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

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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:

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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

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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 = 4sin�/�, 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.

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

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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.

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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-1V (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

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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 program is 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 from PROFILE. 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. From this 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. How many 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

26

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 maximum to

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 maximum is

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(-C7x

2/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) 2C21/2/(H�1/2)(1+4C2x

2/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)x2cot�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.

34

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 from the 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 = (R2cos2� + 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 Vj2 / �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)/Yio1/2, 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

�k2=(M-1)kkS

2 (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=wi1/2(Yoi-Yi) (8.9)

The least-squares sum (Eq. 2.1) can then be written as Sy=�ai2. 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'=[�jaj2]+[�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-ti2)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-12+ai

2). If aiai-1>0 then

ti=[2(ai-12+ai

2)]1/2/{2+[2(ai-12+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.

46

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 new compounds 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; 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.

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.

52

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.

A-4

A.2 Unit cells The three-dimensional periodicity of crystals means that we can describe a crystal as being built from small boxes, or unit cells, with identical contents in identical orientations. By simple translations along any of the box edges, the same box will be found throughout the whole crystal. Although the periodicity of crystals is a fundamental property, the box itself is not. We may in principle choose any box as long as it does not violate the periodicity. Fig. A.6 illustrates different choices of unit cells covering the same area in a two-dimensional periodic lattice. The unit cells not only repeat the atomic arrangement in a crystal, it will also, properly chosen, repeat the symmetry properties of the atomic arrangement. Our choice of unit cell should also comply with the symmetry elements of the lattice. Fig. A.7 illustrates two cell choices in a two-dimensional lattice, both agreeing with the periodicity of the lattice. However, the lattice also has a two-fold symmetry, and to make these two-fold axes parallel to a cell edge, the only proper cell choice is the right one. A unit cell having lattice points in its corners only is called a primitive cell, while when containing more lattice points it called a centred cell. The proper cell choice in Fig. A.7 is thus the centred cell. Obviously, a proper unit cell is not necessarily the smallest repeated unit of a lattice, but rather the smallest repeated unit that also contain all the symmetry elements of the lattice. Combining the symmetry operators as described above and possible relationships between the lengths of the unit cell edges and the angles between them, we can divide the unit cells into seven crystal systems. Adding a minimum set of centred cells we get a total of fourteen Bravais lattices as illustrated in Fig. A.8 together with their restrictions in cell edges, a, b and c, and their angles, �, � and �. The allowed symmetry operations for the seven crystal systems combine into 32 point groups, which are also given in Fig. A.8. So far we have only viewed a lattice as a set of points, with which we mark the corners and centring of our unit cell. Except for the very simple structures, a unit cell will contain a set of atoms, ions and/or molecules. As a consequence of the periodicity, we only need to specify the contents of one unit cell to know the atomic arrangement in a complete crystal. Furthermore, if we specify the symmetry operations in our unit cell, we can restrict the number of atomic positions to those atomic sites not generated by the symmetry operators, the so called asymmetric unit. The unit cell also provide us with a convenient coordinate system, with which we can assign the coordinates. To become independent of the actual unit cell edges and angles, we use coordinates from zero to unity in the three directions of the cell edges, so called fractional coordinates. The x-coordinate will then run along the a-axis, y along b, and z along c. Note that, unless we are dealing with a cubic system, this type of coordinate systems will not be orthonormal. Coordinates outside the range zero to unity means that we have moved into another unit cell. By adding or subtracting any integer we can always transfer the coordinates into an arbitrary unit cell. If we consider the monoclinic system in Fig. A.9 (top) we have an object in a (fractional) coordinate x,y,z in all of the unit cells. If we place a two-fold rotation axis along the b-axes (Fig. A.9 middle), the objects in x,y,z will get symmetry equivalents in -x,y,-z (note the rotated object) in all of the cells. Now, when we look at all the objects, also in adjacent unit cells, we will find them pair-wise related by two-fold rotations. In Fig. A.9 (bottom) all these “extra” two-fold axes have been added. Note that none of these “extra” axes will generate any new objects. In fact they are not at all “extra”, but a mere consequence of the combination of a single symmetry operator and the periodicity of a crystal lattice. In an analogous way it can be shown that a mirror plane in the ac-plane through y=0 will be accompanied by another, parallel, mirror plane through y=½. We can now use our fractional coordinates to give a more mathematical and general formulation of centring and symmetry operations. Table A.1 gives the equivalent coordinates in cases of centring and Table A.2 for

Figure A.6. Different unit cell choises in a two-dimensional lattice.

Figure A.7. With a centred unit cell we may include more symmetry elements.

A-5

Figure A.8. The seven crystal systems, the fourteen Bravais lattices and the 32 point groups.

A-6

the different types of rotation and rotation-inversion axes. Table A.1. Equivalent positions generated by centred unit cells. _____________________________________________________________________________________________ Lattice Coordinates Lattice Coordinates _____________________________________________________________________________________________ P x, y, z I (body centring) x, y, z; x+½, y+½, z+½ A x, y, z; x, y+½, z+½ F (face centring) x, y, z; x+½, y+½, z; x+½, y, z+½; x, y+½, z+½ B x, y, z; x+½, y, z+½ R (rhombohedral) x, y, z; x+�, y+�, z+� C x, y, z; x+½, y+½, z x+�, y+�, z+� _____________________________________________________________________________________________ Table A.2. Equivalent positions generated by rotation and rotation-inversion axes. _____________________________________________________________________________________________ Operation Coordinates Operation Coordinates Direction _____________________________________________________________________________________________ 1 x,y,z -1 x,y,z; -x,-y,-z Independent 2 x,y,z; -x,y,-z m x,y,z; x,-y,z Along b 3 x,y,z; -y,x-y,z; y-x,-x,z -3 x,y,z; -y,x-y,z; y-x,-x,z; Along c -x,-y,-z; y,y-x,-z; x-y,x,-z 4 x,y,z; -x,-y,z; -4 x,y,z; -x,-y,z; Along c -y,x,z; y,-x,z y,-x,-z; -y,x,-z 6 x,y,z; -y,x-y,z; y-x,-x,z -6 x,y,z; -y,x-y,z; y-x,-x,z; Along c -x,-y,z; y,y-x,z; x-y,x,z x,y,-z; -y,x-y,-z; y-x,-x,-z _____________________________________________________________________________________________

Figure A.9. Fractional coordinates, symmetry and periodicity.

A-7

A.3 Translational symmetry When a unit cell has been fixed, there are more symmetry operations that become defined, namely those involving translations. These symmetries are of course present independent of our choice of the unit cell and they may conversely be used to verify a proper choice of unit cell. There are two types of symmetries involving translations: Glide planes and screw axes. A glide plane combines a translation with a mirror operation. We can think of a solution of “head and tail” molecules crystallizing. An effective way of packing will be as illustrated in Fig. A.10 (top). This arrangement can now be described as a glide plane. There are three types of glide planes: Axial glide planes ( a, b and c), diagonal glide planes (n) and "diamond" glide planes (d). The latter can only be present in I- or F-centred cells. Except for the symbol, a, b, c, n or d, it is necessary to specify the mirroring direction. For example, a b-glide may be mirrored perpendicular to the a- or c-axes. The coordinate transformations of the glide planes are given in Table A.3. An a-glide with the mirror plane perpendicular to the b-axis is illustrated in Fig. A.10 (bottom). Note that the periodicity automatically gives us an “extra” glide plane at y=½. Table A.3 Equivalent positions generated by glide planes __________________________________________________________________________________ a x, y, z + x+½, -y, z m�b or x+½, y, -z m�c b x, y, z + -x, y+½, z m�a or x, y+½, -z m�c c x, y, z + -x, y, z+½ m�a or x, -y, z+½ m�b n x, y, z + -x, y+½, z+½ m�a or x+½, -y, z+½ m�b or x+½, y+½, -z m�c d x, y, z + -x, y+¼, z+¼ m�a or x+¼, -y, z+¼ m�b or x+¼, y+¼, -z m�c __________________________________________________________________________________ Returning to the “head and tail” molecule there will be a problem if the molecules themselves not contain the mirror operation, either within each molecule or they are in a racemic mixture. A screw axis will then fulfil the needs. A screw axis combines a rotation with a translation along the rotation axis and thus do not require a change of absolute configuration. A mn screw axis means a rotation by 2�/m plus a translation by n/m along the rotation axis. The different types of rotation axes are compared in Fig. A.11(a)- (d). To obtain the transformed coordinates we can just add the consecutive translations to the coordinates generated by the simple rotation axes in Table A.2.

Figure A.10. Molecules arranged on a glide plane (top). An a-glide perpendicular to the b-axis (bottom). In addition to the glide plane at y=0, there will be an “extra” at y=½.

