The Kossel method and divergent-beam techniques

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. 510-516

Section 5.3.2.4. The Kossel method and divergent-beam techniques

E. Gałdecka^a

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

5.3.2.4. The Kossel method and divergent-beam techniques

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5.3.2.4.1. The principle

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Another group of methods with photographic recording has been developed in parallel with those discussed in Subsection 5.3.2.3. These are the methods in which the crystal remains stationary and the diffraction conditions are fulfilled, simultaneously for more than one set of crystallographic planes, by the use of a highly divergent beam , dispersed from a point source (Fig. 5.3.2.2 ). The Kossel method (Kossel, 1936, and references therein), the divergent-beam techniques initiated by Lonsdale (1947), and their numerous modifications belong to this group.

Figure 5.3.2.2| top | pdf |

Schematic representation of the origin of the Kossel lines. (a) The Kossel (1936) method. (b) The divergent-beam method developed by Lonsdale (1947). (c) The proton-induced Kossel effect (Geist & Ascheron, 1984). In (b), the divergent X-ray beam is directed onto the sample from a point source while in the remaining cases it is generated within a crystal by (a) electrons or (c) protons.

The excitation of the characteristic X-rays used in these methods can be performed by X-radiation (Lonsdale, 1947), by electron bombardment (Kossel, 1936; Gielen et al., 1965; Ullrich & Schulze, 1972) or by proton irradiation (Geist & Ascheron, 1984) of a single crystal. The source of emitted X-rays may be located either in the sample itself (the Kossel method), on the surface of the sample in a layer of target material (the pseudo-Kossel method), or outside the sample (the divergent-beam techniques). The divergent X-ray beam diffracts from sets of crystallographic planes. The diffracted rays for each Bragg reflection form a conical surface whose semivertical angle is equal to 90° − $[\theta]$ and whose axis is normal to the Bragg plane (i.e. coincides with the reciprocal-lattice vector).

The conical surface of an hkl reflection can be described in the form (Morris, 1968; Chang, 1984): $[x'^2+y'^2=z'^2\tan^2\alpha, \eqno (5.3.2.6)]$ where [(x',y',z')] is an orthogonal coordinate system with its origin at the vertex of the cone and with z′ along the axis of the cone and normal to the plane of interest, and α is the semivertical angle. Since α depends on the Bragg angle, it is possible to combine (5.3.2.6) with the Bragg law [equations (5.3.1.1) or (5.3.1.2)], and so with the lattice parameters. In particular, the dependence can be presented as: $[{r\over z'} = {1\over \sin\theta} = {2d \over n\lambda}, \eqno (5.3.2.6a)]$ where $[r=(x'^2+y'^2+z'^2)^{1/2}]$ .

In another convenient coordinate system (x, y, z) common for all the cones, say with z along the direction of the incident beam, (5.3.2.6a) will take the form: $[{r\over c_xx+c_yy+c_zz} = {2d \over n\lambda}, \eqno(5.3.2.6b)]$ where [c_x,c_y,c_z] are direction cosines of the angles between the [z'] axis and the axes x, y and z, respectively. Since the origin of the coordinate system has not been changed, $[r=(x^2+y^2+z^2){}^{1/2}. \eqno (5.3.2.6c)]$ The Kossel lines (Fig. 5.3.2.3 ) are formed at the intersections of the cones with a flat film placed parallel to the specimen surface (Fig. 5.3.2.2). When the film plane is normal to the z axis, and the focus-to-film distance is equal to Z, putting z = Z in (5.3.2.6b,c) gives the formulae describing the conic section on the film.

Figure 5.3.2.3| top | pdf |

(a) The Kossel pattern from iron and (b) the corresponding stereographic projection (Tixier & Waché, 1970).

A high-precision Kossel camera is described by Reichard (1969) and the generation of pseudo-Kossel patterns by the divergent-beam method has been described by Imura, Weissmann & Slade (1962), Ellis, Nanni, Shrier, Weissmann, Padawer & Hosokawa (1964), and Berg & Hall (1975).

