International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 5.2, pp. 491504
https://doi.org/10.1107/97809553602060000596 Chapter 5.2. Xray diffraction methods: polycrystallineThe determination of lattice parameters using Xray powder methods is reviewed. Topics covered include: wavelength errors, refraction and statistical fluctuations; geometrical and physical aberrations; angledispersive diffractometer methods (using conventional and synchrotron sources); wholepattern methods; energydispersive techniques; camera methods; testing for remanent systematic error; powderdiffraction, intensity and instrumental lineprofileshape standards; and factors determining accuracy. 
Xray powder methods for the accurate determination of lattice parameters can be divided broadly into four groups, depending on the type of dispersion, type of source, and the type of detector. They are:
The geometry, advantages, and some practical details of the methods are given in the sections whose numbers are given in parentheses. The techniques will be discussed in the above order in Sections 5.2.4–5.2.8. More details of systematic errors in diffractometry are given in Wilson (1963, 1965c, 1974). Some general points on checking precision and accuracy were made in Chapter 5.1 . Many of them are treated in greater detail in Section 2.3.5 , and are recapitulated in Section 5.2.13.
The technique of choice will depend on the accuracy required and on the nature and quantity of the material available. At present, the technique most frequently used for the purposes of this chapter is angledispersive diffractometry with a conventional tube source (1). Angledispersive diffractometry with synchrotron radiation (2) is capable of greater precision and accuracy, but access to the synchrotron sources is cumbersome and may involve long waiting periods. Energydispersive methods (3) would ordinarily be adopted only if the required environmental conditions (high or low temperatures, high pressures, ) can be achieved most readily by means of a fixedangle diffractometer. Camera methods (4) are adaptable to small quantities of material, but microdiffractometers (Subsection 2.3.1.5 ) can be used with similar or even smaller quantities.
The relation between the lattice spacing d, the angle of incidence (Bragg angle) θ, and the wavelength λ is Bragg's law: The lattice spacing d is related to the lattice parameters a, b, c, α, β, γ and the indices of reflection h, k, l. In the simple case of cubic crystals, the relation is where a is the single lattice parameter. The general relation is where a, b, c are the edges of the unit cell, and are the functions of the angles of the unit cell given in Table 5.2.1.1.

Differentiation of (5.2.1.1) shows that the errors in the measurement of d are related to the errors in the measurement of λ and θ by the equation Wavelength and related problems are discussed in Section 5.2.2 and geometrical and other aberration problems in Section 5.2.3.
The error in the Bragg angle, , will ordinarily consist of both random and systematic components. The random components (as the name implies) have an expected value zero, but the systematic errors will affect all measurements consistently to a greater or lesser extent. The systematic errors may be, and usually are, functions of θ and/or λ. Such errors would ordinarily reveal themselves in checks of internal consistency: the values of the apparent lattice parameter, plotted as a function of θ, would show a systematic drift, not a random scatter. The success or otherwise of attempts to eliminate or account for them would be subject to statistical tests (Section 5.2.9 and Chapter 8.5 ). There is an exception to the `ordinarily'; if the variation of with θ happens to be of the form , where K does not depend on θ either explicitly or through λ, the resultant fractional error is a constant, and would not be revealed either by systematic drift of the apparent lattice parameter with θ or by statistical tests.
The Bragg angles are determined from the observations by a series of operations that are often quite complex.
For film cameras of diameter 57.3 or 114.6 mm, a simple measurement with a millimetre scale gives θ in degrees (1 mm = 1 or 0.5°). This determination is crude, and ordinarily the lines on the film would be measured with a lowpower travelling microscope or a densitometer. The effective camera diameter is found from measurements of fiducial marks imprinted on the film, or by use of the Straumanis film mounting. References to detailed descriptions are given in Section 2.3.4 .
For Bragg–Brentano (Parrish) and Seemann–Bohlin diffractometers, ratemeter measurements with stripchart recordings have timeconstant errors, and precision measurements require stepscanning (Subsection 2.3.3.5 ). The data may be analysed to give one or more of the following measures of position:
As usual, it is necessary to distinguish between the precision (reproducibility) of a measurement and its accuracy (extent to which it is affected by systematic errors). In principle, it does not matter if the Bragg angle obtained by any of the above operations is affected by systematic errors, as these can be calculated and allowed for, as described in the following paragraphs. The most precise methods are the peaksearch and individual profilefitting computer procedures. They are routinely capable of a precision of about for reasonably sharp reflections, and are free from the subjective effects that may influence, for example, film measurements or the graphical extrapolation of the midpoints of chords. As well as a measure of the peak position, the peaksearch procedure gives a measure of the peak intensity, and the profilefitting procedure gives a measure of the peak intensity and (if desired) a measure of the integrated intensity.
In diffractometry, the errors in wavelength, , are usually entirely systematic; the crystallographer accepts whatever wavelength the spectroscopist provides, so that an error that was random in the spectroscopy becomes systematic in the diffractometry. One or two exceptions to this rule are noted below, as they are encountered in the discussion of the various techniques. Equation (5.2.1.4) shows that such a systematic error in wavelength, arising either from uncertainty in the wavelength scale (affecting all wavelengths) or from a systematic error in one wavelength (possibly arising from a random error in its determination) produces a constant fractional error in the spacing, an error that is not detectable by any of the usual tests for systematic error.
