Chapter 5.2. X-ray diffraction methods: polycrystalline

The symmetrical Bragg–Brentano (Parrish) and the Seemann–Bohlin angle-dispersive 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 X-rays 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 (`flat-specimen 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).

Table 5.2.4.1| top | pdf | Centroid displacement 〈Δθ/θ〉 and variance W of certain aberrations of an angle-dispersive diffractometer; for references see Wilson (1963, 1965c, 1974) and Gillham (1971) For the Seemann–Bohlin arrangement, S and R are given by equations (5.2.4.1) and (5.2.4.2); for the symmetrical arrangement, they are equal to R0. Other notation is explained at the end of the table. Aberration W Zero-angle calibration Constant 0 Specimen displacement 0 Specimen transparency  Thick specimen  Thin specimen See Wilson (1974, p. 547) 2:1 mis-setting Zero if centroid of illuminated area is centred Inclination of plane of specimen to axis of rotation Zero if centroid of illuminated area on equator of specimen for uniform illumination Flat specimen Focal-line width Small Receiving-slit width Small Interaction terms Small if adjustment reasonably good See Wilson (1963, 1974) Axial divergence No Soller slits, source, specimen and receiver equal Narrow Soller slits  One set in incident beam  One set in diffracted beam Replace R by S in the above   Two sets Wide Soller slits Complex. See Pike (1957), Langford & Wilson (1962), Wilson (1963, 1974), and Gillham (1971) Refraction Physical aberrations See Wilson (1963, 1965c, 1970a, 1974) and Gillham & King (1972) Notation: 2A = illuminated length of specimen; β = angle of equatorial mis-setting 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; r1 = width of receiving slit (varies with θ in some designs of diffractometer); s = specimen-surface displacement; f1 = 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.

Lattice-parameter 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 specimen-surface-displacement and flat-specimen 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 profile-fitting or peak-search measurements, and the only significant geometrical aberration is axial divergence.

There are no lines in the synchrotron-radiation 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 10⁶ 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 specimen-surface displacement shift. Mixing reduces the intensity of both patterns and worsens the peak-to-background 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 synchrotron-radiation 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 high-quality single-crystal plate of float-zoned oxygen-free silicon, now widely available. Its lattice parameter is known to about one part in 10⁷ (Hart, 1981), which is much higher accuracy than that of the published lists of X-ray 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 step-scanning 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 lattice-parameter 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 half-widths (full widths at half height) on both sides of the peak, with increments of about 0.1 to 0.2 of the half-width, 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 goodness-of-fit of the calculated profiles and the experimental points (Subsection 2.3.3.8 and Chapters 8.4 and 8.6 ).

The lower-angle peaks generally have higher intensities and are therefore preferred to the higher-angle 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 least-squares 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 zero-angle position should be included as a variable parameter in the least-squares calculation.

A precision of a few parts per million in the lattice parameter of NIST silicon has been reached with the high-precision 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 crystal-structure refinement and analysis has also stimulated interest in lattice-parameter 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 profile-fitting 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 least-squares 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 least-squares residuals.

An alternative to the Rietveld method is the pattern-decomposition method in which the integrated intensities are derived from profile fitting and the data used in a powder least-squares-refinement 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 crystal-structure model is required and only the first stage is used for lattice parameters. The Pawley method was developed for neutron-diffraction data and uses slack constraints to handle the problem of least-squares ill-conditioning due to overlapping reflections, and the positions of the reflections are constrained by the lattice parameters. The refinement also determines the zero-angle calibration correction.

Toraya extended the Pawley method to X-ray 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 pseudo-Voight function for the nearly symmetrical synchrotron-radiation 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 whole-pattern profile fitting and incorporating the peak-shift 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 least-squares 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 zero-angle calibration correction is always determined, and the specimen-surface displacement shift is usually included.

The lattice-parameter 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 high-angle reflections are given greater weight because of their higher accuracy for a given error. As noted in Subsection 5.2.13, higher-order reflections usually have low intensities and much overlapping. Some judgement and critical tests are often required.

There are now two basic energy-dispersive 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 solid-state detector and a multichannel pulse-height 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 angle-dispersive techniques, thereby greatly limiting its applications. The second method uses an incident-beam monochromator, a conventional scintillation counter, and a single-channel pulse-height analyser. The monochromator is step-scanned 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 X-ray optics, the resolution is the same as in angle-dispersive 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 lattice-parameter determination.

