COMPUTER ANALYSIS OF DIFFRACTION DATA Deane K. Smith Department of Geosciences The Pennsylvania State University INTRODUCTION Much of the theory of geometrical crystallography was developed in the 1800's, and much of the theory of diffrac tion was developed in the early 1900's shortly after von Laue and his associates proved, in 1912, that crystals dif fract X rays. As the computer impacted the field in the 1940's, experiments caught up with the theory, and the field advanced rapidly and is still advancing as our ability to process numbers improves. In the early days of diffraction experiments, computers were used for data interpretation only. Elaborate experiments were cumbersome, time consuming, and required direct super vision by the operator. Today the computer also runs the experiment, and in some cases it can process and interpret the data without any input from the operator. It is not easy to say when the computer had its biggest influence on our use of diffraction experiments, as it has played such a vital role in all aspects of the advancement of the science. This presentation will be concerned with only the data analysis aspects of computer applications. The modern auto mated powder diffractometer (APD) is controlled by a com puter which not only directs the instrument to collect the desired data, it usually tests the instrument to assure the operator that the system is working properly. These systems have data analysis programs as part of the software which will do most of the steps usually desired by the user. These programs are so integrated with the APD system that they are usually not separated from the package and distrib uted independently and are not available to the user who has not purchased the APD. However, many of the APD analysis programs have been derived from stand-alone routines written earlier for specific applications, and there are many other programs which have not been incorporated that are also available to the researcher who does not desire to be depen dent on a specific APD package. These latter routines are the subject of this presentation. References to programs will be handled in two ways. Where there is a literature reference describing the pro gram, there will be a year given in parentheses with the authors. Where there is only a program manual or no general documentation, the authors will be listed with no date. The list of program sources at the end of this Chapter should then be searched for the individual or company to contact for further information. If the program name is marked with (*) or (**), the program is known to require some payment for a copy which may range from supplying the medium to a commercial price. Even when requesting freeware, a reques tor should supply the medium and ask if a small remittance would be appropriate. This presentation will follow the sequence of analysis that would usually be followed in the interpretation of dif fraction data. Initially, the discussion will consider the many sources of crystallographic and diffraction information which are available in computer-readable form. Then it will consider programs to be used when nothing is known about the material under study followed by programs for specific pur poses when some information is known. Finally, it will con sider the routines used for preparing presentation material for publications. In each category, there will be no attempt to compare all the routines which are available. Only the general aspects of the particular analysis will be considered, pointing out differences in concepts rather than actual practice. Some sections will be deferred to other chapters in this volume where much more detail will be pre sented. The routines will be listed here only for complete ness. The reader must consider that this list of programs is not complete. The programs included are based in part on the author's experience with the many aspects of diffraction analysis; but many are the result of a request to over 100 mineralogists and diffractionists throughout the world. Without their help some programs would have been overlooked. There is still the probability that some significant pro grams have been missed, and an apology is due to the authors. This list has been supplied to the Commission on Powder Diffraction of the International Union of Crystallog raphy to initiate a program information center. Without the exchange of programs, this field would advance much more slowly than it currently does with the ready and willing distribution of routines that we enjoy today. During this discussion, the reader must keep in mind that there are three types of information in a diffraction pattern: the position, the intensity, and the shape of the diffraction peaks. The positions of the diffraction peaks are determined by the geometry of the crystal lattice, i.e., the size and shape of the unit cell. The intensities of the peaks in a pattern are related to the specific atoms in a crystal and their arrangement in the unit cell of the crys tal structure. The diffraction profile is related to the size and perfection of the crystallites under study and to various instrumental parameters. The peak positions are fairly easy to determine accurately by diffraction methods; whereas the intensities and the profiles are considerably more difficult to measure with accuracy. Consequently, most analytical methods emphasize the position variables (the d-spacings) and rely less on the other two parameters unless the intensities (e.g., quantitative analysis or structure refinement) or the profile parameters (e.g., crystallite size and strain) are specifically required. TYPES OF NUMERICAL DIFFRACTION DATA The raw diffraction data consist of a record of inten sity of diffraction versus diffraction angle. In the early days, this information was recorded on film or on a chart record from which the user measured the desired information. This information was usually the peak position which was converted to a d-value and the peak height which was consid ered the intensity. Peak positions of complex profiles were estimated by methods which ranged from "guesstimation" to graphical curve fitting. Intensities were derived from charts by planimetry, counting squares or cutting out the peak and weighing the cutout. Intensities from films were estimated visually or obtained by using calibrated film strips for matching. Only very elaborate experiments pro vided more precise d-values and integrated intensities. These data were adequate for the identification of phases and refinement of lattice geometry. Structure determination and the determination of structure-related properties require the resolution of all peaks and the determination of integrated intensities and details of the diffraction pro file. Extensive use of this information waited until APD's or densitometers were available to collect digitized dif fraction data which could be processed by computer methods. As early as 1839 with the Dana System of Mineralogy, 1922 with the Barker index of crystals, and Structurbericht (Niggli et al., 1931), crystallographers recognized the need to collect and categorize crystallographic data. Collecting data directly applicable to powder diffraction began in the 1930's, and these early efforts have lead to several cur rently available databases. The data banks will be described in terms of the type of information which is incorporated. Most diffraction pattern databases use dis crete d-I information. New ones are being developed which will employ the digitized trace information. Some of the databases use derived crystallographic data. The data analysis portion of this presentation will be divided into two sections. The first will deal with those programs which use only d-spacings to determine information on the geometrical lattice or those which use the d's with intensities for phase comparison. These programs are useful for any d-I data including those derived by APD systems. The second will examine those routines which use the digit ized trace as the starting point in the analysis. These routines may yield d-I information for use with the preced ing routines, or they may do more complex projects such as structure refinement, quantification, or determination of crystal perfection. REFERENCE NUMERIC DATABASES FOR POWDER DIFFRACTION ANALYSIS Many philosophers have described the impact of the com puter on society as being not the "Age of Computers" but rather the "Age of Communication and Information Transfer". Now that the cost of data storage is no longer the most expensive component in the computer system, vast amounts of data are being stored for direct access by the user. Actu ally, the two primary diffraction oriented databases both have their roots outside computer influences. The Powder Diffraction File (PDF) began in the late 1930's following the publication of the material which became the first set by the Dow Chemical Company (Hanawalt et al., 1938). Crys tal Data was first published by Donnay et al. (1954) as a Memoir of the Geological Society of America. Both projects have continued; the PDF under the auspices of what is now the JCPDS-International Centre for Diffraction Data (ICDD) and Crystal Data under what is now the National Institute for Science and Technology, formerly the National Bureau of Standards. The PDF derives its financial support entirely from the sales of the PDF. Crystal Data is partly supported by sales and partly by funding from the U. S. Government. Detailed information on the present status of the many crys tallographic databases may be found in Allen et al. (1987). The Powder Diffraction File The Powder Diffraction File (PDF) contains diffraction data primarily in the form of tables of d's and I's coupled with derived crystallographic information and documentation on the sample used, the experimental conditions, and on the source of information (Jenkins and Smith, 1987). Much of the information has been derived from the open literature and reviewed and edited by a group of experts in diffraction analysis. The literature information has been augmented by experimental projects sponsored by the ICDD to collect accu rate data on the most important compounds and by calculation of theoretical powder diffraction patterns using crystal structures determined by single-crystal methods. The edit ing procedure selects the best data for each compound where there are multiple data sets reported; thus, the PDF is con tinually being revised to keep it as