Ordinarily, the wavelength to be inserted in (5.2.1.1) is not known with high accuracy. The emission wavelengths given by spectroscopists – the exact feature to which they refer is usually not known, but is probably nearer to the peak of the wavelength distribution than to its centroid – are subject to uncertainties of one part in 50 000 [see, for example, Sandström (1957, especially p. 157)], though this uncertainty is reduced by a factor of ten for some more recent measurements known to refer to the peak defined by, say, the extrapolated midpoints of chords (Thomsen, 1974). Energydispersive and synchrotron devices are usually calibrated by reference to such Xray wavelengths, and thus their scales are uncertain to at least the same extent. Use of a standard silicon sample (Sections 5.2.5 and 5.2.10) will ordinary give greater accuracy. There are a few wavelengths determined by interferometric comparison with optical standards where the uncertainty may be less than one part in a million (Deslattes, Henins & Kessler, 1980); see Section 4.2.2 .
The wavelength distributions in the emission spectra of the elements ordinarily used in crystallography are not noticeably affected by the methods used in preparing targets. There is a slight dependence, at about the limit of detectability, on operating voltage, takeoff angle, and degree of filtration (Wilson, 1963, pp. 60–63), and even the fundamental emission profile is affected somewhat by the excitation conditions (Chevallier, Travennier & Briand, 1978). Effective monochromators, capable of separating the and components (Barth, 1960), produce large variations. However, (5.2.1.1) depends only on the ratio of d to λ, so that relative spacings can be determined without regard to the accuracy of λ, provided that nothing is done that alters the wavelength distribution between measurements, and that the same identifiable feature of the distribution (peak, centroid, midpoint of chord, ) is used throughout.
Xrays, unless incident normally, are refracted away from the normal on entering matter, and while inside matter they have a longer wavelength than in vacuo. Both effects are small, but the former leads to a measurable error for solid specimens (that is, specimens without voids or binder) with flat surfaces (single crystals or polished metal blocks). This effect becomes prominent at grazing incidence, and may lead to total external reflection. For the usual powder compacts (Section 2.3.4 ), refraction leads to a broadening rather than a displacement (Wilson, 1940, 1962; Wilkens, 1960; Hart, Parrish, Bellotto & Lim, 1988; Greenberg, 1989). The greater wavelength within the powder grain leads to a pseudoaberration; the actual wavelength ought to be used in (5.2.1.1), and if the in vacuo wavelength is used instead the lattice spacing obtained will be too small by a fraction equal to the amount by which the refractive index differs from unity. The difference is typically in the fourth decimal place in the lattice parameter expressed in Å. The need for any refraction correction for very fine powders has been questioned.
Statistical fluctuations in the number of counts recorded are not aberrations, but random errors. They influence the precision with which the angles of diffraction, and hence the lattice parameters, can be determined. The fluctuations arise from at least two sources: emission of Xray quanta from the source is random, and the number of crystallites in an orientation to reflect varies with position within the specimen and with the relative orientations of the specimen and the incident beam. The theory of fluctuations in recording counts is discussed in Chapter 7.5 ; their effect can be reduced as much as is desired by increases in the counting times. Fluctuations in particle orientation are more difficult to control; use of smaller particles, larger illuminated volumes, and rotation of the specimen are helpful, but may conflict with other requirements of the experiment. The section on specimen preparation in Chapter 2.3 should be consulted.
Among the many papers relevant to the problem are Mack & Spielberg (1958), Pike & Wilson (1959), Thomsen & Yap (1968a, b), Wilson (1965a, b, c, 1967, 1968, 1969, 1971), Wilson, Thomsen & Yap (1965), and Zevin, Umanskij, Khejker & Pančenko (1961). The formulae are complicated, and depend on the measure of location that is adopted for the diffraction profile. In general, however, the variance of the angle is inversely proportional to the number of counts accumulated.
The systematic errors are generally called aberrations; they differ from random errors in that in principle they can be calculated for any particular experimental arrangement and the observations corrected for them, leaving only the random error. In practice, of course, the calculation may be difficult. Aberrations can be divided broadly into two classes: (i) geometrical and (ii) physical. The geometrical aberrations are those that depend on the dimensions of the source, specimen, and detector (or of the slits that limit their effective dimensions). In angledispersive techniques, the physical aberrations depend on the intensity distribution in the range of wavelengths used, and in both angle and energydispersive techniques they depend on the response characteristics of the detector and associated circuits.
The aberrations shift and distort the diffraction maxima. The study of their effects can be divided into four stages, corresponding to four levels of mathematical difficulty, and the stage to which it is necessary to carry the calculation depends on the purpose in view and the identifiable feature (Subsection 5.2.2.1) of the wavelength distribution that it is intended to adopt as a measure of the position of the line profile. The three usual features are:
The first of these, the centroid, requires only the first stage of the calculation for the geometrical aberrations and the first and second for the physical; the second, the peak, logically requires all four stages, but approximations can be obtained at the second stage; and the third, the best overall fit, requires all four stages.
