Rietveld_Compendium.pdf

POWDER DIFFRACTION
AND
THE RIETVELD METHOD
by
Kenny Ståhl
Department of Chemistry
Technical University of Denmark
DK-2800 Lyngby, Denmark
kenny@kemi.dtu.dk
Lyngby, February 2008

CONTENTS
1. INTRODUCTION 3
2. POWDER DIFFRACTION 5
3. THE RIETVELD METHOD 17
4. HOW TO GET STARTED, 2 -ZERO, UNIT CELL AND SCALE FACTOR 21
5. BACKGROUND 23
6. PEAK FUNCTIONS 27
7. STRUCTURAL PARAMETERS AND RESTRAINTS 35
8. RESIDUAL VALUES AND STATISTICS 43
9. CONSECUTIVE DATA SETS 49
11. PUBLICATION REQUIREMENTS, CIF 51
12. REFERENCES 53
APPENDIX A SYMMETRY AND CRYSTALS
APPENDIX B X-RAY DIFFRACTION
APPENDIX C COMPUTER PROGRAMS

3
1 INTRODUCTION
Powder diffraction has from the very beginning in the 1910's been an indispensable crystallographic
technique for materials identification and characterization. In addition to information on the atomic structure, a
powder pattern will also contain information on other properties like particle sizes, size distributions, residual stress
and strain, and texture. From mixed samples it is possible to obtain quantitative information on the different phases.
As a powder diffraction pattern can be rapidly recorded using a position sensitive device, it can be used for in situ
structure studies during temperature, pressure and/or environmental variations. Powder diffraction methods are not
limited to academic research. It is the most used diffraction method in industry, as it can be highly automated and is
well suited for production control.
A complete structure characterization requires a set of accurately measured 2 -values to determine the unit
cell. Information from extinct reflections together with intensity statistics will suggest a plausible space group.
Finally, integrated intensities from the three-dimensionally indexed Bragg reflections is used to solve and refine the
crystal structure. The final step, the extraction of intensities from diffraction data, is straight forward using single-
crystal methods. Powder diffraction data will in general not directly provide these data. As a powder diffraction data
set is a one-dimensional projection of the three-dimensional reciprocal space it gets increasingly overlapped at
increasing 2 . Due to this overlap problem, crystal structure solutions and refinements is preferably done from
single-crystal data. Unfortunately not all crystalline materials can be obtained with a crystal size and/or quality
suitable for single crystal work. For such materials, powder diffraction is the only crystallographic method available,
and consequently we have to learn how to handle these problems.
However overlapped, a powder diffraction pattern does contain the same information as a single crystal
data set. The problem is just how to deconvolute it. A breakthrough in handling this problem, or rather how to
circumvent it, came when Hugo Rietveld in the late 1960s introduced a whole-pattern-fitting structure refinement
method, now known as the Rietveld method [1, 2]. With this method we no longer need to deconvolute a powder
diffraction pattern in order to get to the individual intensities. Instead we fit all the reflections directly to the pattern.
The Rietveld method was developed for constant wavelength neutron diffraction. Over the years several
modifications have been made to the method, and it is now widely applied to data collected from constant
wavelength conventional and synchrotron X-ray sources, as well as different white radiation from X-ray
synchrotrons and neutron spallation sources. To date several thousands of structures have been refined and published
following this method. The method offers a way to refine a crystal structure, but it will not solve an unknown crystal
structure. A reasonably good model is still needed to get the refinements going.
The purpose of the present course is not to cover all aspects of powder diffraction as we know it today. As
may be guessed from its title, it will focus on the Rietveld method, i.e. how to refine crystal structures from powder
diffraction data. A few other aspects, like methods and strategies for data collection, are covered to some extent, while
for instance indexing, profile decomposition methods and structure solution methods are left out. This paper is
intended as a practical guide and introduction to the Rietveld method. For a more comprehensive treatment of powder
diffraction and crystallography in general, see for instance:

4
R.A. Young (Ed.): The Rietveld method; IUCr Monographs on Crystallography no 5; Oxford University Press; New
York; 1993.
D.L. Bish and J.E. Post (Eds): Modern Powder Diffraction, Reviews in Mineralogy Vol 20, Mineralogical Society of
America, Washington D.C., 1989.
C. Giacovazzo (Ed.): Fundamentals of Crystallography, Second Edition, IUCr Texts on Crystallography 7, Oxford
University Press, 2002.
These notes are organized in the following way: First a presentation of powder diffraction in general. Next the
Rietveld method itself is presented with emphasis on the computing aspects. From thereon the different components
of the method are covered in an order that one would encounter them trying to solve a real problem: Getting started;
background; peak functions; structural parameter; multiphase refinements; consecutive data sets; residual values; and
publication requirements. In addition these notes contain some appedecis covering very briefly the fundamental
aspects of crystallography, X-ray diffraction and computer programs.

5
2 POWDER DIFFRACTION
Powder diffraction is an indispensable crystallographic tool for materials identification and
characterization. A powder diffraction pattern contains in principle the same amount of information as a single-crystal
diffraction data set does. However, it is a one-dimensional representation, or projection, of the three-dimensional
reciprocal space, which creates a fundamental problem with overlapping reflections (Fig. 1.1). A powder diffraction
pattern is nevertheless unique for a given compound. Its uniqueness forms the basis for phase identifications. Also
mixtures of phases can be rapidly identified and quantified using databases with effective search-match procedures. If
we have a reasonable good idea about the crystal structure, we can use powder diffraction data for structure
refinements. If the crystal structure is completely unknown there will be a problem. To solve an unknown structure
we need a set of uniquely indexed reflections and their intensities. This is harder to get from powder data due to the
overlap problem. Nevertheless, programs have been developed to extract such information and solve crystal structures
from powder diffraction data. A powder diffraction pattern may also provide information about particle sizes and
particle size distributions, as well as residual stress, which forms the basis for technologically important applications.
We will in the following look at powder diffraction in general, some commonly used techniques for
powder diffraction data collection, error sources and how to treat data.
2.1 Powder diffraction in general
If we put a crystal in an X-ray beam we have to be a little bit lucky in order to observe any diffraction: The
crystal has to be placed in the beam so as to fulfill the Bragg condition for diffraction. With a single-crystal
diffractometer we can rotate the crystal in the beam about different axes and we can thereby always position any
lattice plane correct for diffraction. Another approach to this problem is to put not one, but an infinite number of
crystals into the beam. Thereby there will always be many crystals in the right position for diffraction from any of the
lattice planes. This is the simple condition for powder diffraction. In practice we never have an infinite number of
crystals, but it is essential for the method that we have enough crystals to fulfill the condition of always having many
crystals in position for diffraction for each lattice plane. The process is best illustrated using an area detector: Each
lattice plane will, according to Bragg’s law, scatter at a distinct 2 -angle to the primary X-ray beam. If we position a
crystal for diffraction and then rotate it about the primary beam it will still be in position for diffraction, and the
diffraction spot will describe a circle with the primary beam as its center. With an ideal powdered sample we do not
need to rotate the sample as there will always be lots of crystals in the correct orientation for diffraction from all
lattice planes. The result will look like in Fig. 2.1: A set of concentric rings, each representing a specific lattice plane.
Due to the circularly symmetric pattern we do not need to collect the powder diffraction data with an area detector: It
is sufficient to record the intensities on a line radially from the center and out. However, if our sample does not fulfill
the conditions for an ideal powder sample the resulting rings may consist of distinct diffraction spots. In such a case
an arbitrary radial line will not be representative for the powder sample. We can help the situation by spinning our
sample during measurement and thereby improve the “powder average”. In some cases like when using high-pressure
cells with very little sample, it may be necessary to record the full circles with an area detector and then integrate
around the rings to get the true powder diffractogram. For normal laboratory powder data collection it is essential that Figure 2.1. A simple powder diffraction experiment.

6
the sample is properly prepared, for instance properly ground with mortar and pestle. A typical diffractogram is
shown in Fig. 2.2.
Figure 2.2. Powder
diffractogram from the
zeolite brewsterite. The
vertical bars represent
the Bragg positions.
Fig. 2.2 illustrates a general problem with powder diffraction data: The increasing reflection overlap at increasing 2 -
angle. A powder diffractogram is a one-dimensional projection of the three-dimensional reciprocal space. As the
number of reflections increases by (sin / )3
the overlap problem is very hard to avoid. There are some exceptions: In
cases with very small unit cells and high symmetry we may not experience any overlap in the sin / range we can
measure; with the extremely high angular resolution that can be achieved at some synchrotrons we can significantly
increase the range of separated reflections. A laboratory powder diffractometer has a typical peak width of 0.1°, while
at the best synchrotron beam lines we can get down to 0.001°.
2.2 Resolution
Resolution in powder diffraction may refer to different properties. Most commonly it refers either to the
reciprocal space or to the direct space. The resolution in reciprocal space we just touched upon: Peak resolution, or
peak separation. Most frequently it is expressed in terms of the minimum Full Width at Half Maximun (FWHM) that
can be achieved with a diffractometer. As the term indicates it is measured as the width of a reflection at half the peak
height. FWHM is not constant in 2 , but typically show a minimum between 20 and 40º(c.f. Fig. 2.3). The minimum
FWHM is determined by different components: Diffraction geometry; wavelength distribution; and the sample. The