A-8

Figure A.11.d. Six-fold screw axes. A.4 Space groups The way the symmetry elements combine with each other in the fourteen Bravais lattices is limited to the 230 space groups as found by Federov and Schöenflies in 1890. The space groups are all described in The International Tables for Crystallography, Vol. A. Fig. A.12 shows two pages from International Tables, for space group no. 10, P2/m, that we will use as an example: Starting from top left, the table tells us that P2/m is space group number ten. It is monoclinic, the lattice is primitive, and the unique axis is b, i.e. the angle different from 90° is �. The full symmetry symbol is P 1 2/m 1, which means that the two-fold axis is along b, and the mirror plane is perpendicular to the b-axis. The symmetry operations are shown in three projections. We can recognize two-fold axes, mirror planes and, where the two-fold axes are intercepted by the mirror planes, inversion centres. The fourth picture shows how a molecule in a general position is transformed. An open circle denotes a molecule. When the circle is divided by a vertical line it specifies two molecules on top of each other. One of the molecules will then be a mirror image of the other, which is denoted by a comma. A plus or minus sign indicates if the molecule is above or below the ac-plane. Note that there are only four molecules inside the unit cell, the rest of the molecules indicated are formally in neighbouring unit cells. The symmetry operations are also expressed analytically on the right hand side. The first line starting with 4 o tells us how a general position x, y, z is transformed. There are three more positions generated by the symmetry operations to give four-fold general positions. The next lines indicate the transformations for atoms in special positions. In P2/m the special positions are on mirror planes (m), on the two-fold axes (2) and where the mirror planes and axes intercept (2/m). An atom in a mirror plane will only be affected by the two-fold axes, the mirror plane will transform the atom into itself. An atom on a two-fold axis will only be affected by the mirror planes. Positions on m or 2 are thus only two-fold positions (2 i - n). Atoms in the centres of symmetry (2/m) are unaffected by the symmetry operations, and are single-fold positions (1 a - h).

Figure A.11a. Two-fold screw axis.

Figure A.11.b. Three-fold screw axes.

Figure A.11.c. Four-fold screw axes.

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Figure A.12. Spacegroup P2/m from International Tables Vol. A.

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A.5 Miller indices A periodic lattice can be viewed as built from equi-distant planes. We can assign a set of indices to these planes based on our unit cell. These Miller indices, hkl, can be found in the following way:

1. Determine the intercept with each axis. If the plane intercepts at the origin, change the origin. 2. Take the reciprocals of the intercepts. If an axis is not intercepted by a plane, the corresponding index will be zero.

3. Clear fractions through multiplications with integers. Such a set of planes will be parallel, equi-distant and contain all of the lattice points. Some two-dimensional plane sets are shown in Fig. A.13. The distance between the planes in a plane set can for the general triclinic case be calculated from 1 / dhkl

2 = [h2sin2� / a2 + k2sin2� / b2 + l2sin2� / c2 + 2kl(cos�cos�-cos�) / (bc) + 2hl(cos�cos�-cos�) / (ac) + 2hk(cos�cos�-cos�) / (ab)] / [1-cos2� - cos2� - cos2� + 2cos�cos�cos�] (A.1) where a, b and c are the cell edges and �,� and � are the cell angles. In higher symmetries, for example in the orthorhombic case, Eq. A.1 simplifies to 1 / dhkl

2 = h2 / a2 + k2 / b2 + l2 / c2 (A.2) The crystal planes have great practical interests. For instance, natural crystal faces and cleavage planes are always parallel to such plane sets of low indices. As a rule of thumb the sum of indices of a natural crystal face is less or equal to six. Crystals for semiconductor and laser optics applications are always oriented and cut parallel to low index planes. As the distance between planes are in the same order of magnitude as X-ray wavelengths, they will give rise to interference when irradiated with X-rays, forming the basis for X-ray diffraction and this course. A.6 Reciprocal space If we imagine all possible planes in a crystal, i.e. (hkl) with � < h,k,l < �, these planes can be mapped as points in a three-dimensional lattice, Fig. A.14. We can further relate this new lattice to our normal crystal lattice through the vector relations: a* = b × c / Vcell b* = c × a / Vcell c* = a × b / Vcell (A.3) where Vcell is the unit cell volume. This lattice is referred to as the reciprocal lattice and spans the reciprocal space, while our crystal, or direct, lattice spans the direct space. is referred to as the direct space. The cross product means that a* will be normal to the direct bc-plane, b* is normal to the ac-plane and c* is normal to the ab-plane.

Figure A.13. Plane sets in a two-dimensional lattice (top) and in a three-dimensional cubic lattice (bottom).

Figure A.14. Reciprocal lattice.

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Figure A.15. The relationship between the direct and reciprocal unit cells in a triclinic (left) and a monoclinic (right) system. Conversely will the a-axis be normal to the b*c*-plane, the b-axis normal to the a*c*-plane and the c-axis normal to the a*b*-plane. We can obtain the reciprocal repeat distances in Å-1 units and the reciprocal cell angles from a* = bc sin� / V b* = ac sin� / V c* = ab sin� / V cos�* = (cos�cos� - cos�) / (sin�sin�) cos�* = (cos�cos� - cos�) / (sin�sin�) cos�* = (cos�cos� - cos�) / (sin�sin�) (A.4) In the orthogonal crystal systems, Eq. A.4 reduces to a* = 1/a, b* = 1/b, c* = 1/c and �*= �* = �* = 90°. One of the properties of a reciprocal lattice is that a distance from the origin to a point, hkl, is the inverse of the distance between corresponding planes in the direct lattice: dhkl* = 1 / dhkl. In Fig. A.16 (bottom), the length of the reciprocal lattice vector d32* is according to Pythagoras |d32*|2 = (3a*)2 + (2b*)2 (A.5) In an orthogonal system, where a* = 1/a and b* = 1/b, we get exactly the same expression as if we applied Eq. A.2 to the direct lattice in Fig. A.16 (top).

Figure A.16. Direct (top) and reciprocal (bottom) lattice with the 32 lattice plan marked.

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B-1

APPENDIX B

X-RAY DIFFRACTION Contents: B.1 X-rays 2 B.1.1 X-ray tubes 2 B.1.2 Synchrotron sources 4 B.2 X-ray diffraction geometry 6 B.2.1 Laue’s equations 6 B.2.2 Bragg’s law 7 B.2.3 The Ewald construction 7 B.3 Diffracted intensity 8 B.3.1 Scattering from a free electron 8 B.3.2 Scattering from an atom 9 B.3.3 Scattering from a unit cell 10 B.3.4 Scattering from a crystal 11 B.3.5 Anomalous dispersion 12 B.3.6 Thermal motions 13 B.3.7 The Lorentz factor 14 B.3.8 Absorption 14 B.3.9 X-ray intensities and the structure factor 15 B.4 The structure factor 16 B.5 The Fourier transform 21 B.5 Data collection principles 23 B.5.1 Single-crystal, monochromatic radiation 23 B.5.2 Single crystal, white radiation 24 B.5.3 Powder, monochromatic radiation 25 B.5.4 Powder, white radiation 26

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X-RAY DIFFRACTION

B.1 X-rays X-rays are electromagnetic radiation in a wavelength range between �-rays and UV, i.e. covering wavelengths between 0.1 and 100 Å, or in energy terms 124 to 0.124 keV. The wavelength range 0.5 to 2.5 Å is of particular interest as it corresponds to chemical bond distances and repeat distances in a crystal. A crystalline material will for these wavelengths act as a grating ideal for diffraction studies. X-rays are as most other things quantized. One quantum, a photon, has an energy E = h c / � (B.1) where h is Planks constant, c the speed of light and � the wavelength. We can rewrite Eq B.1 approximately as E = 12.4 / � or � = 12.4 / E (B.2) with E expressed in keV and � in Å. We all know from medical applications that X-rays have an ability to penetrate matter, especially soft tissues. On their way through our bodies they will interact with the tissues and be absorbed. Due to its high energy we can expect a wide spectrum of reactions to take place, from simple heating to ionization and formation of free radicals. The latter two processes are dangerous and may in serious cases lead to burns and possibly cancer. X-RAYS ARE ALWAYS A POTENTIAL HEALTH HAZARD According to Danish law, everyone working with X-rays shall have proper guidance and instructions on the safety rules that apply to the X-ray equipment. There are several ways of generating X-rays. We will consider here the two main methods: the X-ray tube; and the synchrotron. B.1.1 X-ray tubes A cross section of an X-ray tube is shown in Fig. B.1. The cathode filament (d) is heated by a current. Electrons from the cathode are accelerated by a voltage (5- 60 kV) between the cathode and the anode (g). The anode is made from a pure metal, commonly silver, molybdenum, copper, cobalt, iron or chromium. Copper and molybdenum are by far the most used materials. As the electrons hit the positively charged anode, their kinetic energy is given off by one of three different processes: 1. The larger part of the energy is transferred to thermal vibrations in the anode. It is therefore necessary to water cool the anode (Fig. B.1 f);

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2. The accelerated electrons can interact with the electrons in the anode material giving rise to a voltage dependent, continuous X-ray spectrum (Fig. B.2a). The shortest wavelength given off can be calculated from Eq. B.2; 3. If the kinetic energy is sufficient, the accelerated electrons may kick an inner shell electron out off the anode material. In Fig. B.3 the energy needed corresponds to Kabs. The empty position will rapidly be filled by successive in-jumping of outer electrons. As an outer electron jumps into an empty orbital it will emit characteristic radiation corresponding to the energy difference between the orbitals. In Fig. B.3 the emitted energies correspond to K�1 and K �2. This process gives rise to a line spectrum that will be overlaid the continuous X-ray spectrum (Fig. B.2b). As the orbital energies are different in different elements, each element will produce its own characteristic line spectrum. The characteristic wavelengths of the most common anode materials are listed in Table B.1.