The first stage of the calculation is the determination of the effect of the aberration on the centroid of the diffraction maximum, and ordinarily this gives rise to no insurmountable difficulty (Spencer, 1931, 1935, 1937, 1939, 1941, 1949; Wilson, 1950; Ladell, Parrish & Taylor, 1959; Pike & Wilson, 1959). It is all that is required for the correction of centroid positions for geometrical aberrations, which should be strictly additive. There is some limitation for physical aberrations (Edwards & Toman, 1970; Wilson, 1970b).
The second stage is the calculation of the meansquare broadening (variance). This can be used to obtain a reasonable approximation to the correction of peak positions over a wide range of Bragg angle (Wilson, 1961; Gale, 1963, 1968). To this approximation, the position of the observed peak is given by where is the centroid and W the variance of the geometrical aberrations and I′′ and I′′′ are second and third derivatives of the observed line profile evaluated at its maximum. The physical aberrations of the centroid depend on the variance of the part of the wavelength distribution used in determining the centroid (Wilson, 1958, 1963; Wilson & Delf, 1961). Those of the peak depend on the ratio of the peak intensity I to its second derivative I′′ (Wilson, 1961, 1963, 1965c).
Often an aberration can be expressed in the form where the function F gives the angular variation of the aberration and K depends only on dimensions etc. that are fixed for a particular experiment but whose actual measurement is too difficult or tedious. The constant K can then be treated along with the lattice parameters as an adjustable parameter in leastsquares refinement (analytical extrapolation; see Subsection 5.2.3.2).
The third stage is the calculation of the line profile corresponding to each geometrical aberration. These aberration profiles can be combined by convolution (folding), either directly or by Fourier methods, and, in the fourth stage, the combined aberration profile can be convoluted with the emission profile of the Xray source (or the emission profile as trimmed by a monochromator, pulseheight analyser, filter etc.) and with the diffraction profile corresponding to the state of strain, crystallite size, etc. of the specimen. This calculation of the composite line profile would be a necessary preliminary to an exact use of peak positions or of overallprofile fitting in latticeparameter determination.
Such calculations were proposed many years ago (for example, by Alexander, 1948, 1950, 1953, 1954), and have been used by Beu and coworkers (see Section 5.2.9), and also by Boom and Smits (Boom & Smits, 1965; Boom, 1966). With the development of more powerful computer methods, such calculations can now be carried out routinely (e.g. Cheary & Coelho, 1992, 1994; Kogan & Kupriyanov, 1992; Timmers, Delhez, Tuinstra & Peerdeman, 1992). However, not all the relevant instrumental parameters can in general be determined with sufficient accuracy and overall instrumental line profiles are normally obtained by means of a suitable standard material, for which sample broadening is negligible (Section 5.2.12). There is, however, a related common approach, empirical rather than fundamental, based on the proposal of Rietveld (1967, 1969). Its use in structure determination is treated in detail in Chapter 8.6 , and its use in latticeparameter determination in Section 5.2.6. There seems to be no detailed published study of the accuracy attainable for lattice parameters, but the estimated standard deviations quoted (see, for example, Young, 1988) are comparable with those obtained for simpler structures giving resolved reflections.
Equation (5.2.1.4) indicates that for a given error in θ the fractional error in the spacing d approaches zero as θ approaches 90°. The errors in θ – expressed as in (5.2.3.2) – arising from any specified aberration may increase as θ increases, but ordinarily this increase is insufficient to outweigh the effect of the factor. In the simple cubic case, one can write where K is a proportionality factor and represents the angular variation of the systematic errors in the lattice parameter. The functions F in (5.2.3.2) and (5.2.3.3) are not exactly the same; they are transformed into one another by the use of (5.2.1.4). Functions suitable for different experimental arrangements are quoted in the following sections; see, for example, equation (5.2.8.1) for the Debye–Scherrer camera and Tables 5.2.4.1 and 5.2.7.1 for diffractometers. Simple graphical extrapolation is quick and easy for cubic substances, and by the use of successive approximations it can be applied to hexagonal (Wilson & Lipson, 1941), tetragonal, and even orthorhombic materials. It is, however, very cumbersome for noncubic substances, and impracticable if the symmetry is less than orthorhombic.
Analytic extrapolation seems to have been first used by Cohen (1936a,b). It is now usual even in the cubic case: programs are often included in the software accompanying powder diffractometers, and many others are available separately. Some programs that are frequently referred to are described by Appleman & Evans (1973), Mighell, Hubbard & Stalick (1981), and Ferguson, Rogerson, Wolstenholme, Hughes & Huyton (1987); for a comparison, see Kelly (1988). If the precision warrants it, the single function may be replaced by a sum of functions , one for each of the larger aberrations listed in Tables 5.2.4.1, 5.2.7.1, and 5.2.8.1. Two – the zero error and a function corresponding to specimensurface displacement and transparency – must be used routinely; one or two more may be added if the precision warrants it.
The symmetrical Bragg–Brentano (Parrish) and the Seemann–Bohlin angledispersive diffractometers are fully described in Chapter 2.3 . The centroid and peak displacements and the variances of the aberrations of the symmetrical diffractometer have been collected by Wilson (1961, 1963, 1965c, 1970a, 1974). For the Seemann–Bohlin type, they are collected in Table 5.2.4.1, mainly from Wilson (1974). They are expressed in inverse powers of the source–specimen distance S and the specimen–detector distance R, and tend to be larger for the Seemann–Bohlin arrangement than for the symmetrical arrangement. For the latter, S and R are constant and equal to the radius, say , of the diffractometer, whereas, for the former, and where is the constant angle that the incident Xrays make with the specimen surface. In the Seemann–Bohlin case, S will be constant at a value depending on the choice of angle , but usually less than , and R will vary with , approaching zero as θ approaches . There will thus be a range of for which the Seemann–Bohlin aberrations containing R become very large. Mack & Parrish (1967) have confirmed experimentally the expected differences in favour of the symmetrical arrangement for general use, even though the effective equatorial divergence (`flatspecimen error') can be greatly reduced by curving the specimen appropriately in the Seemann–Bohlin arrangement. The aberrations for the symmetrical arrangement are found by putting , in the expression in Table 5.2.4.1; they are given explicitly by Wilson (1963, 1965c, 1970a).
Notation: 2A = illuminated length of specimen; β = angle of equatorial missetting of specimen; γ = angle of inclination of plane of specimen to axis of rotation; Δ = angular aperture of Soller slits; μ = linear absorption coefficient of specimen; r_{1} = width of receiving slit (varies with θ in some designs of diffractometer); s = specimensurface displacement; f_{1} = projected width of focal line; h = half height of focal line, specimen, and receiving slit, taken as equal; 1 − δ = index of refraction; p = effective particle size.

Latticeparameter determination with synchrotron radiation has a number of advantages over focusing methods (Parrish, Hart, Huang & Bellotto, 1987; Parrish, 1988; Huang, 1988). The doublet problem does not arise; the symmetrical single profiles greatly simplify the accurate angular measurement of peaks. The higher intensity and low uniform background out to the highest θ values give a higher statistical counting precision, an important factor in accurate measurements. Short wavelengths (0.65 to 1.4 Å) can be used to increase greatly the number of reflections without compromising the accuracy of the peak measurements. If desired, the patterns can be recorded with two or more wavelengths of about the same intensity, instead of being confined to the and lines (Popović, 1973). The large specimensurfacedisplacement and flatspecimen errors associated with most other methods do not occur, so that systematic errors are small or absent. The wavelength can be selected to obtain the desired dispersion, to avoid fluorescence, and to reduce specimen transparency.
The reflections are virtually symmetrical narrow peaks (Subsection 2.3.2.1 ), with widths of the order of 0.02–0.04°(2θ) when an analysing crystal is used instead of a receiving slit, and of the order of 0.05° when a long Soller slit is used as a collimator. These increase with increasing because of wavelength dispersion and small particle size. The angular positions of the peaks can be determined with high precision by the use of profilefitting or peaksearch measurements, and the only significant geometrical aberration is axial divergence.
There are no lines in the synchrotronradiation spectrum, and this creates the problem of determining the wavelength selected by the monochromator. If a highly accurate diffractometer were used for the monochromator and the monochromator d spacing were known accurately, the wavelength could be determined directly from . The angular accuracy of the diffractometer would have to be 0.0002° to achieve an accuracy of one part in 10^{6} in the wavelength at λ = 1.54 Å.
In practice, the wavelengths can often be determined by scanning the absorption edges of elements in the specimen or a metal foil placed in the beam. There is no feature of the absorption edge that is accurately measurable, and the wavelengths are usually listed to one or two decimal places fewer than those for the emission lines.
The wavelength problem could be avoided by using the ratio of the lattice parameter of the specimen to that of an accurately known standard measured with the same experimental conditions (Parrish et al., 1987). The standard may be mixed with the specimen or measured separately, as there is no specimensurface displacement shift. Mixing reduces the intensity of both patterns and worsens the peaktobackground ratio. The limitation is the accuracy of the lattice parameter of the standard. The only widely available one is the National Institute of Standards and Technology [NIST, formerly National Bureau of Standards (NBS)] silicon powder 640b (see Section 5.2.10). This accuracy may not be sufficient for measuring doping levels, stoichiometry, and similar analyses now possible with synchrotronradiation methods and the wavelength is normally determined directly from data for a standard whose lattice parameter is known with a high degree of precision, such as NIST SRM silicon 640b.
The most promising method is to use a highquality singlecrystal plate of floatzoned oxygenfree silicon, now widely available. Its lattice parameter is known to about one part in 10^{7} (Hart, 1981), which is much higher accuracy than that of the published lists of Xray wavelengths. Several orders of reflection (for example 111, 333, 444) should be used to improve the accuracy of the measurement.
Data are usually collected by stepscanning with selected constant angular increments and count times. To avoid interruptions due to refilling of the synchrotron ring, it is better to make a number of short runs rather than one long one. The data can then be added together and treated as a single data set. A shift in the orbit may cause a change in the wavelength reflected by the monochromator, and it is important to be aware of this in accurate latticeparameter determination. The peaks are narrow, and the angle increments should be small enough to produce at least a dozen points in each peak. In practice, the scans may be made to cover a range of one to two halfwidths (full widths at half height) on both sides of the peak, with increments of about 0.1 to 0.2 of the halfwidth, in order to record a sufficient number of data points for accurate profile fitting. The count time, which depends on the intensity, should be checked by determining the goodnessoffit of the calculated profiles and the experimental points (Subsection 2.3.3.8 and Chapters 8.4 and 8.6 ).
The lowerangle peaks generally have higher intensities and are therefore preferred to the higherangle peaks because of the better counting statistics. If the diffractometer can scan to negative angles, the number of strong peaks can be doubled by measuring the reflections on both sides of the zero position. The specimen can be used in either reflection or transmission, but reflection generally gives higher intensity. The lattice parameters are determined by a leastsquares analysis of the peak angles determined by profile fitting, and it is therefore necessary to measure a sufficient number of reflections to give a statistically valid result. The zeroangle position should be included as a variable parameter in the leastsquares calculation.
A precision of a few parts per million in the lattice parameter of NIST silicon has been reached with the highprecision diffractometer in the Daresbury Laboratory (Hart, Cernik, Parrish & Toraya, 1990). This instrument has an accurate gear and an incremental encoder driven by a DC servomotor with a feedback servoloop capable of positioning the detector arm within 0.36′′. A large number of repeated measurements showed a statistical accuracy of 0.0001°(2θ), corresponding to 1 in the fifth decimal place of d for λ = 1 Å and 2θ = 20°.
The recent large increase in the use of powder samples for crystalstructure refinement and analysis has also stimulated interest in latticeparameter determinations, which are derived during the course of the calculation. The most frequently used method is that of Rietveld (1967, 1969) described in Chapter 8.6 . In outline, a profilefitting function containing adjustable parameters to vary the width and shape with is selected. The parameters corresponding to the atomic positions, multiplicity, lattice parameters, etc. of the selected structure model are varied until the best leastsquares fit between the whole observed diffraction pattern and the whole calculated pattern of the model is obtained. There is no detailed published study of the accuracy of the lattice parameters that is attained but the estimated standard deviations quoted in a number of papers (see, for example, Young, 1988) appear to be comparable with those published for simple structures having no overlapped reflections. In this type of calculation, the accuracy of the lattice parameters is tied to the accuracy of the refined structure because it includes the model errors in the leastsquares residuals.
An alternative to the Rietveld method is the patterndecomposition method in which the integrated intensities are derived from profile fitting and the data used in a powder leastsquaresrefinement program. The reflections may be fitted individually or in small clusters (Parrish & Huang, 1980) or the whole pattern can be fitted (Pawley, 1981; Langford, Louër, Sonneveld & Visser, 1986; Toraya, 1986, 1988); unlike the Rietveld method, no crystalstructure model is required and only the first stage is used for lattice parameters. The Pawley method was developed for neutrondiffraction data and uses slack constraints to handle the problem of leastsquares illconditioning due to overlapping reflections, and the positions of the reflections are constrained by the lattice parameters. The refinement also determines the zeroangle calibration correction.
Toraya extended the Pawley method to Xray powder diffractometry. He first determined the profile shapes and peak positions of several standard samples by individual profile fittings to generate a curve relating the peak shifts to . A pair of split Pearson VII profiles was used for conventional patterns to handle the doublets and the profile asymmetries, and a pseudoVoight function for the nearly symmetrical synchrotronradiation profiles. The program is set up so that the parameters of the fitting function are varied with to account for the increasing widths and the peak shifts and the whole pattern is automatically fitted. The positions of the individual reflections are a function of the calculated lattice parameters, which are refined together with the integrated intensities as independent variables. This method also permits simultaneous refinement of several phases present in the pattern. Unit cells calculated from wholepattern profile fitting and incorporating the peakshift corrections had estimated standard deviations an order of magnitude smaller than those not using the systematic error correction. It is also possible to use an internal standard and to make the corrections by refining the cell parameters of the sample and holding constant the parameters of the standard.
Good results can also be obtained using selected peaks rather than the whole pattern (Parrish & Huang, 1980). Peak search or profile fitting is used to determine the observed peak positions. The leastsquares refinement is used to minimize (observed − calculated). It also determines the average and the standard deviation of all the d's and 's. In principle, all the aberrations causing shifts can be incorporated in the refinement. There are, however, large correlations between aberrations with similar angular dependencies. In practice, the zeroangle calibration correction is always determined, and the specimensurface displacement shift is usually included.
The latticeparameter determination requires an indexed pattern in which the peak angles have been determined by peak search or profile fitting. Reflections known to have poor precision because of very low intensity or close overlapping should be omitted. The estimated standard deviation is dependent on the number of reflections used and it is better to use all the well measured peaks. There is the question of using a weighting scheme in which the highangle reflections are given greater weight because of their higher accuracy for a given error. As noted in Subsection 5.2.13, higherorder reflections usually have low intensities and much overlapping. Some judgement and critical tests are often required.