7
geometric factors are typically beam divergence and beam size; in effect these factors are determined by the slit
system. The wavelength distribution as determined by the source and the monochromator(s). The sample effects are
due to particle size and defects and will discussed further below. The direct space resolution tells us how accurate
results we can expect from a structure refinement. It is determined by the maximum sin / we measure. It is also
known as Fourier resolution as it determines the number of terms in the Fourier transformation. We can express it as
/(2sin ), or equivalently dmin, which is directly related to the general laws of optics, i.e. we can not expect to resolve
objects smaller than the resolution limit. A routine powder diffraction measurement with CuK -radiation in the range
0-100º in 2 will give (sin / )max = 0.5 Å-1
, or a resolution of 1 Å. For comparison will a routine single-crystal
measurement up to = 30º and MoK -radiation give (sin / )max = 0.7 Å-1
, or a resolution of 0.7 Å. While the single-
crystal resolution represents the true physical resolution, we have to be cautious with the interpretation of the powder
diffraction resolution due to the overlap problem at higher 2 .
2.2 Data collection methods.
The basic principle for powder diffraction is that the sample contains so many small randomly oriented
crystals that there will always be several in the correct position for diffraction according to Bragg's law. To fulfill this
condition we have to grind our sample carefully to obtain crystallite sizes in the order of 5 - 10 µm. The sample will
in most cases be kept spinning during data collection to increase the probability to have correctly oriented crystallites.
The diffracted beams will come out as a set of cones, creating a corresponding set of circles (Fig. 2.1) when recorded
on a film or area detector. The ring pattern gives several possibilities in terms of how to collect a powder diffraction
pattern. We may simply place a film or an area detector behind the sample and record it all. However, there are some
problems in the evaluation of such a ring pattern. The dominant way of collecting powder diffraction data is instead to
record the intensities radially, across the rings. The diffraction pattern can be recorded using a film strip or a position
sensitive detector. Another possibility is to use a scintillation detector that is moved, scanned, radially across the
powder rings. With such a powder diffractometer, the diffraction pattern is directly obtained digitized and the
evaluation of the powder pattern can be highly automated. When photographic film is used we need to develop it and
either evaluate it manually or use a film scanner.
There are several ways of arranging the sample with respect to the X-ray source and the recording device.
There are of course advantages and disadvantages with each of these methods in terms of required sample amounts,
recording time, peak resolution etc. It is therefore not possible to recommend any particular method, but they have to
be chosen according to the requirements of the particular problem at hand.
2.2.1 The Debye-Scherrer method
The arrangement for the Debye-Scherrer method is shown schematically in Fig. 2.4. The sample is loaded
in a glass capillary, 0.1-0.5 mm in diameter, with a wall thickness of about 0.01 mm. The original method used a film
strip for intensity recording. The film is placed all around the circle, which gives the diffraction pattern in the 2 -
range +180°. In the modern version, the film is replaced by a position sensitive detector covering 0 - 120°. The pattern
can then be directly viewed on a monitor as it grows up. Most modern diffractometers are equipped with a focusing
Figure 2.3. FWHM as a function of 2 for a standard
Si sample measured on the Huber diffractometer.
Figure 2.4. Debye-Scherrer geometry.

8
monochromator, which allow for the removal of the K 2 radiation and on general gives sharper peaks.
The advantage with Debye-Scherrer method is that it requires a very small amount of sample, from 50µg to
a few mg depending on the size of the capillary. It is also convenient to handle air or moisture sensitive samples as the
top of the capillary can be easily sealed. The data collection time can be made very short, it is often sufficient with a
few minutes for phase identifications. When heavily absorbing samples are used, only a small part close to the
capillary surface will contribute to the diffracted intensity. The data collection time will then have to be increased and
the absorption will also cause some systematic peak position shifts (see below on Absorption effects). The
background level is in general higher with this method as compared to for instance the Bragg-Brentano geometry due
to scattering from air and the capillary.
2.2.2 The Guinier method
The geometry of the Guinier method is shown schematically in Fig. 2.5. Its focusing monochromator
makes it possible to remove the K 2 line, so as to have pure K 1 radiation. The sample is used in transmission mode
and placed in a very thin layer on a plastic tape. Alternatively capillaries can be used as for the Debye-Scherrer
method. Normally only a few hundred µg are used. The sample holder is kept spinning during data collection.
Originally the powder pattern is recorded with a film strip placed around the focusing circle. The latest version uses
an imaging plate strip and is equipped with an integrated readout system. The method has been and to some extent
still is the work horse for phase identifications and unit cell determinations. As with the Debye-Scherrer geometry the
background is relatively high due to air and tape scattering.
2.2.3 The Bragg-Brentano method
The Bragg-Brentano geometry is the most commonly used geometry for powder diffractometers. It is
schematically shown in Fig. 2.6 in two 2 -settings. In Fig. 2.6 the source and the detector are each moved by an angle
, while the sample is fixed horizontally. Alternatively the source is fixed and the sample is rotated by and the
detector by 2 . The sample is used in reflection mode and a comparably large amount of sample is needed, typically
0.5 cm3
. Due to the large irradiated sample surface, the sample is not always rotated during data collection. The
standard version uses filtered radiation and a monochromator in the diffracted beam. In this way fluorescence
radiation is effectively removed, and in general the background level is very low. The secondary monochromator will
not remove the K 2 contribution and the peaks gradually split up in two at higher 2 -angles. The Bragg-Brentano
geometry is very good with medium to highly absorbing samples. Low sample absorption will allow the primary
beam to penetrate the sample, causing profile broadening and asymmetry (see below on Absorption effects). Sample
preparation is crucial for a good result. Uneven sample grinding may result in micro-absorption at the surface with
strongly absorbing samples. It is often difficult to avoid preferred orientation when packing the sample in the sample
holder.
Figure 2.5. The Guinier geometry.
Figure 2.6. Ideal Bragg-Brentano geometry seen with
to different 2 -angles.

9
2.3 The synchrotron advantage
The introduction of synchrotron radiation has revolutionized powder diffraction. The main advantages with
synchrotron radiation are:
1. Very high intensity, 100 – 1000 000 times a conventional X-ray tube
2. Wavelength tunable
3. Very high collimation
4. Polarized radiation
We can use these properties to obtain for example:
1. Very high angular resolution, FWHM down to 0.001º
2. Very short measuring time
3. Wavelengths free of choice
Fig. 2.7 illustrates the effects of improving the angular resolution. This example is fairly old, and today one can
achieve a factor of ten more narrow reflections. Fig. 2.8 illustrates the reduction in measuring time when using
synchrotron radiation.
Figure 2.7. The same capillary sample measured with conventional and a synchrotron source. Figure 2.8. Diffraction patterns collected with Huber
G670 diffractometer in the laboratory and at a
synchrotron source.

10
2.4 Error sources
To be successful in phase identification or indexing it is an absolute necessity to have very accurate data. In
indexing an accuracy of 0.02° or better in the 2 -values is most often required. Most software for phase identification
and indexing have a tolerance level that may be increased to in fact any value, but the number of suggested solutions
will increase accordingly, and eventually the methods become useless. A powder diffraction pattern will inevitably
contain some systematic errors and others may be added by improper practicing. In general, a very good way of
overcoming the different types of errors is to use a standard, i.e. a material with a very well determined unit cell.
When we have determined the peak positions from the standard, we can compare them to its calculated values. From
the comparison we can compute a calibration function that we then apply to our sample peak positions.
2.4.1 Geometrical factors
Diffraction from a powder sample appears as a set of cones. We normally measure it in only one dimension
using a film, a position sensitive detector or by scanning radially with a scintillation detector. What we actually record
is then, due to the finite width of our film or detector, the intensity of a strip cut out from the diffraction cones. The
effect is seen very clearly on a film, where the reflections will show up as slightly bent, especially at 2 close to 0 and
180° (Fig. 2.9). With a position sensitive detector, a scintillation detector or when we evaluate a film with a scanner
we will only record the projection of the strip of intensities entering the detector. As a result the intensity profiles will
be asymmetric and their centers of gravity systematically shifted towards lower 2 -values.
Figure 2.9. True powder diffraction
pattern.
A comparable effect is caused by the extension of the sample (Fig. 2.10). Each point of the sample will
generate its own set of diffraction cones. When they add up on the detector it will show up as an asymmetric profile
broadening, and a centre of gravity shift to lower 2 -values (Fig. 2.11). The effects can be limited by slits to reduce
the detector opening and the illuminated width of the sample, but it will be at the expense of the recorded intensity.
2.4.2 Sample misalignment
Misalignment of the sample should of course be avoided. The time spent on doing a proper sample
alignment is normally just a few minutes and it is simply stupid not to spend the necessary time doing it.
A misaligned rotating capillary sample will only cause a corresponding broadening of the profiles. In a
severely overlapped pattern it may result in unnecessary problems in determining proper peak position. A misaligned
flat sample will cause a systematic shift of the whole pattern towards higher 2 -values if the sample is above and
Figure 2.10. Axial divergence. The extension of the
sample (left) will add up to an asymmetric peak at low
2 -values.
Figure 2.11. The asymmetry effect from axial
divergence on low-angle reflections.

11
towards lower 2 -values if it is below the correct position. In general such systematic peak shifts are much more
serious problem than random errors. In some cases it can be treated as 2 -zero error and be corrected for.
2.4.3 Absorption effects
Sample absorption will affect a powder diffraction pattern differently in different geometries. The Guinier
flat-plate geometry is virtually unaffected by absorption, while the Debye-Scherrer and Guinier capillary geometries
are strongly affected. The effect on peak position and asymmetry is shown in Fig. 2.12. The diffracted intensity is also
reduced by sample absorption, and both the positional and the intensity effects will vary with 2 . In the Bragg-
Brentano geometry low sample absorption will cause the largest unwanted effects as illustrated in Fig. 2.13. Also the
intensities are affected by absorption. In particular the capillary geometries. Knowing the absorption coefficient, ,
and the capillary radius it is possible to correct for it. Note that should be multiplied by about 0.5 to take account for
the packing efficiency.
2.4.4 Non-linearity of film and detectors
A powder diffraction pattern recorded on a film may suffer from uneven film shrinking during
development or small irregularities in the film holder. Position sensitive detectors will despite careful manufacturing
never be perfectly linear. The non-linearity is rarely larger than about one percent. However, if it is left uncorrected, it
may cause severe problems when the data is used for phase identification, indexing or Rietveld refinements. The best
way to determine and correct for the non-linearity is to use a standard as discussed above (2.4.1). One may choose
either an internal standard mixed with the sample or a separate data collection with the standard sample.
2.4.5 Preferred orientation
By preferred orientation is meant that the crystallites tend to arrange themselves according to their habitus.
Flat crystallites tend to be stacked and the needle shaped tend to line up in the needle direction. A proper powder
diffraction pattern requires a random crystallite orientation. Any preferred orientation will show up as an incorrect
intensity distribution, but the peak positions will remain the same. Clay minerals and other layer structures are
notorious in this respect. In serious cases, the powder patterns will show only a few peaks corresponding to the
strongest 00l reflections. Phase identification may in such cases be impossible. The problem can be reduced by
careful sample preparation. In general, capillary sample will give much less problem than flat samples. Other ways to
reduce this problem in reflection geometry is to use side-loading or back-loading, i.e. the scattering surface is covered
by a plate and the sample loaded from the side or from the bottom. After loading, the plate is removed and the sample
used as usual.
Figure 2.12. Absorption effects with capillary
geometry. With strong absorption only the outer part
of the capillary will diffract and shifts the peak
position to higher 2 .
Figure 2.13. (Lack of) Absorption effects in reflection
geometry. Low absorption will shift the peak position
to lower 2 .