Figure B.1. Cross section of conventional X-ray tube. (a) Metal block (b) X-rays (c) Vacuum (d) W filament (e) Glas tube (f) Cooling water (g) Anode (h) Electrons (i) High-voltage connection (j) Metal screen (k) Be window. The X-ray intensity that can be obtained from a conventional X-ray tube is limited by the cooling rate of the anode. If the anode instead of being a static block is rotated during operation, the heat load will be spread out around the anode and the intensity can be increased be at least a factor of five. These types of X-ray sources are called rotating anode generators. The X-ray spectrum will qualitatively be the same as from an X-ray tube.

Figure B.2a. Continuous X-ray spectra at different accelerating voltages.

Figure B.2b. Continuous X-ray spectra with overlaid characteristic peaks from Mo and Cu anods. The K� peaks are actually double peaks (c.f. Fig. B.3).

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Table B.1 Characteristic X-ray wavelengths (Å) of some common anode materials. K<��> is the weighted average of K��1 and K��2. ______________________________________________________________________________________________ Representation K�2 K�1 K<��> K�1 Kabs Transition K - LII K - LIII K - MIII K - � ______________________________________________________________________________________________ Ag 0.563775 0.559363 0.56083 0.49701 0.4858 Mo 0.713543 0.70926 0.71069 0.62099 0.6198 Cu 1.54433 1.54051 1.54178 1.39217 1.380 Co 1.79278 1.78892 1.79021 1.62075 1.608 Fe 1.93991 1.93597 1.93728 1.75653 1.743 Cr 2.29351 2.28962 2.29092 2.08480 2.070 ______________________________________________________________________________________________ B.1.2 Synchrotron sources A charged particle in an accelerating field will, according to classical electro-dynamics, emit electromagnetic radiation. As a consequence it will of course loose some of its kinetic energy. This phenomenon caused some problems in the early days, when people where trying to find a model for the atom. On an atomic level, the electrons are bound to orbitals that prohibits the classical behaviour. However, on a macroscopic level the phenomenon is real, and it is the working principle of a synchrotron. A simple synchrotron scheme is shown in Fig. B.4. Electrons are generated and accelerated by an accelerator (a) and enter the synchrotron through an injection magnet (b). They are forced into a closed orbit by strong electromagnets (c). Each time the electrons are subject to the centripetal acceleration from the magnets they will give off (synchrotron) radiation. If the electrons are relativistic, i.e. moving close to the speed of light, the radiation will come out in a narrow cone, tangential to the electron orbit. The energy lost due to radiation is restored in the radio frequency cavity (d). The electrons can be kept in orbit for many hours, but will eventually be lost due to scattering in the incomplete vacuum. Synchrotron radiation is said to be white radiation, i.e. it has a continuous spectrum, although it drops of rapidly at the high energy (short wavelength) side as illustrated in Fig. B.5a. The intensity and the lower wavelength limit depends on the electron energy and the magnetic field. One usually defines a critical wavelength, �c in Å, at which half the integrated energy is above and the other half is below �c = 18.6 / (B E2) (B.3) where the magnetic field, B, is expressed in T, the electron energy, E, in GeV. With 1 T bending magnets and an electron energy of 2.5 GeV the critical wavelength becomes 3 Å. That means that X-rays well below 1 Å will still

Figure B.3. Simplified energy diagram of an anode material. n = � corresponds to ionization.

Figure B.4. A simplified synchrotron source. (a) Electron accelerator (b) Injection magnet (c) Bending magnets (d) Radio frequent cavity (e) Synchrotron radiation.

B-5

have useful intensity. One can improve the performance of the synchrotron by introducing various magnetic lattices in the straight sections between the bending magnets. Most common are wigglers and ondulators. They both consist of a set of magnets that forces the electrons into a wave path. As a result the synchrotron intensity from each turn will add to the total intensity of the beam, Fig. B.5. With relatively strong magnetic fields the effect is a simple addition of intensities, a so called wiggler. With a weaker tunable magnetic field the beam from different bend may constructively interfere with photons from the other bends and thus create a huge intensity increase at the resonance wavelengths, a so called ondulator.

Figure B.5. The electron path in a wiggler or ondulator. If we compare the intensity from a synchrotron to our conventional X-ray tubes we find a difference of several orders of magnitude, Fig. B.5b. The other important advantage of synchrotron radiation is that it has a continuous spectrum. We are no longer limited to the line spectra of X-ray tubes, but we can tune to a desired wavelength. It is no exaggeration to say that X-ray synchrotrons have revolutionized X-ray diffraction. Are there no disadvantages? Indeed, there is one: The price; an X-ray tube costs about 20 000 DKR, a small synchrotron about 1000 000 000 DKR. Figure B.6. Wavelength distribution from synchrotron with different electron energy (left). Development of brilliance as a function of time (right).

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B.2 X-ray diffraction geometry When electromagnetic radiation is travelling through matter its electric component will interact with charged particles. If an atom is irradiated, its electrons will be accelerated by the electric field. In such an accelerating field, the electrons themselves will emit radiation in all directions (cf. Sect. B.3.1). This scattered radiation will have some important properties: Its wavelength (and energy) will be the same as for the incident radiation, in other words, the scattering is an elastic process; the scattered radiation will have a fixed phase difference of 180° with respect to the incident radiation; and the scattered radiation will be given off in random directions, independent of the direction of the incident radiation. What will be the effect of this scattering? We start by considering a one-dimensional array of point atoms that is hit simultaneously by radiation (straight lines in Fig. B.7). The electrons in all atoms will be accelerated and thereby radiate (semi-circles) simultaneously. Each atom will thus act as a radiation source with the important properties stated above. In Fig. B.7 the positions of the lines (incident radiation) and semi-circles (scattered radiation) represent the same phase of the waves, for instance their maxima. If we connect positions on the semi-circles with identical phases we can draw straight lines representing wave fronts. There will be a zeroth order front, a first order front, a second order front and so on. In the propagation directions of these waves we have constructive interference between the individual atomic waves. In all other directions the addition of waves will result in a reduced amplitude or even cancellation of the scattered waves. In other words, if we observed an irradiated one-dimensional crystal from the outside, we would see scattered radiation coming out with maximum intensity from these distinct directions only. B.2.1 Laue’s equations We may expand the reasoning by looking at an arbitrary incident direction (Fig. B.8). The geometrical condition for constructive interference can be formulated as a (cos��1 - cos��2) = h� (B.4) where a is the distance between atoms, ��1 is the angle to the incident radiation, �2 to the scattered radiation, � is the wavelength and h is any integer. With a given incident direction, there will only be scattered radiation coming out in certain directions corresponding to the different h:s. Due to the spherical symmetry of the atomic scattering, these directions will actually describe cones in three-dimensional space. One-dimensional structures are not very common, so in order to make something useful out of eq. B.4 we have to expand it into two and three dimensions. In a two-dimensional lattice we will get a (cos��1 - cos��2) = h� b (cos��1 - cos��2) = k� (B.5) where b is the repeat distance in the second direction. We can graphically represent Eq. B.5 as two sets cones, one along the a-axis one along the b-axis. The intersections will represent pairs of lines along which may observe the scattered radiation. In three dimensions, we will have a triple condition as

Figure B.7. A wave front is traveling perpendicular to a linear array of point atoms. The excited atoms will behave as independent wave generators creating new wave fronts of ordet zero, one, two etc.

Figure B.8. Diffraction from a one-dimensional lattice with spacing a.