There are now two basic energydispersive techniques available. In both, the specimen and detector are fixed in a selectable θ–2θ setting. The method (Giessen & Gordon, 1968) first described and most widely used requires a solidstate detector and a multichannel pulseheight analyser (Section 2.3.2 and Chapter 2.5 ). The resolution of the pattern is determined by the energy resolution of the detector and is considerably poorer than that of conventional angledispersive techniques, thereby greatly limiting its applications. The second method uses an incidentbeam monochromator, a conventional scintillation counter, and a singlechannel pulseheight analyser. The monochromator is stepscanned to select a gradually increasing (or decreasing) single wavelength (Parrish & Hart, 1987). This method permits much higher count rates, thereby reducing the time required for the experiment. Since the resolution is determined by the Xray optics, the resolution is the same as in angledispersive diffractometry (Subsection 2.3.2.4 ). The method has, however, the disadvantage that the widths of the profiles vary with energy, and unless care is taken with the step size there may be too few points per reflection to define the profile adequately. The method is particularly applicable to synchrotron radiation, but there have been no publications to date on its use for latticeparameter determination.
Energydispersive techniques (Section 2.2.3 and Chapter 2.5 ) are not ordinarily the method of choice for latticeparameter determination. Relative to angledispersive techniques, they suffer from the following disadvantages:
Nevertheless, the advantage of stationary specimen and detector may outweigh these disadvantages for special applications.
A diffractometer can be converted from angledispersive to energydispersive by (i) replacing the usual counter by a solidstate detector, (ii) replacing the usual electronic circuits by a multichannel pulseheight analyser, and (iii) keeping the specimen and detector stationary while the counts are accumulated. When so used, the geometrical aberrations are essentially the same as those of an angledispersive diffractometer, though the greater penetrating power of the higherenergy Xrays means that greater attention must be paid to the irradiated volume and the specimen transparency (Langford & Wilson, 1962; Mantler & Parrish, 1977). As Sparks & Gedcke (1972)^{1} emphasize, spacing measurements made with such an arrangement are subject to large specimensurface displacement and transparency aberrations, and the corrections required to allow for them are difficult to make. Fukamachi, Hosoya & Terasaki (1973) and Nakajima, Fukamachi, Terasaki & Hosoya (1976) showed that this difficulty can be avoided if the Soller slits are rotated about the beam directions by 90°, so that they limit the equatorial divergence instead of the axial; this was, of course, the orientation used by Soller (1924) himself. Any effect of specimensurface displacement and transparency is then negligible if ordinary care in adjustment is used, and the specimen may be placed in the reflection, or the symmetrical transmission, or the unsymmetrical transmission position (Wilson, 1973). The geometrical aberrations are collected in Table 5.2.7.1, and apply to the original orientation of the Soller slits; in the Sparks & Gedcke (1972) orientation, the usual ones apply. In general, the physical aberrations are the same for both orientations. The most difficult correction is that for the energy distribution in the incident Xray beam; aspects of this have been discussed by Bourdillon, Glazer, Hidaka & Bordas (1978), Glazer, Hidaka & Bordas (1978), Buras, Olsen, Gerward, Will & Hinze (1977), Fukamachi, Hosoya & Terasaki (1973), Laguitton & Parrish (1977) and Wilson (1973). Only the last of these is directly relevant to the latticespacing problem. The best results reported so far seem to be those of Fukamachi, Hosoya & Terasaki (1973) (0.01% in the lattice parameter).
Notation: A and B are the angular apertures (possibly equal) of the two sets of Soller slits; E is the energy of the detected photon; f (E) is the variation of a response (energy of the continuous radiation, absorption in the specimen etc.) with E; g(θ) is an angledependent response (Lorentz factor etc.); I(E −E_{1}) dE is the counting rate recorded at E when the energy of the incident photons is actually E_{1}; R is the diffractometer radius; V is the variance and μ_{3} is the third central moment of the energyresolution function I; 2X, 2Y, 2Z are the effective dimensions (possibly equal) of the source, specimen, and detector; the primes indicate differentiation; the averages <(Δθ)^{2}> etc. are over the range of Bragg angles permitted by the slits etc.

Okazaki & Kawaminami (1973) have suggested the use of a stationary specimen followed by analysis of the diffracted Xrays with a singlecrystal spectrometer. This would give some of the advantages of energydispersive diffractometry (easy control of temperature etc., because only small windows would be needed), but there would be no reduction in the time required for recording a pattern.
The types of powder camera frequently used in the determination of lattice parameters are described in Section 2.3.4 . The main geometrical aberrations affecting measurements made with them are summarized in Table 5.2.8.1. At high angles, most of them vary approximately as (π − 2θ)^{2}, and one would thus expect to obtain an approximately straightline extrapolation if the apparent values of the lattice parameter were plotted against a function something like (π − 2θ)^{2}. A function that has been found very satisfactory in practice was suggested by Nelson & Riley (1945a, b) [see also Taylor & Sinclair (1945a, b)]: This function gives linear plots down to quite small values of θ.
^{†}For van Arkel and Bradley–Jay arrangements. For Straumanis–Ievins', + or − and , respectively.

