Triple-crystal spectrometers

Gałdecka, E.

doi:10.1107/97809553602060000597

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 530-531

Section 5.3.3.7.2. Triple-crystal spectrometers

E. Gałdecka^a

^a Institute of Low Temperature and Structure Research, Polish Academy of Sciences, PO Box 937, 50-950 Wrocław 2, Poland

5.3.3.7.2. Triple-crystal spectrometers

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Higher precision than that obtained with the double-crystal arrangements (§5.3.3.7.1) can be achieved by means of triple-crystal diffractometers. Arrangements specially designed for the determination of lattice-parameter changes are described by Buschert (1965) and Skupov & Uspeckaya (1975), and reviewed by Hart (1981).

The principle of the triple-axis 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 [R_1] and [R_2] . The principle of the measurement is as follows. First, the crystals S, [R_1] , and [R_2] are set to their diffraction (peak) positions using two detectors [D_1] and [D_2] . Then the standard crystal S is replaced by the sample and the new peak position is found by means of [D_1] when the sample is turned from its original position to its reflecting position. The angle of rotation of the sample $[\Delta\omega_s]$ depends on the lattice-parameter difference $[\Delta d]$ between the sample and the standard. The relation is given by (Hart, 1981) $[\Delta\omega_s=-\tan\theta\Delta d/d. \eqno (5.3.3.40)]$ Next, the second reference crystal [R_2] is turned through the angle $[\Delta\omega_R]$ to its diffracting position, the intensity being controlled with the second detector [D_2] . From the geometry of the arrangement, $[\Delta\omega_R=2\Delta\omega_s. \eqno (5.3.3.41)]$ Because the origin of the $[\omega_s]$ scale is lost during the crystal exchange, this second angle of rotation $[(\Delta\omega_R)]$ is used to determine $[\Delta d]$ rather than the first one $[(\Delta\omega_s)]$ , by using (5.3.3.41) and (5.3.3.40).

Figure 5.3.3.11| top | pdf |

Schematic representation of the triple-crystal spectrometer developed by Buschert (1965) (after Hart, 1981).

The diffraction profiles observed in the second detector, described by Hart (1981), $[h(\theta)_R=\int\limits^\infty _{-\infty}\,R^2(\theta') R(\theta'-\theta)\,{\rm d}\theta, \eqno (5.3.3.42)]$ are not symmetric but can be as narrow as 0.1–1′′, so that a precision of 2 parts in 10⁸ 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).

Triple-crystal spectrometers are often applied as lattice-spacing comparators, when very small changes of lattice parameters $[(10^{-8}\le|\Delta d|/]$ $[d\le10^{-6})]$ 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 lattice-parameter change (Skupov & Uspeckaya, 1975).

The basis of the accurate lattice-parameter comparison proposed by Bowen & Tanner (1995) is the use of a high-purity 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 diffraction-peak 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 diffraction-peak 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 Bragg-angle difference and then of the unknown interplanar spacing. The method uses a standard double-crystal 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 lattice-parameter determination as high as a few tens of parts in $[10^{6}]$ , and a much greater relative sensitivity are reported.

By combining a triple-axis 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 triple-crystal 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)].

References

Baker, J. A., Tucker, T. N., Moyer, N. E. & Buschert, R. C. (1968). Effects of carbon on the lattice parameter of silicon. J. Appl. Phys. 39, 4365–4368.Google Scholar

Bond, W. L. (1960). Precision lattice constant determination. Acta Cryst. 13, 814–818.Google Scholar

Bowen, D. K. & Tanner, B. K. (1995). A method for the accurate comparison of lattice parameters. J. Appl. Cryst. 28, 753–760.Google Scholar

Buschert, R. C. (1965). X-ray lattice parameter and linewidth studies in silicon. Bull. Am. Phys. Soc. 10, 125.Google Scholar

Hart, M. (1981). Bragg angle measurement and mapping. J. Cryst. Growth, 55, 409–427.Google Scholar

Pick, M. A., Bickmann, K., Pofahl, E., Zwoll, K. & Wenzl, H. (1977). A new automatic triple-crystal X-ray diffractometer for the precision measurement of intensity distribution of Bragg diffraction and Huang scattering. J. Appl. Cryst. 10, 450–457.Google Scholar

Skupov, V. D. & Uspeckaya, G. I. (1975). The combined X-ray spectrometer for deformation measurements in single crystals. (In Russian.) Prib. Tekh. Eksp. No. 2, pp. 210–213.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 530-531