B-7

a (cos��1 - cos��2) = h� b (cos��1 - cos��2) = k� c (cos��1 - cos��2) = l� (B.6) which are the famous Laue equations, derived by von Laue in 1912. The condition for constructive interference will be fulfilled for integer values of h, k and l only, but it will also require some restrictions on the direction of the incident radiation. ��1, ��1 and ��1 are no longer independent as they are defined with respect to our three-dimensional lattice. In other words, our crystal has to be aligned in certain directions with respect to the incident beam, if we want to observe the scattered, or diffracted, beam. B.2.2 Bragg’s law Shortly after von Laue had derived his equations, Bragg showed a simplified geometric construction to demonstrate the conditions for diffraction (Fig. B.9). The diffraction process is treated as reflection in different crystallographic planes. When the incoming and outgoing beams make the same angle to a crystal plane, we will have constructive interference as soon as the difference in path lengths between the upper and lower beams is equal to an integer number of wavelengths. For any given crystallographic plane (hkl), we can express Bragg's law as 2dhkl sinhkl = n� (B.7) where dhkl is the distance between the (hkl) planes. As demonstrated in Fig B.9 we can omit the integer, n, in Bragg's law by multiplying the plane indices by n. Although the derivation of the Laue equations is based on a true picture of the diffraction process they are rarely used. Instead Bragg's law has become the dominant way of describing and treating the diffraction conditions. We will in the following also rely on Bragg's law. B.2.3 The Ewald construction When we discussed the (direct) unit cell we also defined a reciprocal unit cell and a reciprocal lattice (2.6). We consider a two-dimensional structure with a correspondingly two-dimensional reciprocal lattice as shown in Fig B.10. We then draw a line representing the incident radiation through the origin of the reciprocal lattice, O. At a distance 1/� along the incident radiation we place our crystal, C. The vector (hk - O) is then the reciprocal vector r*, which is normal to the (direct) crystal planes hkl, and |r*| = 1/dhk. The scattering vector, s, is in the direction of the vector (hk - C) making an angle 2 to the incident radiation. Thus, as a reciprocal lattice point, hk, is on the circle centred at C and with the radius 1/�, the conditions for Bragg’s law will be fulfilled. This construction, named the Ewald construction after its inventor, tells us the direction of the diffracted beam and the orientation of the crystal. We can then easily imagine what will happen when we rotate the crystal. The reciprocal lattice points will rotate accordingly, but about its origin, O. As it rotates, new reciprocal lattice points will cross the Ewald circle, each time giving rise to diffraction (a reflection) in a new direction. Accidentally, more than one point can be on the circle simultaneously, but they will then diffract in different directions. Rotating the crystal, and thereby the reciprocal

Figure B.9. (a) First order reflection from the (100) plane. (b) and (c) Second order reflection from the (100) plane will be equivalent to a first order reflection from the (200) plane.

B-8

lattice, a complete turn, all reciprocal lattice points having 1/d less than 2/� will have diffracted. Those with larger reciprocal distances we will not be able to observe. However, by using a shorter wavelength, corresponding to a larger 1/�, we would be able to observe more reflections. The directions of the diffracted beam, 2, with respect to the incident beam will vary according to the d-spacing, dhkl. Thus, the directions of the diffracted beam contain information about the unit cell dimensions. Not only the directions but also the intensity of the diffracted beam will vary with reflection indices. The intensity variations contain information about the atomic arrangement within the unit cell, and will be the subject of the next two sections. B.3 Diffracted intensity The intensity of the diffracted beam will basically depend on the interactions between the X-rays and the electrons in our crystal. We can decompose it into four levels of interactions or contributions: 1. The interaction with a free electron. 2. The combined contribution from all electrons in a single atom. 3. The combined contribution from all atoms in a unit cell. 4. The combined contribution from all unit cells in a crystal. In addition, the diffracted intensities are affected by a few other processes that will be treated in the following subsections: 5. Anomalous dispersion. 6. Thermal motions. 7. A geometric factor, the so called Lorentz factor. 8. Absorption. B.3.1 Scattering from a free electron If we consider a free electron in a plane electromagnetic field (Fig. 3.10) it will be subject to an oscillating force F = e E0 exp(it) (B.8) where e is the electronic charge, E0 the amplitude of the incident radiation, its angular frequency and t the time. According to classical electro-dynamics, a charge particle in an accelerating field will itself be a source of electromagnetic radiation. The radiation will be emitted with a phase lag of 180° in all directions. The intensity of this emitted radiation is given by

Figure B.10. Ewald construction to demonstrate diffraction conditions. The incident radiation enters at X. The diffracted beam leaves at b along the scattering vector. The crystal is placed in C, and the origin of the reciprocal lattice is in O. The radius of the Ewald circle is 1/�. Any reciprocal lattice point hk on the Ewald circle will be in position for diffraction. The reciprocal lattice vector (O - hk) will be normal to the diffracting plane in the crystal.

B-9

I = I0 sin2� e4 / (m2r2c4) (B.9) I0 is the intensity of the incident radiation, m the electron mass, r the distance to the observer and c the speed of light. The angle � is formed between the direction of acceleration (along z in Fig. B.11) and the direction of observation. In effect sin� is a polarization term, we can observe only the component normal to the direction of propagation and parallel to the direction of observation. As a result the intensity distribution will be donut-shaped about the z-axis (Fig. B.11, bottom). If the incident radiation is not planarly polarized we may separate it into two planar components, one polarized in the xz-plane and one in the xy-plane, and then express the polarization term p() = sin2� as p() = K + (1-K) cos2(2) (B.10) where 2 is now the angle between the incident and scattered radiation, K is the fraction of polarization in the xz-plane and (1-K) in the xy-plane. From a conventional X-ray tube the radiation is unpolarized and K = ½. From a synchrotron the radiation is polarized in the plane of the ring and K will be unity if the scattering occurs perpendicular and zero if it occurs parallel to the ring plane. B.3.2 Scattering from an atom If we consider the scattered wave from a free electron we can express it as fe = f0 exp(i�e) (B.11) where f0 is given by �I from Eq. B.9 and includes the polarization function, and �e is the phase of the wave. Due to the phase, fe is a complex number, or, when viewed in an Argand diagram, a vector. If we place a free electron in the origin, O, and look at another electron in P (Fig. B.12) subject to the same incident radiation, we will find a difference in the traveling distances of the incident and scattered waves coming through the electron in the origin and the electron in P. The difference in traveling distance can, referring to Fig. B.12, be expressed as �d = r � s - r � s0 (B.12) where the scalar products with the unit vectors s and s0 represents the distances OB and AP respectively. The resulting phase difference between these scattered waves is then � � = r � (s - s0) 2�/� (B.13) where � is the wavelength of the incident and scattered waves. We can then express the scattered wave from the electron in P as fP = f0 exp(i�e -2�i/� r � (s - s0))

Figure B.11. (tpo) Scattering from a free electron. The electrical component of the incident radiation is in the xz-plane. The scattered radiation makes an angle � to this plane. (bottom) The intensity distribution about the z-axis due to polarization.

Figure B.12. Scattering from two electrons, one in O and one in P. The difference in traveling distance (O -B) - (A - P) result in a phase difference.

B-10

or, if we normalize to a free electron: fP = exp(-2�i/� r � (s - s0)) (B.14) To obtain the total scattering from an atom we now have to add the contributions from all electrons in an atom. As the electrons are not stationary, we have to average over time and their distributions (orbitals). The result, the atomic scattering factor, will be a function of the scattering angle, , and the wavelength, �, as shown in Fig. B.13. Due to the normalization, the value of the functions at sin/� = 0 will be equal to the number of electrons in the atom. B.3.3 Scattering from a unit cell Proceeding from an assembly of electrons to an assembly of atoms we can use an analogous reasoning. We can again use the scattering of a free electron as a reference, but the principle scatterer will now be the atoms. An atom j is placed at the fractional coordinates xj, yj, zj, corresponding to the vector rj (Fig. B.14). Normalizing to the free electron, the scattered wave can be written as (c.f. Eq. B.14) fj = fj exp(2�i/� r � (s - s0)) (B.15) where fj is the atomic scattering factor of the j:th atom. We can substitute the reciprocal vector, r*, for (s - s0)/�. The vector r* has a length of 2sin/�, and corresponds to the reciprocal lattice indices h, k and l. The scalar product r � r* then evaluates as (hxj + kyj + lzj), which we use to rewrite Eq. B.15 as fj = fj exp{2�i(hxj + kyj + lzj)} (B.16) where xj, yj and zj are the fractional coordinates of atom j. As fj is normalized to a unit scatterer in the origin, we can now simply add up the (complex) contributions from all N atoms in the unit cell to get the total scattering factor of a given reflection as Fhkl = �N fj exp{2�i(hxj + kyj + lzj)} (B.17) Fhkl is denoted the structure factor of reflection hkl, and contains information on the atomic coordinates, i.e. the structure. Note that we are summing complex numbers in Eq. B17. We can illustrate it as a vector summation in an Argand diagram (Fig. B.14). We will deal with this very important expression of the structure factor in more detail in Sect. B.4. One conclusion to draw from Eq. B.17 is that the structure factor of reflections hkl and -h-k-l will have the same absolute value, |Fhkl| = |F-h-k-l| and their phases will be related as � hkl = -� -h-k-l. Such a pair of reflections is called a Friedel pair. In the case of a centrosymmetric structure, each atom in x,y,z will have a centro-symmetrically related atom in -x,-y,-z and the resulting phase will be either 0 or �. The corresponding Friedel pairs will thus be identical, also with respect to their phases.

Figure B.13 Atomic scattering factors of Si, Na, O

and Li as a function of sin/�.

Figure B.14. Scattering from an atom in a unit cell.