Energy-dispersive techniques (Section 2.2.3 and Chapter 2.5 ) are not ordinarily the method of choice for lattice-parameter determination. Relative to angle-dispersive 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 angle-dispersive to energy-dispersive by (i) replacing the usual counter by a solid-state detector, (ii) replacing the usual electronic circuits by a multichannel pulse-height 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 angle-dispersive diffractometer, though the greater penetrating power of the higher-energy X-rays 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 specimen-surface 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 specimen-surface 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 X-ray 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 lattice-spacing problem. The best results reported so far seem to be those of Fukamachi, Hosoya & Terasaki (1973) (0.01% in the lattice parameter).

Table 5.2.7.1| top | pdf | Centroid displacement and variance W of certain aberrations of an energy-dispersive diffractometer [mainly from Wilson (1973), where more detailed results are given for the aberrations marked with an asterisk] The Soller slits are taken to be in the original orientation (Soller, 1924). For the notation, see the footnote. Aberration W Specimen displacement Included in equatorial divergence Specimen transparency* ? Equatorial divergence* for narrow Soller slits Axial divergence Refraction* Probably negligible at the present stage of technique Response variations  Centroid ?  Peak ? Interaction of Lorentz etc. factors and geometrical aberrations 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 angle-dependent response (Lorentz factor etc.); I(E −E1) dE is the counting rate recorded at E when the energy of the incident photons is actually E1; R is the diffractometer radius; V is the variance and μ3 is the third central moment of the energy-resolution 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 X-rays with a single-crystal spectrometer. This would give some of the advantages of energy-dispersive 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 straight-line 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.

Table 5.2.8.1| top | pdf | Some geometrical aberrations in the Debye–Scherrer method [increase in θ = +, decrease = −] Source of aberration Effect on θ Angle variation of Δd Remarks Specimen displacement        towards exit − Minimized by accurate construction and centring  towards entrance + Extrapolates to zero  sideways † †   Beam divergence        perpendicular to axis + or Minimized by reducing collimator dimensions  parallel to axis + or − Complex See Langford, Pike & Beu (1964) Film shrinkage + Affects only van Arkel arrangement Knife-edge calibration + or − Affects only Bradley–Jay arrangement. Partly eliminated by usual extrapolation Specimen absorption + or Minimized by reducing specimen diameter or dilution. Extrapolates to zero †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 lattice-parameter project (Parrish, 1960) showed a standard deviation of 1 in 30 000 in inter-laboratory 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 lattice-parameter 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 Technology² (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 Wong-Ng & Hubbard (1987).

The current silicon-powder 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 (mass-weighted 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 5.2.10.1| top | pdf | NIST values for silicon standards (λ = 1.5405929 Å, T = 298 K for 640, 640a and 640b, T = 295.6 K for 640c, a0 ± 0.000035 Å, no refraction correction) Standard Year issued a0 (Å) Cu Kα1 111 (°2θ) 444 (°2θ) 640 1974† 5.43086 28.4427 158.6382 640a 1982‡ 5.430806 28.4430 158.6443 640b 1987 5.430922 28.4424 158.6315 640c 2000 5.4311946 28.4410 158.6031 †Hubbard, Swanson & Mauer (1975). ‡Hubbard (1983).

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 high-angle 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 Kα₁ = 1.5405929 Å are given in Table 5.2.10.2 and in Table 5.2.10.5 for a number of other wavelengths.

