Title: Indexing Xray powder diffraction patterns
1Indexing Xray powder diffraction patterns using
the homology method
Do we always need to apply brute force programs
for automatic indexing, or in many cases the
problem will be solved with our crystal chemistry
knowledge and a pocket calculator? Could we
extract useful information from the intensities
of the reflections, which will help us to index
the pattern? Could we get an idea about the
structure model while determining unit cell
parameters? Lets try to use an elegant
ancient technique, which people developed
before the computer era.
homologos  similar, conformable (Greek)
2Everybody can manually index XRD pattern of a
cubic structure
indexation is based on analyzing Q series
cubic lattice with a 10 Å
3Distortions of cubic lattice
4Sublattice
Lets assume that your new structure with (a, b,
c, a, b, g) is a derivative from some more
simple structure, which we will call the
substructure with (asub, bsub, csub, asub, bsub,
gsub)
 Usually the substructure is a highly symmetric
structure,  belonging to one of the simple prototypes
(perovskite, rock  salt, fluorite, rutile, shpinel, pyrochlore etc.)

2. We will call a reflection a sublattice
reflection if its h,k,l are related to hsub,
ksub, lsub by a unit matrix with det1. All other
reflections are the superlattice reflections.
Note that the subcell symmetry can be lower than
the true symmetry of the structure
3. If the deviation from the substructure is not
really huge, the superlattice reflections will be
the brighter ones on the XRD pattern. In most
cases we can easily recognize them.
Distortion of the substructure with respect to
the prototype structure causes the sublattice
reflections to split. We can analyze the
splitting type in order to determine the
distortion character for the substructure.
5Sublattice
6Tetragonal distortion
Cubic Tetragonal
100 ? 100 010 ? 010 001 ? 001
100 ?2 001 ?1
tetragonal distortion
110 ?1 101 ?2
110 ? 110 011 ? 011 101 ? 101
111 ? 111
111 ?1
7Orthorhombic distortion
Cubic Orthorhombic
100 ? 100 010 ? 010 001 ? 001
orthorhombic distortion
110 ? 110 011 ? 011 101 ? 101
111 ? 111
8Monoclinic distortion
Cubic Monoclinic
100 ? 100 010 ? 010 001 ? 001
monoclinic distortion
110 ? 110 011 ? 011 101 ? 101 101
? 101
111 ? 111 111 ? 111
9Very simple case tetragonally distorted
perovskite
110
Cubic perovskite asub ? 3.8Å
200
111
100
La0.8Sr0.2Cu(O,F)3
10Indexing of the XRD powder pattern of
La0.8Sr0.2Cu(O,F)3
 Analyzing Q series for the subcell gives
sublattice type cubic primitive  Splitting type corresponds to tetragonal
distortion of the subcell  Intensity ratios between split reflections allow
to assign multiplicity  factors and perform indexation
 Refinement of the lattice parameters
 a 3.7893(7)Å, c 4.048(1)Å
11More complicated case ordered double perovskite
110
Cubic perovskite
200
111
100
Ca3ReO6
12Indexing of the XRD powder pattern of Ca3ReO6
 Sublattice type cubic primitive
 Splitting type corresponds to monoclinic
distortion of the subcell  (111 reflection is split)
 3. Indexation of the reflections according to
their multiplicity  factors
 ?
 monoclinic unit cell (bsetting)
 with ac
 Refinement of the lattice parameters
 a 4.0099(8)Å,
 b 3.9917(4)Å
 c 4.0095(7)Å
 b 92.695(6)o
13Indexing of the XRD powder pattern of Ca3ReO6
For the monoclinic unit cell with ac we always
can find an orthorhombic unit cell of larger
volume
monoclinic P ? orthorhombic B
14Indexing of the XRD powder pattern of Ca3ReO6
Lattice parameters of the new Bcentered
orthorhombic unit cell a 5.5363(3)Å, b
3.9921(3)Å c 5.8022(4)Å What to do with
other unindexed reflections? We will treat them
as superlattice reflections.
15Superstructure
1. Cation ordering
2. Anion ordering
3. Ordering of anion vacancies
4. Ordering of cation vacancies
5. Atomic displacements
16Superstructure
Tetragonal altc
1. Superstructure is related to increasing unit
cell volume (determinant of the transformation
matrix gt 1)
2. Intensities of superlattice reflections depend
strongly on the degree of deviation from the
substructure.
3. Transformation matrix shows the relationships
between the sublattice and superlattice vectors
4. Transformation matrix does not allow you to
calculate indexes of the superlattice
reflections.
17Indexing of the XRD powder pattern of Ca3ReO6
Searching for the superstructure is a
nontrivial step. Low angle reflections are the
most important. Check possible multiplication of
subcell axes lengths. Check different
possibilities of simple hkl for low angle
reflections. Check relationships between the
low angle and sublattice reflections. In our case
doubling of bparameter Q010 628/4
157 Q011 157 1188/4 454 first low angle
reflection Q110 157 1305/4 483 second
low angle reflection Final refinement of the
lattice parameters a 5.5366(3)Å, b
7.9845(4)Å c 5.8022(3)Å
18What do we get extra?
Transformation matrix
Transformation of the atomic coordinates 
inverse transformation matrix
You immediately get a starting set of atomic
coordinates in the superstructure unit cell for
further work, for example, to use as a first
model for the Rietveld refinement. You can limit
the choice of possible space groups by the
subgroups of the space groups of the prototype
structure.
In addition to standard figureofmerit criteria,
the correctness of indexation is confirmed by a
clear crystallographic relationship with the
prototype structure Indexing the subcell does
not require the peak positions to be very precise
you can do it even if a zero shift is present.
Then use the subcell reflections as a reference
to refine the zero shift, correct your data and
continue.
19Primitive cubic lattice
P
R
C
20Facecentered cubic lattice
21Bodycentered cubic lattice
22Polymorphic forms of cryolite K3AlF6
220
400oC elpasolitetype Fcentered cubic
sublattice with a 8.579Å
400
222
200
111
311
RT
202 220
004 400
222
113 311
002 200
111
23Polymorphic forms of cryolite K3AlF6
 Features of this XRD pattern
  absence of lowangle superlattice reflections
 very dense set of weak unindexed reflections,
which disappear at phase transition, and, hence,
also belong to this phase.  Manual search for superstructure seems to be
impossible.  Automatic indexing (TREOR90) fails.
 Do not waste time, electron diffraction will give
straight answer.
24Polymorphic forms of cryolite K3AlF6 the
superstructure
We can apply the same concept and separate the
reflections into the sublattice ones (brighter)
and superlattice ones (less bright).

4 2 0sub 10 0 0ss 2 4 0sub 0 10 0ss 0 0 2sub
0 0 8ss
25Polymorphic forms of cryolite K3AlF6
Applying the transformation matrix to the lattice
vectors of the substructure and then refining
the lattice parameters using the LeBail fit we
get (space group I41/a) a 18.8489(2) Å c
33.9827(4) Å V 12073 Å3
 Summarizing
 Time you spend for indexing pattern with homology
method  is proportional to unit cell volume
 is inversely proportional to the symmetry of the
structure, level of your crystallographic and
crystal chemistry expertise and your experience 