accurate and as current as possible. The present PDF contains data for over 50,000 X-ray diffraction patterns. Minerals are carefully edited for conformity with the nomenclature of the International Mineralogical Association Commission on New Minerals and Mineral Names, and indexes are provided for both mineral names and crystal-chemical mineral groups. The PDF was initially published on 3"x5" cards, and this format is still the image used for data presentation in all hard copy forms of the PDF currently produced. All informa tion on these card images is now in computer-readable form, and there are two products, known as PDF-1 and PDF-2, being distributed by the ICDD on various computer media. A third product, presently designated PDF-3, containing the full diffraction data trace along with all the derived and docu menting information is under development. PDF-1** The first computer-readable version of the PDF contained only the d's, I's, the chemical formula and name, and the PDF number. These data were used by the developers of the phase-identification programs which compared the d-I sets to experimental data. These SEARCH/MATCH programs will be described in the section on the analysis of d-I data. The PDF-1 version was the basis for on-line search services as well as being distributed on magnetic tape for installa tion at individual computer facilities. Currently, PDF-1 is the basis for most APD search packages. There are several routines available to convert the d-I data into a graphical image for visual comparison to the experimental data. Such simulations allow the user to view the subtleties of the "pattern" as a whole and usually assist the user in confirming the identification of the sample under study. Simulations may use single bars, iso sceles triangles, or analytical profiles to represent each diffraction peak. Simulations may be made to any scale on either a graphics terminal or a hard-copy device and have even been made on transparent bases for direct comparisons with diffractometer charts. Film images have also been sim ulated, complete with the line density and curvature charac teristic of the Debye-Scherrer pattern. Some routines even allow the user to combine several single patterns to simu late a mixture. Programs which provide simulations include: DISPLAY Goehner and Garbauskas (1983) SIMUL/COMBIN* Smith PEAK** Sonnefeld and Visser (1975) PPDP* Okamoto and Kawahara XRAYPLOT* Canfield DEBYE* Blanchard DEBYE* Millidge DEBYE Bideaux DEBYE Morton PDF-2** The weakness of the PDF-1 version is that it only contains the key data for identification but none of the supporting information. Supporting information must be obtained from a hard copy of the PDF. In the days of lim ited storage on the computer, this situation was the only reasonable alternative. With the advent of inexpensive sto rage, it is now feasible to have all the data in computer- readable form. PDF-2 contains all the information which appears on a PDF card image, so that the user can display these data on the computer screen after determining the desired phase. Not only are the crystal unit cell and the Miller indices available, the symmetry, some physical prop erty information, and the reference are also accessible. Most of the information can be directly searched to locate compounds by means other than through the d-I search proce dures. PDF-2 is available from the ICDD on several computer media. Probably the most interesting medium to the indepen dent user is the CD-ROM, which can be installed on most PC's and on the microVAX# series of computers. This medium is here to stay, and it will replace the hard-copy forms of the PDF in most laboratories in the not-too-distant future. The CD-ROM does not replace PDF-1 as a search database because data retrieval from the CD-ROM is slow, but it can provide all the necessary d-I information. It is ideal, though, for other types of searches which index on specific data. ------------------------------------------------------------ # microVAX is a trademark of Digital Equipment Corporation ------------------------------------------------------------ Searches by chemistry, symmetry, color, etc. are feasible and provide many new ways of using the PDF effectively. Perhaps the most important information which can be located is the reference to the source of the information for a given compound. The PDF is an excellent entry into the lit erature. There are two major CD-ROM-directed programs in the pub lic domain which are available through the ICDD. These pro grams are by Holomany and Jenkins and by Toby, Harlow and Holomany (1989). Many commercial systems also exist inclu ding Philips; Materials Data, Inc.; Socabim; INEL; and A. Wassermann. PDF-3** With the growing importance of utilizing the full digitized diffraction trace, the ICDD is researching the feasibility of providing a database of diffraction pat terns. The value of such a database will become clear in the discussions below. PDF-3 is presently envisioned to contain all the information in PDF-2 along with the digit ized trace. The current storage capabilities are no longer a limitation. Once a standard file structure which meets the needs of the user is agreed upon, a pilot project will begin. One of the problems is setting up a universal file transfer system, and considerable attention is being given to JCAMP-DX (McDonald, 1988). This program was devised by the optical spectroscopists for establishing communication between instruments and among users. Prototype PDF-3 type databases are now in use for some of the programs to be described in later sections. NIST Crystal Data** The Crystal Data File (CDF) is a compendium of informa tion on crystalline materials for which the crystal unit cell is reported (Stalick and Mighell, 1986; Mighell et al., 1987). Along with the unit cell, many other properties of the crystal are reported including the chemistry, status of the crystal structure knowledge, and the source of the information. The philosophy of editing data for the CDF is different from that used for the PDF in that all literature is retained either as an independent entry or through cross referencing from related compounds. Consequently, the CDF is useful for locating literature references as well as for identifying and characterizing materials. The CDF currently has over 136,000 entries and is available in two forms, either on magnetic tape or on CD-ROM. Both are available from the ICDD. To use the CDF effectively with powder diffraction information, one must determine the unit cell from the d-spacings. With these data and a knowledge of the rules for defining cells, one can search the CDF for possible matching cells without regard to any chemical information. Either the compound of interest or an isostructural compound might be located. These topics are discussed in the section on "Unit-cell Identification". NIST/Sandia/ICDD Electron Diffraction Database The Electron Diffraction Database (EDD) is a new data base which has been derived from the PDF and CDF specifi cally for use with electron diffraction data (Carr et al., 1989). By using the crystal cell data in both the PDF and the CDF, d-spacings have been calculated for all the com pounds present. In electron diffraction, the intensities are not as useful as in X-ray diffraction because of sensi tivity to the instrument conditions and multiple diffrac tion, so the emphasis for identification is placed on the d-spacings of the first 10 lines along with chemical infor mation. One advantage of electron diffraction is that the region of the sample producing the pattern is more likely to be single phase than for the larger sample used in X-ray diffraction. The main responsibility for developing this database is with the Crystal Data group at the National Institute for Science and Technology. Details on the use of this database will be given in the section on "Unit-cell Identification". There are over 70,000 entries in this database. FIZ-4 Inorganic Crystal Structure Database The Inorganic Crystal Structure Database (ICSD) is a product of earlier independent databases developed by G. Bergerhoff and I. D. Brown (Bergerhoff and Brown, 1987). It is presently based at the Fachsinformationzentrum-4 in Karlsruhe in conjunction with the Gmelin Institute in Frank furt, FRG. This database contains over 24,000 entries of crystal structure information for inorganic compounds other than metals. In addition to the crystal cell and chemistry, all the atom locations are given. Thus, from this informa tion, it is possible to calculate the powder diffraction pattern which can then be compared with experimental data. Many programs for calculating patterns will be described in the section on "Crystal Structure Analysis". This file is also an entry into the literature. At present this file is available only through on-line subscription, but it will soon be available on CD-ROM also. Programs are available to search on various names and chemistry, and it is possible to display the structures graphically. NRCC Metals Structure Database** The Metals Structure Database (MSD) is similar to the ICSD in purpose (Rodgers and Wood, 1987). It contains the same type of information as is found in the ICSD except for metals and their corrosion products. It is based at the National Research Council Canada in Ottawa. It augments the information in the ICSD, so between them, the mineralog ist will find most of the minerals whose structures are known. It also provides data for the calculation of powder diffraction patterns. The MSD also may soon be available on CD-ROM. Mineral Data Databases** There are many new PC-based databases whose goal is to provide complete descriptive information for all known min erals. MINERAL by Nichols and Nickel (ALEPH) is derived from the Fleischer (1983) compendium and contains crystal information along with other information. This file pro vides an entry to the PDF by references to the PDF file num bers, and the crystal cell information is included so the user can calculate d-spacings. Many other such databases are appearing now that PC systems are so common. ANALYSIS OF d-I DATA There are many routines which use discrete d-I data for various purposes. The d-I data may be determined by the diffractionist from the film or chart or provided by the APD. Prior to use for interpretive purposes, the data should be calibrated against an internal standard, if used, or by external calibration. Then the data are ready for study. By far