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B.3.4 Scattering from a crystal When we derived the expression for the structure factor (Eq. B.17), we did only include the atoms in one unit cell. For a more complete treatment we should add up all atoms in a crystal. Introducing the lattice vector v = n1a + n2b + n3c, where the n:s are integers and a, b and c the unit cell vectors, Eq. B.15 may be rewritten as fj = fj exp{2�i( r + v) � r*} (B.18) Summing over all atoms in a crystal we get Fcryst = �Crystexp{2�i v � r*} Fhkl (B.19) The term in front of the structure factor, �Crystexp{2�i v � r*}, is denoted the interference function, J(r*). Assuming our crystal being built from N1, N 2 and N 3 unit cells in the a, b and c directions respectively, the interference function can be separated into three terms J(r*) = �n1exp{2�i n1 a � r*}�n2exp{2�i n2 b � r*}�n3exp{2�i n3 c � r*} (B.20) Each of these terms may be evaluated as a power series �N1exp{2�i n1 a � r*} = (1 - exp{2�i N1 a � r*}) / (1 - exp{2�i a � r*}) (B.21) To obtain the diffracted intensity from the structure factor expression must be squared. The interference function then becomes J(r*) = {sin2(� N1 a � r*) / sin2(� a � r*)} {sin2(� N2 b � r*) / sin2(� b � r*)} {sin2(� N3 c � r*) / sin2(� c � r*)} (B.22) A function sin2(� N x ) / sin2(� x) has maxima of N2 at integer values of x. This function normalized to N2 is illustrated in Fig. B.16. As N increases, the maxima become more and more narrow. When N approaches infinity, the function approaches the delta function, �(x-h), with integer h. If we consider a real crystal, the number of unit cells in any direction will be very large. A 0.1×0.1×0.1 mm3 crystal with a unit cell of 10×10×10 Å3 will have N1 = N2 = N3 = 10 5, and the interference function will consequently form very narrow peaks. The positions of the peaks corresponds to the reciprocal lattice points as a � r* = h, b � r* = k and c � r* = l. The interference function is thus an alternative way of demonstrating the geometrical conditions for diffraction, which directly corresponds to the Ewald construction (Sect. B.2.3). In-between the reciprocal lattice points the diffracted intensity will be practically zero. On the other hand, if we are dealing with powdered samples of for example clays and other layered materials, the thickness of the crystallites may be just a few unit cells in the c-direction. Consequently, the diffracted intensity will show an elongated distribution in the l-direction in reciprocal space, and reflections in or close to this direction will appear considerably broadened.

Figure B.15. Adding complex numbers as vectors in an Argand diagram.

Figure B.16. The function sin2(� N x ) / sin2(�x) / N2 for different N: Solid broad curve N = 2; broken line N = 5; and solid narrow curve N = 100.

B-12

B.3.5 Anomalous dispersion The derivation of the atomic form factor (Sect. B.3.2) was based on the assumption that an atom is composed of free electrons. However, in a real atom, each electron is associated with an orbital with a definite energy level. We can describe the true scattering situation as having an atom built from a set of (electronic) oscillators, each with a characteristic frequency n, under the influence of an oscillating field (the incident X-ray beam) with a frequency . The scattering power of the n:th electron can then be expressed as fn = 2 / (2 - n

2 - i�n) (B.23) where �n is the oscillator damping factor. If we separate this complex expression into its real and imaginary components and sum over all electrons in the atom we get f = f’ + � �n gn / (n

2 - 2) + i� [dg / d ]/ 2 (B.24) where f’ is the atomic scattering factor as derived in Sect. B.3.2, the second term is a real correction to the atomic scattering factor, and the third is an imaginary correction term. The g is referred to as the oscillator strength, and its derivative with respect to as the density of oscillator states. For daily use we may simplify Eq. B.24 to f = f’ + �f’ + i �f” (B.25) with �f’ and i �f” being the real and imaginary anomalous scattering corrections respectively. As the origin of the anomalous scattering, or anomalous dispersion, is absorption rather than scattering, these corrections are independent of scattering angle. On the other hand they depend heavily on the X-ray wavelength and the orbital energies of the atoms in the structure. In general, lighter atoms are much less affected than heavier atoms. The corrections may become appreciable when the X-ray energy is close to an ionization energy as illustrated for the d-elements and Cu radiation (� = 1.54 Å) in Fig. B.17. The way �f’ and i �f” add to f’ is shown with vectors in Figs B.18 (top) and (bottom) for a non-centrosymmetric and a centrosymmetric structure respectively. Note that the complex term always adds counter-clockwise as it causes a delay of the reemission of the beam. In the non-centric case, Friedel pairs will now have different structure factor amplitudes as well as phases. In the centrosymmetric case the Friedel pairs will be affected in the same way, but the phase of the resulting scattering factor will deviate from 0 or �. The anomalous dispersion has some important applications. We can use the difference between Friedel pairs in non-centrosymmetric structures to determine the absolute configuration. In a non-centrosymmetric structure we are left with an ambiguity: There are always two possible absolute configurations or enantiomorphs. When the anomalous dispersion effects are strong enough, the two forms will not fit equally well to the diffracted intensities due to the anomalous dispersion. By inverting the signs of the coordinates we can test which enantiomorph gives the best fit to our diffraction data. We can also use the anomalous dispersion close to an absorption edge as an additional contrast. If a specific element substitutes in small quantities we may still locate it if we compare data collected just above and just below an absorption edge of that element. The additional contrast can also be used in phase determinations in

Figure B.17. Anomalous dispersion corrections for the d-elements. Higher Z corresponds to larger n in Eq. B.24.

Figure B.18. Anomalous dispersion correction of Friedel pairs: (top) non-centric; (bottom) centric structure.

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particular for protein structure. The contrast is then used either to locate heavy atoms or to directly start the phasing process. B.3.6 Thermal motions An atom in a crystal is never at rest. Even at 0 K it will have so called zero point motions corresponding to its lowest vibrational state. If we assume atom j being displaced by uj from its position at rest, rj, we can write the structure factor expression as Fhkl = �j fj exp{2�i (rj + uj) � r*} = �j fj exp{2�i rj � r*} � exp{2�ih uj � r*} (B.26) As uj will vary from unit cell to unit cell as well as over time, we will only observe an average of u j, and we can rewrite Eq. B.26 as Fh = �j fj exp{2�i rj � r*} � exp{2�i<uj � r*>} (B.27) The second exponential term may be expanded in a series exp{2�i<uj � r*>} = 1 + 2�i<uj � r*> + ½(2�i<uj � r*>)2 + ...... (B.28) If the vibrations are symmetric the odd exponential terms will vanish and we will have exp{2�i<uj � r*>} = 1 - 2(�<uj � r*>)2 + ....... �� exp{-2�2 <uj � r*>} (B.29) which is the harmonic approximation to the thermal vibrations. If we introduce that |r*| = 2sin/�, and Bj = 8�2<uj

2> we will get fj

T = fj exp{-Bj sin2/�2} (B.30) where the exponential part is called the temperature factor and Bj [Å

2] the isotropic temperature factor coefficient. The effect of thermal motions on the atomic form factor is illustrated in Fig. B.19. In general, the thermal motions will not be equal in all directions, but will reflect the bonding conditions. We can take this anisotropy into account by expanding B into a 3×3 symmetric tensor fj

T = fj exp{-h2��11 - k2��22 - l

2��33 - 2hk��12 - 2hl��13 - 2kl��23} (B.31) where the ��ij:s are the anisotropic temperature factor coefficients. In the harmonic approximation an isotropic temperature factor means that the probability to find the atom will be constant on a spherical surface, while in the anisotropic case, the probability will be constant on the surface of an ellipsoid.

Figure B.19. The effect of thermal motions on the atomic form factor of Si. B is given in Å2.

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B.3.7 The Lorentz factor Referring to the two-dimensional Ewald construction (Fig. B.10), the condition for obtaining diffraction is that a reciprocal lattice point is on the Ewald circle. We may then observe a reflection. To get an arbitrary reciprocal lattice point with d* < 2/� to be on the circle we have to rotate the crystal in C (and thereby the reciprocal lattice about O). The reciprocal lattice points will then pass in and out of the circle. The total intensity we record when a reciprocal lattice point passes the circle will be proportional to the time it takes to cross it. At a given angular velocity, , a reciprocal lattice point with a long reciprocal lattice vector will move faster than one with a short, Fig. B.20. v = � d* = 2sin � (B.32) A reciprocal lattice point will always have a finite size. In addition we also have finite distribution of wavelengths in our X-ray beam, which we can describe as a finite thickness of the Ewald circle. Taken separately or together it means that a reciprocal lattice point have to travel a certain distance to ensure we capture all the diffracted intensity. This distance is proportional to cosine of the angle with which a point crosses the Ewald sphere. l �~ 1 / cos (B.33) Combining the traveling velocity and distance gives us a relative expression for the time to pass the Ewald circle, which is used to correct the recorded diffraction intensities. The correction factor is referred to as the Lorentz factor: L() = 1 / sin2 (B.34) B.3.8 Absorption When an X-ray beam travels a distance t through a crystal, its intensity, I0, will be reduced due to absorption as I = I0 � exp(-µt) (B.35) where µ is the linear absorption coefficient. The absorption will introduce systematic errors to our measured diffraction intensities that need to be corrected for. If we imagine a crystal in the shape of a thin plate, diffraction in a plane perpendicular and parallel to the crystal plate will by differently affected by absorption. Knowing the crystal dimensions, orientation and absorption coefficient we can calculate a correction for each reflection. From the composition of our crystal we can calculate the absorption coefficient from tabulated values of the elemental absorption factors, µmi, µ =�i �igiµmi (B.36) where �i is the density and gi is the mass fraction of element i. We may also use the X-ray cross sections, �i,

Figure B.20. Rotation about O will make a reciprocal lattice point cross the Ewald sphere.

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µ =�j�j / V (B.37) where V is the unit cell volume and the summation is over all atoms in the unit cell. From Eq. B.35 we can also compute the thickness necessary to reduce the X-ray intensity by a factor of two. Some examples are given in Table B.2. The linear absorption coefficient, µ, depends on the absorbing elements and the X-ray wavelength and is proportional to the imaginary part of the anomalous dispersion correction (Eq. B.24). As a function of the wavelength it shows characteristic jumps at X-ray energies corresponding to atomic ionization energies, which shows up as absorption edges as illustrated in Fig. B.21. The edges are named after the electron shell from where the electron is ejected (c.f. Fig. B.3 and Table B.1). When we move to a wavelength slightly longer than at an edge, the X-ray energy is not sufficient for ionization, and µ drops abruptly. An ionized atom will radiate in the same way as an anode material in an X-ray tube. This secondary radiation, or fluorescence, will increase the background for our diffracted intensity. Table B.2 Path lengths (mm) that will reduce the intensity by a factor of two in different absorbers. _________________________________________ � (Å) Air Al Cu _________________________________________ 2.0 284 0.025 0.0071 1.0 2292 0.17 0.0059 0.5 12200 1.3 0.040 _________________________________________ Figure B.22. The effect of a �-filter. Figure B.22. The effect of a � �-filter. Absorption edges are frequently used in connection with X-ray tubes as so called ��-filters. For instance is the K absorption edge of Ni in-between the K �� and K �� lines of Cu radiation. By simply placing a Ni foil in front of a Cu tube we will get a rather clean CuK�� radiation, Fig. B.22. In the same way a Zr foil can be used to filter Mo radiation. B.3.9 X-ray intensities and the structure factors We can now summarize sections B.3.1 - 8 by writing down an expression for the relationship between the measured diffraction intensity, Ihkl, and the structure factor, Fhkl: Ihkl = K � Ahkl � L() � p() �| �Fhkl|�

2 (B.38) where Ahkl is the absorption correction, L() is the Lorentz factor (Eq. B.34) and p() is the polarization

Figure B.21. The linear absorption coefficient as a function of wavelength.

B-16

correction (Eq. B.10). Fhkl will in this expression include also the temperature factor (Eq. B.30 or B.31) and the anomalous dispersion correction. K is a scale factor that depends on the incident intensity, crystal size etc. K is common to all reflections, while the other factors will depend on and/or hkl. When we measure a diffracted intensity we can only record the number of X-ray photons, not their phases. The scattering factor thus enters Eq. B.38 with its absolute value. The lost phase information is a serious problem known as the phase problem in crystallography. It is serious because the phases carry information about the atomic coordinates (cf. Eq. B.17). B.4 The structure factor We will in this section take a closer look at the expression for the structure factor as given in Eq. B.17, without taking thermal motions and anomalous dispersion into consideration: Fhkl = �j fj � exp{2�i(hxj + kyj + lzj)} As it stands, it is a sum over all atoms in the unit cell irrespective of any symmetry relations between the atoms. Clearly, symmetry relations between the atoms will affect the total sum and also relate reflections of different indices. We already observed in Sect B.3.3 that any structure will have |Fhkl| = |F-h-k-l| and their phases related as �hkl = -�-h-k-l. In a centrosymmetric structures, i.e. structures where an atom in xyz has a symmetry equivalent atom in -x-y-z, will in addition to |Fhkl| = |F-h-k-l| also have that �hkl = �-h-k-l = 0 or �. Each atom in such a pair will have the same fj, but their phases will have opposite signs. The sum of the two atomic scattering vectors will thus be a vector along the real axis in the complex plane, i.e. their combined phase must be either 0 or �. Other symmetry operations will give similar relationships: A mirror plane perpendicular to the b-axis will have atoms to by two related as xyx and x-yz. Rewriting the summation in Eq. B.17 into such pairs of atoms and comparing reflections of type hkl and h-kl (note the signs on the kyj term): Fhkl = �½j fj � [exp{2�i(hxj + kyj + lzj)}+ exp{2�i(hxj - kyj + lzj)}] and Fh-kl = �½j fj � [exp{2�i(hxj - kyj + lzj)}+ exp{2�i(hxj + kyj + lzj)}] (B.39) we find that �Fhkl� = �Fh-kl�. If we instead consider an a-glide perpendicular to the b-axis the expressions will look like: Fhkl = �½j fj � [exp{2�i(hxj + kyj + lzj)}+ exp{2�i(h(xj+½) - kyj + lzj)}] and Fh-kl = �½j fj � [exp{2�i(hxj - kyj + lzj)}+ exp{2�i(h(xj+½) + kyj + lzj)}] (B.40) Applying the rules for exponentials we can rewrite them as Fhkl = �½j fj � [exp{2�i(hxj + kyj + lzj)}+ exp{2�i(hxj - kyj + lzj)}�exp{�ih}] and Fh-kl = �½j fj � [exp{2�i(hxj - kyj + lzj)}+ exp{2�i(hxj + kyj + lzj)}�exp{�ih}] (B.41) Depending on h, the last exponential term is either 1 (h=2n) or -1 (h=2n+1). In the former case it is equivalent to Eq. B.39. In the latter case the phases will be reversed, but �Fhkl� = �Fh-kl� is still valid.

B-17

Similarly, a two-fold axis, or a two-fold screw axis along the b-axis will make �Fhkl� = �F-hk-l�. In other words: The symmetries in real space (our crystal) will be reflected in reciprocal space. However, as �Fhkl� = �F-h-k-l�, irrespective of crystal symmetry, the reciprocal space will in addition always have a centre of symmetry. If we take our 32 point groups, which can be either centrosymmetric or non-centrosymmetric, and add a centre of symmetry, they will reduce to the 11 Laue classes listed in Table B.3 together with the point groups and their symmetry equivalent reflections. Table B.3. The 11 Laue classes and their equivalent reflections ______________________________________________________________ Laue classes Point groups Equivalent reflections* ______________________________________________________________ -1 1, -1 hkl 2/m 2, m, 2/m hkl=h-kl mmm 222, mm2, mmm hkl=-hkl=h-kl=hk-l 4/m 4, -4, 4/m hkl=hk-l=k-hl=-khl 4/mmm 422, 4mm, -42m, hkl=-hkl=h-kl=hk-l= 4/mmm khl=-khl=k-hl=kh-l -3 3, -3 hkil# -3m 32, 3m, -3m hkil=hik-l# 6/m 6, -6, 6/m hkil=-hki-l# 6/mmm 622, 6mm, -6m2, 6/mmm hkil=hki-l=hikl=hik-l# m3 23, m3 hkl=-hkl=h-kl=hk-l# m3m 432, -43m, m3m hkl=-hkl=h-kl=hk-l= hlk=-hlk=h-lk=hl-k# ______________________________________________________________ Notes: * In addition are hkl = -h-k-l # In addition cyclic rotation of first three indices hki or hkl ______________________________________________________________ By observing the symmetry of a diffraction pattern (Fig. B.23) we can thus determine to which Laue class our crystal belongs. From a single-crystal data collection we can compute the diffraction pattern for such a comparison. However, more commonly one compares the intensities from the equivalent reflections according to Table B.3 and will that way get a statistical argument for the choice of Laue class. In structures with a translational symmetry we will have additional effects. We can illustrate it in the case of a C-centering. For each atom in x,y,z there will be another in x+½, y+½, z. Applying Eq. B.17 we will get Fhkl = �½N fj � [exp{2�i(hx + ky + lz)} + exp{2�i(h(x+½) + k(y+½) + lz)}] (B.42)

Figure B.23. A Laue diffractogram of zircon taken along the c-axis. The four-foldness and mirror planes indicate Laue class 4/mmm.

