Double-crystal spectrometers

Gałdecka, E.

doi:10.1107/97809553602060000597

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 528-530

Section 5.3.3.7.1. Double-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

5.3.3.7.1. Double-crystal spectrometers

| top | pdf |

Detailed information concerning the double-crystal spectrometer, which consists of two crystals successively diffracting the X-rays, 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 lattice-parameter 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 $[h(\theta)]$ , 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 $[h_C(\theta)]$ [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): $[h_C(\theta)=K\textstyle\int\limits^\infty_{-\infty}\,R(\theta')R(\theta'-\theta)\,{\rm d}\theta', \eqno (5.3.3.37)]$ where $[R(\theta)]$ is an individual reflectivity function of one crystal and K is a coefficient of proportionality. Its half-width 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 $[h_\lambda(\theta)]$ , modified by the first crystal, is extremely narrow. Additional advantages of the diffraction profile are: its symmetry, because $[h_C(\theta)]$ 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.

Figure 5.3.3.8| top | pdf |

Schematic representation of the double-crystal spectrometer.

When there is a small difference in the two lattice spacings, so that one has a value d and the other $[d+\delta d]$ , if $[\delta\lambda/\lambda]$ 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 $[\theta_0]$ , the peak position shifts to $[\theta_0-\tan\theta\delta d/d]$ . The measurement of this shift rather than the absolute position of the rocking curve is the basis of all the double-crystal methods. An example of the application of a double-crystal 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 double-crystal 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 double-crystal spectrometer have been considered by Bearden & Thomsen (1971).

The double-crystal diffractometer, because of the small width of the diffraction profile, is a very suitable tool for local measurements of lattice-parameter differences, for example between an epitaxic layer and its substrate. Hart & Lloyd (1975) carried out such a measurement on a standard single-axis 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, [D_1] and [D_2] , 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 double-crystal 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 10⁴).

Figure 5.3.3.9| top | pdf |

Schematic representation of the double-crystal arrangement of Hart & Lloyd (1975) for the examination of epitaxic layers. (a) Experimental set-up. (b) Diffraction profiles recorded by detectors D₁, D₂, and D₃.

The very important advantage of this method, from the point of view of local measurements, is that single- or double-crystal diffraction can be selected, simultaneously if needed, on exactly the same specimen area. Other examples of strain measurements by means of a double-crystal 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 double-crystal 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 10⁵, 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 double-crystal X-ray diffraction with high-frequency ultrasonic excitation. Since ultrasound has a wavelength a little less than the X-ray 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 10⁶ 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 double-crystal spectrometer is to be used for absolute lattice-parameter determination, since such an arrangement combines the high precision and sensitivity of a double-crystal 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 germanium-crystal 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 double-crystal arrangements is formed by those in which white X-radiation 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 X-ray diffractometer and can be operated with either an $[\omega]$ or a $[\theta]$ – $[2\theta]$ scan. As diffraction phenomena appear for both the specimen and the analyser (in general of different materials) whose interplanar spacings are equal to [d_s] and [d_A] , respectively, the following relation results from Bragg's law: $[d_s\sin\theta_s=d_A\sin \theta_A, \eqno (5.3.3.38)]$ where $[\theta_s]$ and $[\theta_A]$ are the respective Bragg angles. Since [d_A] and $[\theta_s]$ are kept constant, a change in [d_s] as a function of temperature is determined from a change in $[\theta_A]$ . The relative error $[\delta d/d]$ resulting from (5.3.3.38) with $[\theta_A\approx90^\circ]$ is $[\eqalignno{ {\delta d_s \over d_s} = \cot\theta_A\delta\theta_A &= \tan(\pi/2-\theta_A)\delta\theta_A \cr &\approx (\pi/2-\theta_A)\delta\theta_A. &(5.3.3.39)}]$ 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 10⁷ 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 $[2\theta]$ – it is possible to combine $[\omega]$ and $[2\theta]$ scans and obtain a two-dimensional 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 rotating-anode high-power X-ray source (examples reported above) and a sealed-tube X-ray source. In the latter case (Irie, Koshiji & Okazaki, 1989), to increase the efficiency of the X-ray tube, the distance between the X-ray 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 lattice-parameter changes as high as 1 part in $[10^{9}]$ – $[10^{10}]$ 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.

Figure 5.3.3.10| top | pdf |

Schematic representation of the double-crystal arrangement of Okazaki & Kawaminami (1973a); white incident X-rays are used.

An original method for the measurement of lateral lattice-parameter variation by means of a double-crystal 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 10⁷ was reported. The method has been applied for the measurement of growth striations in silicon.

The main disadvantage of double-crystal 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 double-crystal 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 pseudo-non-dispersive 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).