Since about 1930, it has been claimed that the lattice parameters of cubic substances could be measured within one part in 50 000. Precision (that is, reproducibility of measurements by one technique within one laboratory) of this order is achieved, but accuracy (agreement between determinations by different techniques or by the same technique in different laboratories) is lower. The IUCr latticeparameter project (Parrish, 1960) showed a standard deviation of 1 in 30 000 in interlaboratory comparisons, with some outlying values differing from the mean by one or two parts in 10 000. At that time, therefore, precision was considerably better than accuracy (absence of significant remanent systematic error). Testing for remanent systematic error is thus valuable as an occasional test of methodology, though not undertaken as routine. The principle is outlined here, and more details are given in Chapters 8.4 and 8.5 .
When refinement of parameters is performed by least squares, weighted in accordance with the reciprocal of the estimated variance, the expected value of the weighted sum of squares is where n is the number of terms summed and p is the number of parameters determined. The standard deviation of the sum S is expected to be approximately (Wilson, 1980), so that if the actual value of S exceeds (where k = 2 or 3), one can reasonably conclude that there are defects in the model (remanent systematic errors). If S is less than this value, one can reasonably conclude that any defects in the model (systematic errors) are at worst of the same order of magnitude as the statistical fluctuations; the sensitivity of the test increases rather slowly with n − p. The method was advocated by Beu and his collaborators (Beu, Musil & Whitney, 1962, 1963; Beu, 1964; Beu & Whitney, 1967; Langford, Pike & Beu, 1964; see also Mitra, Ahmed & Das Gupta, 1985) because tests of the hypothesis `no remaining systematic error' based on likelihood were available; they assumed a normal distribution of errors, possibly without realizing, and certainly without emphasizing, that the method was then equivalent to least squares. Their application of the method to testing for remanent systematic error in latticeparameter determination was successful: the aberrations of the counter diffractometer were found to be adequately accounted for: additional aberrations were found for the Bond method (see Chapter 5.3 ); Boom (1966) used it in testing the accuracy of the Debye–Scherrer method.
In statistical literature, the weighted sum of squares S is often called the scaled deviance, and is called the excess. The test for the absence of significant systematic error is then that the excess should be less than three.
The use of properly characterized materials is an important step in determining the performance characteristics of instruments and methods. The best documented and most widely used standards for powder diffraction are those from the [US] National Institute of Standards and Technology^{2} (Dragoo, 1986).
Such standards are used as specimens in diffractometers and cameras for angular calibration to determine systematic errors in the observed 2θ's for profile shapes and in intensities for quantitative analysis and for determining instrumental line profiles. The standard may be used separately as an independent specimen (`external standard'), or mixed with the powder to be investigated (`internal standard'). Some examples of the use of standards are given by Hubbard (1983) and WongNg & Hubbard (1987).
The current siliconpowder standard for 2θ calibration is Standard Reference Material (hereinafter abbreviated SRM) 640c; SRM 640, SRM 640a and SRM 640b are no longer available, but data for all four are listed in Table 5.2.10.1 for the use of workers who may still have stocks of the earlier standards. The median particle size (massweighted distribution) is about 5 µm, and 95% of the particles are < 10 µm. There is a wide range of particle sizes in SRM 640, and sieving is necessary to remove the larger particles. The agreement between SRM's 640 and 640a and between 640 and 640b is one part in 10^{−5}, and between 640a and 640b is two parts in 10^{−5}. The accuracy is given as for each. All were calculated by the use of the Deslattes & Henins (1973) Cu wavelength of 1.5405981 Å, without refraction correction, and corrected to 298 K. Because this wavelength was later found to have a systematic error (see Section 4.2.2 ), and a more accurate value, 1.5405929 (5) Å (see Table 4.2.2.1 ), is now available, this wavelength was used for SRM 640c, with the temperature adjusted to 295.6 K. The data for the earlier SRMs have also been adjusted to reflect this more accurate wavelength.

Table 5.2.10.2 lists the reflection angles for silicon 640c, silver and tungsten calculated from the adjusted NIST lattice parameters and the Table 4.2.2.1 value for the Cu wavelength. Table 5.2.10.3 lists the reflection angles of silicon 640c calculated from the Table 4.2.2.1 wavelengths for Mo , Cr and other wavelengths selected for synchrotron radiation users. The highangle reflections of silicon for Mo are listed in Table 5.2.10.4. NIST does not provide a tungsten standard, but reflection angles calculated from a = 3.16523 (4) Å at 298 K for Cu Kα_{1} = 1.5405929 Å are given in Table 5.2.10.2 and in Table 5.2.10.5 for a number of other wavelengths.




For calibration at small diffraction angles, NIST provides fluorophlogopite, a synthetic mica, as SRM 675. The (001) lattice spacing, adjusted for the revised wavelength of Cu , is 9.98101 (7) Å at 298 K. Table 5.2.10.6 lists the diffraction angles for Cu . NIST advises mixing it with silicon because the higherangle reflections may be in error because of specimen transparency. SRM 675 was purposely prepared as large particles (up to 75 µm) to encourage preferred orientation of the mica flakes; only the 00l reflections are then observed. The first reflection with Cu radiation for SRM 675 occurs at 8.853° (2θ) (Table 5.2.10.6) and a material that extends the coverage of NIST SRMs down to very low angles is silver behenate (Huang, Toraya, Blanton & Wu, 1993). The long spacing for this material, obtained with synchrotron radiation and by using SRM 640a as an internal standard, is d_{001}= 58.380 (3) Å and, for Cu radiation, there are 13 well defined and evenly spaced 00l reflections in the range 1.5 to 20°(2θ) (Table 5.2.10.7). This material is suitable for use as an external or an internal lowangle calibration standard for the analysis of materials with large unitcell dimensions and modulated multilayers with large layer periodicity.


