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. 5.3, pp. 516534

Although, theoretically, the limit of accuracy in all methods based on the Bragg law [equation (5.3.1.1)] is given by the accuracy of the wavelength measurement , with photographic recording this limit is not attained. Surprisingly high accuracy may be offered by accurately applied Kossel or divergentbeam techniques. In practice, however, even in this case the accuracy achieved is poorer by an order of magnitude.
The use of Geiger–Müller, proportional, or scintillation counters together with a stepscanning motor makes it possible to record the diffraction profile in a quantitative numerical form convenient for data processing, to locate it with better accuracy and precision and, as a consequence, to obtain better accuracy and precision for the Bragg angle and thus for the lattice parameter. To make the most of this possibility, theoretical papers concerning methods of peak location, estimation of systematic and statistical errors, and optimization of the measurement were developed in parallel with constructional and experimental methods.
Methods of latticeparameter determination with counter recording form a large and heterogeneous group. As well as measurements on two or fourcircle standard diffractometers, a separate method developed by Bond (1960) and a variety of nondispersive (Xray and optical interferometry) and pseudonondispersive methods (two and threecircle spectrometers, multiplebeam techniques, and combined methods) are included in this group.
The determination of lattice parameters by the use of a standard diffractometer is based, as in the case of photographic methods, on (5.3.1.1) and (5.3.1.2), and the main task is to measure a sufficient number of reflections (the values for various hkl indices) for determining and solving the equations and for calculating the unknown parameters. The reflections can be chosen arbitrarily or in a special way (high angle, axial or nonaxial reflections).
The characteristic feature of measurements performed on a diffractometer is, however, that to satisfy the Ewald condition for a given reflection the crystal and the detector are rotated or, depending on the geometry (equatorial or inclination), shifted round their axes as well. Basic and more detailed information about the geometry of diffractometers is given elsewhere (Arndt & Willis, 1966, Chap. 3; Stout & Jensen, 1968, Section 6.3; Kheiker, 1973, Chap. 4; Luger, 1980, Chap. 4; Section 2.2.6 of this volume). For calculating the setting angles for given hkl reflections, the lattice parameters (at least preliminary values) have to be known, and conversely, if the setting angles are known, it is possible to calculate or to refine lattice parameters. Therefore, not only the values (given by the angle of rotation of the detector about the goniometer axis) but also the values of the remaining setting angles (i.e. , , and χ of the crystal rotation in equatorial geometry, or μ and for the crystal and v for the detector in inclination geometry) can be used for latticeparameter determination. This problem can be treated by a matrix analysis.
In the case of an automated fourcircle (equatorial geometry) diffractometer, the setting angles are calculated by means of the orientation matrix U, i.e. a matrix such that where is the reciprocalaxis system with metric and is the crystalfixed orthonormal system. As can be proved (Busing & Levy, 1967; Hamilton, 1974; Luger, 1980, Section 4.1.1; Gabe, 1980), the reciprocalcell parameters are related to the orientation matrix by the following equation: where is given by (5.3.3.1b). It is thus possible to calculate the lattice parameters from the terms of the orientation matrix.
The determination of the orientation matrix is usually the first step in measurements performed on the fourcircle diffractometer. This task can be accomplished when the preliminary latticeparameter values are known, and even when they are unknown. In the first case, the setting angles of two reflections, and, in the second, of three reflections, have to be determined. The procedure (Busing & Levy, 1967; Hamilton, 1974) is usually accomplished by the software of the fourcircle diffractometer. Leastsquares refinement of the lattice and orientation parameters may be performed when the setting angles of several reflections have been observed (Clegg, 1984). Appropriate constraints, resulting from the presence of symmetry elements in the given crystal structure, to be introduced during the refinement, are discussed by Bolotina (1989).
In a particular case, the fourcircle diffractometer can be used for latticeparameter measurements performed in the plane perpendicular to the main goniometer axis (say, the horizontal plane), for which χ = 0°, so that, in practice, only and values are used for latticeparameter determination (see also §5.3.3.4.1). The equations to be solved can be simplified if only axial reflections are taken into account. In an example described by Luger (1980, Section 4.2.2), the axis of a monoclinic crystal is oriented in the direction of the main axis. Then each of the two axial lengths, and (see Fig. 5.3.3.1 ), can be obtained from only one measurement: whereas values of two reflections are used to determine the angle between and axes, since This method is more suitable for orthogonal systems than for nonorthogonal ones, because of the difficulties in obtaining the proper orientation in the case of the monoclinic and, particularly, the triclinic system. In the latter case, the crystal has to be set three times.
Latticeparameter determination by the use of the twocircle (inclination) diffractometer, the socalled `Weissenberg diffractometer', is more troublesome than by means of the fourcircle one, because only two rotations [ (or ) of the crystal, and (or γ) of the detector] are motordriven under computer control, while two inclination angles (μ for the crystal and ν for the detector) must be set by hand.
The problem of application of the popular twocircle (Euleriancradle) diffractometer for measurements similar to those presented in §5.3.3.2.1 was discussed by Clegg & Sheldrick (1984). The main idea of their paper was to introduce equations combining setting angles, obtained for selected reflections, with reciprocalcell parameters, for calculating the latter. The authors started with zerolayer reflections for which, for a crystal mounted about the c axis, where and is a zeropoint correction.
The remaining parameter c had to be determined from the inclination angle μ, measured by hand. The use of zerolayer reflections was advantageous, apart from the simplicity of the formulae (5.3.3.4a,b,c,d), because they were less affected by crystal misalignment than were upperlayer reflections. However, a zeropoint correction for had to be performed. For this purpose, the value was treated as an additional parameter in offline leastsquares refinement.
As the next step, the authors introduced equations for a general crystal orientation instead of an aligned crystal (cf. §5.3.3.2.1) and derived equations defining the setting angles for an arbitrary reflection useful for data collection from a randomly oriented crystal if preliminary latticeparameter values had been assumed. This made possible measurements of reflections on a range of layers; only one crystal mounting was required. The matrix formulae suitable for Euleriangeometry diffractometers are also given by Kheiker (1973, Chap. 3, Section 9) and Gabe (1980).
In order to perform precise refinement of all six cell parameters, Clegg & Sheldrick (1984) used least squares with empirical weights: where is the width of the hkl reflection. An additional (third) motor to control the μ circle was proposed.
The authors point out that the twocircle diffractometer, owing to its simpler construction in comparison with the fourcircle one, is well suited to operations that require additional attachments; for example, for lowtemperature operation.
Every measurement is based on a certain model of its object. By `model' we understand here^{2} all the systematized a priori knowledge concerning the given measurement, necessary for planning and performing the experiment and for estimating parameters being determined. The use of an incorrect model results in a bias, i.e. an additional systematic error that may appear aside from physical and geometric aberrations. Therefore, the choice of a well founded model is essential in accurate measurements.