References

Azároff, L. V. (1974). X-ray spectroscopy, Chap. 2. New York: McGraw-Hill.Google Scholar

Barla, K., Herino, R., Bomchil, G. & Pfister, J. C. (1984). Determination of lattice parameter and elastic properties of porous silicon by X-ray diffraction. J. Cryst. Growth, 68, 727–732.Google Scholar

Bearden, J. A. & Henins, A. (1965). Precision measurement of lattice imperfections with a photographic two-crystal method. Rev. Sci. Instrum. 36, 334–338.Google Scholar

Bearden, J. A. & Thomsen, J. S. (1971). The double-crystal X-ray spectrometer: corrections, errors, and alignment procedure. J. Appl. Cryst. 4, 130–138.Google Scholar

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

Compton, A. H. & Allison, S. K. (1935). X-rays in theory and experiment. New York: Van Nostrand.Google Scholar

Fewster, P. F. (1982). Absolute lattice-parameter measurements of epitaxial layers. J. Appl. Cryst. 15, 275–278.Google Scholar

Fukahara, A. & Takano, Y. (1977). Determination of strain distributions from X-ray Bragg reflexion by silicon single crystals. Acta Cryst. A33, 137–142.Google Scholar

Godwod, K., Kowalczyk, R. & Szmid, Z. (1974). Application of a precise double X-ray spectrometer for accurate lattice parameter determination. Phys. Status Solidi A, 21, 227–234.Google Scholar

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

Hart, M. & Lloyd, K. H. (1975). Measurement of strain and lattice parameter in epitaxic layers. J. Appl. Cryst. 8, 42–44.Google Scholar

Irie, K., Koshiji, N. & Okazaki, A. (1989). High-angle double-crystal X-ray diffractometry (HADOX): combination with a sealed-tube X-ray source. Jpn. J. Appl. Phys. 28, 1504–1506.Google Scholar

James, R. W. (1967). The optical principles of the diffraction of X-rays. London: Bell.Google Scholar

Korytár, D. (1984). Lateral lattice parameter variation measurement by means of a double crystal X-ray method with oscillating slit. Czech. J. Phys. B34, 1277–1281.Google Scholar

Kurbatov, B. A., Zubenko, V. V. & Umansky, M. M. (1972). The use of the monochromator crystal with anomalous transmission of X-rays in precise lattice parameter measurements. (In Russian.) Kristallografiya, 17, 1058–1060.Google Scholar

Ohama, N., Sakashita, H. & Okazaki, A. (1979). Improvement of high-angle double-crystal X-ray diffractometry (HADOX) for measuring temperature dependence of lattice constants. II. Practice. J. Appl. Cryst. 12, 455–459.Google Scholar

Okazaki, A. & Kawaminami, M. (1973a). Accurate measurement of lattice constants in a wide range of temperature: use of white X-ray and double-crystal diffractometry. Jpn. J. Appl. Phys. 12, 783–789.Google Scholar

Okazaki, A. & Kawaminami, M. (1973b). Lattice constant of strontium titanate at low temperatures. Mater. Res. Bull. 8, 545–550.Google Scholar

Okazaki, A. & Ohama, N. (1979). Improvement of high-angle double-crystal X-ray diffractometry (HADOX) for measuring temperature dependence of lattice constants. I. Theory. J. Appl. Cryst. 12, 450–454.Google Scholar

Okazaki, A. & Soejima, Y. (2001). Ultra-high-angle double-crystal X-ray diffractometry (U-HADOX) for determining a change in the lattice spacing: theory. Acta Cryst. A57, 708–712.Google Scholar

Ridou, C., Rousseau, M. & Freund, A. (1977). Détermination précise des paramètres cristallins au voisinage de changement de phase cubique quadratique dans RbCaF₃. J. Phys. (Paris), 38, L-359–L-363.Google Scholar

Sasvári, J. & Zsoldos, É. (1980). Accurate lattice parameter measurements of epitaxial layers. Invited paper at International Symposium on Industrial Applications of X-ray Spectrometry and Diffractometry, Turawa, Poland, 15–18 April 1980.Google Scholar

Soejima, Y., Tomonaga, N., Onitsuka, H. & Okazaki, A. (1991). Two-dimensional intensity distribution in high-angle double-crystal X-ray diffractometry (HADOX). Z. Kristallogr. 195, 161–168.Google Scholar

Takano, Y. & Maki, M. (1972). X-ray measurement of lattice strain of oxygen diffused silicon. Acta Cryst. A28, S171.Google Scholar

Zolotoyabko, E., Sander, B., Komem, Y. & Kantor, B. (1993). Improved strain analysis in semiconductor crystals by X-ray diffractometry enhanced with ultrasound. Appl. Phys. Lett. 63, 1540–1542.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 5.3, pp. 528-530