Multiple-beam methods

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, p. 531

Section 5.3.3.7.3. Multiple-beam methods

E. Gałdecka^a

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

5.3.3.7.3. Multiple-beam methods

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The other possibility of recovering the crystal-angle scale in differential measurements with a double-crystal 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 $[(\bar h\bar k\bar l)]$ planes] by means of a double-beam arrangement and to measure them simultaneously.

The second X-ray beam may come from an additional X-ray source (Hart, 1969) or may be formed from a single X-ray source by using a beam-splitting crystal (Hart, 1969, second method; Larson, 1974; Cembali, Fabri, Servidori, Zani, Basile, Cavagnero, Bergamin & Zosi, 1992). In particular, two beams with different wavelengths $[(K\alpha_1,K\beta_1)]$ separated with a slit system can be used for this purpose (Kishino, 1973, second technique). The principle of the double-beam 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 double-crystal diffraction profile is recorded first in one detector and then in the other. The angle $[\Delta\theta]$ of crystal rotation between the two rocking curves is given by (Baker & Hart, 1975): $[\Delta\theta=(\theta_1-\theta_2)=\tan\theta\Delta d/d. \eqno (5.3.3.43)]$ This formula leads to the lattice-parameter changes Δd.

Figure 5.3.3.12| top | pdf |

Schematic representation of the double-beam comparator of Hart (1969).

A double-beam diffractometer can be used for the examination of variations in lattice parameters of about 10 parts in 10⁶ 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 10⁹) can be achieved in the double-source double-crystal X-ray 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 multiple-beam X-ray 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 X-ray 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 [(h_1k_1l_1)] and [(h_2k_2l_2)] , the lattice parameter [a_0] of a cubic crystal can be determined using the formula (Kishino, 1973) $[a_0=\textstyle{1\over2}\{(L\lambda_1)^2+[(M\lambda_2-L\lambda_1\cos\theta_{1- 2})/\sin\theta_{1 - 2}]^2\}^{1/2}, \eqno (5.3.3.44)]$ where $[L=(h^2_1+k^2_1+l^2_1){}^{1/2}]$ , $[M=(h^2_2+k^2_2+l^2_2){}^{1/2}]$ , and $[\theta_{1 - 2}]$ is the difference between the two Bragg angles for the specimen crystal, estimated from the measurement of $[\Delta\theta=|\theta_{1 - 2}-\theta'_{1 - 2}|]$ if the difference $[\theta_{1 - 2}']$ 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 $[K\alpha_1]$ and $[K\beta_1]$ 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 10⁷. The same precision is reported by Fukumori, Imai, Hasegawa & Akashi (1997), who introduced a precise positioning device and a position-sensitive proportional counter to their instrument.

As in the other multiple-crystal 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.

References

Baker, J. F. C. & Hart, M. (1975). An absolute measurement of the lattice parameter of germanium using multiple-beam X-ray diffractometry. Acta Cryst. A31, 364–367.Google Scholar

Baker, J. F. C., Hart, M., Halliwell, M. A. G. & Heckingbottom, R. (1976). Precise lattice parameter determination of dislocation-free gallium arsenide. I. X-ray measurements. Solid-State Electron. 19, 331–334.Google Scholar

Buschert, R. C., Meyer, A. J., Stuckey Kauffman, D. & Gotwals, J. K. (1983). A double-source double-crystal X-ray spectrometer for high-sensitivity lattice-parameter difference measurements. J. Appl. Cryst. 16, 599–605.Google Scholar

Cembali, F., Fabri, R., Servidori, M., Zani, A., Basile, G., Cavagnero, G., Bergamin, A. & Zosi, G. (1992). Precise X-ray relative measurement of lattice parameters of silicon wafers by multiple-crystal Bragg-case diffractometry. Computer simulation of the experiment. J. Appl. Cryst. 25, 424–431.Google Scholar

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International Tables for Crystallography (2006). Vol. C. ch. 5.3, p. 531