In the case of latticeparameter determination, the object of direct measurements is a diffraction profile, already mentioned in Subsection 5.3.1.1, and the quantity that is directly determined from the experiment is the Bragg angle .
The a priori information about the diffraction profile should define: (i) the way in which the Bragg angle is related to the measured profile , i.e. a measure of location; (ii) the mean values of the measured intensities within the profile; and (iii) their variances.
The analysis of the variance of a chosen measure of location permits a combination of the precision of the Braggangle determination, and so of the latticeparameter determination [equation (5.3.1.4)], with the scanning range = [see definition (1), §5.3.3.3.1] or truncation level α [see definition (5)], the number of measuring points n (usually ), the parameters of the profile (number of counts H in the peak position, the halfwidth ), and its shape. It is convenient to present the profile in a standardized form (Thomsen & Yap, 1968) as: where are standardized angle values and is the shape function, not dependent on the parameters H and . For each measure of location [definitions (1)–(5) of §5.3.3.3.1(i)], there is the dependence: where is the peak intensity, T is the total counting time, and F is a dimensionless factor that depends on the measure of location and the shape of the profile.
Since, in the case of fixedtime counting, the total counting time T is proportional to the number n of measuring points: where is the counting time, and since the number of counts h is proportional to the intensity I: and, in particular, the number of counts H in the peak position is proportional to the peak intensity : the dependence (5.3.3.11) can be presented as Thus, for a given measure of location and given shape of the profile [equations (5.3.3.10), (5.3.3.10b)], the variance depends on the ratio of the profile parameters and decreases with an increase of the number of points, n.
In particular, the variance of the peak [definition (3), §5.3.3.3.1] of the leastsquares parabola has been estimated (Wilson, 1965) as where is the second derivative of in the peak position and p is a number such that n = 2p + 1 (n 2p, if p is sufficiently large).
Taking into account the standardization performed [equations (5.3.3.10), (5.3.3.10a,b)], equation (5.3.3.15) can be rewritten in the form: where X is the standardized scanning range and is the second derivative of the shape function in the peak positions. By comparing (5.3.3.16) and (5.3.3.14), we find the factor F in this case to be From (5.3.3.14) and (5.3.3.16b), the variance of the peak of the leastsquares parabola decreases with an increase of the scanning range. On the other hand, the bias of the peak position, resulting from the asymmetry of the profile, is proportional to (Wilson, 1965): where and are the second and third derivatives of a function describing the profile at its peak position . These two aspects should be taken into account in choosing the scanning range. Yet, as shown by Gałdecka (1993b; Section 5), (5.3.3.16c) may be applied to reduce the bias by extrapolating to the results obtained within various scanning ranges.
In the case of polynomials of higher (and even) degrees (m = 4, 6, 8) and , the factor F can be expressed by a semiempirical dependence (Thomsen, 1974; Gałdecka, 1993b): but it is difficult to evaluate the bias. Therefore, as shown by Gałdecka (1993b), polynomials of higher degrees have no advantage over a leastsquares parabola.
To minimize the bias, a reasonable shape function may be used rather than a polynomial (Gałdecka, 1993a, b). The function should be continuous (including its derivatives), not negative and closely related to known physical models of the diffraction profiles. Since the measured diffraction profiles are, as a rule, asymmetric, the proper selection of a description of asymmetry is of primary importance. The use of the socalled `split functions', consisting of two `half' functions of the same (or different) shape and different halfwidths, leads to a noticeable bias, so such functions must not be used for accurate latticeparameter determination.
The variance of a single midpoint of a chord [definition (5), §5.3.3.3.1] has been estimated by Bačkovský (1965) as where and are the variances of the coordinates and h, respectively, and is the first derivative at the ith point. If it is assumed that is small in relation to the second component of (5.3.3.17) and if (5.3.3.9) and the standardizations (5.3.3.10), (5.3.3.10a,b) are taken into consideration, (5.3.3.17) can be rewritten in the form: where is the first derivative of the shape function in the ith position.
Comparison of (5.3.3.18) and (5.3.3.14), with n = 2, leads to For an arbitrary shape function describing the diffraction profile, it is thus possible to find such a truncation level [§5.3.3.3.1, definition (5)], for which F is a minimum. If the shape function is the Cauchy function, the optimum truncation level is , and the resulting F factor, .
In spite of a large bias introduced by the midpoint of a single chord (the difference between its position and the peak position), this measure of location is preferred by Barns (1972), because the calculations are less timeconsuming than those for other points of the profile. Barns takes α = 0.5 [F = 1 for the Cauchy function; equations (5.3.3.18a), (5.3.3.19)] and compensates the bias at this level by determining an effective value of the wavelength based on a silicon standard.
The estimators of the variance for the centroid and the median given by Wilson (1967, 1968, 1969), or estimators of both the variance and the bias of the extrapolatedpeak position given by Gałdecka (1994) can also be the basis of the choice of the scanning range if these measures of location are applied.
The other possibility of affecting the precision of the measurements is to change the shape and the parameters of the profile [see equations (5.3.3.14), and (5.3.3.16b) or (5.3.3.18a)] by changing the apparatus parameters [the influence on , equation (5.3.1.6)], or the Xray source profile , or the crystal profile .
An example of the first possibility is the optimization of the parameters of inplane collimation in the case when the peak of the leastsquares parabola is used as the measure of the location (Urbanowicz, 1981a). Since both the shape and the parameters of the profile depend on the collimation parameters, the task is to choose collimatorslit dimensions to minimize the value [cf. equation (5.3.3.16)]. As a result of detailed considerations, under the assumption given by (5.3.3.9), the optimum exists and is defined by the following formula: where and are the widths of the slits, L is the collimator length, and is the halfwidth of the original profile [cf. equations (5.3.1.6), (5.3.1.7), and (5.3.1.8)]. Systematic errors connected with collimation have been discussed separately (Urbanowicz, 1981b).
The width of the original profile can be reduced by means of spectrally narrow sources or by the use of additional crystal(s) in multiplecrystal methods (Subsection 5.3.3.7). The latter also affects the crystal profile .
A diffractometer in which both and scans are available, intended for precise and accurate latticeparameter determination, is sometimes called a onecrystal spectrometer, by analogy with a similar device used for wavelength determination. This name has been used by Lisoivan (1982), who in his review paper described various properties and applications of such a device.
Braggangle determination with the onecrystal spectrometer can be performed in an asymmetric as well as in a symmetric arrangement (Arndt & Willis, 1966, pp. 262–264). In the asymmetric arrangement (Fig. 5.3.3.3a ), the angle is the difference between two detector positions, related to the maximum intensity of the diffracted and the primary beam, respectively. Braggangle determination in such an arrangement is subject to several systematic errors; among these zero error, eccentricity, and absorption are of great importance. As shown by Berger (1984), the latter two errors can be eliminated when Soller slits are used.