the most common use of powder diffraction data is the confirmation of materials or the identification of unknowns. Identification usually involves the interrogation of the PDF database using experimental d's and I's for com parisons. Such programs are described in the section on "Identification of Mineral Phases". If identification is not successful by this approach, the d's may be used in an indexing program to attempt to find a unit cell which is compatible with the observed d-spacings using one of the routines described in the section on "Indexing of Unknown d-spacing Data". A successful indexing may provide a cell which can be matched against the CDF database for an identi fication. Cell reduction programs may be necessary to locate a match. There are other applications for discrete d-I data. Accurate d-values can be refined to provide accurate unit cell parameters which are sensitive to subtle differences in composition or structural ordering, so cell refinements are often used in the studies of materials as described in the section on "Unit-cell refinements". Discrete intensity data may be used for applications such as quantitative phase ana lysis, texture analysis, and crystal structure determination as discussed in succeeding sections. Processing of d-I Data Sets In order to achieve accuracy with diffraction data, it is necessary to eliminate or minimize systematic errors in the experimental data. There are several important steps in achieving accurate data, the most important of which is alignment of the instrument. Analysis of the systematic errors is then possible. By far the most common method of reducing systematic errors in experimental data is to include a known phase in the sample and to use the known phase to calibrate the measurement scale for the data. Com mon calibrators are Si, W, Ag, CaF2, MgAl2O4, SiO2, fluorophlogopite, and LaB6 (Hubbard, 1980, 1983a,b). Si, fluorophlogopite, and LaB6 are available from the Office of Standard Reference Materials (OSRM), National Institute of Standards and Technology in Washington, DC. Adding an internal-standard with known lattice parameters allows for the correction of all the systematic errors at the same time. There are several routines available that make this correction: INTCAL Snyder PPLP Gabe et al. CALIBR Hubbard These programs use the precise angle values from the care fully calibrated standards to correct the angular diffracto metric measurements prior to the calculation of the d-spac ings. Careful calibration can provide data accurate to 0.005 degrees 2theta. Some comments on the preparation and use of internal standards are appropriate here. For internal standards other than those available from the OSRM, the user will usu ally have to calibrate their own supply against one of these certified standards. Quartz is nearly ubiquitous in most geological samples, and its unit cell varies little from sample to sample. The variation is less than the accuracy necessary in many experiments, so it may be employed without calibration in many studies. Spinel is easily made by fir ing a stoichiometric mixture of components to 1400-1600C with regrindings and pelletizing. Do not use a commercial ly-grown single-crystal boule as a source of spinel because they are always off stoichiometry. Chemically pure CaF2 can be fired at 1000C to improve crystallinity. Corundum, rutile, zincite, and some other commonly used standards are usually better for intensity references rather than as d-spacing standards because they usually have small crystal lite sizes and exhibit line broadening. Any standard pre pared by the user should be calibrated by preparing a mix ture with one of the certified standards, and refining the corrected d-spacing measurements using one of the routines mentioned in the section on unit-cell refinements. Identification of Mineral Phases Diffraction data are not a collection of independent d's and I's but a very systematic set of interrelated values which reflect the lattice and the crystal structure of the material. Consequently, the set of d's and I's are a pat tern of values characteristic of the material. Thus, it is possible to use this pattern of data as a means of identifi cation, and many routines have been developed to achieve this end. Most of these routines were developed on the large mainframe computers, and some have seen over 20 years of development (Johnson and Vand, 1967; Frevel et al., 1976; Edmonds, 1980). Now, there are many routines on PC-based systems. The early limitations on the developments were the storage restrictions which prevented the database being directly accessible, and magnetic tapes were used to run through the database. Very ingenious search schemes were employed to locate candidate answers. Direct access to the database is now the norm, and the search schemes are much more straightforward. PC-based search programs are also now feasible, and several are available. The philosophy of identification of unknowns using a database of known phases is to ask the question whether the pattern of the known is present in the pattern of the unknown. The diffraction pattern of a mixture is a compos ite of the patterns of each phase in the mixture, so the approach is to determine a figure-of-merit for each known and then to rank the known phases for user interpretation. The database of knowns may be prescreened using chemical information or other information such as the knowledge that the specimen under study is a mineral. Prescreening creates a much smaller set of knowns which needs to be examined. The search process then takes considerably less time to execute, and the results are usually much easier to inter pret. Although the computer is a tireless comparer of numbers, the results are only as good as the data allow and are highly dependent on the algorithm used. The major problem in searches is the degree of accuracy of both the unknown and the known data, and how well the known data represent the sample under study. Mineral identification can be very difficult when the mineral exhibits extensive solid solu tion. Ionic substitutions can alter both the lattice dimen sions and the characteristic intensities of the mineral, and it is unlikely that an exact match is present in the refer ence database for such minerals. Thus, it is usually up to the user to recognize the probable answer from the short list of possible phases provided by the algorithm. Logic and an understanding of the geologic environment are often the key to selecting the right answer. Diffraction patterns of mixtures are generally more dif ficult to unravel than those for single phases. The com puter can be very useful for mixtures, but it still has lim itations where the quantity of the phase is small and its pattern is only weakly represented in the experimental data. Logic, additional information such as chemical or optical data, perseverance, and luck are needed to identify such phases. Mini and Main Frame Routines All of the early search programs were developed on large computers. Most of these programs interrogate the PDF as the primary database. Most also modify the primary data to structure the data files for efficiency according to the algorithm used. Consequently, it is necessary to acquire these files or the ability to build them when implementing one of the programs. Because the PDF is a private database, it is also necessary to obtain lease permission from the ICDD prior to using it for identification. Mini and main frame programs include: SEARCH/MATCH GJohnson and Vand (1967) PDIDENT/PDMATCH Goehner and Garbauskas (1983a,b) XRDQUAL Clayton SEARCH Toby et al. (1989) FSRCH Carr et al. (1986) ZRD-SEARCH Frevel et al. (1976) FAZAN Burova and Schedrin SEARCH/MATCH Lin et al. (1983) SEARCH/MATCH Marquart et al. (1979) Some of these routines are available from ICDD, but most would have to be obtained from the authors. PC-Based Search Routines All of the PC-based search programs are commercial. They follow similar philosophies to the mini and main frame programs which are now possible with new data storage devices designed for PC's. These pro grams include: U-PDSM** Marquart (1986) Micro-ID** QJohnson SEARCH** O'Connor These programs must be obtained from the authors. Indexing of d-Spacing Data for an Unknown Where one has absolutely no knowledge of any crystallo graphic information on a material and is unable to identify the phase using the search procedures mentioned above, the next step is to try to determine a cell which will explain the d-spacing data observed. This procedure requires the material to be single phase and the experimental data to be very accurate. The procedure is usually known as 'indexing a pattern'. Indexing procedures use only the position variables of the pattern and try to find a solution to the relations: d(hkl) = f(h,k,l,a,b,c,(8Ma,b,c(8U) where h, k, and l are the Miller indices and a, b, c, (8Ma, b, c(8U are the parameters which describe the lattice. The form of the equation in real crystal space is very complex for the triclinic crystal system and quite simple for the cubic system where it takes the form: 1 h2 + k2 + l2 Q(hkl) = ------- = ---------------. d2(hkl) a2 Most indexing procedures use the equation in reciprocal crystal space which takes the form: d*2(hkl) = (h2 + k2 + l2)a*2 in the cubic system and: d*2(hkl) = h2a*2 + k2b*2 + l2c*2 + hka*b*cos(8Mc(8U* + hka*c*cos(8Mb(8U* + klb*c*cos(8Ma(8U*, or Q = h2A + k2B + l2C + hkD + hlE + klF in the general case of the triclinic system. This approach linearizes the problem to determining six unknowns, A, B, C, D, E, and F, that have integer coeffi cients in every d-spacing equation. Finding the proper val ues for the lattice parameters so that every observed d-spacing satisfies a particular combination of Miller indices is the goal of indexing. It is not easy even for the cubic system, but it is very difficult for the tri clinic system. Because one knows nothing about the mate rial, one does not even know which form of the equation to use. There are two general approaches to indexing (Shirley, 1980). One can assume in the beginning that the material is cubic and increase the a value until a solution is found to the above equation. That such a solution can be found is not proof of success, e.g. if the a value is ridicu lously large, then it is apparent that it is not accept able. One then tries the equations for hexagonal and tetragonal trying to determine reasonable values for a and c. Beyond orthorhombic, this