Although the reflection angles are given to three decimal places in the tables in this section, the accuracy is lower by an amount that is not known with certainty. The lower accuracy arises from three factors: uncertainties in the lattice parameters of the W and Ag internal standards, the experimental precision, and the methods used. The wavelength given in Table 4.2.2.1 is far more accurate than these factors. The tables can probably be used to two places of decimals, the 2θ errors increasing with increasing 2θ.
In using an external standard for calibrating an instrument (without a wide receiving slit), it is essential to minimize specimensurface displacement, which shifts the measured position of the reflection (Subsection 5.2.3.1). The amount of the shift and even its direction may vary when the specimen is remounted, and it is advisable to make several measurements after removal and replacement, in order to determine the degree of reproducibility. Specimen transparency is equivalent to a variable specimensurface displacement, since the effective depth of penetration varies with the angle of incidence of the beam. The maximum shift occurs at 2θ equal to 90°, and it vanishes at 0 and 180°. For example, for silicon, the linear absorption coefficient is 133 cm^{−1} for λ = 1.54 Å and 15 cm^{−1} for 0.7 Å, shifting the 422 reflection by −0.01° at 88° and −0.05° at 37°, respectively. It should be noted that SRM silicon 640b, as supplied by NIST, exhibits measurable sample broadening (van Berkum, Sprong, de Keijser, Delhez & Sonneveld, 1995) and is thus not suitable for determining instrumental line profiles.
The measurement of intensity falls within the scope of Parts 6 and 7 . However, powder methods are much used in quantitative analysis, and the National Institute of Standards and Technology provides five standards for use as internal standards for this purpose and for checking the accuracy of diffractometer and camera intensity measurements. The five materials, certified as SRM 674, are αAl_{2}O_{3} (corundum), ZnO, TiO_{2} (rutile), Cr_{2}O_{3}, and CeO_{2}. Table 5.2.11.1, taken from the NIST certificate, is a partial list of pertinent data. The lattice parameters have an uncertainty of 3 parts in 10^{5}, which must be increased by a factor of 2 or 3 because of uncertainty in internal standards and thermal expansion. The five materials have a wide range of absorption coefficient and the crystallite size (about 2 µm) causes a small profile broadening. The table gives the intensities of the second and thirdstrongest lines relative to the strongest = 100, and the final column gives the ratio of the strongest peak to the strongest peak of Al_{2}O_{3}.

The need for standard reference materials to determine instrumental line profiles arose from the increased use in recent years of wholepattern methods (Section 5.2.6) in several applications of powder diffraction. Instrumental lineprofile standards are required to determine resolution, as a check that alignment has been optimized, or to compare the performance of different diffractometers, and to obtain sample contributions from observed data in lineprofile analysis. Different standards may therefore be required if samples of interest do not have a high absorption coefficient for the radiation used.
In addition to the usual requirements for SRMs, suitable substances for instrument characterization clearly should not exhibit any measurable sample broadening, even when used with highresolution diffractometers. Various materials were considered by the Technical Committee of the JCPDS–ICDD, in association with NIST, and lanthanum hexaboride [LaB_{6}: a_{0} = 4.15695 (6) Å at T = 299 K] was selected for use as an instrumental standard (Fawcett et al., 1988). This was subsequently marketed by NIST as SRM 660 and it also serves as a line position standard. Other materials used as instrumental standards include BaF_{2} (Louër & Langford, 1988) and KCl (Scardi, Lutterotti & Maistrelli, 1994). Both are lowcost materials, are available in large quantities, and can readily be annealed to minimize sample broadening. Although KCl introduces a measurable contribution to line breadth owing to sample transparency, it can be used to advantage for correcting data from materials having a similar absorption coefficient, such as many ceramics. van Berkum, Sprong, de Keijser, Delhez & Sonneveld (1995) selected a 5–10 µm size fraction from silicon SRM 640b, deposited about 1.5 Mg m^{−2} uniformly on a (510)oriented singlecrystal silicon wafer and annealed the whole assemblage to produce an instrument lineprofile standard. The resulting lineprofile widths were found to be slightly less than for LaB_{6} at angles below about 100°(2θ) with Cu Kα radiation.
Many factors influencing accuracy in latticeparameter determination have been mentioned in passing or discussed at length in this and previous chapters. This section attempts to summarize them and put them into perspective. Accuracy in the range of 1 to 0.1% can now be achieved routinely with average care. Increasing the accuracy to 0.01% requires considerable care in specimen preparation, data collection, instrument alignment, and calibration. The range 0.001 to 0.0001% is rarely reached and each determination is virtually a research project. The more important factors are:
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