Determination of the Bragg angle by means of the onecrystal spectrometer using (a) an asymmetric or (b) a symmetric arrangement. The zero position of the detector arms must be known in (a), but not in (b). After Arndt & Willis (1966). 
To eliminate the zero error, a symmetric diffractometer may be used, in which each measurement of the Bragg angle is performed twice, for two equivalent diffracting positions of the sample, symmetrical in relation to the primarybeam direction (Fig. 5.3.3.3b). The respective positions of the counter (or counters, since sometimes two counters are used) are also symmetrical. Such an arrangement may be considered to be (Beu, 1967), in some ways, the diffractometer counterpart of the Straumanis film method (Straumanis & Ieviņš, 1940). From geometric considerations, the absolute value of the angle between the two counter positions is and the absolute value of the angle between the two sample positions, and , is 180° − , so that both and scans can be used for the Braggangle determination.
As was mentioned in §5.3.2.3.4(vi), the idea of calculating the angle from the two sample positions has been used with photographic methods (Bragg & Bragg, 1915; Weisz, Cochran & Cole, 1948). Bond (1960), in contrast, was the first to apply this to measurements on the counter diffractometer, and proved that, owing to the geometry, not only the zero error but also the eccentricity, absorption, and several other errors can be reduced.
Although the Bond (1960) method, based on a symmetric arrangement presented in §5.3.3.4.3, makes possible higher accuracy than that obtained by means of a standard diffractometer, an asymmetric arrangement proves to be more suitable for certain tasks connected with latticeparameter measurement, because of its greater simplicity. The more detailed arguments for the use of such a device result from some disadvantages of the Bond method, discussed in §5.3.3.4.3.4.
One of the earliest and most often cited methods of latticeparameter determination by means of the counter singlecrystal diffractometer (in an asymmetric arrangement) is that of Smakula & Kalnajs (1955). The authors reported unitcell determinations of eight cubic crystals. The systematic errors due to seven factors were analysed according to the formulae derived by Wilson (1950) and Eastabrook (1952) for powder samples, and valid also for single crystals. The lattice parameters computed for various diffraction angles were plotted versus ; extrapolation to ° gave the lattice parameters corrected for systematic errors. Accuracy of 4 parts in 10^{5}, limited by the uncertainty of the Xray wavelength, and precision of 1 part in 10^{6} were achieved.
A more complete list of factors causing broadening and asymmetry of the diffraction profile, and so affecting statistical and systematic errors of latticeparameter determination, has been given by Kheiker & Zevin (1963, Tables IV, IVa, and IVb). Since the systematic errors due to the factors causing asymmetry (specimen transparency, axial divergence, flat specimen) are, as a rule, dependent on the Bragg angle and proportional to , , or , they can be removed or reduced – as in the method of Smakula & Kalnajs (1955) – by means of extrapolation to °. The problem has also been discussed by Wilson (1963, 1980) in the case of powder diffractometry [cf. §5.3.3.3.1(i)]. When comparing the considerations of Kheiker & Zevin and Wilson [the list of references concerning the subject given by Kheiker & Zevin (1963) is, with few exceptions, contained in that given by Wilson (1963)], it will be noticed that some differences in the formulae result from differences in the geometry of the measurement rather than from the different nature of the samples (single crystal, powder).
As in the photographic methods, the accurate recording of the angular separation between and diffraction lines can be the basis for latticeparameter measurements with a diffractometer (Popović, 1971). The method allows one to reduce the error in the zero setting of the scale and the error due to incorrect positioning of the sample on the diffractometer, since the angular separations are independent of the zero positions of the and scales.
An example of a contemporary method of latticeparameter determination is given by Berger (1984). As has been mentioned in §5.3.3.4.1, the characteristic feature of the device is the Soller slits, which limit the divergence of both primary and diffraction beams and, at the same time, eliminate errors due to eccentricity and absorption. On the other hand, systematic errors due to refraction, vertical inclination, vertical divergence, and Sollerslit inaccuracy, as well as asymmetry of profiles and crystal imperfection, have to be analysed.
Since, in this case, the angle between the incident and the reflected beam is measured, the inclinations of both beams must be considered. As a result of the analysis [analogous to that of Burke & Tomkeieff, 1969; referred to in §5.3.3.4.3.2(4)], the following expression for the angular correction (to be added to the measured value of ) is obtained: where α and γ are the vertical inclinations of the incident and reflected beams, respectively. The correction for vertical divergence is presented in §5.3.3.4.3.2(3).
The Sollerslit method, the accuracy and precision of which are comparable to those obtained with the Bond method, is suitable both for imperfect crystals, since only a single diffracting position of the sample is required, and for perfect samples, when an exactly defined irradiated area is required. It is applicable to absolute and to relative measurements. Examples are given by Berger, Rosner & Schikora (1989), who worked out a method of absolute latticeparameter determination of superlattices; by Berger, Lehmann & Schenk (1985), who determined latticeparameter variations in PbTe single crystals; and by Berger (1993), who examined point defects in II–VI compounds.
An original method, based on determining the Bragg angle from a twodimensional map of the intensity distribution (around the reciprocallattice point) of highangle reflections as a function of angular positions of both the specimen and the counter, was described by Kobayashi, Yamada & Nakamura (1963) and Kobayashi, Mizutani & Schmidt (1970). A finely collimated Xray beam, with a halfwidth less than 3′, was used for this purpose. The accuracy of the counter setting was ±0.1°, the scanning step °. Systematic errors depending on the depth of penetration and eccentricity of the specimen were reported, and were corrected both experimentally (manifold measurements of the same planes for different diffraction ranges, and rotation of the crystal around its axis by 180°) and by means of extrapolation. The correction for refraction was introduced separately. The method was used in studies of the antiparallel 180° domains in the ferroelectric barium titanate, which were combined with optical studies.
The determination of variations in the cell parameter of GaAs as a function of homogeneity, effects of heat treatments, and surface defects has been presented by Pierron & McNeely (1969). Using a conventional diffractometer, they obtained a precision of 3 parts in 10^{6} and an accuracy better than 2 parts in 10^{6}. The systematic errors were removed both by means of suitable corrections (Lorentz–polarization factor and refraction) and by extrapolation.
A study of the thermal expansion of αLiIO_{3} over a wide range of temperatures (between 20 and 520 K) in the vicinity of the phase transition has been reported by Abrahams et al. (1983). Latticeparameter changes were examined by means of a standard diffractometer (CAD4); absolute values at separate points were measured by the use of a Bondsystem diffractometer.
An apparatus for the measurement of uniaxial stress based on a fourcircle diffractometer has been presented by d'Amour et al. (1982). The stress, produced by turning a differential screw, can be measured in situ, i.e. without removing the apparatus from the diffractometer. An example of latticeparameter measurement of Si stressed along [111] is given, in which the stress parameter ζ is calculated from intensity changes of the chosen 600 reflection.
By the use of the symmetric arrangement presented in §5.3.3.4.1 (Fig. 5.3.3.3b), it is possible to achieve very high accuracy, of about 1 part in 10^{6} (Bond, 1960), and high precision (Baker, George, Bellamy & Causer, 1968) but, to make the most of this, some requirements concerning the device, the sample, the environmental conditions, the measurement itself, and the data processing have to be fulfilled; this problem will be continued below.
Bond (1960) in his notable work used a large, highly pure and perfect single crystal (zonerefined silicon) in the shape of a flat slab. The scheme of the method is given in Fig. 5.3.3.4 . The crystal was mounted with the reflecting planes accurately parallel to the axis of the shaft on a graduated circle (clinometer), the angular position of which could be read accurately (to 1′′). The Xray beam travelling from the tube through a collimator (two 50 µm slits, 215 mm apart, so that the halfwidth of the primary beam was 0.8′) fell directly upon the crystal, set in one of the two diffracting positions. The diffracted beam was intercepted by one of two detectors [Geiger–Müller (G–M) counters], which were fixed in appropriate positions. The detectors were wide open, so that their apertures were considerably wider than the diffracted beam, which eliminated some systematic errors depending on the counter position. The crystal was rotated step by step through the reflecting position to record the diffraction profile. Next, the peak positions of both profiles were determined by the extrapolatedpeak procedure [§5.3.3.3.1, definition (4)] to find the accurate positions of the sample, and , from which the Bragg angle was calculated by use of a formula that can be written in a simple form as Before calculating the interplanar distance [equation (5.3.1.1)] or, in the simplest case, the lattice parameter directly, the systematic errors have to be discussed and evaluated. Sometimes, corrections are made to the parameters themselves rather than to the values. The reader is referred to §5.3.3.4.3.2, in which present knowledge is taken into account, rather than to Bond's original paper.
Bond performed measurements at room temperature (298 K) for reflections 444, 333, and 111 and, after detailed discussion of errors, reported the values (in kXU), which related to these measurements (standard deviations are given in parentheses), as 5.419770 (0.000019), 5.419768 (0.000031), 5.419790 (0.000149). These values are referred to λ = 1.537395 kXU. These results were then tested by Beu, Musil & Whitney (1962) by means of the likelihoodratio method to test the hypothesis of `no remaining systematic errors'. They proved that the estimate for this sample of silicon is accurate within the stated precision (1 part in 390 000).
The results reported in Bond (1960) – very high accuracy and remarkable reproducibility (low standard deviation), obtained by use of a relatively simple device, which can be realized on the basis of a standard diffractometer – encourage experimenters to perform similar measurements. However, many problems arise with the adaptation of the Bond method to other kinds of samples and/or to other purposes than those described by Bond (1960) in his original paper. Both theoretical and experimental work have increased the accuracy and the precision of the method during the last 35 years.
As mentioned above (§5.3.3.4.1), some systematic errors that affect the asymmetric diffractometer are experimentally eliminated in the Bond (1960) arrangement. According to Beu (1967), who has supplemented the list of errors given by Bond, the following systematic errors are eliminated at the level:

As well as these errors there are other systematic errors, due to both physical and apparatus factors, which should be eliminated by suitable corrections.

The Bond (1960) method, in its first stage, was meant for large, specially cut and set samples. In principle, only one lattice parameter can be determined in one measuring cycle. As has been shown, the method can also be adapted to other samples, with noncubic symmetry, and to geometries of the illuminated area, different from those used by Bond. This task needs, however, some additional operations and often some additional corrections for systematic errors.
The basic application of the Bond (1960) method, because its geometry reduced several systematic errors, was to absolute latticeparameter measurements. The method also proved useful in precise investigations of latticeparameter changes.
Bondsystem diffractometers were most often realized in practice on the basis of standard diffractometers under computer control (Baker, George, Bellamy & Causer, 1968; Segmüller, 1970; Pihl, Bieber & Schwuttke, 1973; Kucharczyk, Pietraszko & Łukaszewicz, 1993). Some were designed for special investigations, such as highprecision measurements, (Baker, George, Bellamy & Causer, 1966; Grosswig, Härtwig, Alter & Christoph, 1983; Grosswig et al., 1985; Grosswig, Härtwig, Jäckel, Kittner & Melle, 1986); local measurements at chosen points of a specimen (Lisoivan & Dikovskaya, 1969; Lisoivan, 1974, 1982); examination of latticeparameter changes over a wide temperature range (Łukaszewicz et al., 1976, 1978; Okada, 1982); or the effect of high pressure on lattice parameters (Mauer, Hubbard, Piermarini & Block, 1975; Leszczyński, Podlasin & Suski, 1993).
By introduction of synchrotron radiation to a Bondsystem diffractometer (Ando et al., 1989), a highly collimated and very narrow beam has been obtained, so latticeparameter measurements can be accomplished reliably and quickly with a routinely achieved precision of 2 parts in 10^{6}; these can be combined with Xray topographs made in selected areas of the sample.
(1) Crystals with different symmetry. Cooper (1962) used the Bond (1960) diffractometer and method for absolute measurements of lattice parameters of several crystals belonging to various orthogonal systems. Special attention was paid to preparing the samples, i.e. cutting and polishing, to obtain crystal surfaces parallel to the planes of interest. One sample of a given substance was sufficient to find the lattice in the case of cubic crystals but two samples were required for tetragonal and hexagonal systems, and three were necessary for the orthorhombic system. This difficulty increases when nonorthogonal lattices have to be examined. This problem was resolved by Lisoivan (1974, 1982), who used very thin singlecrystal slabs, which made possible measurements both in reflection and in transmission. Lisoivan (1981, 1982), developing his first idea, derived the requirements for a precision determination of all the interaxial angles for an arbitrary system. The coplanar lattice parameters can also be determined in one crystal setting when only reflection geometry is used (Grosswig et al., 1985).
Superlattices can be determined using the system proposed by Bond; a simple method for this purpose was derived by Kudo (1982).
(2) Different sample areas. A separate problem is to adapt the Bond method for measurement of small spherical crystals, commonly used in structure investigations. A detailed analysis of this problem is given by Hubbard & Mauer (1976), who indicate that the effect of absorption and horizontal divergence has to be taken into account if the sample dimensions are less than the cross section of the primary beam. As has been mentioned above (§§5.3.3.4.1, 5.3.3.4.3.2), these factors, as well as eccentricity and uncertainty of the zero point, could be neglected in Bond's (1960) experiment. Kheiker (1973) considered systematic errors resulting from the latter two factors when small crystals are used. He proposed a fourfold measurement of the sample position (rather than a twofold one used by Bond), in which `both sides' of a given set of planes are taken into account, so that measurement by the Bond method is performed for two pairs of specimen positions: and , and and . The corresponding positions of the counter are also determined and used in calculations of the Bragg angle (cf. §5.3.3.4.1). The mean value of the angle is not subject to the errors mentioned. A similar idea has been presented by Mauer et al. (1975).
In many practical cases, it is necessary to determine lattice parameters of thin superficial layers. One of the possibilities is to use the Bond method for this purpose. Wołcyrz, Pietraszko & Łukaszewicz (1980) used asymmetric Bragg reflections with small angles of incidence, to reduce the penetration depth of Xrays. This rather simple method permits high accuracy if proper corrections (the formulae are given by the authors) resulting from the dynamical theory of diffraction of Xrays are carefully determined. This method was used to estimate the gradient of the lattice parameter inside diffusion layers. The penetration depth was changed by rotation of the sample. Golovin, Imamov & Kondrashkina (1985) achieved a penetration depth as small as about 1 to 10 nm, using Xray totalreflection diffraction (TRD) from the planes normal to the surface of the specimen. The sample was oriented in such a way that the conditions for total external reflection were satisfied when the Xray beam fell on the sample at a small angle of incidence, about 0.5°.
The homogeneity of the crystal in a direction parallel to its surface may be examined by means of local measurements, described by Lisoivan & Dikovskaya (1969) and Lisoivan (1974), in which the goniometer head was specially designed so that the sample could be precisely set and displaced.
(3) Determination of latticeparameter changes. Baker, George, Bellamy & Causer (1968) have shown that a carefully manufactured and adjusted Bondsystem diffractometer (mentioned above) with good stability of environmental conditions (temperature, pressure, power voltage) may be a suitable tool for the investigation of latticeparameter changes. A static method of thermalexpansion measurement is proposed, in which changes in angle of an in situ specimen due to changes in the lattice parameter with temperature are quickly determined. If it is assumed that the intensity and the shape of the peak have not altered with the change of conditions (cf. the method based on doublecrystal diffractometers in §5.3.3.7.1), the change in angle can be determined by intensity measurement alone. The reported precision of the relative measurement is 1 part in 10^{7}. Since the shape of the profile may change with the change of conditions, the whole profile must be determined accurately and precisely, so that the whole experiment, consisting of a series of measurements, is timeconsuming. The optimization problems resulting from this inconvenience have been discussed above (§5.3.3.3.2; Barns, 1972; Urbanowicz, 1981a,b).
In particular, thermalexpansion studies can detect phase transitions and the resulting changes in crystal symmetry (Kucharczyk, Pietraszko & Łukaszewicz, 1976; Kucharczyk & Niklewski, 1979; Pietraszko, Waśkowska, Olejnik & Łukaszewicz, 1979; Horváth & Kucharczyk, 1981; Pietraszko, Tomaszewski & Łukaszewicz, 1981; Keller, Kucharczyk & Küppers, 1982; Åsbrink, Wołcyrz & Hong, 1985a,b).
Another group of applications of the Bond method is connected with singlecrystal characterization problems (homogeneity, doping, stoichiometry) resulting from technological operations (epitaxy, diffusion, ion implantation) producing changes in lattice spacings, δd/d = 10^{−2} to 10^{−5}. The examples cited below show a variety of applications.
Stępień, Auleytner & Łukaszewicz (1972) and StępieńDamm, Kucharczyk, Urbanowicz & Łukaszewicz (1975) examined γirradiated NaClO_{3}. The effect of Xray irradiation on the lattice parameter of TGS crystals in the vicinity of the phase transition was studied by StępieńDamm, Suski, Meysner, Hilczer & Łukaszewicz (1974). Pihl, Bieber & Schwuttke (1973) dealt with ionimplanted silicon, using a Bondsystem diffractometer for local measurements. The effect of silicon doping on the lattice parameters of gallium arsenide was studied by Fewster & Willoughby (1980). Crystalperfection studies by the Bond method were reported by Grosswig, Melle, Schellenberger & Zahorowski (1983), and Wołcyrz & Łukaszewicz (1982). In the latter paper, the measurements were performed on a superficial singlecrystal layer by the use of the geometry described above [paragraph (2)] (Wołcyrz, Pietraszko & Łukaszewicz, 1980). Lattice distortion in LiF single crystals was examined by Dressler, Griebner & Kittner (1987), who used the method of Grosswig et al. (1985) [cf. paragraph (1)]. The use of anomalous dispersion in studies of microdefects was considered by Holý & Härtwig (1988).
The significant advantages of the Bond (1960) method, such as:

make this method one of the most popular at present.
The method, however, has the following limitations:

Nevertheless, the geometry proposed by Bond (1960), owing to its advantages, is commonly used in precise and accurate multiplecrystal spectrometer methods (§§5.3.3.7.1, 5.3.3.7.2).
Other limitations concerning the precision and accuracy of the method are common to it and to all the `traditional' methods (Subsection 5.3.3.5).
As `traditional' are considered the methods that depend on a comparison of the lattice spacings to be determined with the wavelength values of characteristic Xradiation that comes directly from laboratory (Bremsstrahlung) sources. The emission lines are wide and asymmetric, which limits both the accuracy and precision of latticeparameter measurements (as discussed in Subsection 5.3.1.1). One of the limiting factors is the uncertainty of the wavelength value. For many years, the wavelength values determined by Bearden (1965, 1967) with an accuracy of 5 parts in 10^{6} were widely used. At present, owing to remarkable progress in the measurement technique, it is possible to achieve an accuracy in wavelength of an order better, and nowadays remeasurements of some characteristic emission Xray wavelengths are reported [cf. §5.3.3.3.1(iii) and Subsection 5.3.3.8]. Yet, even after reducing the uncertainty in wavelength, and after introducing all necessary corrections for systematic errors, the highest accuracy of traditional methods does not exceed 1 part in 10^{6} (cf. Subsection 5.3.3.8).
The accuracy of an order better is possible with Xray and optical interferometry. This nondispersive method (cf. Subsection 5.3.3.8) is used for accurate latticespacing determination of highly perfect standard crystals; the standards are next used for both latticeparameter determination with a doublebeam comparison technique (Baker & Hart, 1975; see also §5.3.3.7.3) and for the accurate wavelength determination mentioned above.
Another problem is the limited precision attainable by traditional methods. As was discussed in Subsection 5.3.1.1, the width of the diffraction profile depends on the spectral distribution of the wavelength, (5.3.1.6), (5.3.1.7), (5.3.1.8), and cannot be less than this owing to the wavelength dispersion. However, much has been done to approach this limit and to attain the precision and accuracy of the diffraction profile location (cf. Subsection 5.3.3.3). The highest precision of latticeparameter determination that it is possible to achieve with traditional methods is about 1 part in 10^{7}. For some problems connected with singlecrystal characterization, such as the effect of irradiation, stress, defect concentration, including local measurement (topography), better precision is required.
From (5.3.1.9), the other possibility of increasing precision, besides choosing optimum parameters for the measurement and improvement of profilelocation methods, is to influence the original profile . This aim can be attained either by applying spectrally narrower Xray sources or by reducing the width of the original profile by means of arrangements with additional crystals playing the role of monochromator and reference crystal. This second possibility is applied in double or triplecrystal spectrometry, in multiplebeam methods, or in combined methods. These methods are called `pseudonondispersive' methods, since the width of the diffraction profile is considerably limited in them owing to considerable limitation of the width of the original profile. A similar situation to that in ncrystal spectrometers, in which the beam reflected from one set of crystal planes is the source of radiation for the second (or the next) diffraction phenomena, arises in multiplediffraction methods; this is described in Subsection 5.3.3.6.
A systematic and well illustrated review of pseudonondispersive and other differential methods is given by Hart (1981), who is the author of numerous papers on this subject.
Multiple diffraction occurs when two or more sets of planes simultaneously satisfy the Bragg law for a single wavelength λ. The beam diffracted from one set of planes becomes the incident beam within the crystal for the next diffraction. In the reciprocalspace representation, this means that three or more reciprocallattice points lie simultaneously on the Ewald sphere (Fig. 5.3.3.5 ). These points can be detected by successive rotations of the crystal, as described below. This phenomenon, known also as simultaneous reflection or (after Renninger, 1937) `Umweganregung', may be observed in both Xray and neutron experiments. In the first case, it occurs both in methods with counter recording, initiated by Renninger (1937), and in methods with photographic recording in which a highly divergent beam is used (§5.3.2.4.2). The intersections of conic sections encountered in the methods developed by Kossel (1936) and Lonsdale (1947) are the cases of multiplediffraction phenomena in photographic methods.
Simultaneous reflection, undesirable in some cases (`forbidden' reflections in measurements of intensities) can be very useful in others. Its various applications have been reviewed by Terminasov & Tuzov (1964) and Chang (1984). Only the utility of multiple diffraction in latticeparameter determination will be discussed here.
The principle of Renninger's (1937) experiment, which is also the basis of the method described by Post (1975), is shown in Fig. 5.3.3.6 . The crystal in the shape of a slab is at first set in a position to diffract the primary Xray beam. A primary reflection whose intensity is very low or which is forbidden by the space group of the crystal is usually selected. Its intensity determines the background intensity of the pattern, which should be low. The detector, with a wideopen window, is situated in the appropriate position and remains fixed throughout the experiment while the crystal is rotated around the axis perpendicular to the crystal planes (and its surface) to record successive reflections.