approach is difficult even for a computer. This approach has been termed exhaustive indexing or trial and error. The other approach is analytical indexing where the crystal is assumed to be triclinic, and a final solution is approached by finding sub-solutions called zones using part of the data and then trying combinations of the sub-solu tions to find the full solution. This method is known as the ITO method (Ito, 1950; deWolff, 1957, 1962). Once a lattice is found which satisfies the experimental d-spacing data, the geometry of the lattice is examined for possible higher metric symmetry (International Tables, Vol. I, pp. 530-533, 1965). Both of these approaches require very accurate d-spac ing data. The smaller the errors, the easier it is to test solutions because there are often missing data points due to intensity extinctions related to the symmetry or the structural arrangement or due to lack of resolution of the d-spacings themselves. The earliest approaches were of the exhaustive type and were done by graphical fitting or numerical table fitting. One of the earliest computer rou tines was by Goebel and Wilson (1965) who programmed the cubic and hexagonal/tetragonal equations for trial and error testing. The method used the Table of Quadratic Forms (International Tables Vol. II, pp. 124-146, 1965) and assumed the first observed d-spacing was among the low est allowed Miller indices. The method was extended to the orthorhombic system by Louer and Louer (1972) and to mono clinic by Smith and Kahara (1975). Related routines were developed by Louer and Vargas (1982), Shirley and Louer (1978), Werner (1964), Werner et al. (1985), and Taupin (1968); and routines by these latter authors are commonly used today. The ITO analytical approach was used by Visser (1969). This routine is the basis of other derivative programs available today. Whereas the exhaustive approaches are not limited by data sets with few values in the high symmetry cases, the analytical approach is. At least 20 data points are required to be successful and more are desirable. How ever, it is usually not advisable to use too many data points because it becomes harder to assign hkl's to spe cific points in the high-angle part of the data set due to the larger number of acceptable assignments. The practical limit is around 40. Indexing programs which were identified include: CUBIC/HEX-TET Goebel/Wilson (1965) CUBIC Snyder ITO Visser (1969) TREOR Werner (1964) DICVOL Louer and Louer (1972) POWDER Taupin (1968) CELL Takeugi, et al. KOHL Kohlbeck and Hoerl (1976) INDEXING Paszkowicz (1989) AIDED Setten and Setten (1979) PC versions include: PC-ITO* Visser UNITCELL Garvey (1986a) Micro-INDEX** QJohnson and Snyder Indexing programs usually provide the user with several answers and a figure-of-merit which is based on the lattice fit. It is up to the user to decide on the best answer. There are two figures-of-merit used: M20, deWolff (1968, 1972) and FN, Smith and Snyder (1979). Q20 M20 = ------------, 2|(8MD(8UQ|N20 where Q20 is the Q value for the 20th observed line and N20 is the number of possible lines which could be observed out to Q20. 1 Nobs FN = -------.--------. |(8MD(8U2O| Nposs where Nposs is the number of possible lines out to the Nth observed line. There is some controversy concerning which of these proposed figures-of-merit is the most appro priate. Shirley (1980) claims that M20 is more sensitive to the correctness of indexing and FN is more a measure of the accuracy of the diffraction data. Both figures-of- merit are now included in all PDF data sets which are indexed. Any lattice with an acceptable figure-of-merit needs to be examined further to determine if it explains all the data including physical property measurements on the mate rial. Several suggested solutions may actually be the same because there are different ways to describe the same lat tice. At this stage it is necessary to reduce the cell by the methods described in the next section and then examine it for possible higher symmetry using the 43 Niggli mat rices (International Tables, Vol. I, pp. 530-534, 1965; Azaroff and Buerger, 1958). Cell Reduction and Lattice Evaluation As indicated above, once a unit cell is found by an indexing procedure, it must be examined for other possibilities of different metric symmetry. For any given lattice there are many ways to describe the periodicity by selecting any combination of three non-coplanar vectors whose unit-cell volume does not enclose another lattice point. The problem is to determine if there is a better choice of lattice vectors. The proce dure follows two steps. The first step is to determine what is called the reduced cell (Azaroff and Buerger, 1958). This cell is unique for every lattice and is found by following specific rules in the selection of the lattice vectors. This process is known as cell reduction. The cell edges of this cell are usually the three shortest non- coplanar vectors in the lattice with the interaxial angles defined in a specific manner. This cell is usually of low metric symmetry. (Metric symmetry means the symmetry of the shape of the unit cell only without regard to the posi tions of other lattice points.) Once the reduced cell is found, it is straightforward to test relationships between the vector magnitudes and directions to seek non-reduced cells with higher metric symmetry. Unfortunately, the met ric symmetry may not be the true symmetry. The true sym metry cannot be confirmed without further tests such a physical property measurements, single crystal X-ray dif fraction, or structure determination. There are several programs other than the indexing pro grams which allow the user to determine the reduced cell. These programs include: Cell Reduction Lawton NBS*LATTICE Himes and Mighell (1985) PC-R-TRUEBLOOD Blanchard CREDUC Gabe, Lee and LePage PC-LEPAGE Spek NEWLAT Mugnoli (1985) There is a related routine which looks at the position coordinates in a crystal structure description to determine higher symmetry which is: MISSYM Gabe, Lee and LePage Once the cell is selected, there are two further steps toward identification of the material which are refinement of the cell and then lattice matching against a database. Unit-Cell Refinements To determine the 'best' unit cell, initial cell parame ters are usually refined using the entire set of experimen tally measured and corrected d-spacings. The most common procedure is to use a least-squares technique which mini mizes the squares of errors between the calculated and measured data. As done in indexing, this calculation is usually done in Q = (1/d2) form because it linearizes the equations and simplifies the mathematics. 1 4(sin2O) Q(hkl) = --- = ---------- . d2 (8Mk(8U2 It is the sum of the squares of Q(obs)-Q(calc) which is minimized in the calculation. A good general discussion of refinement may be found in Klug and Alexander (1974) and Wilson (1963). Unit-cell refinement is usually done with weighting factors based on the experimental reliability of the measurement of the data. Weighting factors may include contributions from the ability to detect the peak usually based on the intensity of the peak and the ability to resolve the peak both from adjacent peaks including the 8U2 component. Where resolution is impossible at low angles, the appropriate wavelength must be used in the refinement. Systematic error functions may also be included for instru mental and sample errors that could not be eliminated experimentally. The method for this error analysis is known as the Cohen (1936a,b) method and involves added terms in the Q equation for position compensation func tions. This method is described in Cullity (1978). There are several routines available for cell refine ment, some with systematic errors. These routines require that you assign hkl's to each data point and select the symmetry and error functions prior to the refinement. The routines include: ARGONNE Mueller and Heaton (1961) LCLSQ Burnham PODEX Foris FINAX Hovestreydt (1983) PC-CELRF Prewitt INDEXING Novak and Colville (1989) There is another approach that may be used during refinement which is to start with a cell as input parame ters and allow the program to determine the hkl's to be assigned to each data point. The input cell must be close to the correct cell or the results may stray, but this approach allows the user to determine a cell from the low- angle low-accuracy peaks and to use this cell as the starting point. The calculation usually follows the sequence: 1) Use the starting cell to index points within a defined error and 2O range; 2) Refine the accepted points to yield a better cell; 3) Reduce the defined error, open the 2O range and repeat step one. This procedure usually converges in less than 10 cycles if it is going to converge at all. It is possible to force the program to use spe cific hkl's for some points, and this assignment is often necessary when cells are pseudosymmetric. Because of the nature of the algorithm, it is not possible to refine sys tematic errors at the same time, but weighting factors are used. All the programs which use this technique have evolved from the routine written by Appleman et al. (1973). Pres ently available routines include: APPLEMAN Appleman et al. (1973) NBS*LSQ85 Hubbard PPLP Gabe et al. PC-APPLVANS Benoit PCPIRAM Werner and Brown LSUCRIPC Garvey (1986b) Because of the importance in determining the best data for inclusion in the PDF and CDF databases, one of the steps in the evaluation of the experimental data includes refinement of the unit cell and the comparison of the meas ured and calculated 2O values and their figures-of-merit. There is a routine which is used specifically for this pur pose which includes a modified APPLEMAN component along with many other checks. This routine is known as NBS*AIDS83 which was prepared under the direction of Mig hell et al. (1983). In addition to refining the unit cell and calculating the figures-of-merit, it warns the user of poorly fitting data values and the presence of systematic errors. Any laboratory planning to prepare data for either of the databases or for publication should use this program to screen the data for errors prior to submission. There are many applications for refined cell data. Comparing refined unit cells is the most sensitive measure of subtle differences between similar materials. All data which are to be published in any form should be refined. Unit-cell data may be the best way to study