Table 5.2.10.3| top | pdf | Silicon standard reflection angles (°) (NIST SRM 640c a0 = 5.431195 Å, T = 295.6 K) h k l   d (Å) I Mo 1.000000 Å 1.250000 Å 1.500000 Å 1.750000 Å Cr 0.709317 Å 2.289746 Å 1 1 1   3.13570 100.0 12.988 18.350 22.994 27.676 32.406 42.829 2 2 0   1.92022 71.1 21.287 30.186 37.990 45.981 54.217 73.202 3 1 1   1.63757 43.5 25.016 35.556 44.873 54.516 64.597 88.714 4 0 0   1.35780 11.8 30.283 43.215 54.813 67.059 80.245 114.955 3 3 1   1.24600 17.4 33.074 47.317 60.213 74.016 89.215 133.514                         4 2 2   1.10864 22.3 37.314 53.616 68.635 85.142 104.232   5 1 1 1.04523 8.7 39.670 57.157 73.447 91.704 113.678   3 3 3 1.04523 2.9 39.670 57.157 73.447 91.704 113.678   4 4 0   0.96011 6.0 43.356 62.768 81.229 102.734 131.386   5 3 1   0.91804 9.8 45.452 66.000 85.812 109.563 144.772                           6 2 0   0.85871 7.1 48.789 71.221 93.411 121.713     5 3 3   0.82825 2.9 50.707 74.268 97.981 129.788     4 4 4   0.78393 1.5 53.797 79.258 105.739 146.162     7 1 1 0.76052 1.9 55.594 82.211 110.532 160.918     5 5 1 0.76052 1.9 55.594 82.211 110.532 160.918                             6 4 2   0.72577 5.7 58.506 87.090 118.893       7 3 1 0.70708 2.4 60.209 90.004 124.237       5 5 3 0.70708 1.2 60.209 90.004 124.237       8 0 0   0.67890 0.5 62.987 94.866 134.030       7 3 3   0.66353 0.8 64.620 97.797 140.757                               6 6 0 0.64007 0.7 67.297 102.735 155.085       8 2 2 0.64007 1.3 67.297 102.735 155.085       7 5 1 0.62714 1.7 68.876 105.740 170.531       5 5 5 0.62714 0.2 68.876 105.740 170.531       8 4 0   0.60723 0.9 71.473 110.855                                 9 1 1 0.59615 0.4 73.013 114.009         7 5 3 0.59615 0.8 73.013 114.009         6 6 4   0.57897 0.7 75.551 119.447         9 3 1   0.56934 0.6 77.061 122.854         8 4 4   0.55432 0.5 79.555 128.846                                 9 3 3 0.54586 0.2 81.042 132.692         7 7 1 0.54586 0.2 81.042 132.692         7 5 5 0.54586 0.2 81.042 132.692         10 2 0 0.53257 0.4 83.509 139.717         8 6 2 0.53257 0.8 83.509 139.717                                 9 5 1 0.52505 0.4 84.982 144.460         7 7 3 0.52505 0.2 84.982 144.460         9 5 3   0.50646 0.3 88.897 161.678         10 4 2   0.49580 0.5 91.340           11 1 1 0.48971 0.1 92.808                                 7 7 5 0.48971 0.1 92.808           8 8 0   0.48005 0.1 95.258           11 3 1 0.47453 0.2 96.729           9 7 1 0.47453 0.2 96.729           9 5 5 0.47453 0.1 96.729

Table 5.2.10.4| top | pdf | Silicon standard high reflection angles (°) (NIST SRM 640c a0 = 5.431195 Å, T = 295.6 K, λ = 0.709317 Å) h k l   d (Å) 2θ 10 6 0 0.46572 99.198 8 6 6 0.46572 99.198 11 3 3 0.46067 100.686 9 7 3 0.46067 100.686 12 0 0 0.45260 103.183           8 8 4 0.45260 103.183 11 5 1 0.44796 104.694 7 7 7 0.44796 104.694 12 2 2 0.44053 107.235 10 6 4 0.44053 107.235             11 5 3 0.43624 108.777 9 7 5 0.43624 108.777 12 4 0   0.42937 111.378 9 9 1   0.42540 112.961 10 8 2   0.41903 115.642             9 9 3 0.41533 117.279 11 7 1 0.41533 117.279 11 5 5 0.41533 117.279 13 1 1 0.41533 117.279 12 4 4   0.40939 120.064             11 7 3 0.40595 121.773 13 3 1 0.40595 121.773 9 7 7 0.40595 121.773 12 6 2   0.40039 124.694 13 3 3   0.39717 126.497             9 9 5   0.39717 126.497 8 8 8   0.39196 129.600 13 5 1 0.38894 131.530 11 7 5 0.38894 131.530 10 10 0 0.38404 134.882           10 8 6 0.38404 134.882 14 2 0 0.38404 134.882 13 5 3 0.38120 136.990 11 9 1 0.38120 136.990 12 8 0   0.37659 140.703             11 9 3 0.37390 143.079 9 9 7 0.37390 143.079 12 6 6 0.36955 147.363 10 10 4 0.36955 147.363 14 4 2 0.36955 147.363             13 7 1 0.36701 150.191 11 7 7 0.36701 150.191 13 5 5 0.36701 150.191 12 8 4   0.36289 155.551 11 9 5 0.36048 159.376           15 1 1 0.36048 159.376 13 7 3 0.36048 159.376 14 6 0   0.35658 168.113