Schematic representation of the multiplediffraction method. (a) Experimental setup (after Cole, Chambers & Dunn, 1962; Post, 1975). (b) Geometric representation in reciprocal space. 
The multiplediffraction pattern (an example is shown in Fig. 5.3.3.7 ) has next to be indexed. The principle of the method of indexing, known as the referencevector method (Cole, Chambers & Dunn, 1962; Post, 1975; Chang, 1984), is shown in Fig. 5.3.3.6(b). Directions of the primary and diffracted beams are marked by vectors and K. The ends of the vectors lie on the Ewald sphere, the radius of which is equal to 1/λ. The reciprocal vector , being the difference between the vectors K and , represents the first diffraction phenomenon, which is observed for setting angles equal to and μ (usually ). Let us assume that the reciprocal vector , observed for setting angles and ν, represents the next diffraction. The vector components of H, parallel and normal to P, are denoted by and , respectively. For a given wavelength, the lengths of respective vectors are functions of lattice parameters and diffraction indices.
The task is to find the relationship between the difference β of angles of rotation (or between two values of the setting angles, β and ν) and the lengths of the reciprocal vectors. The following relations result from Fig. 5.3.3.6(b): Taking these into consideration, and remembering that , we finally obtain
Since cos(−β) = cosβ, the appearance of successive reflections does not depend on the direction of rotation. A detailed discussion of (5.3.3.34) is given by Cole, Chambers & Dunn (1962) and Chang (1984). When preliminary values of the lattice parameters are known, (5.3.3.34) can be applied for indexing multiplediffraction patterns. A computer program (Rossmanith, 1985) can be very useful in rather complicated calculations and in the graphical representation of the multiplediffraction pattern. Formula (5.3.3.34), since , can be presented in another form: where two setting angles, μ and β, are taken into account. When the indices are known, both (5.3.3.34) and (5.3.3.35) can be used for the determination or refinement of lattice parameters.
Another analytical method for indexing multiplediffraction patterns, based on the determination of the Lorentz point, has been described by Kshevetsky, Mikhailyuk, Ostapovich, Polyak, Remenyuk & Fomin (1979).
Formulae (5.3.3.34) and (5.3.3.35) are valid for all crystal systems. In practice, however, the rather complicated method is used mainly for cubic crystals, and a special approach proved to be needed in order to adapt the method to other (rectangular) systems (Kshevetsky, Mikhalchenko, Stetsko & Shelud'ko, 1985). In the case of a cubic lattice, it is convenient to substitute into (5.3.3.34) and (5.3.3.35) rather than R = 1/λ used in the general case, so that the lengths of the reciprocal vectors, being now functions of the indices only, are: The lengths of the components of H can be determined from (5.3.3.35b,c,d) taking and . After introducing the alterations [equation (5.3.3.35a) and the resulting equations (5.3.3.35b,c,d)], (5.3.3.35) now describes a simple dependence between the ratio a/λ, the indices, and the setting angles.
The accuracy of the latticeparameter determination resulting from (5.3.3.35) in the cubic case can be assumed to be: this thus depends on the values of the setting angles β, μ and their accuracies , . The latter depend on various systematic errors.
Since the differences between the two angular settings at which a given set of planes diffracts are measured rather than their absolute values, the systematic errors due to absorption, specimen displacement, and zerosetting are eliminated. In contrast, errors due to vertical divergence, refraction and the change of wavelength of the incident radiation (when it enters the crystal), alignment, and dynamical effects should be taken into account. In the case described by Post (1975), when a fine focus (effective size 0.4 × 0.5 mm) and collimation limiting the beam divergence to 2′ were used, the vertical divergence causing the relative error in d of about could be ignored.
The errors due to the real structure (inhomogeneity, mosaicity and internal stress) were discussed by Kshevetsky et al. (1979).
The accuracy possible by this method (from 1 to 4 parts in 10^{6}) is comparable with that obtained with the Bond (1960) method. The advantages of this method from the point of view of latticeparameter determination are as follows:

A disadvantage, on the other hand, is the complicated interpretation (indexing) of multiplediffraction patterns, so that this method is less popular than the Bond (1960) method.
The Post (1975) method has been applied to the accurate latticeparameter determination of germanium, silicon, and diamond single crystals (Hom, Kiszenick & Post, 1975).
Detailed information concerning the doublecrystal spectrometer, which consists of two crystals successively diffracting the Xrays, can be found in James (1967, pp. 306–318), Compton & Allison (1935), and Azároff (1974). This device, usually used for wavelength determination, may also be applied to latticeparameter determination, if the wavelength is accurately known. The principle of the device is shown in Fig. 5.3.3.8 . The first crystal, the monochromator, diffracts the primary beam in the direction defined by the Bragg law for a given set of planes, so that the resulting beam is narrow and parallel. It can thus be considered to be both a collimator (or an additional collimator, if the primary beam has already been collimated) and a wavelength filter. The final profile , obtained as a result of the second diffraction by the specimen when the first crystal remains stationary and the second is rotated, is narrower than that which would be obtained with only one crystal. The final crystal profile [cf. equation (5.3.1.6)] is due to both crystals, which, if it is assumed that they are cut from the same block, can be described by the autocorrelation function (Hart, 1981): where is an individual reflectivity function of one crystal and K is a coefficient of proportionality. Its halfwidth is 1.4 times larger than that related to only one crystal. In spite of this, the recorded profile can be as narrow as, for example, 2.6′′ (Godwod, Kowalczyk & Szmid, 1974), since the profile due to the wavelength , modified by the first crystal, is extremely narrow. Additional advantages of the diffraction profile are: its symmetry, because is symmetric as an effect of autocorrelation, and smoothness, as an effect of additional integration. The profile can thus be located with very high accuracy and precision.
When there is a small difference in the two lattice spacings, so that one has a value d and the other , if is small enough, it can be assumed that the profile does not alter in shape but in its peak position [cf. §5.3.3.4.3.3, paragraph (3)]. If for two identical crystals this were located at , the peak position shifts to . The measurement of this shift rather than the absolute position of the rocking curve is the basis of all the doublecrystal methods. An example of the application of a doublecrystal spectrometer with photographic recording has been given in §5.3.2.3.5 (Bearden & Henins, 1965).
The basic requirements that should be fulfilled to make the most of the doublecrystal spectrometer are: limitation of the primary beam by means of a collimator, parallelism of the two axes [precision as high as 1′′ obtained by Godwod, Kowalczyk & Szmid (1974)], and high thermal stability (0.1 K; Godwod, Kowalczyk & Szmid, 1974). Alignment procedure, errors, and corrections valid for the doublecrystal spectrometer have been considered by Bearden & Thomsen (1971).
The doublecrystal diffractometer, because of the small width of the diffraction profile, is a very suitable tool for local measurements of latticeparameter differences, for example between an epitaxic layer and its substrate. Hart & Lloyd (1975) carried out such a measurement on a standard singleaxis diffractometer (APEX) to which a simple second axis, goniometer head, and detector were added (Fig. 5.3.3.9 ). The diffracted beam was recorded simultaneously by three detectors. A symmetric arrangement with two detectors, and , with no layer present, makes possible the determination of the absolute value of the lattice parameter of the substrate, as in the Bond (1960) method. The third detector makes it possible to record the doublecrystal rocking curve, which usually fully resolves the layer and substrate profiles. The changes in the lattice parameter between the two components can be used for determination of strain (at 1 part in 10^{4}).