systematic changes in compounds with crystallochemical substitutions or with environmental changes such as temperature, pressure and/or atmosphere. If nothing else is known about the material, the cell may also be the basis for identification by comparing it to one of the crystallographic databases. Unit-Cell Identification Where the compound under study is not represented in the PDF, d-I pattern searching is usually not successful unless there is an isostructural compound present. All is not lost at this point, as it is very effective to use the CDF database and search the cell directly against the cells in this database. NBS*LATTICE, developed by Himes and Mighell (1985), uses the reduced cell and derivative cells formed by doubling or halving input parameters to locate possible cells in the database which are within specified error limits. This database has cell information for over 70,000 inorganic compounds which is a large portion of the known inorganic materials. This large selection of unit cells provides a good chance of finding a cell match and thus an identification. Along with the identification comes additional information on the compound including its composition, true symmetry, and the reference to the paper from which the cell was abstracted. Because of the large amount of information in the CDF database, other programs have been written to access the file other than through the cell. BOOLEAN/LATTICE, written by Harlow et al., allows searches on symmetry, chemistry, authors, journals, density, optics, and many other proper ties which are included in the data entries. This type of search is useful for examining the coverage of the CDF database. It allows the user to select compounds which may show specific properties, e.g., piezoelectricity, which are symmetry sensitive. It also allows the user to locate a specific compound to obtain the unit cell for generation of allowed d-values prior to an experiment. Although designed for electron diffraction, the Elec tron Diffraction Database is a useful source for the X-ray diffractionist. The EDD was produced to allow identifica tion where the intensity values are generally very unre liable and the emphasis is on the d-value measurements only. The database contains the first 10 lines generated from the unit cell, and it is the d-values of these lines which are searched in the D-SPACING program written by Carr et al. (1989). There is a lattice matching routine avail able for this database also known as UNIT-CELL written by Himes and Mighell (1987). It works like the NBS*LATTICE described above. Quantitative Analysis There are other uses for d-I data in the analysis of materials. One use is for the quantitative analysis of the phases present in a mixture. The diffraction pattern of a mixture is a composite of the individual patterns of the component phases. The integrated intensities from each phase in the mixture are proportional to the amount present in the mixture modified by absorption effects due to the phases in the mixture. The theory of analysis developed by Alexander and Klug (1948) is the basis for several techniques which use the intensities of specific peaks from the diffraction pattern usually after creating a calibration suite of simi lar samples. The fundamental equation is KiJ xJ IiJ = -------- (8Mq(8UJ (8Ml(8U* where i represents the ith line and J the Jth phase. The weight fraction of the Jth phase is xJ, and its density is 8UJ. The factor 8U* is the mass absorption coefficient of the whole sample. Thus, the intensities are not independent of the other phases in the mixture. Methods have been devised to get around this problem most commonly by including a known amount of an added phase as an internal standard or by preparing calibration mixtures. The calculations are straightforward, so that most programs are subsets of the general routines within the system, but there is one program for the PC worth mentioning which is: PC/PEAKS Hill and Foxworthy. Tests for Preferred Orientation One of the problems with most diffraction experiments involving intensities of powdered samples is that the crys tallites in the sample are not randomly oriented, i.e., the crystallites do not assume all possible orientations with equal probability. Considerable effort must be expended in sample preparation to achieve the random state, and it is often difficult to determine when randomness is attained unless one has a reference set of intensities which are known to represent the random sample. However, if one does have such a reference, it is possible to test the exper imental intensity measurements and to determine the type of orientation which is present in the sample. This procedure is known as texture analysis. The most common procedure for determining texture is to tilt the specimen and rotate it while following the variation in intensity of a single diffraction peak. Such an experiment involves elaborate equipment to quantify the measurements. When the texture is only needed qualitatively, the standard powder pattern may be examined for its intensity deviations from the ran dom sample values. Preferred orientation in a sample is usually due to the shape of the particles in the sample and to the lack of sufficient numbers of particles to allow randomness to be attained. This situation is often referred to as "particle statistics". The shape is usually due to crystallite shapes or to cleavages when the sample is being crushed. Where there is one well-developed cleavage, the peaks corre sponding to the cleavage plane will be enhanced and all others diminished. Where several symmetry-related cleav ages occur, the orientation becomes more complicated. If two cleavages result in a needle shaped particle, the orientation effect is similar to the plate effect. There are several ways to describe the orientation effect on intensity values by applying an analytical function to cal culated intensities; these are reviewed by Dollase (1987). Program PREF/SUPREF has been written by Snyder to evaluate preferred orientation in a sample by comparing the exper imental intensities to the expected intensities from that material. POWD12 by Smith et al. (1983) includes orienta tion functions to model preferred orientation. Orientation functions are an option in most Rietveld programs. Structure Analysis using Integrated Intensities True structure analysis, i.e., the determination of atomic coordinates in the unit cell for an unknown struc ture, is rarely done today using only powder diffraction data. Usually the structures are too complex, and it is not possible to resolve the individual diffraction peaks sufficiently to measure the needed integrated intensities. Peaks must be considered as clusters. Most of the inten sity modeling programs were written to calculate the inten sities from a proposed structural arrangement to test the model. There is one program which uses the individually resolved intensities from a powder pattern for structure analysis which is POWDER Rossel and Scott. Any single-crystal structure analysis program may be used when the intensity values can be fully resolved. Usually the individual intensities cannot be fully resolved, and the methods in the next section must be employed. ANALYSIS OF DIGITIZED DIFFRACTION TRACES With the availability of modern computer-controlled diffractometers, many of the experiments utilize the digit ized diffractometer trace directly in the data analysis. It is now routine to collect the full trace and determine the peak positions and intensities by algorithmic processes rather then by visual estimation. Such processes lead to more accurate measurements and improve the utility of the experiment considerably. Some experiments use these derived values with the analytical programs already described above. For these studies, the d's and I's are often extracted by routines which are part of the APD sys tem software, but there are some non-APD programs available also. These programs will be discussed in the next two sections. Subsequent sections will be devoted to programs which use the digitized trace either directly or only slightly modified from the raw data, e.g., removal of the background and/or the 8U2 component. Although it is not the intent of this paper to discuss APD systems, the programs by Rogers and Lane (1987) and Madsen et al. (1988) for controlling a diffractometer using a PC should be men tioned. They provides the do-it-yourself user with a lim ited budget a means to control the data collection system. Processing of Raw Digitized Data There are several programs designed to process raw dif fraction traces and extract d-I information. The process ing includes options for smoothing of the data and for stripping out the background and the 8Ma8U2 component of the diffraction peak and then locating the peaks and inten sity values. Experience has shown that there is no perfect algorithm for peak recognition due to the complexities of the shapes. It is usually necessary for the diffractionist to examine, preferably using display graphics, the selec tions and to add or subtract peaks where necessary. Most APD packages allow the user to test the positions with the screen cursor and edit the data file as necessary. The background stripping is usually accomplished by selecting points on the trace which are the lowest and fitting either a polynomial or spline function to these points. Steen strup (1981) discussed the use of orthogonal polynomials for background fitting. Fitting problems often occur at the ends of the 2O range when using analytical functions. Linear fits within selected regions are also used. Routine BKGRD Smith is available as a separate program for these operations. The smoothing is either done with a Savitsky and Golay (1964) procedure or by Fourier filtering (Armstrong, 1989) The 8Ma8U2 stripping uses the Rachinger (1948) method which uses one-half the 8Ma8U1 profile to image the 8Ma8U2 shape at its proper position or the Ladell (1965) algorithm which uses experimentally determined profiles for each component. Programs for the reduction of digitized data include: ADR Mallory and Snyder (1979) PC/DRX Vila1 ISIP Rogers and Lane (1987) There are also programs which digitize and reduce the data on film patterns. These programs include: GUFI* Dinnebier and Eysel SCAN/SCANPI** Werner and Brown Location of Peak Positions and Intensities There are two general procedures used to determine the positions of peaks in a diffraction trace: derivative meth