Table 5.2.10.5| top | pdf | Tungsten reflection angles (°) (a0 = 3.16523 Å, T = 298 K) h k l   d (Å) I Mo         Cr 0.709317 Å 1.000000 Å 1.250000 Å 1.500000 Å 1.750000 Å 2.289746 Å 1 1 0   2.23816 100.0 18.235 25.817 32.431 39.157 46.027 61.531 2 0 0   1.58262 18.1 25.899 36.834 46.521 56.575 67.130 92.672 2 1 1   1.29220 37.0 31.860 45.528 57.851 70.958 85.241 124.747 2 2 0   1.11908 11.1 36.953 53.076 67.903 84.164 102.868   3 1 0   1.00093 14.4 41.505 59.938 77.279 97.059 121.896                           2 2 2   0.91372 3.3 45.679 66.352 86.316 110.333 146.518   3 2 1   0.84594 14.2 49.574 72.464 95.262 124.894     4 0 0   0.79131 1.3 53.255 78.376 104.339 142.810     3 3 0 0.74605 2.0 56.768 84.164 113.805       4 1 1 0.74605 4.0 56.768 84.164 113.805                               4 2 0   0.70777 3.1 60.145 89.893 124.027       3 3 2   0.67483 2.5 63.411 95.621 135.687       4 2 2   0.64610 2.0 66.586 101.406 150.632       5 1 0 0.62075 1.6 69.687 107.312         4 3 1 0.62075 3.2 69.687 107.312                                 5 2 1   0.57789 2.2 75.717 119.815         4 4 0   0.55954 0.5 78.668 126.656         5 3 0 0.54283 0.8 81.589 134.172         4 3 3 0.54283 0.8 81.589 134.172         6 0 0 0.52754 0.2 84.488 142.810                               4 4 2 0.52754 0.7 84.488 142.810         6 1 1 0.51347 0.6 87.373 153.695         5 3 2 0.51347 1.2 87.373 153.695         6 2 0   0.50047 0.5 90.251 175.042         5 4 1   0.48841 1.0 93.129                                   6 2 2   0.47718 0.4 96.016           6 3 1   0.46669 0.8 98.919           4 4 4   0.45686 0.1 101.845           5 5 0 0.44763 0.2 104.802           7 1 0 0.44763 0.3 104.802                                 5 4 3 0.44763 0.7 104.802           6 4 0   0.43894 0.3 107.800           5 5 2 0.43073 0.3 110.851           6 3 3 0.43073 0.3 110.851           7 2 1 0.43073 0.6 110.851                                   6 4 2   0.42297 0.6 113.963           7 3 0   0.41562 0.3 117.150           7 3 2 0.40198 0.5 123.837           6 5 1 0.40198 0.5 123.837           8 0 0   0.39565 0.1 127.376                                   7 4 1 0.38961 0.5 131.091           8 1 1 0.38961 0.3 131.091           5 5 4 0.38961 0.3 131.091           8 2 0 0.38384 0.3 135.029           6 4 4 0.38384 0.3 135.029                                   6 5 3   0.37832 0.6 139.257           8 2 2 0.37303 0.3 143.877           6 6 0 0.37303 0.1 143.877           7 4 3 0.36795 0.6 149.106           7 5 0 0.36795 0.3 149.106                                 8 3 1 0.36795 0.6 149.106           6 6 2   0.36308 0.4 155.271           7 5 2   0.35839 1.1 163.450

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 higher-angle 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 d₀₀₁= 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 low-angle calibration standard for the analysis of materials with large unit-cell 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 specimen-surface 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 specimen-surface 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 α-Al₂O₃ (corundum), ZnO, TiO₂ (rutile), Cr₂O₃, and CeO₂. 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 10⁵, 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 third-strongest lines relative to the strongest = 100, and the final column gives the ratio of the strongest peak to the strongest peak of Al₂O₃.

The need for standard reference materials to determine instrumental line profiles arose from the increased use in recent years of whole-pattern methods (Section 5.2.6) in several applications of powder diffraction. Instrumental line-profile 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 line-profile 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 high-resolution diffractometers. Various materials were considered by the Technical Committee of the JCPDS–ICDD, in association with NIST, and lanthanum hexaboride [LaB₆: a₀ = 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 BaF₂ (Louër & Langford, 1988) and KCl (Scardi, Lutterotti & Maistrelli, 1994). Both are low-cost 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 single-crystal silicon wafer and annealed the whole assemblage to produce an instrument line-profile standard. The resulting line-profile widths were found to be slightly less than for LaB₆ at angles below about 100°(2θ) with Cu Kα radiation.

Many factors influencing accuracy in lattice-parameter 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: