SUGGESTIONS ON WHICH INDEXING PROGRAM TO USE
Mainly by Robin Shirley
Additions by Deane Smith
Edited by Jerry Johnson
INTRODUCTION
We now have available a number of powerful and mature programs, all of
which usually succeed in at least 50 percent of the cases, and many achieve
better than 80 percent. Because they use quite different and complementary
methods, their limitations tend to affect different problems, so that
with several of them available, one will seldom be unable to index a
properly measured pattern. By knowing the right order to apply them, which
broadly means trying the speedy deductive methods first (VISSER'S ITO),
then working down from high symmetry with the more powerful exhaustive
and semi-exhaustive methods, computing costs should not be a problem.
If the problem is important enough, run all the programs, using each of
the programs as a verification of the correct solution. Note that if
different programs give different solutions, they may appear to be
different, but the use of cell reduction techniques may confirm a common
result.
Indexing Diffraction Data
-------- ----------- ----
Once the crystal lattice is known, it is a simple exercise to
calculate all of the possible d-spacings which may diffract x-rays.
d(hkl)=f(h,k,l,a,b,c,alpha,beta,gamma)
Specific equations:
Cubic: a = b = c alpha = beta = gamma = 90
-----
1 h*h + k*k + l*l
-------- = ---------------- = (h*h + k*k + l*l) A
d(hkl)**2 a*a
Orthorhomic: a .NE. b .NE. c alpha = beta = gamma = 90
-----------
1 h*h k*k l*l
-------- = --- --- --- = h*h*A + k*k*B + l*l*C
d(hkl)**2 a*a b*b c*c
Triclinic: a .NE. b .NE. c alpha .NE. beta .NE. gamma .NE. 90
---------
1
--------- = h*h*A + k*k*B + l*l*C + h*k*D + h*l*E + k*l*F
d(hkl)**2
Thus the above equation is the simplest form of the general d-spacing
equation. The special forms for each crystal system develope because of the
symmetry constraints on the crystal axes. Other forms of the general
equation:
1 4 2
Q(hkl) = --------- = --------- sin (theta(hkl))
d(hkl)**2 lambda**2
2
hence the equations may be written in terms of sin (theta)'s
Given A to F, one can generate all allowed hkl's and calculate all
allowed d-spacings. The problem is to determine A to F, given a set of
measured d-spacings.
Computers are especially appropriate for indexing because they do not
tire doing the many trials which are necessary.
Aids in Indexing
---- -- --------
* Large d-spacings have low hkl.
* Small number of regularly spaced peaks suggests high crystal symmetry.
* Large number of lines in a pattern suggests low crystal symmetry.
* Density of low-angle peaks can suggest cell volume.
* Low-angle lines can suggest trial values for A, B, and C.
* Zones (h, k, or l=0) may be reconizable in the data set.
Complication in Indexing
------------ -- --------
* High crystal symmetries cause peak multiplicities
* Some peaks may be unobserved due to weak intensities
* Many peaks may be unobserved due to symmetry and lattice type
* Some peaks may be unresolved in the experimental data
* Experimental measurements always have random and systematicic errors
Indexing Strategies
-------- ----------
Assume low symmetry
-------------------
1
--------- = h*h*A + k*k*B + l*l*C + h*k*D + h*l*E + k*l*F
d(hkl)**2
Examine Q's for zones
Zones have h, k, or l = 0
1
--------- = h*h*A + k*k*B + h*k*D
d(hkl)**2
1
--------- = h*h*A + l*l*C + h*l*E
d(hkl)**2
1
--------- = k*k*B + l*l*C + k*l*F
d(hkl)**2
Combine zones with common rows
Rows have h and k = 0 or
h and l = 0 or
k and l = 0
Accept combinations which fit all Q's in first 20 with good figure-of-merit
Indexing Figures of Merit
-------- ------- -- -----
Q(sub 20)
M(sub 20) = --------------------
2 |average delta Q|
This DeWolff figure-of merit (FOM) is more sensitive to the correctness of
a solution. (DeWolff (1968,1972)
Good pattens > 10
1 N (lines observed)
F(sub n)= ------------------------ * ----------------------
average | delta 2 theta| N (lines possible)
This Smith-Snyder FOM is a measure of the accuracy of measurements of the
d-vales (2 theta's). (Smith , G. S. and Snyder 1979)
Good patterns > 50
Rank lattice by the figure-of-merit and success (measured by the following)
number of indexed lined out of first 20
success = ---------------------------------------
20
Examine lattice for metric symmetry to determine true crystal system
Assume high symmetry
--------------------
Try cubic first (one paramter)
1
-------- = (h*h + k*k + l*l) A = s*s*A
d(hkl)**2
use the first peak to suggest the A value
hkl=100,110,111,200, etc
Try tetragonal-hexagonal (two parameter)
1
-------- = (h*h + k*k)*A + l*l*C (tetragonal)
d(hkl)**2
1
-------- = (h*h + h*k k*k)*A + l*l*C (hexagonal)
d(hkl)**2
Try orthorhombic (three parameters)
1
--------- = h*h*A + k*k*B + l*l*C
d(hkl)**2
Try monoclinic (four parameters)
1
--------- = h*h*A + k*k*B + l*l*C + h*l*E
d(hkl)**2
Indexing Criteria for Acceptance
-------- -------- --- ----------
* Figure of merit M(sub 20) > 10
* Completeness of indexing
All lines indexed regardless of intensity
* Logical Indexing
Very few missing INDICES
Lattice type accounts for most "absences"
* Cell symmetry fits other know data
Crystal optics
Isostructure compounds
* Metric symmetry checked for higher celkl
Hexagonal cell may appear as orthorhombic
Cubic cell may appear as tetragonal
* Volume is logical
Cell found may be double, trible, half, sqrt(2) of true cell
1. RESOLUTION
Probably the most exacting requirement for successful indexing is the
need to observe most of the first 20-30 lines (that are not too weak) as
distinct individuals, rather than merged groups.
