Section 5.2.10. Powder-diffraction standards

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.