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2.4.6 Statistics
All experimental measurements will suffer from some sort of uncertainties, also powder diffraction. On top
of the error sources mentioned above we have the counting statistics. The diffraction process is governed by Poisson
statistics saying that the uncertainty, or standard deviation, in the measured intensities is the square root of the
intensity itself. In other words, the variance of the intensity is the intensity itself. The relative error we can express as:
(I) / I = 1 / sqrt(I) (2.1)
One immediate effect of Eq. 2.1 is that we need to increase the measuring time by a factor of four in order to reduce
the relative error by a factor of two. The uncertainty will show up as a general noise level, or “ripples” added to the
diffractogram. The noise level will for instance limit the detection of impurity phases in a sample. By increasing the
measuring time we can reduce the relative noise level and improve (reduce) the detection limit. Equipped with
sufficient patience we can in principle reach any detection level. When our powder diffractogram is used in a least-
squares procedure as the Rietveld method the variances of the intensities are used for weighting. It is therefore
essential that any manipulation of the data (background subtraction, corrections etc) also produce the correct
variances according to the rules of statistics. As a rule, one should use the raw data as input to refinement programs
and let the programs internally handle the corrections, i.e. adding corrections to the calculated data.
2.5 Data formats
Powder diffractograms comes in many formats; typically every manufacturer and each synchrotron has
their own format. However, most manufacturers offer the possibility to transfer the data into a set of generally
accepted formats. The simplest of them is the xy-format; one column with 2 -values and one with recorded
intensities. There are some variations of that simple theme, for instance by starting the file with information on
wavelength, measuring time etc. Many programs will be able to read the data anyway, but sometimes it is necessary
to delete those initial lines. Another common and more compact format is to start with a line giving start, stop and
step values in 2 and in the following lines giving the recorded intensities with ten intensity values per line. One
disadvantage with rewriting into the general formats is that the information on the measurement like time,
wavelength, diffractometer settings etc, are lost in the translation. Make sure to keep the original data as well.
2.6 Phase identification
The first thing to do with an unidentified powder diffraction pattern is to compare it to known powder
patterns. The most extensive collection of known patterns is the Powder Diffraction File, PDF. It is available through
ICDD (International Centre for Diffraction Data: 'www.icdd.com') on CD, or in books. The latest edition contains
about 175 000 powder diffraction patterns (2007). An example of the stored information is shown in Fig. 2.14.

13
Figure 2.14. Powder database information on scolecite. Please note that the reflection list is truncated.
The database comes with primitive search-match software, where for instance the d-values of the strongest reflections
in an unknown pattern can be rapidly compared to the known ones. It is also possible to include knowledge of for
instance specific elements or groups, density and unit cells in the search. More effective though is to use dedicated
search-match software. This software will import your powder pattern directly, subtract background and find peak
positions and intensities. Also the search-match programs have possibilities to narrow the search using restrictions on
for example elements, crystal system, symmetries, colors etc. Restriction should be used with cautions when dealing
with unknown samples. In addition it is possible to identify several phases in a mixture. Identified phases may be
subtracted to facilitate successive identifications. Four to five phases can normally be identified given some patience.

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2.7 Indexing
When our sample can not be identified in a database the next step will be to try and index it. Indexing
means to find the unit cell, from which we then can assign indices to all reflections in our diffraction pattern.
Unknown cubic and to some extent tetragonal and hexagonal structures may be indexed by hand. In the cubic case we
have
1 / dhkl
2
= (h2
+ k2
+ l2
) / a2
(2.2)
Combining with Bragg's law we can rewrite it into
sin2
hkl = (h2
+ k2
+ l2
) 2
/ (4a2
) (2.3)
We start by calculating a set of sin2
hkl values from our unknown sample. By comparing the sin2
values, one will find
that several of them are related by some integer factors, which in turn are related to the index sum in Eq. 2.3. Going
through the list it will be possible to assign index sums to all the reflections. When the indices are found, the common
factor, 2
/ (4a2
), will give us the unit cell.
Turning to symmetries lower than cubic the amount to work that goes into manual indexing is rapidly
increasing. Special indexing programs have instead been developed to automate the process. The algorithms are based
on trial-and-error, where indices are systematically, but still intelligently varied, starting from the low order
reflections. The general reciprocal cell relationship is rewritten as
Qhkl = h2
X1 + k2
X2 + l2
X3 + hk X4 + hl X5 + kl X6 (2.4)
where Qhkl = sin2
hkl, and Xj contains the reciprocal cell parameters and the wavelength. Starting from the six first
reflections and assigning them indices, Eq. 2.4 will give us a set of linear equations. The indices are assigned by
intelligent trial-and-error and systematically varied. Solving the linear equations for each set of trial indices will
produce a long list of unit cells. Tests of the internal consistency with all reflections are then used to produce figure of
merits, from which the best fitting solutions are chosen. The different crystal classes are tested one by one, starting
from the cubic and ending by the triclinic. Provided the input data are phase pure and accurate enough, the success
rate is in the order of 90 %.
2.8 Structure solution
Structure solution from powder diffraction data is still just as much an art as a science. The standard
method, the direct method, is the same for powders as for single-crystals. Other methods for structure solution is
being developed as for instance direct space methods, Monte Carlo methods, Patterson methods and multiple-
wavelength phasing. However, before starting the actual structure solution we need to find a unit cell from an
indexing procedure. We then need to extract the individual intensities from the powder diffraction pattern. The
number of extractable reflections will be much less than for single-crystal data due to the overlap problem. Unless we

15
have a rather well resolved diffraction pattern we can not hope to be successful in solving an unknown structure.
Synchrotron radiation has revolutionized powder diffraction in this respect. The much improved angular resolution
has significantly increased the number of resolvable reflections and thereby narrowed the gap between powders and
single-crystals for structure solution. At ESRF, Grenoble, it has recently been possible to solve structures of small
proteins! Today the vast majority of structure solutions from powder samples are based on synchrotron data.
2.9 Structure refinements
After structure solution, or when a structure model has been found by other means, the next step will be to refine the
structure. There are two approaches for this: The Pawly method, where the intensities are first extracted based on a
refinement of the unit cell and profile parameters, and then the structure is refined as in the single-crystal case. The
other method is the Rietveld method, where all parameters, unit cell profile and structure parameters are refined in
one process. We will deal with the Rietveld method in the following chapters.
2.10 The Debye equation
A different approach to the scattering process was demonstrated by Debye already in 1915. It relies on the knowledge
of all the atomic positions in the sample. From them we can compute all interatomic distances and from them the
diffraction pattern from
I(Q) = N fi(Q) fj(Q) sin(Q rij) / (Q rij) (2.5)
where Q = 4 sin / , f is the atomic form factor and the summation is over all interatomc distances in the sample. It is
seemingly simple, but the computational cost goes as the number of atoms squared. The method is not restricted to
crystalline materials, but we can use amorphous materials or crystals with defects as stacking faults and dislocations.
We “only” need to find all interatomic distances and feed them into Eq. 2.5. Fig. 2.15 shows an example where
powder diffraction patterns have been simulated for a hypothetical structure containing from 1 to 1000 atoms in a
crystal. Note the successive sharpening of the reflections and how relatively few atoms are needed in a crystal in order
to give a diffraction pattern.

16
Figure 2.15. Debye simulation of a hypothetical structure with 1 to 1000 atoms per crystallite.

17
3 THE RIETVELD METHOD
The Rietveld method is a least-squares procedure, which minimizes the quantity
SY= iwi(Yi-Yci)2
(3.1)
where Yi is the observed intensity at point i of the observed powder pattern and Yci is the calculated intensity. The weight,
wi, is based on the counting statistics, wi=Yi
-1
, although at different stages of the refinements it may be advantageous to use
for instance wi=Yci
-1
. The contribution to Yci from Bragg reflections, diffraction optics effects and instrumental factors is
expressed as
Yci=s HLMH FH
2
(2 i-2 H)PHA+Ybi (3.2)
where s is the overall scale factor,
H represents the Miller indices for the Bragg reflection,
L contains the Lorentz and polarization factors,
MH is the multiplicity,
FH is the structure factor for Hth Bragg reflection, and
(2 i-2 H) is a profile function, where 2 i is corrected for the 2 zero error,
PH is a preferred orientation function,
A is the absorption factor,
Ybi is the background intensity at step i.

18
The Bragg reflections contained in the summation at each point of the powder pattern are determined from a sorted list of
the possible reflections and their profile widths at 2 i. The structure factor as usual contains the structural information
FH= fjgjexp-2 i(hxj+kyj+lzj)exp(-Bjsin2
/ 2
) (3.3)
where fj is the scattering factor, or in the case of neutron data the scattering length, of atom j, gj is the occupancy factor, xi,
yi and zi are the fractional coordinates, and Bj the temperature factor coefficient. We can obtain the parameters from Eq.
3.1 by putting its derivatives with respect to its parameters to zero. It gives us a set of non-linear equations, which are as
Taylor series, where only the first term is retained. From the so derived normal equations we may in matrix form write
Mx=V (3.4)
where M is an p p matrix, p being the number of refined parameters, and with elements
Mkl= iwi( Yci/ pk)( Yci/ pl). The summation is performed over all observations, i.e. profile steps. x is a p-dimensional
vector with the parameters shifts, pk, as its elements. V is also a p-dimensional vector with elements Vk= iwi(Yci-
Yi)( Yci/ pk). Inverting M and multiplying with V gives the solution to the parameter shifts
x=M-1
V (3.5)
The solution thus gives us the parameter shifts relative to the starting parameters, which is why a reasonably good starting
model is required. After applying the shifts to the original parameters, the procedure is repeated until convergence. Due to
the summation over the pattern steps, which each may have contributions from a large number of overlapping reflections,
the computational efforts with the Rietveld method are much larger than for a single-crystal structure refinement.
The beauty of the Rietveld method lies in that it allows simultaneous adjustments of structural parameters,
contained in FH, profile parameters, unit cell, background etc. Thus, an improved profile fit will enable a correction of the

19
structure model as the intensity extraction is improved, and vice versa, which is then fed back during the cause of
refinements. The Rietveld refinement program that is introduced here, WINPOW, is essentially the LHPM1 program
written by Hill and Howard. LHPM1 was developed from the DBW3.2 program by Wiles and Young, which in turn was
developed from Hugo Rietveld´s original code. It has been extensively modified to enable the use of unequal step data
from a position sensitive detector, Chebyshev polynomials for background fitting, restraints, a split pseudo-Voigt profile
function, asymmetry treatment according to Finger, Cox and Jephcoat. In addition, the Rietveld program has been
combined in a Windows graphics user interface with programs for distance and angle calculations, Fourier calculations,
profile, Fourier map and pattern plotting, and various output and report possibilities. The WINPOW programis available
for Windows 2000/XP.
The main structure of WINPOW (as taken from LHPM1) is shown in Fig. 3.1. The INPTR routine controls the reading of
input data. REFGEN interprets the space group symbol, generates the Bragg reflections, and stores them in a sorted list
together with profile widths, Lp-factor etc. ASSIGN goes through the pattern point by point to determine which are the
contributing reflections at each point. The serial numbers, as stored by REFGEN, of the first and last reflection
contributing to a point is stored for later use. Note that ASSIGN is called only once in the beginning of a refinement and
the assignment is not updated as parameters that are affecting the assignment are refined. This may cause problems when
halfwidths and/or unit cell parameters show large variations. ASSIGN will also check to see that the number of peaks
present at a given point in the pattern is not more than the program dimensioning allows. If the limit is exceeded, the
program will stop with a message like Excess peak overlap . Reducing the number of halfwidths in a peak will help in
such a case. ITER controls the actual refinement cycles. CALCUL generates the structure factors and derivatives with
respect to the refined parameters. SUMMAT then calculates the intensity at each point of the pattern using the structure
factors from CALCUL and profile information fromPROFILE. SUMMAT also calculates the derivatives at each point and
adds them to the least-squares matrices. DPINV inverts the least-squares matrix. CHISQ evaluates residuals and OUTPTR
calculates new parameters and generates output from each refinement cycle. ITER repeats this process until convergence
or for a chosen number of cycles. EXPUT completes the calculations after the last cycle, generates an output parameter
file, structure factor file for Fourier calculations, pattern files for plotting and generates the final output. WINPOW can Figure 3.1. The WINPOW program structure.

20
also be used for pattern calculation without refinements.
The instructions necessary for the program is given in more details in the program manual. One may also start
with default values as defined in the different dialog boxes for parameter editing. The dimensioning of the program may
vary. Representative limits are 20000 observations, 8000 Bragg reflections, 1024 overlapping reflections in any point, ten
different phases, 500 atoms and 400 refined parameters.
WINPOW, or the original LHPM1, is far from the only available Rietveld program. A more complete list of
Rietveld programs and powder diffraction programs in general has been collected on different web sites. Through the IUCr
homepage, www.iucr.org, you will find links to lots of crystallographic programs. Follow the link for Crystallography
News” and “Software”
Some additional features included in WINPOW:
1. Plotting. Usually the best indication of successful or unsuccessful refinements is seen in the pattern plot. It is a
good idea to check the profile parameters during refinement by plotting them as a function of 2 . If they do not
behave, fix them.
2. Distance and angle calculations. It is always a good idea to check distances and angles during structure
refinements. Even thought R-values are reduced during refinements, the structure may turn into nonsence.
3. Table output. Summarizes the refinements. It is advisable to print the table once in a while to keep track and
records of different refinement models attempted.
4. Fourier calculation. When atoms are missing in the model a difference Fourier calculation can be performed. It
produces a list of suggested atomic positions that can be included in the distance and angle calculations.
Chemically reasonable atoms can then be added to the atom list and included in the refinement.
5. CIF-output.
6. ATOMS-input file. Free format input for structure plot program ATOMS.

21
4 HOW TO GET STARTED, 2 ZERO, UNIT CELL AND SCALE FACTOR
The most effective way to start up a new refinement project is without doubt to calculate a diffraction pattern
based on the starting model and plot this pattern together with the observed powder pattern. To simplify and speed up the
process one should restrict the upper angular limit to 20-50° depending on the complexity of the pattern. Fromthis starting
point we can manually adjust the scale factor to make the intensities of the observed and calculated pattern comparable. By
measuring the full-width at half maximum (FWHM) of some reflections we will get a starting value for the constant
FWHM parameter. This W-parameter is entered as the squared FWHM. Simple inspection will also give us a starting
parameter for the first (constant) background parameter. The so determined parameters are then fixed for the time being.
Before starting the actual refinements, at least the first few peaks in the calculated and measured pattern must coincide.
The easiest way to achieve this is to manually adjust the 2 -zero parameter. We are then ready to start the refinements. In
the first cycles we refine the unit cell parameters together with the 2 -zero parameter. Releasing three to six background
parameters is often rewarding at this stage. During this initial stages, it is essential to keep the scale factor fixed. If the
scale factor is released too early, a misfit may result in a close to zero scale factor and meaningless refinements. If
everything behaves properly, we can now release more and more parameters: FWHM, the peak shape parameters,
asymmetry parameter and scale factor. If the initial 2 -range was too restricted should it also be increased at this stage.
The improvements of the refinements can, almost too, conveniently be monitored through the different residual
values given by the program. However, the initial stages of the refinements are best followed by plotting the measured or
calculated pattern together with the difference pattern. Such a plot will show the performance at each and every point of
the 2 range. The R-values will only give the average performance and may hide gross errors at minor peaks. The plot will
also directly tell us if the residual is mainly due to intensity differences, is due to misfits of profiles, or poor background
fitting. In the former case one would allow the structure model to vary more freely, while in the latter case, one might
increase the number of profile or background parameters. A successful refinement start is shown in Fig. 4.1. The top curve
is the measured powder pattern, the bottom one is the difference between the measured and calculated patterns.
Figure 4.1. A successful refinement start.

22
The 2 -zero, unit cell (monoclinic) and one background parameter was refined. From the difference curve we
can see that 2 -zero and the unit cell is essentially correct, but the background and structure model need adjustments.
Adding a few more background parameters would immediately improve the background fit. In this particular case, a
refinement of the water occupancy factors would improve the profile fit.
In general, it is an advantage to first release the global parameters as the 2 -zero and some background
parameters together with the unit cell, one FWHM parameter and the scale factor. With these parameters refined it is time
to gradually increase the 2 range. To start structure parameter refinements it is necessary to have expanded the
refinements to a reasonable range in 2 . Reasonable in this context means a range that contains enough Bragg reflections,
to ensure a certain degree of over-determination to the structural parameters varied. A too wide a range in 2 will
unnessecarely increase the computing time, and may also cause unpredictable behavior of some parameters. Small steps
when increasing the 2 range and the parameter number is strongly recommended.
There are a set of options in WINPOW that may help in the initial stages of refinements or when larger changes
in the model is attempted:
- Damping factor. Multiplies the diagonal elements in the LS-matrix and thereby reduces the correlation effects.
Convergence will as a consequence be slower. Values between 1.02 and 1.1 give clear effects.
- Groups of parameters (coordinates, thermal parameters, profile parameters and others) can be dampened or completely
blocked by giving them factors between one and zero.
- Using only every n:th point in the pattern will considerably speed up the calculations.
- Restraints. This will be discussed in more details later, but in short one enters known bonding distances as observations.
- Backup/Restore. These options in the Edit Project menu allows you to save a successful refinements result and restore it
when continued refinements completely fails. The default file names are easily changed to indicate different stages or
models during the refinements.

23
5. BACKGROUND
The background is by definition the non-Bragg intensity present in the powder pattern. It can be divided into
three contributions; air scattering; non-Bragg scattering from sample and sample holder; and electronic noise. Air
scattering is a problem especially for film and PSD data, as the diffracted beam in those cases cannot easily be collimated.
The air scattering is caused by the primary beam and it can be reduced substantially with screens and adjustment of the
beam stop. It can be further reduced if the diffractometer enclosure is evacuated or filled with He. Vacuum or a He
atmosphere will in addition give less absorption and thereby increase the diffracted beam intensity. Figs. 5.1 and 5.2
illustrates simple measures to reduce the background from a Debye-Scherrer diffractometer.
The remaining part of the air/He scattering as well as the sample holder background can be measured and
subtracted from the powder pattern, although it is not trivial to do so. With capillary samples two additional data sets have
to be recorded: one with an empty beam path, A, and one with an empty capillary, B. In order not to add unnecessary
contributions to the counting errors they should be treated with simple smoothing before proceeding. The difference
pattern, Ci=Bi-Ai, will approximately correspond to the capillary scattering. This part has to be corrected for absorption
when a sample is present. With knowledge of the sample absorption coefficient this can be done numerically. The
absorption corrected capillary scattering, E, is then added to A to give a corrected air and capillary background function, F.
Fi=c2(c1Ai+Ei) (5.1)
where c1 is a correction factor on the air scattering part due to sample absorption and c2 is a scale factor equal to the ratio
between the data collection time of the sample and the background patterns. The curve F is then subtracted from the
sample powder pattern. It is clear from Fig. 5.3 that if the background from the empty capillary, B, was used directly for
background subtraction, it would result in an overcorrection. When using the Guinier-Hägg sample holder, the background
curve is obtained from a sample holder with an empty scotch tape. This pattern is also smoothed and then subtracted from
the sample pattern with the appropriate scale factor. As pointed out before, it is necessary to retain the original pattern for
weighting.
Figure 5.1. A Debye-Scherrer diffractometer equipped
with an additional screen and an adjustable beamstop.
Figure 5.2. Background curves: (top) no screen,
beamstop in back position; (middle) with screen; and
(bottom) with screen and adjusted beamstop.

24
Non-Bragg scattering from the sample itself can of course not be reduced by screening. It can be divided into
different parts. Amorphous scattering from the sample we simply have to live with. Compton scattering in the case of X-
rays can in principle be corrected for. However, it is a serious problem mostly at very high energies and can be disregarded
with Cu-radiation. Fluorescence scattering is a serious problem whenthe sample contains elements to the right of the anod
material in the periodic table, i.e. first row transition elements in the case of Cu-radiation. If we use a secondary
monochromator this radiation will be removed due to its longer wavelength. With a position sensitive detector this
scattering will add to the background. Due to the longer wavelength we can reduce it with a suitable filter. Pure absoption
will reduce the fluorescence scattering more than the Bragg scattering. For neutron diffraction, spin-incoherent scattering
from certain elements and isotopes will give considerable contributions to the background. A commonly present incoherent
scatterer is hydrogen, which is one important reason to use deuterated samples in neutron powder studies. Yet another
contribution to the background is TDS (thermal diffuse scattering, or phonon scattering). This contribution piles up under
the Bragg peaks and will increase the observed intensities. In favourable cases the TDS contribution can be calculated, and
it will be reduced on cooling. Wether or not we manage to reduce or correct for the background there will in most cases be
a remaining background we have to model in our Rietveld refinement.
In a structure refinement the background is generally a nuisance, whatever origin it has. The aim of the
background fitting in the Rietveld method is just to find a function that can describe it. The most commonly used
background correction functions are variations of simple polynomials.
Yib= mBm(2 i)m
, m=0,1,2,.... (5.3)
Most programs also allows for an additional term with m=-1, which is useful to take care of the relatively sharp increase in
background intensity when approaching low 2 -values. When going to higher order polynomials one often runs into
rounding of errors. A more effective set of functions for least-squares methods is the Chebyshev Type I functions. These
are normalized orthogonal functions, defined in the interval -1 to +1, which can only take values between -1 to +1.
Figure 5.3. Background correction curves: (top) empty
capillary; (bottom) air scattering; and (middle)
absorption corrected capillary scattering.

25
T0[x]=1
T1[x]=x or Tn[x]=cos(n arccos(x)) (5.4)
Tn+1[x]=2xTn[x]-Tn-1[x]
In order to fit into the -1 to 1 range in x, the powder pattern has to be normalized into that range:
Yib= nBnTn[2(2 i-2 min)/(2 max-2 min)-1], m=0,1,2,..... (5.5)
The normalization is to some extent a disadvantage, as the functions will not automatically fit when the angular range of
the calculations is changed. On the other hand they converge very rapidly, often one cycle is enough. Howmany terms one
should use can be deduced from a comparison of the refined parameters and their e.s.d.s: when the parameter is less than
2-3 e.s.d.s it does not significantly contribute to the overall fit. It is often very instructive to plot the background function
together with the recorded powder pattern. This may reveal unwanted features towards the end points of the pattern. Due to
correlations with for instance thermal parameters it may otherwise pass unobserved. When the background behaves
unrealistic, the standard recipe is to reduce the number of parameters until a smooth background is obtained.
In some powder experiments the background may supply additional, at least qualitative, information. Variations
in the background of samples heated to certain temperatures can reveal the onset of a structural collapse. When a liquid
medium as for instance water is used, it will give rise to additional amorphous or liquid scattering, which is easily
monitored during a heating experiment (Fig. 5.4). In this particular case, the zeolite laumontite submerged in water, it was
shown that the release of crystallographic water started well below the temperature where the excess water was boiled off.
Figure 5.4. Part of the diffraction patterns of laumontite
at (top to bottom) 50, 60, 70, 80, 90 and 100 ºC

27
6. PEAK FUNCTIONS
The key to success of the Rietveld method is the analytical functions that describe the diffraction peaks and how
they vary with 2 . There are several factors affecting the peak shapes. We can divide them into two groups: instrumental
and sample contributions.
Instrumental contributions: Radiation source (1)
Monochromator (2)
Slit systems (3)
Axial divergence (4)
Misalignment (5)
Sample contributions Sample rocking curve (mocaicity) (6)
Absorption effects (7)
Crystallte size effects (8)
Strain broadening (9)
- (1 and 2) The source image is a problem particularly with sealed tube or rotating anode sources. The wavelength
distribution is difficult to describe with an analytical function. It is close to Lorentzian, but not completely symmetric. In
the case of synchrotron radiation the source is white raiation and peak shape contribution depends solely on the
monochromator. A primary monochromator will in general improve the peak shapes also with X-ray tubes.
- (3) Slit systems will in general just add a rectangular contribution and thus only contribute to broadening effects.
However, when slits are used in combination with a primary monochromator to remove the K 2 line, it has to be done with
care. Too wide slits will allow some K 2 contributions through, but too narrow slits will truncate the wavelength
distribution and result in cumbersome reflection profiles.
- (4) When a powder pattern is collected as a one-dimensional strip of the diffraction cones (cf. Fig. 2.1) the intensity seen
by the detector will look like in Fig. 2.9. The recorded reflections will become wider and more asymmetric when going to

28
lower angles. Reducing the vertical aperture of the detector will improve the low-angle profiles, but at the expense of
intensity. Alternatively, it should be possible to use a variable vertical slit that accepts a constant angle of the diffraction
cones to make the effect constant over the pattern. A related effect is due to the finite size of the sample. Each and every
point of your sample will ideally produce its own set of diffraction cones. Adding them up, Fig. 2.10, will increase
broadening and asymmetry. Reducing the sample size, for instance by reducing the vertical entrance slit will reduce this
problem, but again, at the expense of intensity.
- (5) Misalignment may add to the profile width and asymmetry, and give erroneous peak positions. It should be eliminated
at its source.
- (6) The rocking curve of the sample itself will of course also add to the width of an observed reflection. With the
extremely high instrumental resolution that can be achieved with the new generation of X-ray synchrotrons, the sample
rocking curve may in fact dominate the recorded profiles.
- (7) Sample absorption will act differently in different geometries. Highly absorbing samples in the Bragg-Brentano
geometry will scatter only from the surface and will not contribute to profile broadening. With low absorbing samples the
radiation will penetrate the bulk of the sample and cause 2 -dependent sample broadening, and shift the peak maximumto
a lower 2 , Fig. 2.13. In capillary geometry a zero-absorbing sample will generate undistorted and 2 -independent
diffraction profiles. As the absorption increases the distortion and 2 -dependence will increase, and the peak maximumis
shifted to higher 2 , Fig 2.12. However, if the detector is calibrated with a well-known material with the same as the
sample itself, the peak shifts due to absorption will cancel. In the flat plate Guinier geometry, peak width and asymmetry is
in principle independent of sample absorption as long as the sample is thin compared to the slit width. The flat plate
Guinier method is therefore the preferred method for accurate unit cell determinations.
- (8) The diffraction theory tells us that the size broadening, , in radians, of the first order diffraction peak will be
= k / (T cos ) (6.1)
where T is the sample thickness. This is the famous Scherrer equation. The factor k is a geometric facotor depending on
the shape of the crystallites. A value of 0.9 – 1. is normally sufficient. With =1.54 Å and 2 =90° a thickness of 5000 Å

29
will give a broadening of 0.02°. Reducing the particle size to 50 Å will give a broadening of 2.5°. This is approximately
the limits for what can be determined from conventional diffraction data. A comparison of the 2 dependence of reflections
in different directions may also reveal the crystallite shape. Fig. 6.1 illustrates the anisotropical size broadening effect
during of hematite. This anisotropic effect can easily be understood when we view the diffraction from thin plates in the
Ewald construction, Fig. 6.2. According to the interference function thin plates will give rise to needle shaped reciprocal
lattice points.
Figure 6.2. When reflections of type 00l pass the Ewald sphere they will leave a brooad trace (left) , while reflections of
type hk0 will leave a narrow trace (right).
- (9) Crystallites with frequent lattice distortions, or strain, will appear as having a distribution of unit cells. The
distribution is in general asymmetric due to the asymmetry of the bond energy curve and in many cases anisotropic. Such a
distribution of unit cells will be directly reflected in the profile widths. The evaluation of strain in metals and alloys is an
important technical application of powder diffraction. The strain, , can be obtained from
Figure 6.1 Anisotropic size broadening in hematite.

30
=k tan (6.2)
When both particle size and strain effects are present they may be difficult to distinguish on top of the general -
dependence of the peak shape unless we are dealing with very well-resolved peaks.
The way the different contributions adds up is through consecutive convolutions (Fig. 6.3). The resulting peak
function will be very complicated and useless for practical applications. We are to some extent saved by the fact that a
convolution of a large number of distributions tend to give a resulting Gaussian distribution (the central limit theorem). A
simple Gaussian was also the peak shape chosen by Rietveld in his original neutron powder work.
Figure 6.3. Convolution of different contributions to the reflection profiles.

31
However, X-ray data and also well-resolved neutron data show significant deviation from the simple Gaussian
and have in addition pronounced asymmetry. Thus we need a more elaborate peak shape function. To be useful it should
be mathematically simple, to allow simple evaluation of its integral and derivatives with respect to its variables. The
dominant functions used today are the Voigt, the pseudo-Voigt and the Pearson VII functions, and variations of these. The
Voigt function is a convolution of a Lorentzian and a Gaussian
V(x,HL,HG)= L(x',HL)G(x-x',HG)dx' (6.3)
where x=(2 B-2 i) and HL and HG are the Lorentzian and Gaussian halfwidths respectively. It can be evaluated numerically
from
V(x,HL,HG) C1/HG Re( (C2x/HG+iC3HL)) (6.4)
where Re( (....)) denotes the real part of the complex error function. The Voigt function offers the possibility to refine
anisotropic peak shapes as shown in Figs 6.1 and 6.2. The pseudo-Voigt function is an analytical approximation to the
Voigt function
pV(x,H, )= C4/[H (1+C5x2
/H2
)] + (1- )C6exp(-C7x2
/H2
)/H (6.5)
where is the mixing parameter, =1 for a pure Lorentzian and =0 for a pure Gaussian (Fig. 6.4). The Pearson VII
function has the form
P(x,H, )= ( )/ ( -1/2) 2C2
1/2
/(H 1/2
)(1+4C2x2
/H2
)-
(6.6)
where C2=21/
-1 and is the gamma function. = will give a pure Gaussian, while =1 will result in a Lorentzian peak
shape.
Figure 6.4. Pure Gaussian (solid line) and pure
Lorentzian (broken line) peak shapes.

32
Many of the peak broadening factors discussed above will for geometrical reasons vary with 2 . In addition
there will always be broadening due to the wavelength dispersion. The variations are taken into account by allowing the
halfwidths and the mixing parameter, , to vary with 2
H=(Utan2
+ Vtan + W)1/2
and = 1 + 2(2 ) + 3(2 )2
(6.7)
To successfully apply these parameters it is necessary to have reached a certain 2 range in the refinements. It is also
advisable to check the correlation between these parameters to find out how many of the parameters are really needed.
The last component to be considered is the asymmetry. Convolutions of asymmetric distributions will generally
not cancel the asymmetry. The original way to obtain an asymmetric peak function was by multiplying the symmetric
function, point by point, with the function
(x)=1-Asign(x)x2
cot B (6.8)
where A is the refinable asymmetry parameter. An improved correction, introduced by Howard, uses a convolution of the
symmetric peak function and
f( )=1/(2 M
1/2 1/2
) for M< <0 and f( )=0 elsewhere (6.9)
where M=-Acot2 B and A is the refinable asymmetry parameter. The convoluted function is approximated through a
numerical integration as a sum of five symmetric peak functions. A more elaborate way to describe the asymmetry is the
split Pearson VII function. In essence it uses two sets of parameters to independently describe the function to the left and to
the right of the maximum. WINPOW has the possibilty to use a split pseudo-Voigt function. Such a function will have six
parameters to describe the half-widths and six to describe the Lorentzian contribution, three each for the left and right hand

33
side of the peaks (c.f. Eq. 6.7). Needless to say, one has to exercise some caution when releasing all of those parameters.
Typically one would couple several of the left and right side parameters to the same values. A more elaborate way of
treating the peak asymmetry was introduced by Finger, Cox and Jephcoat: It uses numerical calculations of the true axial
divergence based on the actual diffraction geometry. It is included in WINPOW, but will normally be very time
consuming.

35
7 STRUCTURAL PARAMETERS AND RESTRAINTS
The main result from a Rietveld refinement is in most cases the structural parameters: unit cell parameters,
fractional coordinates, occupancy factors and temperature factor coefficients. The structural parameters are somewhat
special in a whole-pattern-fitting method. They do not refine directly against the observed quantities, the step intensities,
but rather against the extracted Bragg intensities. The number of structural parameters that can be refined is therefore not
related to the number of observations as in a single-crystal structure refinement. This is also the reason why the Rietveld
method was met with a great deal of skepticism and resistance from single-crystal crystallographers when it was first
introduced. In single-crystal structure refinements an overdetermination of a factor five to ten, i.e. five to ten times as many
reflections as refined structural parameters, is commonly considered appropriate. Depending on how severe the reflection
overlap is, a structure refinement from powder data may require an overdetermination of anything between 10 and 50.
To turn on the structural parameters, we have to use a wide enough angular range to ensure a sufficient number
of Bragg reflection in the Rietveld refinement. For the low angle part of the pattern, an initial overdetermination down to
five may be acceptable if the reflections are reasonably well resolved. As the angular range is increased more and more
parameters can be turned on. However, as 2 increases so will reflection overlapping, and the overdetermination need to be
raised accordingly. In addition, the temperature factor, and for X-ray data the form factor, will exponentially reduce the
reflection intensities with higher sin / . At some point there will be no benefits from increasing the angular range, only
increasing computing time. A simple way of finding this upper limit is to watch the standard deviations of the structural
parameters. If they do not decrease as the angular range is increased, the upper limit is reached with a given structure
model.
7.1 Unit cell
The best unit cell parameters are no doubt obtained with the Guinier method. The peak resolution is superior
and sample absorption effects are minimal. With a manual determination of peak positions fromthe sample and an internal

36
standard, systematic effects like the axial divergence is compensated for. However, the Rietveld method has one great
advantage in that it does not depend on fully resolved peaks in order to refine the unit cell parameters. If the unit cell
parameters are the prime concern, their precision can be enhanced by excluding the low angle part, where reflections are
strongly affected by axial divergence and sample absorption, and the high angle region, where reflection overlap gets
severe. It is essential to lock dependent cell parameter in high symmetry cases like for instance hexagonal, tetragonal and
cubic symmetry.
7.2 Fractional coordinates
To turn on the refinement of coordinates is in most cases straight forward provided a sufficient number of Bragg
reflections is included. Compared to single crystal data one can expect in the order of five to ten times larger e.s.d.s from
conventional powder data, and two to five times larger with good synchrotron data. A comparison between neutron powder
and single crystal data for HIO3 and DIO3 is given in Table 7.1
Table 7.1. Bonding distances (Å) in HIO3 and DIO3 obtained with neutron powder (P) and
single-crystal (S) data.
HIO3 (P) HIO3 (S) DIO3 (P) DIO3 (S)
I-O(1) 1.797(6) 1.812(1) 1.821(7) 1.810(2)
I-O(2) 1.898(6) 1.896(1) 1.909(7) 1.889(2)
I-O(3) 1.786(6) 1.783(1) 1.778(7) 1.780(2)
O(2)-H/D 0.968(9) 0.996(3) 1.009(7) 0.989(2)
The main reason for the poorer performance of a Rietveld refinement is again the profile overlap. The individual Bragg

37
intensities are less well determined than from single-crystal data. In addition, the increasing overlap will set a limit on the
sin / range that can be effectively used, which is much lower than in single-crystal data collection. A powder data
collection using Cu K radiation out to 2 =120° corresponds to a maximum sin / of 0.56 Å-1
, while a single-crystal data
collection using Mo K radiation out to 2 =80° corresponds to a maximum sin / of 0.91 Å-1
. The improved peak
resolution from synchrotron data allows higher 2 and/or shorter wavelengths to be used, and places it in between
conventional powder and single-crystal data.
With X-ray data from compounds containing atoms with widely different atomic weights, one may have problems
refining the positions of the lighter elements. Hydrogen positions can only in exceptional cases be refined from X-ray
powder data. Nevertheless, it is often worthwhile including the hydrogens if their positions are known approximately, or
can be calculated from geometrical considerations. The hydrogen coordinates are then coupled to the adjacent atom
coordinates.
7.3 Restraints
Occasionally the refinements are not giving the expected results in terms of bonding distances and angles. Well
known, rigid structure fragments may come out with very unrealistic geometries. The reason is in most cases a poor model
of the complete structure, but sometimes the diffraction data is simply not good enough. To get around such problems it is
possible to include geometrical information in the refinements. One of the ways is to use the geometrical information as
constrains, i.e. to refine the positions of rigid groups instead of individual atoms. Thereby we can reduce the number of
parameters in a refinement. However, constraints are inflexible and also cumbersome to compute. A more flexible solution
is offered by restraints. With these, the geometric information is used as additional observations in the refinement process.
If we recall our least-squares expression, Eq. 3.1, we can include the geometric information and form a new residual
SYR=SY+cwSR with (7.1)
SR= wk(Rok-Rck)2
(7.2)

38
where cw is a common weighting factor, Rok is the expected distance, Rck is the calculated distance and wk is a weight,
usually 1/ 2
of the expected distance. The geometrical information is simply added to the least-squares matrix, Eq. 3.1,
with the appropriate weights. By tuning the common weighting factor, cw, we can control to what degree we want to trust
the diffraction data or the geometric information. Initially, cw may be quite high to ensure a stable start of a refinement. As
the refinements progress, it may be possible to reduce cw, and at the end the restraints may not be needed at all.
If the number of "known" bond distances is larger than the number of refinable coordinates, the crystal structure can
in fact be refined without any diffraction data. This method is known as DLS, distance least-squares, which was actually
the way restraints were initially introduced.
A typical example when restraints are very helpful is when framework structures like zeolites are refined. There is a
limited number of frameworks, and it is most often the extra-framework structure that is of interest to solve and refine.
However, when the extra-framework structure is poorly known, the framework atoms tend to "cover up” by moving into
channels and cavities when freely refined. Consequently, difference Fourier maps may not be very helpful in locating the
extra-framework atoms. As the framework building units, the SiO4 and AlO4 units, are well-known and very rigid, their
geometries as well as Si-Si and Si-Al distances can be entered as restraints. They will then have much less tendencies to
move into the channels and cavities, which increases the chances of locating extra-framework positions in a difference
Fourier map. As the extra-framework structure becomes better and better determined the restraints can gradually be
released and finally removed completely.
To use the restraints option, the connected atoms have to be specified in some way. In WINPOW the restraints can
be set up using special dialog box. It will write the restraints instructions to the parameter file. An atom is specified by the
number it appears in the atom list, a three digit translation vector relative 555, and the symmetry operation number (three
digits) as they appear in the output of the symmetry operator (ORTEP notation). Atoms in their original position can be
specified by the atom number only. For a single bond two atoms and a distance is given, for tetrahedra, five atoms and one
distance, for octahedrons, seven atoms and one distance are given. Note that the polyhedra are restraint to become regular.
For known irregular polyhedra it is necessary to enter individual bond distances. All distances are entered with weights.
Normally they are related to the standard deviations of the bond distances; w = 1 / (d)2
. Using such weights for the
individual distances, the overall restraints weight should ideally be unity.