2. ACCURACY
Measurement accuracy comprises precision (random errors) and correction
(systematic errors), and powder indexing responds very differently to
these two contributions.
a) Indexing is surprisingly tolerant of imprecision. Random
variation will increase the computation time, but probably
not prevent the solution from being found.
b) Systematic error is much more serious. Quite small increases
in these errors will cause computing time to increase rapidly
and depress the figure of merit of the correct solution so much
that it becomes hard to recognize the solution with any
confidence.
3. SELF-CALIBRATION
A quick search through the the line listing should be made to reveal
any higher order relations. If such exists, then the pattern can be
'corrected' to give more accurate input.
4. SPURIOUS LINES
It is very desirable to eliminate spurious lines which do not belong to
the phase under investigation, because they introduce additional unknown
parameters, as well as serious methodological difficulties. If we permit
observed lines to be discarded as "not indexed", we are then changing
our data to fit the proposed model--a dubious scientific practice in itself
and very hazardous when multiple solutions are to be expected. The only
safe course is to remove all doubtful lines on a priori grounds.
5. NUMBER OF DIGITS SPECIFIED
A surprisingly common source of data degradation is simply heavy-handed
rounding of d-spacing when a measured pattern is reported. It can be shown
that two decimal places are virtually never enough, three are often
insufficient, and even four may not be overgenerous.
6. COMPUTER TIME
In general exhaustive methods are excellent with high symmetry down to
orthorhombic, but times run high somewhere between three an five unknown
parameters. For monoclinic, and certainly for triclinic, we should first
try deductive programs (although these are less tolerant of data errors)
and then semi-exhaustive ones. (We have run cases using LOUER that have
run for greater than 14 hours of microVAX II CPU time for a monclinic cell
of about 2000 cubic A.)
7. TOLERANCE OF DATA ERRORS
The following should be regarded as only a guide:
Tolerance of Spurious lines
---------------------------------
Sensitive Insensitive
Sensitive TAUPIN VISSER'S ITO
(option 35) (option 2)
WERNER
Tolerance of Inaccuracy (option 22)
-----------------------
Insensitive LOUER KOHLBECK
(option 18) (option 16)
GOBEL-WILSON
(option 7)
---------------------------------------------------------------------------
---------------------------------------------------------------------------
Personal comments (GGJJ) April 14, 1990
---------------------------------------
I usually run the programs in the following order (assuming that I have
good data and I keep the number of
lines at 30 even if there are more !):
VISSER (It is quick and gives good results
on about 60-70% of the cases. This
code does require at least 20 lines !)
WERNER (Will usually 'miss lines' that others
do not, ie a-Al2 O3. Look at solutions
that have a small number of misses !)
TAUPIN (Had good success with it, but there is
too much output)
LOUER (Very good if volume is small, have run
with monoclinic cell of 2000 cubic A.,
and took 14 hours of cpu time on a
microVAX II).
KOHLBECK (Only low symmetry cases--a-Al2 O3 did
index, but as orthorhombic, but the with
the correct volume. It was up to me
to transform the cell to hexagonal.)
GOBEL-WILSON (Only high symmetry cases--error window
tends to be too large. This is very old
code, and I only trust it for one and
two parameter cells)
Additional personal comments (GGJJ) June 16, 1990
-------------------------------------------------
added at JCPDS-ICDD Clinic
run at MRL/PSU for students.
Table 1.
Phase Symmetry # of parameters Quality PDF #
---------------------------------------------------------------------------
MgAl2O4 Cubic 1 * 21-1152
Al2O3 Hex 2 I 10-173
BeH2 Ortho 3 * 39-1208
La2Ti2O7 Mono 4 * 28-517
Table 2.
Phase Lattice Parameters
------------------------------------
MgAl2O4 a=8.0831
Al2O3 a=4.758 c=12.991
BeH2 a=9.082 b=4.160 c=7.707
La2Ti2O7 a=13.015 b=5.5456 c=7.817 beta=98.64
Table 3.
Author Program Name Systems Revision
------------------------------------------------------------------------------
Visser (Netherlands) ITO13 All 6/87
Werner (Sweden) TREOR (trial and error) All 3/90
Taupin (France) TAUPF4 All 3/90
Louer (Belgium) DICVOL (dichotomy of volume) CTHOM 9/82
Kohlbeck (Austria) TMO OMA 4/90
Gobel-Wilson (USA) INDEX CTHO 1965
Table 4.
Dependence of Run (CPU) times
Program # of lines Tolerence Symmetry Volume
----------------------------------------------------------------------
Visser None Yes None None
Werner Some Yes Yes None
Taupin Some Yes Yes Some
Louer Some Yes Yes Great
Kohlbeck Some Yes Yes None
Gobel-Wilson None Yes Yes None
Table 5.
Results using Werner's Treor (1990 version) on a microVAX II running
VAX/VMS 5.3-1 and FORTRAN 5.4
Phase Time (CPU) Results
---------------------------------------------------------------------
MgAl2O4 2.15 seconds JCPDS-ICDD value
Al2O3 14.85 seconds JCPDS-ICDD values
BeH2 53.20 seconds JCPDS-ICDD values
La2TiO7 215.85 seconds Volume correct
b correct
c correct
a and beta different
automatic transformation
using Inter. Tables
# 35 to
JCPDS-ICDD values
Table 6.
Correctness is to obtain exact JCPDS-ICDD values (Note to index all of the
lines is certainly a solution !)
Phase Visser Werner Taupin Louer Kohlbeck Gobel-Wilson
--------------------------------------------------------------------------------
MgAl2O4 Yes Yes Yes No No Yes
29/29 25/25 25/25 20/20
(1/sqrt(2))
Al2O3 Yes Yes Yes Yes No No
39/40 22/25 25/25 20/20 (volume
correct)
BeH2 Yes Yes No Yes Yes No
40/40 25/25 25/25 20/20 20/20
(volume
X 3)
LaTi2O7 No Yes No Yes Yes No
40/40 25/25 20/20 20/20
(triclinic
with 2 angles
within 0.01
of 90.00)
-------------------------------------------------------------------------
CONCLUSION
Powder indexing requires either that the correct values of the cell
constants (or powder constants, etc.) are found in a continuous parameter-
space of up to six dimensions, or that the appropriate values of the Miller
indices for each line are located in a 3N-dimensional integer-valued index-
space, where N is the number of observed lines to be indexed. These two
objectives are equivalent, and are both fulfilled when the correct solution
is found.
Index- and parameter-space methods are considerably different in
approach, and can act as useful complements (or checks) for each other. In
general, parameter-space methods lend themselves more to the inclusion of
cell information and constraints, while index-space programs are the
stronger in the presence of unhelpful accidental or systematic absences.
Programs may also be classified according to whether they adopt a
mainly deductive or exhaustive approach. Broadly, deductive methods try to
infer the values of lattice parameters from coincidences and relations
between the observed lines, to achieve speed at the cost of rigor. By
contrast, exhaustive methods systematically search the relevant solution
space, gaining rigor at the expense of speed. Deductive methods try to
work out quickly what the solution ought to be, but may fail with poor data
or a tricky problem. Exhaustive methods eliminate everywhere that the
solution cannot be a procedure which must succeed but is slow in low-
symmetry cases.
The classification is not rigid, because programs often incorporate
some aspect of both approaches. For example, deductive programs usually
try some kind of systematic combination of the deduced possibilities, while
most of the new programs are semi-exhaustive, making judicious deductions
to limit the solution field to be searched, in order to gain speed.
Indexing Classification of Techniques
-------- -------------- -- ----------
Analytical
----------
* Assumes triclinic
* uses general equations
* Tries to find zone relations within set of d-values
* Then intersects zones to find lattice
* Lattices are tested for higher symmetry
* Calculations are in "parameter space"
Exhaustive
----------
* Assumes a crystal system
* Tries all possible hkl combinations for first n lines
* Then tries another crystal system
* Calculations are in index space
Semi-exhaustive
---------------
* Excludes unlikely calculation regions
* Then uses either index or parameter space calculations
IMPORTANT NOTICE
--------- ------
The following important items should be remembered:
1) Different results, with the same or different programs, may be
be the same cell with different orientation. The various cell
reduction programs of the MRL/PSU XRD system will allow the
determination of the same reduced cell.
2) The indexing programs should (must) be followed with the running
of least squares program to get the best metric parameters. In the
running of the least squares, the user may notice relationships
between the hkl's that illustrate 'screw', 'glide', or 'centering'
from the allowed hkl's of the solution.
The following table show the programs available at MRL/PSU:
Parameter-space | Index-space
------------------------------------------------------------
| |
| |
Deductive | VISSER'S ITO |
(Analytical) | (option 2) |
| |
---------------------------------------------------------------
Semi-exhaustive | | WERNER
| | (option 22)
| | KOHLBECK
| | (option 16)
---------------------------------------------------------------
Exhaustive | LOUER | TAUPIN
| (option 18) | (option 35)
| | GOBEL-WILSON
| | (option 7)
June 20, 1990