B-18

Rewriting Eq. B.42 using the rules of exponentials we will have Fhkl = �½N fj � exp{2�i((hx + ky + lz)} � [1 + exp{�i(h + k)}] (B.43) The factor [1 + exp{�i(h + k)}] can only take one out of two values: [1 + exp{�i(h + k)}] = 2 if h + k = 2n [1 + exp{�i(h + k)}] = 0 if h + k = 2n + 1 (B.44) When (h + k) is odd, the diffracted intensity will be zero. In effect, every second reflection will be absent (Fig. B.24). The conditions limiting possible reflections from centred lattices are summarized in Table B.4. The other types of translational symmetry, screw axes and glide planes will also give rise to symmetry extinct reflections. We can take as an example a c-glide perpendicular to the b-axis. For each atom x,y,z there will a symmetry related atom in x,-y,z+½. Inserting into Eq. B.17 gives Fhkl = �½N fj � [exp{2�i(hx + ky + lz)} + exp{2�i(hx - ky + l(z+½))}] (B.45) We can rearrange Eq. B.45 as Fhkl = �½N fj � exp{2�i(hx + lz)} � [exp{2�iky} + exp{2�i(- ky + ½l)}] (B.46) The expression [exp{2�iky} + exp{2�i(- ky + ½l)}] can be further reduced if we restrict our interest to the reflections having k = 0, i.e. the h0l reflections [1 + exp{�il}] = 2 if k = 0 and l = 2n [1 + exp{�il}] = 0 if k = 0 and l = 2n+1 (B.47) With a c-glide perpendicular to the a-axis, the l-condition will be the same, but would apply to another group of reflections, the 0kl reflections. An example of a diffractogram from a compound with screw axes is shown in Fig. B.25. If we record all possible reflections, we can locate the systematically extinct reflections and their indices. From Tables 3.4 we can determine the translational symmetry operations. When we then combine the translational symmetries with the Laue class, we will have a very limited set of possible space groups. In favorable cases we can uniquely determine the space group as shown in Table B.5.

Figure B.25. (left) Possible reflections from a primitive unit cell. The reciprocal unit cell is indicated with solid lines. (right) Possible reflections from the same unit cell, but now C-centred. The reciprocal unit cell is indicated with solid lines, a primitive unit cell is indicated with broken lines.

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Table B.4. Conditions limiting possible reflections Set of reflections Conditions Lattice P None I h + k + l = 2n C h + k = 2n A k + l = 2n B hkl h + l = 2n F h + k = 2n k + l = 2n h + l = 2n Robv -h + k + l = 2n Rrev h - k + l = 2n Screw-axis c 21, 42, 63 l = 2n 31, 32, 62, 64 00l l = 3n 41, 43 l = 4n 61, 65 l = 6n Screw-axis a 21, 42 h00 h = 2n 41, 43 h = 4n Screw-axis b 21, 42 0k0 k = 2n 41, 43 k = 4n _____________________________________________________________________ Screw-axis (110) 21 hh0 h = 2n

Set of reflections Conditions Glide-plane �c a h = 2n b hk0 k = 2n n h + k = 2n d h + k = 4n Glide-plane �a b k = 2n c 0kl l = 2n n k + l = 2n d k + l = 4n Glide-plane �b a h = 2n c h0l l = 2n n h + l = 2n d h + l = 4n Glide plane in (110) c l = 2n b hhl h = 2n n h + l = 2n d h + l = 4n

_____________________________________________________________________

B-20

Table B.5. The 230 space groups. Bold faces indicate space groups that can be uniquely determined from systematic absences and Laue class.

B-21

B.5 The Fourier transform. We will in this section take a different view of the structure factor as given in Eq. B.17: Fhkl = �j fj � exp{2�i(hxj + kyj + lzj)} According to Eq. B.17 the structure factor is the sum of the contributions from all atoms in the unit cell. In a more general way we can define the structure factor as a sum of contributions from all points in the cell: Fhkl = �j �(xi,yi,zi) � exp{2�i(hxj + kyj + lzj)} V x y z (B.48) where �(xi,yi,zi) is the electron density [electrons/Å3] at a point xi,yi,zi with extensions x y z. If now the points are evenly distributed and they become infinitely small we get: Fhkl = V �x�y�z �(x,y,z) � exp{2�i(hx + ky + lz)} dxdydz (B.49) where the integrations are from 0 to 1 in each coordinate. As the electron density is a periodic function in a crystal, it can be expressed in a Fourier series: �(x,y,z) = �h’�k’�l’ C(h’,k’l’) exp{2�i(h’x + k’y + l’z)} (B.50) with x, y and z in fractional coordinates and the summations from -� to +�. If we apply Eq. B.50 to Eq. B.49 we get: Fhkl = V �x�y�z �h’�k’�l’ C(h’,k’l’) � exp{2�i((h+h’)x + (k+k’)y + (l+l’)z)} dxdydz (B.51) We can change the order of integration and summation to get: Fhkl = V �h’ �k’�l’ C(h’,k’l’) �x exp{2�i((h+h’)x}dx �y exp{2�i((k+k’)y}dy �z exp{2�i((l+l’)z)}dz (B.52) Due to the integral terms Fhkl will non-zero contributions only if h’ = -h, k’ = -k and l’ = -l, which gives a solution: C(-h,-k,-l) = Fhkl / V (B.53) The expression for the electron density at a given point x,y,z in the unit cell can now be rewritten as �(x,y,z) = �h�k�l Fhkl exp{2�i(hx + ky + lz)} / V (B.54) or if we express Fhkl with its absolute value and phase:

B-22

�(x,y,z) = �h�k�l |Fhkl| exp{i�hkl} exp{2�i(hx + ky + lz)} / V (B.55) In other words: The structure factors in the reciprocal space are the Fourier transform of the electron density in the direct space and the reversed. Using Eq. B.55 we can then calculate the three-dimensional electron density in our structure if we know the structure factors. As we saw from Eq. B.38 we can get the absolute value of the structure factors from the measured diffraction intensities, but we can not get the corresponding phases! This is the famous phase problem. There is no direct way to solve this problem, but different indirect methods have over the year been developed to solve it. Another aspect of Eq. B.55 and Fourier transforms in general is the summation from -� to +�. That is of course practically impossible. We will always be limited to a finite number of measured reflections. Eq. B.55 will still give us an electron density, but the resolution will depend on the number of reflections as illustrated in Fig. B.26.

Figure B.26. Electron densities for C6H5F calculated with different number of reflections: (left) 980 reflections and maximum sin/� = 0.63 Å-1; (right) 99 reflections and maximum sin/� = 0.29 Å-1.

B-23

B.5 Data collection principles We should now possess some basic knowledge on the diffraction process. We will in this section graphically outline some general data collection principles. The data collection principles can be divided into four groups according to two choices: Firstly, as a sample we may choose a single-crystal, or a powder. Secondly, our X-rays may be monochromatic (single wavelength) or white (continuous wavelength distribution). B.5.1 Single-crystal, monochromatic radiation The principle is very well illustrated by the Ewald construction in Fig. B.10. In order to see more than one or two accidental reflections we need to rotate our crystal. The orientation of the crystal to an arbitrary diffraction plane requires three independent rotations. If we choose to record the diffracted intensities one-by-one with a scintillation detector, it too must be able to move (2). A device capable of doing this is called a four-circle diffractometer. With an area detector we can reduce the number of circles to two or three. The principle is illustrated in Fig. B.27.

Figure B.27. Single-crystal, monochromatic radiation. In the left detector setting, all reciprocal lattice points inside the broken circle can be captured on the area detector, A, by rotating the crystal a full turn. In the left detector setting, all reciprocal lattice points between the two broken circles can be captured by a full rotation of the crystal.

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B.5.2 Single crystal, white radiation With white radiation the Ewald construction looks a little different from Fig. B.10. Instead of one sphere we will have two limiting spheres, drawn with solid lines in Fig. B.28. The small sphere represents the maximum wavelength and the large sphere the minimum wavelength in our white beam. All reciprocal lattice points In-between these two spheres will fulfill the requirements for diffraction, and we can measure all of these reflections without rotating the crystal. One should note that the crystal position in the white beam case is not fixed as in the monochromatic Ewald construction. The position will always be at the centre of the Ewald sphere of the wavelength we consider. An Ewald sphere with an intermediate wavelength is drawn with a broken line to illustrate the principle. This data collection method is called the Laue method after its inventor and was used in the first diffraction experiment ever. The diffractogram in Fig. B.23 is a nice example of what can be obtained.

Figure B.28. Ewald construction for single-crystal, white beam case. The small solid circle represents the long wavelength limit, the large solid cirle the short wavelength limit. Note that the crystal is not in a fixed position, but will be at the center of a specific wavelength circle. The diffraction from a lattice point at an intermediate wavelength is illustrated with broken lines.

