Test Data for the Calculation of Powder Patterns for Intermetallic Phases

Powder diffraction patterns are often calculated from structural parameters to assist in the identification of materials. To ensure that powder pattern calculations are correct, it is useful to have data to test the computer program doing the calculations. This paper contains test data for each of the crystallographic point groups and 63 of the 230 space groups. An important feature of the data is that many tests involve two high-symmetry structures (sodium and magnesium) that are set in successively lower-symmetry space groups. Thus, the calculated powder intensities for sodium, for example, are identical whatever the setting is. Though the data were chosen to be especially useful for the calculation of the powder patterns of metals and intermetallic compounds, the data have wider utility. Introduction It is becoming increasingly common to use powder patterns calculated from structural parameters for phase identification. Calculation is facilitated by the ready availability of information regarding critically evaluated crystal structures in structural data bases. An example of the potential application of calculated powder patterns can be obtained by comparing the number of crystallographically characterized phases in Pearson ‘s Handbook of Crystallographic Data for Intermetallic Phases [l] with the number currently in the Powder Dijfraction File (PDF) [2]. In [I], the number of crystallographically characterized phases is about 40,000 to 50,000, while there are about 11,000 to 12,000 metals and alloys powder patterns in the PDF. The recently published Metals and Alloys Indexes [3] to the PDF are very useful for identifying structural similarities among materials in the PDF, and these indexes would be even more useful with complete coverage of intermetallic phases, coverage that could only be obtained by the calculation of large numbers of powder patterns. However, before calculated powder patterns can be routinely used for phase identification or expanding the coverage of the PDF, one has to demonstrate that the patterns are calculated correctly. The data in this paper (a) permit quality checks to be carried out on powder pattern calculation programs and (b) can provide increased assurance that calculated powder pattern data are correct. Important Note: Because this data set does not test for every calculation problem, the authors do not suggest that successfully performing all calculations in this data set ensures that all ensuing calculations are correct. See the companion paper by Dann et al. [4] for more information. Requirements There are two requirements for correctly calculating a powder diffraction pattern: . the structural model must be suitable and . the computer program must give the correct results for the model used. Copyright 0 JCPDS-International Centre for Diffraction Data 1997 Copyright (C) JCPDS-International Centre for Diffraction Data 1997 The first requirement is met if the atomic parameters are derived from an accurate structure refinement, that is, one with R = 0.05 or less. However, R-factors of the order of 0.10 are common from earlier photographic and diffractometer work. In these cases, the parameters from these determinations can be used if calculated interatomic distances are reasonable. In every powder pattern calculation, the atomic coordinates of the original paper must be checked by an interatomic distance calculation because l-3% of published structural papers contain numerical errors [5]. This checking is often the most time-consuming part of calculating a pattern. The second requirement is met if the computer program used has been shown to give correct results. Correct results are verified if (a) the program successfully handles a wide variety of symmetries and structural parameters and (b) the user repeats the verification tests after any software or equipment change. Symmetry Tests An important test of a powder pattern calculation program is determining how well the program handles symmetry calculations. Thus, these data provide for rigorous symmetry tests. To test symmetry calculations, we provide data covering l 11 centric point groups (-1,2/m, mmm, 4/m, 4/mmm, -3, -3m [-3ml& -3 lm], 6/m, 6/mmm, m3, m3m), l 21 acentric point groups (1, 2, m, 222, mm2,4, -4,422,4mm, -42m [-42m & -4m2], 3, 32 [321 & 312],3m [3ml& 31m], 6, -6,622,6mm, -6m2 [-6m2 & -62m], 23,432, -43m), and 0 the alternate settings of the centric and acentric point groups. (The alternate settings are in square brackets above.) The data are listed in Table 1. To simplify data input, two high-symmetry structures (sodium and magnesium) are reset in successively lower-symmetry space groups. Sodium is set in space groups that are derived from m3m, and magnesium is set in space groups that are derived from 6/mmm. Since both structures are centric, when each structure is reset in acentric space groups, the x-ray scattering is still centric [6, 71. When anomalous scattering effects are present, differences between FM and F-h-k.1 may exist, but not if the underlying structure is centric. If the program is handling symmetries correctly, (a) the calculation results will be as predicted [8], and (b) the structure factors and integrated intensities of each class of symmetrically equivalent reflections must be listed separately in an output table. Because of the latter, the program’s rules for choosing one set of hkl to represent a class are particularly important. If the program is handling multiplicities correctly, intensities that are exactly overlapped due to the symmetry of the structure will be correctly summed for peak heights in each lower symmetry space group [9, lo]. In addition, the distinct Fhkl classes must be consistent with the listings for individual space groups [ 111. Other Tests In addition to the symmetry tests discussed above, these data test l different origin choices (i.e., with a centric or noncentric origin) using AlzRu and gamma-plutonium in space group Fddd, No. 70, with origin at O,O,O and then at l/8, l/8, l/8; . alternate settings for orthorhombic space groups using HoSb2 in three settings, C222, A222, and B222; l monoclinic space groups with a or c as unique axes using sodium and AlB2; and l both hexagonal and rhombohedral settings using sodium. In each of these cases, the different settings should produce equivalent results. Copyright 0 JCPDS-International Centre for Diffraction Data 1997 Copyright (C) JCPDS-International Centre for Diffraction Data 1997 The plutonium calculation provides a test of extreme absorption. Use this calculation to check the uncorrected, integrated intensities against the absorption-corrected [ 12-141 values as a function of 2-theta for each experimental method. For plutonium, the difference between diffractometer and Debye-Scherrer intensities is large (especially at high 2-theta angles), while the difference between uncorrected and diffractometer intensities is zero. Three examples (CaF2, BeO, and BPd2) allow checking for weak intensities, and one of these examples (BPd2) has a third of its observed lines with intensity less than 1.0. In addition, there are several data sets with either mixed or partial site occupancy. Calculational Parameters The parameters used in any calculation must be recorded so that the calculation may be reproduced. These parameters are described below. l Structural parameters: Chemical formula; unit cell parameters; space group number and symbol; atomic coordinates; site occupancies; thermal parameters; and scattering and absorption factors including the anomalous components. Small differences exist between (a) different tables of scattering factors and (b) 7or 9-parameter functional representations. These differences have their greatest effect on calculations involving some light atoms. l General parameters: Wavelength; absorption function; specimen size, shape, and assumed geometry; and any special polarization correction. The latter would be especially important any time monochromators are used. l Program parameters: Program name and version; computer model used; profile function used (Gaussian, Cauchy, etc.); the half-width as a function of 2-theta; the range of calculation on each side of the central 2-theta; the peak-determining algorithm and its corollary, the definition of resolved peaks; and the background treatment (if any). Further, the treatment of alpha-2 peaks in unresolved groups should be clearly defined, and the omission (if any) of resolved alpha-2 peaks also specified. Rules for the choice of an hkl (e.g., h>k>l) to represent a group of symmetrically equivalent reflections should be clearly defined so that the comparison of classes can be properly made. How the rules are defined is of particular importance in different settings of space groups in the same point group. Note: The International Centre for Diffraction Data (ICDD) has developed guidelines for accepting calculated powder diffraction patterns [ 151. These guidelines parallel much of what is written here. Copies of the guidelines are available from the ICDD. Output Requirements Ideally the program should have one output that includes all data needed for testing the program. For example, the output should include l Bond distances, the crystallagraphic density, and a reconstruction of the chemical formula based upon input parameters (these are especially useful for screening out incorrect calculations); l Structure factors (F~M and F-h-k-1 ) so that the relationships between different classes of symmetrically related reflections can be checked; Copyright 0 JCPDS-International Centre for Diffraction Data 1997 Copyright (C) JCPDS-International Centre for Diffraction Data 1997 l Integrated intensities, their 2-theta values, and d-spacings, as well as integrated intensities corrected for all common geometries (diffractometer, Debye-Scherrer, Guinier, transmission flat-plate, etc.); l Peak intensities for the same geometries together with their 2-theta values (which will be shifted in overlapped groups) and a clear indication of the hkl reflections which contributed to each unresolved peak; and l Sine-squared-theta, Q-values, the absolute scale factor, and the reference intensity ratio. The treatment of weak peaks in the output should be clear (e.g., less than 1, 100-l as digits, 1.0 to 0.1, or l-999), and as noted before, it is convenient to have outputs for each specific geometry. Using the Table All these tests should be run to get a rigorous test of a powder pattern calculation program. The sequence in which the tests are run is not as important, though one of us (LDC) found it convenient to run all the sodium (or magnesium) examples together to make comparisons easier. Powder Pattern Calculation Programs A compilation of powder pattern calculation programs is available [ 161, and more recent information on available programs is available from the authors of that paper. Corrections or Suggestions Corrections or suggestions for improvement in the test data will be appreciated and may be addressed to any author. Acknowledgments The genesis of this data lies in tests made at the National Research Council of Canada by A. C. Larson [ 171 on a PDP-8 program for calculating powder diffraction patterns. We are also grateful to A. Brown, C. R. Hubbard, R. B. Roof, and W. M. Syvinski for testing earlier versions of this data. Their tests improved the data, and in some cases, errors in the programs tested were found. The publication of this data is dedicated to the late Larry Calvert, and we respectfully suggest that the data be called the Calvert Test Data.