ods and fitting methods. Both have their virtues and their problems. The derivative method is very sensitive to the noise due to counting statistics; peak fitting is slower and sensitive to the peak shape assumed. In the derivative method, both the first and second derivatives are determined using the techniques of Savitsky and Golay (1964). The first derivative changes sign when ever the diffraction trace forms a peak or valley. The sign of the second derivative for the same point identifies which are peaks. The second derivative also locates inflection points in the diffraction trace which indicate half-widths and shoulders of peaks. The noise effects are minimized by the smoothing step, but in the process, some detail is lost and peaks are missed or misplaced by the algorithm. These missed peaks must be reentered by the user usually employing a interactive graphical display. Once the peaks are located, the intensities are determined from the raw trace by either selecting the intensity at the peak point or by fitting a parabola near the peak to define its position and maximum. Some attempts are made to integrate peak intensities which are usually unsatisfactory for experiments requiring accurate measure ments. General programs which use the derivative method include: ADR Mallory and Snyder (1979) PEAK** Sonneveld and Visser (1975) POWDER PATTERN Pyrros and Hubbard (1983) Each diffraction peak is a 'convolution' of many compo nents which can be classified into three groups. The first component is the source profile which is dependent on whether the source is a sealed X-ray tube, a synchrotron, or a nuclear reactor. The sealed tube produces the most complicated spectral distribution, being composed of sev eral different wavelengths each yielding peaks the shape empirically close to a Lorentzian (Cauchy)-like profile with long tails and a sharper than Lorentzian peak. The The 8Ma8U1 and 8Ma8U2 peaks are dominant but there is a small 8Ma8U3 peak which also must be considered when detailed studies are done. This source profile is modified by the aberrations of the diffractometer, known as the instrumen tal profile, and by the sample profile which depends on the perfection of the sample. Ideally, one would like to remove the contributions of the source and the instrument and isolate the sample profile which contains the desired information. Achieving this goal requires extremely accu rate data over the whole profile because of the large amount of intensity in the long tails. Several mathemati cal methods have been devised to deconvolute diffraction peaks (Klug and Alexander, 1974), but the details are beyond the scope of this course. Deconvolution methods are not used for d-I determination. They are used for deter mining crystallite size and strain. Profile fitting to locate peak positions shows consid erable promise to unravel complex peaks and yield accurate locations and intensities, both peak height and integrated. To use profile fitting, one must assume an analytical peak shape and fit all three wavelength components using the known dispersion and known intensity ratios and a changing breadth over the 2theta range. After finding the best fit for each peak in the diffraction pattern, the centroid of the profile is used to determine the d-value and the area or peak height is used for the measure of the intensity. A mathematically simple approach to peak finding is to fit an inverted parabola to several points around each peak position. This parabola then indicates a peak (centroid) position, a peak intensity, and a probable half width. The number of points to be used in the fitting procedure depends on the data collecting strategy (the 2theta interval) and the half widths of the experimental peaks. All the points in the top half of the profile should be used for this method to be successful. This method has great diffi culty with overlapped peaks where weaker peaks appear as shoulders on the sides of stronger peaks. In the more elaborate profile-fitting methods, there are several analytical functions which have been used to represent the experimental profile. Some fit each peak with profiles using many parameters. Others use the same profile throughout the whole pattern varying only the half- width and intensity. These profile models and fitting procedures have been evaluated by Howard and Snyder (1983). Profiles which have been employed for X-ray studies include: the Lorentzian (exponent = -1), the Intermediate Lorentzian (exponent = -1.5), the Modified Lorentzian (exponent = -2), Voigt and pseudo-Voigt functions, the Pearson VII function, and the Edgeworth series. The Gaus sian profile commonly is used for neutron diffraction. The functional forms of these analytical profiles are listed in Table 8-1. The selected function is then best-fit to the diffraction trace, usually by a least squares method; and all the peak positions and intensities are recorded. Note that this process 'is not deconvolution'; it is 'decomposi tion'. The intensities determined by profile fitting are usu ally applicable to all experiments requiring accurate intensities. If time permits, profile fitting is the best procedure for determining pattern parameters. However, profile refinement is hampered by the lack of any a priori information on the expected peak positions and intensities. Any profile-fitting procedure which is not constrained by the crystallography of the material under study may gener ate peaks which are not real or might miss peaks, and all located peaks must be examined to confirm that they are allowed by the unit cell. Routines which are available for profile fitting include: SHADOW Howard Micro-SHADOW** Howard and QJohnson PROFIT** Sonneveld and Visser (Langford et al., 1986) PRO-FIT* Toraya et al. (1983) REGION Hubbard and Pyrros XRAYL Zhang and Hubbard Pi'oPiliPa'a Jones EDINP Pawley (1980, 1981) LAT1 Tran and Buleon (1987) Crystal Structure Analysis The determination of the arrangement of atoms in a crystal structure is the ultimate goal of some diffraction experiments, so the researcher can relate physical or chem ical properties to the structure. Powder diffraction is rarely used in the determination of a totally new structure because it is difficult to separate the diffraction inten sities into the individual contributions with sufficient accuracy to employ the standard structure-solving tech niques. Most structure types are known now, and it is pos sible to test structure models by comparing the theoreti cal diffraction intensities with experimentally measured values. Two main approaches are used; one is to start with a model of a structure, make the substitutions and adjust ments in parameters expected, and calculate the theoretical pattern for comparison to an experimental diffraction pat tern. Structural parameter changes are guessed by examin ing the contributions of individual atoms to the intensi ties, and a new model is calculated to see if the fit is improved. The other is to refine the structure parameters using the differences between observed and calculated pow der intensity data to calculate parameter shifts for the next cycle. Both techniques are extensively used. Structure Modeling Programs to calculate integrated intensities essentially evaluate the equation on intensities I(hkl) = (8Ma(8UmLPA|Ft(hkl)|2 where (8Ma(8U is a scale factor, m is the multiplicity factor, L and P are the Lorentz and polarization factors, A is the absorption factor, and Ft is the temperature-corrected structure factor which describes the types and positions of atoms in the unit cell. N 2(8Mp(8Ui(hxj+kyj+lzj) F(hkl) = (8MR(8U fj e j=1 where fj is the atomic scattering factor and xj, yj, and zj are the position coordinates of the N atoms in the unit cell. The calculation is straightforward. Input consists of the symmetry and size of the unit cell and the atom types and coordinates. Starting with a given structure model, one can substitute other atoms in selected sites and examine the predicted intensity values. Position parame ters or occupancy factors can be adjusted to achieve improvement in the match between experimental and calcu lated intensities until the user is satisfied with the new model. Models do not have to be close to the true struc ture for the calculations to be useful. A series of pat terns for completely hypothetical structures will provide data for determining which member of the series might be the true one. For example, the 12 possible one-layer chlorite structures of Brown and Bailey (1962) have been calculated for comparison with patterns from natural speci mens. The following programs are available for calculating powder intensities: POWD12 Smith et al. (1983) Micro-POWD** QJohnson LAZY PULVERIX Yvon et al. (1977) DISPOW Gabe et al. ENDIX Hovestreydt and Parthe (1988) PC/POWIN Prewitt PC/XDPG Russ EDDA Gerward and Olsen (1987) Of these programs only POWD and its derivatives use the integrated intensities to simulate the diffractometer trace. Each diffraction peak is distributed over one of the selected analytical profiles from Table 1 so that the area conforms to the integrated intensity. Both wavelength components are included, and the half-widths are adjusted with the diffraction angle. All the peaks thus created are then added to generate a single intensity trace which is to be compared with the experimental diffractometer pattern. This trace can be scanned for peak-height intensities which are the values usually incorporated as "C" patterns in the Powder Diffraction File. The POWD family of programs list both the peak-height intensities and the integrated inten sities as output data. Clay minerals and other two-dimensional structures are usually studied in oriented diffraction mounts, so that only the 00l reflections are observed. The programs men tioned above are not appropriate for this type of struc ture. The following programs MOD2** Reynolds INTER Vila1 and Ruiz-Amil (1988) are designed specifically for this calculation. Structure Refinement Structure refinement is the refinement of the structural parameters of a model, which is close to the true structure, by comparing the calculated and experimental intensities of all the reflections in a pattern at the same time. Discrepencies are used to deter mine the parameter changes. Pattern parameters and unit cell values are refined along with the structure parame ters. There are two basic techniques, the profile-fitting technique and the Rietveld technique. The profile-fitting