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B.5.3 Powder, monochromatic radiation A powder sample is ideally an infinite number of infinitely small crystals randomly oriented. An infinite number of randomly oriented crystal means that there will always be many crystals perfectly oriented for diffraction. In the Ewald construction we can illustrate it by rotating the reciprocal lattice about its origin. The result is a set of reciprocal lattice circles as seen in Fig. B.29. All lattice planes up to the 1/d limit determined by the Ewald sphere will simultaneously be in position for diffraction. In three dimensions the reciprocal lattice rings are actually spheres. Their intersections with the Ewald sphere form circles. The diffracted beams will thus appear as cones with an opening angle of 4. We can record the intensity either by stepping a scintillation detector in 2, using a position sensitive detector, or an area detector, Fig. B.30. Figure B.29. Ewald construction for powder sample, monochromatic beam case. Due to the random orientation of the crystallites, all possible orientations will make the reciprocal lattice points join into reciprocal lattice spheres. The reciprocal lattice spheres intersect the Ewald sphere in circles around the incoming beam (c.f. Fig. B.29).

Figure B.30. Scattering from a powder sample.

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B.5.4 Powder, white radiation When we combine the Ewald construction of Fig. B.29 with the white beam construction of Fig. B.28 one may intuitively think that we will achieve a total chaos of reflections. In fact, we would if we did not use a detector capable of resolving the X-ray wavelength, i.e. an energy dispersive detector. We can look at the Ewald construction in Fig. B.31 and imagine our detector at a fixed 2-angle indicated by broken lines. By resolving the energy of the diffracted beam at this fixed 2-angle, we are in effect sampling the reciprocal space along the solid line connecting the limiting circles in Fig. B.31. Figure B.31. Ewald construction for powder sample and white beam. With an enery dispersive detector at a fixed 2-angle, we are sampling the reciprocal space along the solid line. The fixed 2-angle is indicated as broken lines at the limiting Ewald spheres.

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APPENDIX C

COMPUTER PROGRAMS

CONTENTS C.1 WINPOW 2 C.2 WINPREP 5 C.3 WINEXT 6

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1 WINPOW WINPOW is essentially a Windows graphics user interface and a file handler supplied with dialog boxes to edit the various input parameters, present the results, calculate distances and angles, and some plot routines for display of diffraction patterns, profile functions, Fourier maps and, optionally, crystal structures. The essence of the file handling is the following: The input parameters and controls are given in an input records file, the project file with default extension ‘rec’. On opening a project this project file is copied to a parameter file with default extension ‘par’. The parameter file is used internally when editing, calculating distances, plotting, etc. The only part of WINPOW that uses the project file directly is the Rietveld program. In order for new parameters to have any effect on further refinements, the parameter file has to be copied back to the project file, i.e. the project has to be updated. The Rietveld program will as its main result produce a new parameter file. The new parameter file may then be examined and modified before a new update of the project file and a new refinement. It is possible and advisable to occasionally backup the parameter file. Default file names for output files are created from the project file name stripped of extension with a leading % and an extension according to its use. To install WINPOW on a Windows XP/2000 system, copy the winpownt.exe file to your favourite Rietveld refinement program directory, copy the winpow.ini file to the same directory. In the INI file you can instruct the program about text editor, structure plotting program, your preferred starting directory and window size etc. The Rietveld program is based on the LHPM1 program by R.J. Hill and C.J. Howard (ANSTO Report M122, Lucas Heights Research Laboratories, Australia, 1986). It has been extensively modified to allow for variable step data, Chebyshev polynomial background, restraints, split pseudo-Voigt profile function, asymmetry according to Finger, Cox and Jephcoat, etc. The distances and angle calculations program and the Fourier calculations and plot programs are based on DISTAN, FORDUP and FOPLOT by J.-O. Lundgren (Uppsala University Report No. UUIC-B13-4-05, Uppsala, Sweden, 1983). WINPOW also make use of a general text editor. Normally Notepad from Windows is sufficient. A more able editor is Write.exe, also in the Windows or WINNT directory. The full path to this or any other preferred editor should be given in the winpow.ini file. Another option is to include a crystal structure viewer. Both WebLabViewLite and Mercury are good choices. Mercury can be downloaded free of charge from Internet. The full path should be given in the winpow.ini file.

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WINPOW can be set up for multi pattern refinements. MLT - run a set of rec-files through WINPOW Create a text file containing a list of the names of the individual rec-files. Save the file using “mlt” as file extension. Open this mlt-file in WINPOW and start refine. The mlt-file can be edited using “edit all”. N.B. update will be necessary for the changes to take effect. After refinements the program will ask if you want to update the files. You can update individual files, all files or none. (Make sure the “Notes” window is active before responding.) Rerunning the mlt-file will now be from the updated rec-files. EXP - expansion of for example a temperature run With a large number of data sets with small structural changes in between, it is practical to be able to start the next refinement with parameters from the previous. Again, create a text file, now with the extension “exp”. It should contain for each refinement the rec-file name (even if it does not exist yet), the lines in the rec-file that need to be changed from the previous rec-file, plus one blank line: pz35b_450.rec TITL PbNb2O6 pz35b 450 PWDT pz35b_450.xy Starting WINPOW with such an exp-file will always start from the first rec-file and then apply the same set of parameters to the following rec-files. If your data contain abrupt changes, phase transitions etc, it is likely that this procedure will screw up. In this case the exp-file should be split up into several files, each for data sets containing reasonable changes.

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Just a few more unsorted hints:

-A refinement can be softly interrupted with a ctrl-i. In this way the cycle is finished normally, with normal updating of files, and all internal files are properly closed. -Dragging with the mouse you can zoom the pattern plot. Full pattern is obtained with F1. Using the arrow keys you can move an expanded window left or right, zoom out (up) or zoom in (down). -Special positions are not treated in any special way. However, occupancy factors can be reset to maximum values according to their site multiplicities. It is the user’s responsibility to set occupancy factors correct and not to refine special coordinates or restricted anisotropic thermal parameters. -Space group symbols are interpreted according to the International Tables Vol. I and Vol. A. When there is a choice of origin, the program will always assume the one with inversion at the origin. -Do not start refining occupancy factors from zero, or splitting of special positions from exactly the special position. The derivatives usually become zero, and the program stops. -There is a main output file %file.sum. It contains 2�-values, raw data, calculated data, refined background, weights, calculated intensities from individual phases, and Bragg markers from individual phases. This file can be imported and manipulated by Excel to produce a variety of plots. -Scattering factor types can now be directly entered on the atom input record as atomic numbers. The number of scattering sets. NSCAT should be put to zero in this case.

It is always a good idea to check the homepage http://struktur.kemi.dtu.dk/kenny/kshome.html under “Powder XRD” and “Programs” for the newest version of these programs. Bugs are fixed and features added continuously. A more detailed, although not fully updated, input instruction for WINPOW can be found there also.

Files generated by different program units in WINPOW

WINPOW - open project file, name.rec

name.rec is copied to %name.par

Distances and angles Output: %name.dis

Table – refinement summary Output: %name.tab

Fourier calculations Output: %name.prm, %name.plo

CIF – generate CIF Output: %name.cif

ATOMS – generate free format input for ATOMS

Output: %name.inp

Refine Output: %name.par, %name.out,

%name.sum, %name.fou

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2 WINPREP WINPREP is a powder diffraction data handler. It includes different features to modify data and to evaluate data. It has no written instruction (yet), so the user has to find out what can be done from the menus. For the Rietveld course we will use it mainly to look at powder patterns and to convert between different formats.

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3 WINEXT WINEXT is a program for extraction of data from *.rec files. It requires a *.mlt file, where the *.rec files are specified. The instructions below may seem strange, but can in many cases be copied from previous data extractions. 1. Output table file 2. Multiple records file 3. X-value desciptors SPX1,SPX2,NXTYP,NNX1,NNX2 (2A4,3I5) SPX1 - Record specifier SPX2 - Atom specifier (Type two records) NXTYP - Type multiplied by ten times the number of Y-values

1 - Record type 1, value no NNX1 2 - Record type 2 value no NNX1 3 - Value starting in position NNX1 and ending in position NNX2 4 - Unit cell volume

4. Y-value desciptor SPY1,SPY2,NYTYP,NNY1,NNY2 (2A4,3I5) SPY1 - Record specifier SPY2 - Atom specifier (Type two records) NYTYP –

1 - Record type 1, value no NNY1 2 - Record type 2 value no NNY1 3 - Value starting in position NNY1 and ending in position NNY2 4 - Unit cell volume 5 - Distance. Next two records

Atom (A4) Transformation U(3,3),T(3) (12F7.3)

6 - Angle. Next four records Atom (A4) Transformation U(3,3),T(3) (12F7.3) Atom (A4) Transformation U(3,3),T(3) (12F7.3)

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7 - Isotropic average of anisotropic thermal parameters 8 - Distance to first position 9 - Valence sum contribution. R0 and B given *1000 as

NY1 and NNY2 respectively. Next two records Atom (A4) Transformation U(3,3),T(3) (12F7.3)

10 - FWHM, average between NNY1 and NNY2 degrees in two-theta. 11 - Peak shape, average Lorentzian contribution between

NNY1 and NNY2 degrees in two-theta NNY1 - Value number NNY2 - 1 - Relative values

2 - Sum of previous values 3 - 9 Next phases >100 - Value * (NNY2/100)

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