Schematic representation of the doublecrystal arrangement of Hart & Lloyd (1975) for the examination of epitaxic layers. (a) Experimental setup. (b) Diffraction profiles recorded by detectors D_{1}, D_{2}, and D_{3}. 
The very important advantage of this method, from the point of view of local measurements, is that single or doublecrystal diffraction can be selected, simultaneously if needed, on exactly the same specimen area. Other examples of strain measurements by means of a doublecrystal spectrometer are given by Takano & Maki (1972), who measured lattice strain due to oxygen diffusing into a silicon single crystal; by Fukahara & Takano (1977), who compared experimental rocking curves and theoretical ones computed within the frame of the dynamical theory; and Barla, Herino, Bomchil & Pfister (1984), who examined the elastic properties of silicon.
The standard doublecrystal technique does not allow determination of relatively small strains, i.e. ones that affect the lattice parameter by, for example, less than 2–3 parts in 10^{5}, as in the case of (004) Si reflection and Cu Kα radiation. To overcome this difficulty, Zolotoyabko, Sander, Komem & Kantor (1993) propose a new method that combines doublecrystal Xray diffraction with highfrequency ultrasonic excitation. Since ultrasound has a wavelength a little less than the Xray excitation length, it affects the diffraction profile close to the Bragg position and so permits the detection of very small profile broadenings caused by lattice distortions. With this method, lattice distortion as small as 5 parts in 10^{6} can be measured.
As has been shown in the case of the device used by Hart & Lloyd (1975), the symmetric arrangement due to Bond (1960) proves to be very useful when the doublecrystal spectrometer is to be used for absolute latticeparameter determination, since such an arrangement combines the high precision and sensitivity of a doublecrystal spectrometer with the high absolute accuracy of the Bond method. Other examples of a similar idea are presented by Kurbatov, Zubenko & Umansky (1972), who report measurements of the thermal expansion of silicon; Godwod, Kowalczyk & Szmid (1974), who also discuss the theoretical basis of their arrangement; Ridou, Rousseau & Freund (1977), who examine a phase transition; Sasvári & Zsoldos (1980), and Fewster (1982). The latter two papers are concerned with epitaxic layers. A rapid method is proposed by Sasvári & Zsoldos (1980) for deconvoluting the overlapping peaks due to the layer and the substrate. A particular feature of the arrangement proposed in the first of these papers (Kurbatov, Zubenko & Umansky, 1972) is the use of a germaniumcrystal monochromator with anomalous transmission, to obtain a nearly parallel primary beam (the horizontal divergence is 28′′ and the vertical 14′′).
The error analyses given by Godwod, Kowalczyk & Szmid (1974) and Sasvári & Zsoldos (1980) show that systematic errors due to eccentricity, absorption, and zero position are eliminated experimentally, owing to the symmetric arrangement, as in the Bond (1960) method. In contrast, the errors due to crystal tilt, refraction and the Lorentz–polarization factor [their uncertainties in lattice parameters, as evaluated by Sasvári & Zsoldos (1980), are 10^{−6} Å each], axial divergence (2 × 10^{−6} Å), angle reading (10^{−4} Å), and instrument correction and calculations (each to 5 × 10^{−5} Å) should be taken into account. The effect of absorption, discussed by Kurbatov, Zubenko & Umansky (1972), proved to be negligible. The final accuracy achieved for silicon single crystals by Godwod, Kowalczyk & Szmid (1974) is comparable with that obtained by Bond (1960).
A specific group of doublecrystal arrangements is formed by those in which white Xradiation is used instead of characteristic. Such an arrangement makes possible very large values of the Bragg angle (larger than about 80°), which increases the accuracy, precision, and sensitivity of measurement of the lattice parameters and their change with change of temperature. This task is rather difficult to realize by means of traditional methods, in which both the wavelengths and the lattice parameters are fixed, and it is difficult to find a suitable combination of their values.
The principle of the method presented by Okazaki & Kawaminami (1973a) is shown in Fig. 5.3.3.10 . The first crystal (the specimen to be measured) remains fixed during a single measurement, the second (the analyser) is mounted on the goniometer of an Xray diffractometer and can be operated with either an or a – scan. As diffraction phenomena appear for both the specimen and the analyser (in general of different materials) whose interplanar spacings are equal to and , respectively, the following relation results from Bragg's law: where and are the respective Bragg angles. Since and are kept constant, a change in as a function of temperature is determined from a change in . The relative error resulting from (5.3.3.38) with is The method initiated by Okazaki & Kawaminami (1973a) has been developed by Okazaki & Ohama (1979), who constructed the special diffractometer HADOX (the positions of the specimen and the analyser were interchanged) and discussed systematic errors. Precision as high as 1 part in 10^{7} was reported. Examples of the application of such an arrangement for measuring the temperature dependence of lattice parameters were given by Okazaki & Kawaminami (1973b) and Ohama, Sakashita & Okazaki (1979). Various versions of the HADOX diffractometer are still reported. By introducing two slits (Soejima, Tomonaga, Onitsuka & Okazaki, 1991) – one to limit the area of the specimen surface to be examined and the other to define the resolution of – it is possible to combine and scans and obtain a twodimensional intensity distribution in the plane parallel to the plane of the diffractometer, and to determine the temperature dependence of lattice parameters on a selected area of the specimen (avoiding the effects of the surroundings). The HADOX diffractometer may work with both a rotatinganode highpower Xray source (examples reported above) and a sealedtube Xray source. In the latter case (Irie, Koshiji & Okazaki, 1989), to increase the efficiency of the Xray tube, the distance between the Xray source and the first crystal has been shortened by a factor of five. As is implied by (5.3.3.39), one can increase the relative precision of the method by using the analyser angle close to π/2. This idea has been realized by Okazaki & Soejima (2001), who achieved the relative accuracy of determination of latticeparameter changes as high as 1 part in – by extending the Bragg angle from 78° (previous versions) to 89.99° and by elimination of systematic errors due to crystal tilt, crystal displacement, temperature effects and radiation damage.

Schematic representation of the doublecrystal arrangement of Okazaki & Kawaminami (1973a); white incident Xrays are used. 
An original method for the measurement of lateral latticeparameter variation by means of a doublecrystal arrangement with an oscillating slit was proposed by Korytár (1984). This method permitted simultaneous recording of two rocking curves from two locations on a crystal. Precision of 3 parts in 10^{7} was reported. The method has been applied for the measurement of growth striations in silicon.
The main disadvantage of doublecrystal spectrometers, in their basic form (Fig. 5.3.3.8), is that they cannot be used for measurements on an absolute scale. Combination of the doublecrystal arrangement with the system proposed by Bond (1960) makes it possible to recover the origin of the angular scale and thus such an absolute measurement, but the reported precision is rather moderate.
There are two other ways to overcome this difficulty in pseudonondispersive methods: addition either of a third crystal (more accurately, a third reflection) (§5.3.3.7.2) or of a second source (a second beam) (§5.3.3.7.3). Such arrangements require additional detectors. Combinations of both techniques are also available (§5.3.3.7.4).
Higher precision than that obtained with the doublecrystal arrangements (§5.3.3.7.1) can be achieved by means of triplecrystal diffractometers. Arrangements specially designed for the determination of latticeparameter changes are described by Buschert (1965) and Skupov & Uspeckaya (1975), and reviewed by Hart (1981).
The principle of the tripleaxis spectrometer is shown in Fig. 5.3.3.11 . The arrangement consists of one standard crystal S, ultimately replaced by the sample under investigation, and two reference crystals and . The principle of the measurement is as follows. First, the crystals S, , and are set to their diffraction (peak) positions using two detectors and . Then the standard crystal S is replaced by the sample and the new peak position is found by means of when the sample is turned from its original position to its reflecting position. The angle of rotation of the sample depends on the latticeparameter difference between the sample and the standard. The relation is given by (Hart, 1981) Next, the second reference crystal is turned through the angle to its diffracting position, the intensity being controlled with the second detector . From the geometry of the arrangement, Because the origin of the scale is lost during the crystal exchange, this second angle of rotation is used to determine rather than the first one , by using (5.3.3.41) and (5.3.3.40).