structure analysis programs for refinement of crystal structures use one of the analytical profiles from Table 1 for each peak and minimize the fit to the pattern on a peak by peak basis. The minimization procedure is similar to conventional crystal structure ana lysis except for adjustments to proportionate the observed intensities in complex clusters of peaks based on the pro file fits. The equations are essentially the same as for intensity modeling. Each intensity is distributed over the selected profile, and the collection of peaks are summed on the 2O scale and matched to the experimental pattern. Programs which fit into this category include: POWLS Will (1979) EDINP Pawley (1980) SCRAP Cooper et al. (1981) (Rouse et al., 1981) PROFIT Scott WPPF* Toraya PROF Rudolf and Clearfield The Rietveld (1967,1969) technique examines the pattern on a point-by-point basis. It is a very powerful method and is becoming very popular to refine the details of spe cific crystal structures. It requires accurate data col lected over a wide angular range. The technique was first developed for neutron diffraction and then adapted to X-ray analysis. An excellent review of the problems of X-ray Rietveld methods is given by Young et al. (1977, 1980). The Rietveld technique is not a profile-fitting method; it fits the whole pattern considering each digital point on the intensity trace as a separate data point. The function which is minimized is M = (8MR(8Uwj[yi(obs)-kyi(calc)]2 where yi(obs) is the observed intensity at step i in the pattern, yi(calc) is the calculated intensity and wi is a weighting factor usually related to the counting statis tics. Each yi may have several peaks contributing to its intensity, so that yi(calc) = (8MR(8U Ii(hkl) hkl Depending on the 2theta range attributed to a peak profile, a dozen or more peaks may have a contribution to any given intensity point. The details of this technique are presented in Chapter 10, so they will not be given here. The programs which have been used for X-ray refine ments include: DBW3.2 Wiles and Young (1981) PC/WYRIET Schneider RIETVELD Toby and Cox RIETVELD Baerlocher and McCusker RIETVELD Prince and Finger CCSL RAL LANCE-GSAS Larson and Von Dreele (1987) LHPM7 Hill and Howard RIETVELD Nurmela and Sourtti RIETAN* Izumi DBW4.1 Howard FIBLS Busing FIBLS is a new program designed for the refinement of two-dimensional structures, especially polymers. It may have applications to clays, disordered mica structures and asbestos-like minerals. It is expected to be available by the end of 1989. An interesting program which supplements the above refinement routines is DLS-76 Baerlocher et al. (1976) This program does not use intensity data. It examines structure models by constraining the bond distances and minimizing the geometric variations in the arrangement of the atoms. It has been used primarily with zeolites, but may be useful for other framework-type structures including tetrahedral and octahedral linkages. Quantification of Crystallite Parameters More common applications of diffraction analysis deter mine sample parameters such as phase identification or cell dimensions. Identification and cell refinements are usu ally done using derived d-I data, although the whole pat tern may be used in some instances. Information requiring accurate intensities or profile shapes, e.g., phase analy sis, determining percent crystallinity, and measuring crys tallite perfection, usually is more accurate when the digitized data are used directly in the analysis. Quantitative Phase Analysis X-ray diffraction has the unique ability to be sensitive to both the elements present and the physical state of a sample. Compounds with the same structural arrangement but with different elements yield similar patterns of peaks with different intensities. Compounds with the same chemistry but with different struc tures yield distinctly different diffraction patterns. Consequently, X-ray diffraction techniques were developed for determining the phase composition of a multicomponent mixture. The programs which are used for quantitative analysis fall into three categories. The first category integrates selected ranges of the diffraction pattern to determine an intensity value for each range which is then used in one of the procedures mentioned in Chapter 5. The second category consists of programs which fit the digital pattern on a point-by-point basis with weighted individual patterns of the component phases from a database of measured reference patterns. Corrections are usually based on the reference- intensity-ratio method. The third category also fits pat terns on a point-by-point basis using the Rietveld method to calculate the patterns of the component phases. Quanti tative analysis using the integrated intensities of a few characteristic peaks of each of the phases present in the mixture is most applicable in systems involving simple high-symmetry materials which create few peak overlap prob lems in the diffraction pattern. The basic theory was well defined by Alexander and Klug basic theory was well defined by Alexander and Klug (1948). Where some of the peaks do overlap, the extension of Cope land and Bragg (1958) is applicable which considers a clus ter of peaks as a single measurement. DeWolff and Visser (1964) showed how to scale intensities to the relative- absolute scale using intensity ratios to a standard mate rial, and Chung (1974a,b), Hubbard et al. (1976), and Davis (1986) incorporated this reference intensity ratio into quantitative analysis. Programs which apply these methods are: QUANT85 Snyder and Hubbard PC-RIMPAC** Davis (1986) Quantitative analysis using the whole diffraction trace has considerable advantages over the independent-peak method when the patterns are very complex such as those obtained from feldspars and clay minerals. The limitations are the need to have pure samples of the analytes to pro duce the necessary experimental patterns for the reference database. These samples not only need to be pure, they must represent the crystallinity and structural state of the phases in the sample to be analyzed. There is some forgiveness in the fitting procedure, however, which com pensates for mismatch of profiles and for preferred orien tation effects because the procedure averages over the whole pattern. In some instances, reference patterns can be extracted from a mixture if the patterns of the other phases are known by pattern subtraction methods, but this approach is of limited usefulness. Programs which use the whole-trace approach include: GMQUANT Smith, et al. (1987) PFLS Toraya (1986) Up to ten phases have been analyzed by this method. Smith et al. (1989) extended the method to allow constraints on the weight fractions based on chemical information which considerably improved the accuracy of the weight-fraction determinations. Quantification by the Rietveld method is particularly advantageous with complex mixtures and can be accomplished with mixtures of crystalline materials of any complexity. The method will not explicitly treat clay minerals exhibit ing two-dimensional diffraction or amorphous components. Most Rietveld programs allow more than one structure to be refined at the same time, and many have been adapted to provide quantitative estimates of the several phases pre sent. Knowledge of the structures and their abundances permits the calculation of the probable absorption coeffi cient of the bulk sample allowing conversion of the rela tive intensities to provide weight ratios. Examples of this approach were presented in Chapter 5, and the general problem of structure refinement was covered previously, so details will not be presented here. In addition to the general Rietveld programs, the following Rietveld programs are specifically designed for quantitative analy sis: QPDA** Hill and Masden PC/QXRD** Hill DBW4.1 Howard and Bish (Bish and Howard, 1988) Crystallinity The term crystallinity has several inter pretations in crystallography, but it usually is some mea sure of the fraction of a sample which is three-dimension ally organized in a matrix of disorganized amorphous mate rial. The absolute value reported by the many proposed methods is often questionable, but the relative values of a series of related samples is useful when determining which have progressed further in a reaction. Crystallinity indexes are usually based on some ratio of a crystalline diffraction pattern to the scattering intensity of the amorphous portion. This ratio may be a area measurement or some function of peak heights and valleys. Crystallinity in geologic systems is applicable to graphite in coal, crystalline SiO2 to amorphous SiO2, clay minerals, and fibrous materials such as asbestos. Most measurements are taken directly from a graphical display of the diffraction pattern. One program that is available which attempts to determine the ratio of crystalline fraction to amorphous fraction: POLYMER Wims et al. Crystallite Size and Strain In the introduction it was mentioned that the information on crystallite size and per fection was in the shape of the diffraction peaks. Note that "crystallite size" is not synonymous with "particle size", and X-ray diffraction is sensitive to crystallite size. Even perfect appearing crystals may be composed of microdomains which act as the coherent diffracting unit. Strained crystals often are composed of elastically- strained microdomains separated by plastically-deformed regions of high defect density. Crystallite domain size and strain are two measures of crystallite perfection. Very small crystallites are con sidered more imperfect than large ones because the surface interactions affect a larger volume fraction of the crys tal. Crystallite size causes the peaks to broaden as the size gets smaller than ~2000A. Strain causes the peaks to shift from their ideal location for the non-strained state. Usually, there is a distribution of strain in a sample which results in a distribution of shifts which distorts the peak shape considerably. Strain is also invariably accompanied by the breakup of the macrocrystallites into microdomains of different strain values which complicates the diffraction profile by superimposing broadening from the domains on the range of shifts caused by the range of strain values. The analysis of this strained state is not easy. Strain is defined as "change in length per unit length" and is measured as the change in d-spacing of a strained sample compared to the unstrained