhkl to a particular hkl in terms of the reciprocal lattice parameters is:

 

(12‑2)

where

13       Batch mode operation – TC.EXE

The command line program tc.exe provides for batch mode operation. Running tc.exe without arguments displays help information. Running an INP file is as follows:

tc pbso4

Macros can be passed to the command line. One use for this is to pass a file name to an INP file as follows:

1)   Create a TEMPLATE.INP file with the required refinement details, this should look something like the following:

xdd FILE

etc…

2)   TEMPLATE.INP is fed to tc.exe by command line and the word FILE (within TEMPLATE.INP) is expanded to whatever the macro on the command line is. For example,

tc …\file_directory\TEMPLATE.INP “macro FILE { file.xy }”

The macro called FILE is described on the command line within quotation marks. On running tc.exe the word 'FILE' occurring in TEMPATE.INP is expanded to 'file.xy'. Note that more than one macro can be described on the command line.

To process a whole directory of data files, say *.XY file for example, then:

1)       From the file directory execute the DOS command:

dir *.xy > …\main_ta_directory\XY.BAT

      The XY.BAT file will then reside in the main TA directory.

2)   Edit …\main_ta_directory\XY.BAT to look like the following:

tc …\file_directory\template “macro FILE { file1.xy }”

copy …\file_directory\template.out …\file_directory\file1.out

tc …\file_directory\template.inp “macro FILE { file2.xy }”

copy …\file_directory\template.out …\file_directory\file2.out

etc….

After each run of tc.exe a TEMPLATE.OUT file is created containing refined results. This file is copied to another file “file1.out”, “file2.out” etc… in order to save it from being overwritten.

After running XY.BAT a number of *.OUT files is created one for each *.XY file.

In summary TC.EXE receives TEMPLATE.INP to process. Words occurring in TEMPLATE.INP are expanded depending on the macros described on the command line.

 

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