39
7.4 Occupancy factors and thermal parameters
Occupancy factors will in most cases refine without problems. When the same site is occupied by two elements they
can be locked to a total occupancy and the elements refined with opposite shifts. However, it may not be possible to refine
occupancy factors and thermal parameters simultaneous. Due to the often limited sin / range, these parameters becomes
strongly correlated, especially for lighter elements. Thermal parameters themselves often cause problems. As in single-
crystal structure refinements they are strongly correlated with the absorption correction and extinction. With powder data
they also tend to be correlated to the background, peak width and shape, and preferred orientation parameters. Only in very
simple or highly symmetric structures one may hope to refine reasonable anisotropic temperature factor coefficients. Even
when individual thermal parameters appear to refine satisfactory, i.e. improving the overall R-values and profile fit, one
will often find that at the same time distances and angles are getting worse. If there is no chemical reasons for widely
different temperature factors within a set of similar atoms, it is in most cases good practice to couple them, and thereby
minimizing the number of refined parameters. When refining the structure, but at different temperatures, it is often
sufficient to use the overall temperature factor only.
7.5 Preferred orientation
Preferred orientation is one of those things one should try to avoid by careful sample preparation. Also the choice of
diffraction method is important. Reflection geometries are generally worse in this respect. WINPOW offers a possibility to
correct for preferred orientation. It requires input of a vector defining the overrepressented crystal planes. The correction is
calculated from
PH = (R2
cos2
+ sin2
/ R)-3/2
(7.3)
where R is the refinable parameter and is the angle between the preferred orientation and reflection vectors.

40
7.6 Multiphase refinements
Phase mixtures are not uncommon in the real world. It is reasonable simple to introduce (and remove) extra phases
in the refinements. When inserted they are treated no different than the first phase(s). It will have its own scale factor, unit
cell, profile parameters and coordinates. In same cases it can be advantageous to couple profile parameters with the
previous phases. It is important when starting a multiphase refinement to adjust at least the scale factor manually. This will
also give a chance to check that the new phase fits at all. How well a multiphase refinement will behave will mostly depend
on how well the peaks from the different phases are separated. In order to get a reliable scale factor there should be a
couple of reflections from each phase which are without too much overlap. Provided the refinements are reliable it is also
possible to get a reliable estimate of the percentage of the different phases, i.e. a quantitative analysis. There are some
things to be aware of. Firstly the scale factor. The quantitative analysis is based on the refined scale factors. Make sure the
occupancy factors of special positions are set correct with respect to the spacegroup symmetry (use the reset g feature).
Secondly the thermal parameters. There is no reason to believe that two or more phases in the same sample have very
different thermal parameters. Even though resetting to similar values increases the R-values and impoverishes the fit it is
necessary to do so. It will otherwise affect the relative scale factors. Thirdly the profile parameters. Unless it is very
obvious that the different phases show different peak shapes, it is safest to lock to each other. Also when quantitative
analysis is not an issue, the profile parameters should be treated carefully in multiphase refinements. There is an obvious
risk that with a very crowded powder pattern some features are “swallowed” by broadened profiles. Fourthly the preferred
orientation parameter should not be used. If your sample really suffers from preferred orientation to some extent the
quantitative analysis will be unreliable to the same extent. The weight fraction of a phase j can be calculated from
Wj = Sj j Vj
2
/ i Si i Vi
2
(7.4)
where S is the refined scale factor, the density of the phase and V the unit cell volume. Please note that the numbers
coming out from this calculation are the weight fractions of the crystalline and refined part only. Amorphous material and
unaccounted phases will of course not be included.

41
8 RESIDUAL VALUES AND STATISTICS
Being a least-squares method, the Rietveld method attempts to minimize the weighted, squared sum of differences
between the observations and the calculated values (Eq. 3.1). One way of following the progress of the refinements is
naturally to watch the decrease of this least-squares residual. In most cases the residual value is normalized and expressed
as
Rwp=( wi(Yio-Yi)2
/ wi(Yio)2
)1/2
(8.1)
Another popular residual value is the plain pattern residual
Rp= |Yio-Yi|/ Yio (8.2)
The denominator of the R-value expressions is the sum of intensities, i.e. the total area of the pattern. Thus a high
background, which represents a large part of that area, will, when properly modeled, always give low R-values. A residual
value where the background has been subtracted may be a useful indicator at the final stages of a refinement, but initially,
when a poor structural model may interfere with background modeling, this type of residual value may cause a great deal
of confusion. Also related to the least-squares sum is the Goodness-of-fit
GOF= wi(Yio-Yi)2
/(n-p) (8.3)
where n is the number of observations and p the number of refined parameters. Some authors prefer the square-root of
GOF, the S value, familiar from single-crystal refinements. When only random errors remain, the expected value of S is as
usual 1. This "fact" is sometimes used to define an expected R-value, representing the smallest possible Rwp that can be

42
reached with only random errors remaining,
Re=Rwp/S=[(n-p)/ wi(Yio)2
]1/2
(8.4)
Yet another residual value is the one formed from the Bragg intensities
RB= |Iko-Ik|/ Iko (8.5)
RB has to be treated with caution. The Iko:s are not true observed intensities, but the intensities are based on the calculated
positions and peak functions. Overlapping reflections are assigned intensity values based on the ratio between the
calculated structure factors. When the structural model is completely off, the scale factor will refine to close to zero. Due
to the way the Bragg intensities are extracted, a close to zero scale factor will result in a very low RB. The close
relationship between "observed" and calculated intensities will in addition result in rather flat difference Fourier maps.
There are several pitfalls in the use of residual values from Rietveld refinements. The R-space is p-dimensional and
hard to predict. There is always a risk of ending up in a local (false) instead of a global minimum. It can never be
emphasized enough that a residual value is just a single number, giving a measure of the average residual. It does not tell
anything about where, or what in your pattern that causes the residual. To obtain this type of information it is necessary to
plot the difference between the observed and calculated pattern. Occasionally, when a refinement appears to have
converged, the difference curve, or a plot of the background, may indicate severe errors in your model. It will then be
necessary to manually change some parameters, to get out of the false minimum. Due to the general decrease in intensity at
higher angles the total difference plot tend to look quite good at higher angles. An expansion of the high angle region will
give a more accurate picture of the residuals, for instance when a poor profile modeling is compensated for by increased
temperature factors and background. A plot of the weighted difference pattern, (Yio-Yi)/Yio
1/2
, is sometimes used to
enhance the residuals at higher angles. It is also the sum of this squared quantity we attempt to minimize in our least-
squares procedure.

43
The different types of refinable parameters, structural, profile, background etc, will all in their own way try to fit the
observed powder pattern. As a result the may become strongly correlated. An important indicator of refinement problems
is therefore the correlation matrix. Correlation coefficients above about 90 % should be regarded as warning sign. In some
cases one of the correlated parameters should be turned off, or they may be refined jointly. In other cases, an increase in
the angular range may resolve the correlation. In addition it is always a good idea to keep an eye on the refined structural
parameters and closely watch their behavior when adding new non-structural parameters. Even if they slightly improve the
profile fit they may not be physically justified.
At the end of almost every refinement one will find that the residual is due to profile misfits. The residual values
will give no indication of the nature of the remaining misfit. A GOF, or S, value much different from 1 does indicate a
systematic nature of the residuals. A more direct measure of the systematic nature of the difference pattern is obtained from
Durbin-Watson statistics. The Durbin-Watson d-value is defined as
d= ( i- i-1)2
/ ( i)2
(8.6)
where i is the intensity difference at point i of the pattern. The weighted d-value is calculated from the weighted
differences, i/ i. If consecutive residuals are uncorrelated, the d-value will be close to 2. To be more precise, choosing for
instance a 0.1 % significance level, one may calculate a Q-parameter
Q=2[(n-1)/(n-p)-3.0902/(n+2)1/2
] (8.7)
If d<Q<2, successive residuals show positive serial correlation, while if d>4-Q>2, they are negatively correlated.
Estimated standard deviations of the refined parameters are obtained from
k
2
=(M-1
)kkS2
(8.8)
where (M-1
)kk is the diagonal elements of the inverted least-squares matrix (Eq. 3.5). The :s are correct estimates of the

44
standard deviations only if the residuals are randomly distributed. This is rarely the case with Rietveld refinements. Instead
the dominant features in a final difference map are profile misfits. The predominantly systematic nature of the errors has
two important implications. Firstly, the :s will be grossly underestimated. Secondly, increasing the measuring accuracy,
i.e. reducing the statistical variations by increasing data accumulation time and/or decreasing step sizes, will improve the
refinement only up to the point where the systematic errors become dominant. It can be demonstrated that for a given
model, the data collection can be optimized with respect to the obtainable e.s.d.s in terms of data collection time and step
sizes. As a rule of thumb one may use step widths in the same order as the minimum FWHM, and the number of counts in
the highest peak need not exceed about five thousand counts. A comparison of some refinement results for thomsonite
using data collected for different periods of time is given in Table 8.1.
Table 8.1. Refinement of thomsonite using different data collection time.
15 min 1 h 4 h 12 h 48 h
Rp (%) 15.74 17.03 11.67 8.77 8.33
Rwp (%) 22.61 25.34 17.68 11.71 11.31
GOF 1.74 4.95 8.87 11.36 44.53
RB (%) 5.13 4.95 4.62 4.15 4.19
g(Ca) 0.562(10) 0.572(7) 0.571(5) 0.565(3) 0.570(3)
g(W1) 0.958(34) 0.999(24) 1.007(17) 0.989(12) 0.988(11)
g(W2) 1.088(27) 1.076(20) 1.096(14) 1.085(10) 1.091(9)
g(W3+W4) 1.005(29) 1.028(21) 1.015(14) 1.015(10) 1.020(9)
Optimizing an experiment with respect to the standard deviations is a somewhat dubious practice. The purpose of
the refinement is to determine parameters, not standard deviations. Increasing the data collection time may not improve the
standard deviations of a given model, but the improved statistics may help revealing structural details otherwise hidden in

45
the random noise. However, to be able to compare parameters from different refinements and to compare different
refinement models we depend on correct estimates of the standard deviations. Bérar and Lelann have proposed a method to
correct the estimated standard deviations for the systematic nature of the residuals. We may express the normalized
differences as
ai=wi
1/2
(Yoi-Yi) (8.9)
The least-squares sum (Eq. 2.1) can then be written as Sy= ai
2
. Divided by (n-p) it gives us the GOF, which is then used to
obtain our e.s.d.s (Eq. 8.8). Following Berar and Lelann, correlated differences should be added linearly, not quadratic.
Sy'=[ jaj
2
]+[ l( mlaml)2
] (8.10)
where the j summation is over the uncorrelated differences, and the differences within each of the correlated regions,
labeled l, are summed linearly before added quadratic to the total sum. The problem is then to determine which sets of
points are suffering from correlations and which are not. We can rewrite Eq. 8.10 as
Sy'=[ i(1-ti
2
)ai
2
]+[ ( tiai)2
] (8.11)
where ti=1 if aiai-1>p, or else ti=0, and p is an (arbitrarily) chosen level. To avoid the strong dependence of the chosen level
of p, one can make use of the 2
-distribution and make ti proportional to the probability of 2
<(ai-1
2
+ai
2
). If aiai-1>0 then
ti=[2(ai-1
2
+ai
2
)]1/2
/{2+[2(ai-1
2
+ai
2
)]1/2
} (8.12)
or if aiai-1<0, ti=0. The factor Sy'/Sy is then used to multiply the regular e.s.d.s to obtain (increased) e.s.d.s where serial
correlations have been taken into account. This factor is given in the list file from WINPOW before the refined parameters
from each cycle.