state usually expressed as dstrained - dunstrained (8Me(8U = -------------------------. dunstrained Strain measurements are usually converted to a stress using the physical properties of the material. The conversion is complicated because of the dependence on the sample mate rial and configuration and direction of measurement with respect to the sample. Even a cursory discussion is beyond the scope of this presentation. Stress analysis is dis cussed in Cullity (1978) and Klug and Alexander (1974). It is more commonly used in metallurgy than in mineralogy. In the absence of significant strain, such as in a loose aggregate of clay particles, a reasonably good esti mate of the crystallite size is possible. The most com monly employed relation is the Scherrer equation K (8Mk(8U D = ---------- b cosO where K is the shape constant (approximately 0.9), and b is the half-width of the diffraction profile due to the sample (measured breadth minus the instrumental breadth) in radians. For equant crystallites of a single size, the equation works quite well. When there is a distribution of sizes or crystallite shapes which are tabular or elongate, the equation is less accurate, and more elaborate methods similar to those used to determine the strain must be employed. The diffraction peaks must be separated into the components due to the various sources of distortion. This process is called deconvolution and is an involved oper ation. There are several programs which are available to deconvolute diffraction data for estimates of the crystal lite size and strain. WARREN-AVERBACH Cohen UNFOLD Roof DECON Wiedemann et al. (1987) CRYSIZE** Zhang et al. LWL/SIZEDIST Louer and LeBail Profile fitting may also be used to obtain crystallite size and strain measurements (Langford et al., 1986). The method is not strictly deconvolution, but because the fit ting function is analytical, the desired crystallite par ameters can be modeled, or the best-fit profile can be deconvoluted with a profile fit to data from a "perfect" sample which provides the instrumental and spectral contri butions. This approximation to deconvolution of the true experimental data yields parameters usable for evaluating the crystallite size and strain. Profile fitting programs which have been used for determining crystallite size and perfection include: PROFIT Sonneveld and Visser (Langford et al. 1986) SHADOW Howard Micro-SHADOW** Howard and QJohnson WPPF* Toraya Most Rietveld programs use adjustable parameters for broadening, which may be interpreted in terms of crystal lite size, and deviations of the crystal lattice parameters may be interpreted as strain if they are not due to chemi cal variations. Asymmetry parameters determined by the Rietveld method are more dependent on instrumental aberra tions and are difficult to interpret in terms of strain distributions. File Building and Transfer Now that digitized data are utilized for so many pur poses in modern diffraction analysis, it is necessary to be able to read and transfer these data from one system or one laboratory to another. There are several routines which allow the transfer of files from APD's to other computer systems by creating standard file structures for communica tion. These routines include: JCAMP-DX** McDonald and Wilks (1987) GMQUANT Smith et al. (1986) Micro-PEAK** QJohnson VAXCONV Zhou and Snyder JCAMP-DX is designed to be a universal digitized-file transfer system which has been adopted by the spectroscop ists and is being introduced to diffraction patterns. STRUCTURE DISPLAY PROGRAMS Once a structure is solved, the results are often dis played graphically especially for publications. There are many routines to accomplish this task. ORTEP CJohnson (1970) PRETEP Izumi PLORTEP Bandel and Sussman (1983) PLOTMD Luo et al. (1989) NAMOD Beppu MODEL Smith SCHAKAL88 Keller (1989) PLOTMOL Gabe, Lee and LePage STRPLT Fischer (1985) PLUTO Motherwell PLUTO78 Crennell and Crisp (1984) PLORTEP Bandel and Sussman (1983) SNOOPI Davies CRYES** Vila2 and Vegas B&S** Muller CHEM-X/MITIE** Chemical Design Co. Mol. Plot Radhakrishnan (1982) ALCHEMY Tripos The best known program is ORTEP which plots a ball-and- spoke image with the option to represent the atoms as ellipsoids showing the thermal motion. PRETEP is a setup program for ORTEP, which is the main difficulty in using ORTEP. PLOTMD simplifies the labelling of an ORTEP plot. MODEL is a setup program for NAMOD. Most of these programs plot ball-and-spoke type pictures, and most have options to remove hidden lines, so the picture looks clean. STRPLT is designed specifically for plotting polyhedral images commonly used to represent framework silicates in particular and many minerals with octahedral coordination as well. Mineralogists are also interested in displaying the external morphologies of crystals. Program SHAPE** Dowty (1988) is available for this purpose. Construction of ball-and- spoke models is also of interest. Program DRILL Smith and Clark will calculate projection coordinates in orthogonal space for all the atoms in the unit cell and the drilling angles for all the bonds around the atoms for a drilling machine using spherical coordinates. ANALYTICAL PACKAGES Several laboratories have assembled many of the above programs into a package that allows data to be transferred from one program to another with a minimum of data entry. In addition to communicating internally, each of these packages can accept data files from at least one APD sys tem. These packages are generally system dependent. Some of the packages are complete substitutions for the analyti cal component of most APD systems except for the data- collection activities. Packages for powder diffraction analysis include: SPECPLOT Goehner and Garbauskas (1979) AUTO Snyder et al. (1982) ALFRED Mallory and Snyder (1979) ZEOPAK** GJohnson and Smith NRC/VAX Gabe, Lee and LePage PC/NRC White FAT-RIETAN** Rigaku Some packages designed for single-crystal analysis have routines useful to powder analysis. These single-crystal packages include: SHELXTL** Sheldrick XRAY88** Stewart XTAL** Stewart and Hall (1985) MISCELLANEOUS ROUTINES There are many other programs that are available to the user that do not fit into the previous categories. Some of these programs will be mentioned here. Almost every diffraction laboratory will have some rou tine for generating d-spacings for a given unit cell. These programs are known collectively as D-GEN's, and the calculations can be programmed on pocket calculators. Such programs are indispensable in the analysis of a diffraction pattern. There are too many versions to list here. Although the International Tables of Crystallography are usually available in every diffraction laboratory, the program CRYSET Larson is very useful to generate the symmetry positions from the space symmetry information. Any space group symbol can be interpreted, but the program only allows settings with the center of symmetry at the origin. Program TABLES Abad-Zapetro and O'Donnell (1987) will allow the plotting of the symmetry related points for viewing and manipulation in real time to see how the points interact. Occasionally, Laue patterns will be used for determin ing crystal orientation or for distinguishing specific directions in a crystal. Programs STEREOPL Scientific Software STLPLT Canut-Amoros (1970) LAUE Goehner and Garbauskas Laue Laugier and Folhol (1983) will plot both transmission and back-reflection Laue pat terns for a given unit cell and symmetry. Such plots save the time of finding a crystal of known orientation and mak ing a reference photograph. LOTUS-123** should be mentioned as a very useful way of preparing data reduction and comparison tables of diffrac tion data. Spread sheets are also a good way of presenting data and interpretations to requestors of sample analyses and for publication. McCarthy (1986) has developed a series of templates for various diffraction applications. Novak and Colville (1989) use d-templates to refine the unit cells for each of the crystal systems. GENERAL COMMENTS AND DISC LAIMER Over 150 programs for the analysis of diffraction data have been identified in this paper. Many of these programs duplicate the calculations of other routines in the same category, so the diffractionist certainly does not need to acquire every program to have a complete system of rou tines. However, because there are some differences in pro grams for similar purposes, it is usually advisable to have more than one routine from a given category. Given a spe cific set of unit-cell parameters, all D-GENS will calcu late the same d-spacings. On the other hand, search/match and indexing programs may not give identical results because they use different strategies on experimental data which are not free of errors. Selection of programs for ones own use depends on the types of experiments commonly performed and on the type of computer which is available. For the diffractionist who has an APD, the selection of programs is usually to augment the APD software package for a specific application. This paper will help identify the individual who can supply that program. For the diffrac tionist who is starting from nothing, it is a different matter. Perhaps the best way to start would be to contact one of the authors who has a collection of programs already integrated into a basic package. Additional programs can then be added later. The advantage of a package of pro grams is usually that the data entry is relatively simple compared to individual programs. Once the data are entered or transferred from an APD, the data can be reformatted for entry to all the programs in the package. This author makes no claim with respect to the com pleteness in the coverage of programs described in this paper or their success in doing the calculations which are reported. It has proved to be a formidable task to assemble this list, and I wish to thank the many contribu tors who supplied information on programs with which I had no direct experience. I hope I have described each routine properly, but without first-hand use of each program, I have probably misclassified some routines. Obviously, there is an emphasis on those programs which have been in use in the United States, but I have made an attempt to collect programs from overseas where possible. This list has been submitted to the International Union on Crystallography Commission on Powder Diffraction where it is hoped that it will be maintained for future reference.