Schematic representation of the triplecrystal spectrometer developed by Buschert (1965) (after Hart, 1981). 
The diffraction profiles observed in the second detector, described by Hart (1981), are not symmetric but can be as narrow as 0.1–1′′, so that a precision of 2 parts in 10^{8} is possible.
The main experimental problem here is to adjust the tilts of the crystals. The errors resulting both from the crystal tilts and from the vertical divergence were discussed by Skupov & Uspeckaya (1975).
Triplecrystal spectrometers are often applied as latticespacing comparators, when very small changes of lattice parameters are to be detected, in particular for the examination of a correlation between lattice parameter and the dopant or impurity concentration (Baker, Tucker, Moyer & Buschert, 1968). Such an arrangement can also be a very suitable tool in deformation studies, since it allows the separation of the effect of deformation on the Bragg angle from that due to latticeparameter change (Skupov & Uspeckaya, 1975).
The basis of the accurate latticeparameter comparison proposed by Bowen & Tanner (1995) is the use of a highpurity silicon standard (cf. §5.3.3.9 below) with a well known lattice parameter. To compensate an error that may result from a slight misalignment of crystal planes in relation to the axes of the instrument, the authors recommend a twofold measurement of the diffractionpeak position of the reference crystal (for a given diffraction position and after rotating the specimen holder through 180° about the axis normal to its surface) and a similar twofold measurement of the diffractionpeak position of the sample – after replacing the reference crystal by the sample. The mean positions of the reference crystal and of the sample are used in calculations of the Braggangle difference and then of the unknown interplanar spacing. The method uses a standard doublecrystal diffractometer fitted with a monochromator (therefore, a third crystal), which provides a well defined wavelength, and with a specimen rotation stage. The measurement is accompanied by a detailed error analysis. The accuracy of absolute latticeparameter determination as high as a few tens of parts in , and a much greater relative sensitivity are reported.
By combining a tripleaxis spectrometer with the Bond (1960) method, the device can be used for absolute measurements (Pick, Bickmann, Pofahl, Zwoll & Wenzl, 1977). The device described in the latter paper is an automatic triplecrystal diffractometer that permits intensity measurement to be made in any direction in reciprocal space in the diffraction plane with step sizes down to 0.01′′ and therefore can be used for very precise measurements [see also §5.3.3.4.3.2, paragraph (5)].
The other possibility of recovering the crystalangle scale in differential measurements with a doublecrystal spectrometer (cf. §§5.3.3.7.1, 5.3.3.7.2) is to obtain reflections from two crystal planes [for example, from (hkl) and planes] by means of a doublebeam arrangement and to measure them simultaneously.
The second Xray beam may come from an additional Xray source (Hart, 1969) or may be formed from a single Xray source by using a beamsplitting crystal (Hart, 1969, second method; Larson, 1974; Cembali, Fabri, Servidori, Zani, Basile, Cavagnero, Bergamin & Zosi, 1992). In particular, two beams with different wavelengths separated with a slit system can be used for this purpose (Kishino, 1973, second technique). The principle of the doublebeam method is shown in Fig. 5.3.3.12 . The beams are directed at the first crystal (the reference crystal) so that the Bragg condition is simultaneously fulfilled for both beams, and they then diffract from the second crystal (the specimen). As the second crystal is rotated, a doublecrystal diffraction profile is recorded first in one detector and then in the other. The angle of crystal rotation between the two rocking curves is given by (Baker & Hart, 1975): This formula leads to the latticeparameter changes Δd.
A doublebeam diffractometer can be used for the examination of variations in lattice parameters of about 10 parts in 10^{6} within a sample in a given direction. An example was reported by Baker, Hart, Halliwell & Heckingbottom (1976), who used Larson's (1974) arrangement for this task.
The highest reported sensitivity (1 part in 10^{9}) can be achieved in the doublesource doublecrystal Xray spectrometer proposed by Buschert, Meyer, Stuckey Kauffman & Gotwals (1983). The device can be used for the investigation of small concentrations of dopants and defects.
The method can also be applied for the absolute determination of a lattice parameter, if that of the reference crystal is accurately known and the difference between the two parameters is sufficiently small. Baker & Hart (1975), using multiplebeam Xray diffractometry (Hart, 1969, first technique), determined the d spacing of the 800 reflection in germanium by comparing it with the d spacing of the 355 reflection in silicon. The latter had been previously determined by optical and Xray interferometry (Deslattes & Henins, 1973; the method is presented in Subsection 5.3.3.8).
In the case of two different wavelengths and diffraction from two different diffraction planes and , the lattice parameter of a cubic crystal can be determined using the formula (Kishino, 1973) where , , and is the difference between the two Bragg angles for the specimen crystal, estimated from the measurement of if the difference for the first (reference crystal) is known beforehand. The idea of Kishino was modified by Fukumori, Futagami & Matsunaga (1982) and Fukumori & Futagami (1988), who used the Cu Kα doublet instead of and radiation. Owing to the change, they could use only one detector (Kishino's original method needs two detectors), but a special approach is sometimes needed to resolve two peaks that relate to the components of the doublet. A similar problem of separation of two peaks (recorded by two detectors) is reported by Cembali et al. (1992). By introducing a computer simulation of the reflecting curves (using a convolution model), the authors managed to determine the separation with an error of 0.01′′ and to achieve a precision of some parts in 10^{7}. The same precision is reported by Fukumori, Imai, Hasegawa & Akashi (1997), who introduced a precise positioning device and a positionsensitive proportional counter to their instrument.
As in the other multiplecrystal methods, the most important experimental problem is accurate crystal setting. Larson (1974), as a result of detailed analysis, gave the dependence between the angular separation of two peaks and angles characterizing misalignment of the first and second crystals.
The idea of multiplebeam measurement (§5.3.3.7.3) can be applied to other arrangements that combine the features of the doublebeam comparator with those of the triplecrystal spectrometer; there are additional advantages in such a system. The application of the doublebeam technique makes it possible to realize a triplereflection scheme for comparing lattice parameters on the basis of a doubleaxis spectrometer. The arrangement proposed by Ando, Bailey & Hart (1978), shown in Fig. 5.3.3.13 , consists of a sample and a reference crystal, which are made from the same material but differ in purity (or strain, stoichiometry, vacancy concentration, etc). The angular difference in the Bragg angles of the sample and the reference crystal, and , respectively, is measured as the sample angle between the doublereflected peak D and the triply diffracted peak T: Assuming that is entirely due to changes in atomic spacings, the authors use the following relation for determination of the latter: The experimental requirements are simple and inexpensive, owing to simple shapes of both the reference crystal and the sample crystal, so that the measurement can be made quickly. By combining the two reference crystals into a single monolithic reference crystal, excellent stability, difficult to achieve with tripleaxis arrangements (cf. §5.3.3.7.2), is obtained at the same time. The disadvantage of the method is that it covers a smaller range of lattice parameters than the other doublebeam methods (Hart, 1969; Larson, 1974) described in §5.3.3.7.3. A new version of the doublecrystal triplereflection scheme (Häusermann & Hart, 1990) allows one to achieve a precision of 1 part in 10^{8} in 2 min of measurement time, which includes the data analysis; 30 min are needed to change the sample. Errors due to the crystal tilt and thermal drifts are considered.

The doubleaxis latticespacing comparator of Ando, Bailey & Hart (1978); a triplediffracted beam is used. 
Another example of the triplereflection scheme realized by means of the doublebeam technique has been presented by Kovalchuk, Kovev & Pinsker (1975), who realized the triplecrystal arrangement on the basis of a doublecrystal spectrometer by parallel mounting of the two crystals to be compared (the sample and the reference crystal) on one common axis. The advantage of this system is that Bragg angles as high as 80° are available. The device can be applied in studies of the real structure of a single crystal.
Highsensitivity (Δd/d up to ) latticeparametercomparison measurement over a wide range of temperatures can be performed by means of the triplecrystal (more accurately, tripleaxis) Xray spectrometer realized by Buschert, Pace, Inzaghi & Merlini (1980). The arrangement (Fig. 5.3.3.14 ) consists of four crystals. The first is used for obtaining a very wide but extremely parallel exit beam, which is incident on both the standard crystal S and an unknown crystal X, placed side by side on a common axis in the cryostat. The reflected beams from S and X are recorded by partially transmitting detectors and , so that the beams reflect from the third crystal and are detected by the counters and . There is a small, sensitive, angle adjustment to rotate the crystal X with respect to the standard S and it is used to bring the peaks of S and X into approximate coincidence. The angular difference in the peak positions on the third axis is used for determination of latticeparameter changes from (5.3.3.46), so that where and are the differences in peak positions at axes (2) and (3), respectively. The device was used, for example, to study the effect of isotope concentration on the lattice parameter of germanium perfect crystals (Buschert, Merlini, Pace, Rodriguez & Grimsditch, 1988). The measured differences in the lattice parameter, of the order of 1 part in 10^{5}, were compared with those evaluated theoretically, and a very good agreement was obtained.
Another variant of a multiplebeam arrangement, based on a triplecrystal spectrometer, was proposed by Kubena & Holý (1988). The authors compared the distances of lattice planes in a direction perpendicular to the surface of the sample while studying the growth striations. One well collimated and monochromated beam coming from the first crystal was directed into the sample, and then two beams – one transmitted and one diffracted in the sample – diffracted in the reference crystal. Intensities of the diffracted beams were measured by two detectors. The difference of lattice spacings of the sample and the reference crystal was determined from the difference in positions of respective peaks. The accuracy of the latticespacing comparison of 2 parts in 10^{7} and the precision of 1 part in 10^{7} were obtained.
A fourcrystal sixreflection diffractometer (Fewster, 1989) was built to study crystals distorted by epitaxy and defects in nearly perfect crystals. Fig. 5.3.3.15 is a schematic diagram of this device. The twocrystal fourreflection Bartels monochromator (Bartels, 1983) defines a narrow reflectivity profile. The analyser selects the angular range diffracted from the sample. The device may be used for recording both nearperfect rocking curves from distorted crystals (when rotations of the sample and the analyser are coupled) and a diffractionspace map for studying the diffuse scattering (when the two rotations are uncoupled). Various applications of such highsensitivity multiplecrystal Xray spectrometers for reciprocalspace mapping and imaging (topography), which are outside the scope of the present paper, are reviewed by Fewster (1993, and references therein).