47
9 CONSECUTIVE DATA SET
One of the exciting possibilities with synchrotron radiation sources is the ability to rapid data collection. Not just
collecting data rapidly, but also getting data good enough for Rietveld refinements. For comparable samples we can expect
to gain a factor of 50 – 1000 in speed as compared to a conventional source. That means data collection in less than a
minute instead of several hours and with the same or even better data quality. The only disadvantage is that one ends up
with a very large number of data sets for refinements. To effectively handle this problem one will need special software.
WINPOW is to some extent set up to handle this kind of problems. One simple possibility is to create an mlt-file. It is
simply a file containing the file names of the instruction files, rec-files, you want to refine. WINPOW will then
automatically refine them in the given order. However, you need to create all the rec-files first. Also this can be done by
WINPOW by creating an exp-file. This file contains the instructions for how to change the consecutive rec-files relative a
starting rec-file. The next problem that appears is how to extract relevant data from all the resulting rec-files. For this there
is a program called WINEXT that can extract information and produce a table of the data. The data can then be presented
graphically with excel or other plot programs.
When one wants to interpret the structural effects of for instance a temperature ramp it is essential that the
refinement results from each temperature is comparable. Due to the correlation problems in Rietveld refinements it is
essential that the refinements are comparable in terms of parameter sets. We have to find a smallest common denominator,
i.e. the smallest common parameter set. When doing measurements on the same diffractometer on the same sample several
parameters should not vary with a stable sample:
2 -zero point
Asymmetry
2 -dependence of halfwidths
Lorentzian part of the profile function
Preferred orientation
It is therefore a good idea to run through all the data while refining all parameters, and from these results find reasonable
average for the 2 -zero point, fix it and rerun to find the average asymmetry parameter, fix it, and so on. With unstable

48
compounds will of course have to be modified. A common case is when one phase gradually transforms into another. It
will then be necessary to find the best set of parameters (except 2 -zero) for each phase. If the content of one of the phases
goes to zero, it will be necessary to fix all parameters except the scale factor for such a phase.
Temperature measurement is a difficult task. Normally a furnace is controlled by a thermocouple some distance
away from the sample. Fig. 9.1 shows the unit cell variation of Si as obtained from a series measurements and Rietveld
refinements as a function of the thermocouple temperature. When comparing to the known thermal expansion of Si, there is
a very significant difference between the thermocouple temperature and the true temperature as seen in Fig. 9.2.
Figure 9.1. Unit cell variations of Si as a function of thermocouple temperature.
Figure 9.2. Temperature calibration curve based on the
thermal expansion of Si.

49
10 PUBLICATION REQUIREMENTS
The information necessary for publication of a crystal structure refined from powder data does not vary much
between journals. What differs is how this information is divided between the article itself and the supplementary material.
The division and amount of information will also depend on to what extent the measurements and/or refinements required
special procedures. The list given in the following is a combination of the recommendations given by Acta Cryst. and
Powder Diffraction. General structural information, like space group, chemical formula, Z, dx, structure description and
discussion etc, has of course to be included as well. For more specific requirements, consult the appropriate "Notes for
authors". A good way of dealing with this problem is to create a CIF (crystallographic information file). Except for the
structural information, unit cell spacegroup and coordinates, it can manage all type of information like:
Experimental: - Sample and sample preparation. Sample container, if sealed or not.
- Instrument type, data collection geometry, radiation source, -filter, monochromator, Soller slits,
sample rotation etc.
- Wavelength, with e.s.d and calibration procedure if synchrotron or neutron radiation was used.
- Data collection time, temperature, pressure, special atmosphere.
- Data reduction: Calibration procedure for position sensitive detector; background subtraction,
etc.
Refinements: - Computer program(s).
- Least-squares expression and weighting.
- 2 -range used, step size, omitted regions if any.
- Absorption correction, and/or R.
- Peak profile type, asymmetry correction, number of halfwidths in a peak.
- Type of refined background function.
- Number of refined parameters in final cycle. Specify type and number of each kind for:
background, 2 -zero point, unit cell, preferred orientation, halfwidth function, peak shape

50
function, asymmetry, and structure parameters: coordinates, occupancy factors and thermal
parameters. Parameter coupling, constraints and restraints used.
- Starting parameters for structure model.
- Source of scattering factors, f' and f'', or scattering lengths.
- Maximum parameter shift-to-e.s.d. ratio, maximum correlation.
- Refined structure parameters with e.s.d.s.
- Refined non-structural parameters: halfwidth, shape, asymmetry, 2 -zero, background
parameters, preferred orientation.
- Rwp, Rp, GOF (or S) and RB with their definitions given together with number of steps and Bragg
reflections.
- Plot of observed and final difference pattern.
- Note any observations of unusual features like particle size or strain effects, possible impurity
phases etc.
Some information should be prepared for depositioning:
- Step intensities
- Individual Bragg reflections when limited to a few hundreds: indices, 2 , observed and
calculated intensities.
A CIF can be prepared automatically by WINPOW, but it will require substantial editing of information not involved in the
Rietveld refinements. Several journals will require depositioning of powder data from newcompounds with the organic or
inorganic structural databases and sometimes with the powder diffraction data base as well. To be up to date, consult the
requirements at their homepages.

51
11 REFERENCES
Klug, H.P. & Alexander, L.E. X-ray Diffraction Procedures (1954) John Wiley & Sons, New York, USA.
R.A. Young (Ed.): The Rietveld method; IUCr Monographs on Crystallography no 5; Oxford University Press; NewYork;
1993.
D.L. Bish and J.E. Post (Eds): Modern Powder Diffraction, Reviews in Mineralogy Vol 20, Mineralogical Society of
America, Washington D.C., 1989.
C. Giacovazzo (Ed.): Fundamentals of Crystallography, Second Edition, IUCr Texts on Crystallography 7, Oxford
University Press, 2002.
Howard, C.J. J. Appl. Cryst. 15 (1982) 615-620.
Meier, W.M. & Villiger, H. Z. Krist. 129 (1969) 411-423.
Durbin, J. & Watson, G.S. Biometrika 37 (1950) 409-428.
Hill, R.J. & Flack, H.D. J. Appl. Cryst. 20 (1987) 356-361.
Hill, R.J. & Madsen, I.C. J. Appl. Cryst. 17 (1984) 297-306.
Hill, R.J. & Madsen, I.C. J. Appl. Cryst. 19 (1986) 10-18.
Bérar, J.-F. & Lelann, P. J. Appl. Cryst. 24 (1991) 1-5.

A-1
APPENDIX A
SYMMETRY AND CRYSTALS
Contents:
A.1 Crystal symmetry 2
A.2 Unit cells 4
A.3 Translational symmetry 7
A.4 Space groups 8
A.5 Miller indices 10
A.6 Reciprocal space 10

A-2
SYMMETRY AND CRYSTALS
Figure A.1. Illustrations from Steno’s (left) and Hauy’s work on symmetry and unit cells, respectively.
Symmetry is a fundamental property in nature. Symmetry becomes very obvious when one tries to arrange
equally shaped and sized objects. Close-packing of balls gives nice regular three- and six-fold arrangements that
extend for as long as there are balls, Fig. A.2. Many of our metal structures can be viewed as such arrangements of
spherical atoms. We can expand the reasoning to the packing of molecules. When two molecules are brought together
they will arrange themselves as to minimize energy, i.e. to take advantage of electrostatic forces, hydrogen bonding
and van der Waals forces. If more molecules are brought together they will naturally continue to join forming larger
and larger aggregates reflecting the symmetry of the individual molecules and the inter-molecular bonding. Crystals
are by definition such aggregates of atoms, ions and/or molecules showing a three-dimensional periodicity.
A.1 Crystal symmetry
When atoms and molecules are packed in a crystal the symmetry of the packing is determined by the
symmetry of the molecules and the symmetry of the inter-molecular bonding. However, not all types of molecular and
bonding symmetries can be extended in two or three dimensions to give a periodic structure. Think for example of the
packing of pentagons, the five-fold symmetry make them impossible to pack without leaving holes. In fact, there are
only a few rotational symmetries that can be observed from the outside of a crystal, Fig. A.3. The integer number, n,
of the different rotation axes means that a rotation by 2 /n, will bring one crystal face into an symmetry equivalent
face. The case of n = 1 is trivial; a full turn will of course bring a crystal face back to itself. Note that the faces need
not be equal in size; the important thing is the angles between them (cf. Fig. A.1).
In addition to rotational symmetry we may have a centre of symmetry, or an inversion centre. A crystal
face is mirrored through an origin in the crystal (Fig. A.4). In a crystal with a just a centre of symmetry, opposite
faces will be parallel.
Figure A.2. Two-dimensional close-packing of
spheres.
Figure A.3. Crystals showing different rotational
symmetry. The num ber s indicate the type of rotation
axis.

A-3
Figure A.4. Inversion centre illustrated by a pair of hands and two molecules.
Combinations of an inversion centre with the rotational axes give rise to inversion axes. They are represented by a
minus sign in front of or above the integer representing the rotation axes. This symmetry operation means a rotation
by 2 /n followed by the inversion. The two-fold rotation-inversion is equivalent to a mirror plane and is denoted by
m. Crystals showing the different inversion axes are illustrated in Fig. A.5.
Figure A.5. Crystals with different inversion axes. Note that the symbol -2 (=m) is never used in practice.

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