The geometry of the diffractometer used by Fewster & Andrew (1995). The scattering angle, 2ω′, is the fundamental angle for determination of the interplanar spacing and P is the analysergroove entrance. 
As was shown a few years later by Fewster & Andrew (1995), the device can also be used for absolute latticeparameter measurements of singlecrystal and polycrystalline materials with a relative accuracy of a few parts in . The authors checked the angular resolution and the sample centring of their instrument, and discussed systematic errors due to refraction, the Lorenz and polarization factor, the diffractingplane tilt and the peakposition determination.
The accuracy of an absolute measurement can be improved, in relation to that obtained in traditional methods (cf. Subsection 5.3.3.5), either if the wavelength of the radiation used in an experiment is known with better accuracy [cf. equation (5.3.1.3)] or if a highquality standard single crystal is given, whose lattice spacing has been very accurately determined (Baker & Hart, 1975; mentioned in §5.3.3.7.3). The two tasks, i.e. very accurate determination of both lattice spacings and wavelengths in metric units, can be realized by use of combined optical and Xray interferometry. This original concept of absolutelatticespacing determination directly in units of a standard light wavelength has been proposed and realized by Deslattes (1969) and Deslattes & Henins (1973).
The principle of the method is presented in Fig. 5.3.3.16 . The siliconcrystal Xray interferometer is a symmetric Lauecase type (Bonse & te Kaat, 1968). The parallel translation device consists of the stationary assembly (a) formed by two specially prepared crystals, and a moveable one (b), to which belongs the third crystal. One of the two mirrors of a highresolution Fabry–Perot interferometer is attached to the stationary assembly and the second to the moving assembly. A stabilized He–Ne laser is used as a source of radiation, the wavelength of which has been established relative to visible standards. The first two crystals produce a standing wavefield, which is intercepted by the third crystal, so that displacement of the third crystal parallel to the diffraction vector (as suggested by the large arrow) produces alternate maxima and minima in the diffracted beams, detected by Xray detector (c). Resonant transmission maxima of the optical interferometer are detected simultaneously by the photomultiplier indicated at (d). Analysis of the fringes (shown in Fig. 5.3.3.17 ) is the basis for the calculation of the latticespacingtoopticalwavelength ratio (d/λ), which is given by where n and m are the numbers of optical and Xray diffraction fringes, respectively, and α and β are the measured angular deviations of the optical and Xray diffraction vectors from the direction of motion. The measurements are carried out in two steps. First, the lattice parameter of silicon along the [110] crystallographic direction was measured in the metric system, independently of the Xray wavelength used in the experiment. As the next step, a specimen of known lattice spacing, treated as a reference crystal, was used for the accurate wavelength determination of Cu Kα_{1} and Mo Kα_{1}. Accuracy better than 1 part in 10^{6} was reported (see Section 4.2.2 ).

Optical and Xray interferometry. Schematic representation of the experimental setup (after Deslattes & Henins, 1973; Becker et al., 1981). 
The above experiment was a turning point in accurate measurements of both wavelengths and lattice parameters. Owing to the idea of Deslattes & Henins, it became possible to determine the wavelength in nanometres rather than in troublesome XU or Å* units (cf. §4.2.1.1.1 ). However, the results obtained and the method itself needed verification and some adjustments. These were performed by another group of experimenters with a similar but different measuring device (Becker, Seyfried & Siegert, 1982, and references therein; Siegert, Becker & Seyfried, 1984).
An extended series of measurements performed by means of the optical and Xray interferometry (cf. §5.3.3.8) led, among other things, to evaluation of the lattice spacing of a highly perfect silicon sample WASO 4.2.A (Becker et al., 1981). Such silicon samples may be used as reference crystals in successive latticespacing comparison measurements – with a doublesource doublecrystal spectrometer (Windisch & Becker, 1990), for example. The latter measurements provided new excellent latticespacing standards (WASO 9, for example) of the well known latticeparameter values. As shown by the authors, the differences in lattice parameters of different samples of floatzone silicon (due to oxygen or carbon content) were not greater than a few parts in . Finally, the lattice parameter of silicon, a = 5.43102088 (16) Å, has been accepted as the atomic scale length standard (Mohr & Taylor, 2000).
Another reference material reported is crystals of pure rhombohedral corrundum (αAl_{2}O_{3}), i.e. of ruby or sapphire (Herbstein, 2000, and references therein; Shvyd'ko et al., 2002).
With silicon standards, measurements or remeasurements of and/or Xray emission lines and absolute wavelength determinations of most of the 3d transition metals (Cr, Mn, Fe, Co, Ni and Cu) have been performed [Härtwig, Grosswig, Becker & Windisch, 1991; Hölzer, Fritsch, Deutsch, Härtwig & Förster, 1997 (see §4.2.2 )].
The standard crystals may also be used for determination of such physical quantities as the Avogadro constant (Deslattes et al., 1974; Deslattes, Henins, Schoonover, Caroll & Bowman, 1976). The single accurate wavelength values, on the other hand, may be used both in simple measurements of lattice parameters [based directly on the Bragg law, equation (5.3.1.1)] and for accurate scaling of the wavelength spectra, in order to use them, for example, in highaccuracy latticeparameter measurements based on complete convolution models [cf. §5.3.3.3.1, point(ii)].
Unlike the Xrays emitted from an Xray tube, for which the spectral line and the characteristic wavelength are known, there are no such characteristic features in synchrotron radiation. Therefore, special energyselective monochromators should be applied in relative latticespacing measurements using synchrotron radiation. Obaidur (2002) proposes two measurement schemes, using two types of highresolution channelcut monolithic monochromators. The first scheme (see Fig. 5.3.3.18 ) is a modification of the Bond method. The second one (see Fig. 5.3.3.19 ) uses the simultaneous Bragg condition for the indices (5, 1, 3), , (1, 5, 3) and . The latticespacing differences in Si wafers were determined in the subparts in range of 0.6 parts in (in the first scheme) and of 0.2 parts in (second scheme).

Synchrotron radiation, SR, from the bending magnet incident on the Si(111) doublecrystal monochromator and, after four reflections from the monolithic monochromator (0.1410 nm), impinges on sample Si(444). Two diffractions are recorded at the photodiode detectors, PIN1 and PIN2. The ω_{1} and ω_{2} values of the crystal positions are recorded using a Heiden height encoder. 

Synchrotron radiation, SR, from the bending magnet incident on the Si(111) doublecrystal monochromator and, after four reflections from the monolithic monochromator (0.1612 nm), impinges on sample Si(153). Two diffractions are recorded at the photodiode detectors, PIN1 and PIN2. The differences between the two peaks' D values are recorded using a Heiden height encoder. 
Recently, a new atomic scale wavelength standard was proposed by Shvyd'ko et al. (2000), instead of the wavelength of the Cu emission line or of the lattice parameter of a silicon standard. It is the wavelength, λ_{M}, of the ^{57}Fe Mössbauer radiation, i.e. of γ radiation of natural linewidth from nuclear transitions. It has been measured to the subparts in accuracy: λ_{M} = 0.86025474 (16) Å (relative accuracy 0.19 parts in ). Its advantage, in relation to the previous standards, is the high spectral sharpness of the Mössbauer radiation of in relative units, which makes its wavelength λ_{M} extremely well defined. This standard wavelength value, which lies a little outside of scope of the present review (Xray methods), was next used for the latticeparameter determination of sapphire single crystals with a relative accuracy of about 0.5 parts in (Shvyd'ko et al., 2002). Fig. 5.3.3.20 is a diagram of the measurement arrangement.

Experimental setup for measuring lattice parameters. Xrays after a highheatload monochromator (not shown) pass through the vertical slits S_{1} and S_{2} at a distance of 26.6 m. λ is a `λmeter'; F is a ^{57}Fe foil used as a source of the Mössbauer radiation of high brightness; D is a semitransparent avalanche photodiode with 0.7 ns time resolution; Al_{2}O_{3} is a sapphire single crystal in a furnace on a fourcircle goniometer. 
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