The photographs may be in either the transmission or the back-reflection region (Fig. 5.3.2.2). The second arrangement seems to be (Lutts, 1968) more suitable for lattice-parameter determination, since the background is less intensive and the lines on the photographs have greater contrast. Both possibilities are used in practice. Photographs in the back-reflection region have been reported by Imura, Weissmann & Slade (1962), Ullrich (1967), Newman & Weissmann (1968), Newman & Shrier (1970), and Berg & Hall (1975). Examples of the use of the transmission region are given by Yakowitz (1966a), Reichard (1969), and Glass & Weissmann (1969).

The recommended crystal thickness t for work in the transmission region , according to Hanneman, Ogilvie & Modrzejewski (1962), is given by: $[t=1/0.2\mu_L, \eqno (5.3.2.7)]$ where $[\mu_L]$ is the linear absorption coefficient for Kα radiation generated in the crystal. A more detailed study of the effect of sample thickness , as well as operating voltage, on the contrast of Kossel transmission photographs is given by Yakowitz (1966a).

The picture geometry does not depend, in principle, on whether the Kossel, pseudo-Kossel, or divergent-beam technique is applied. Imura (1954) has studied in detail the form of the curves of the light or deficiency type, and recorded both in the transmission and in the back-reflection region. The curves on transmission patterns can be considered to be conics; those recorded in the back-reflection region are related to ellipses, but of higher order. In general, the photograph has to be indexed before performing measurement on the film. For this purpose, the pattern may be compared with a calculated pattern (gnomonic, orthogonal, cylindrical, or stereographic projection). For lattice-parameter determination, various features of the photographs may be used, i.e. intersections or near-intersections of Kossel lines , their near-tangency, lens-shaped figures, and the whole lines approximated with a function.

5.3.2.4.2. Review of methods of accurate lattice-parameter determination

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The basis of lattice-parameter determination involves measurements performed on the film. There are various methods covering most of the different geometrical features of the cones and recorded pictures. These were reviewed by Lutts (1968), Yakowitz (1966b, 1969) and Tixier & Waché (1970). In each case, the wavelength of the excited radiation has to be known. Often, the resolved $[K\alpha_{1,2}]$ doublet and/or Kβ radiation is applied rather than a single (but most pronounced) Kα₁ line. The other data needed (a sufficient number of equations, the solution of which leads to lattice-parameter determination; camera geometry; crystal system; and indices) depend on the method.

Biggin & Dingley (1977) propose a classification of all the methods using a divergent beam based on the information required.

(i) All the kinds of information mentioned above are needed in the earliest method (Kossel, 1936), in which near-tangency of Kossel lines is taken into account.

(ii) As has been shown in successive papers that appeared from 1947 to the early 1970's, information on camera dimensions can be eliminated if the crystallographic system is known and the photograph is indexed. Some dependence between crystal planes and, as a consequence, between lines on the photograph, is then taken into consideration. This is of great importance, since camera dimensions, in particular the distance from the focus to the centre of the photograph, are difficult to measure accurately and negatively influence the precision and accuracy of the determined lattice parameters.

The Lonsdale (1947) method is based on triple intersections of the Kossel lines resulting from multiple-diffraction effects (cf. Subsection 5.3.3.6), which are dependent on the wavelength, so first the particular wavelength has to be determined by an extrapolation. Two or three lines with known indices produced by different wavelengths ( $[K\alpha_1]$ , $[K\alpha_2]$ , and/or Kβ) are used for this task (Schwartzenberger, 1959; Mackay, 1966; Isherwood & Wallace, 1971; Spooner & Wilson, 1973). Similar problems arise when near-tangency of two lines is taken into consideration (Kossel, 1936; Mackay, 1966).

With the use of reciprocal-lattice geometry , the equation of the so-called Kossel plane (Isherwood & Wallace, 1971) for a diffracting plane (hkl) is given by (Spooner & Wilson, 1973; Chang, 1984): $[L_1x^*+L_2y^*+L_3z^*={1\over d}, \eqno (5.3.2.8)]$ where [L_1] , [L_2] , and [L_3] are direction cosines between the reciprocal vector $[{\bf H}=h{\bf a}^*+k{\bf b}^*+l{\bf c}^*]$ and the unit-cell vectors $[{\bf a}^*]$ , $[{\bf b}^*]$ , $[{\bf c}^*]$ , i.e. $[L_1={ha^*\over H}, \quad L_2={kb^*\over H}, \quad L_3={lc^*\over H}', \eqno (5.3.2.8a)]$ where $[H=|{\bf H}|=1/d]$ , $[a^*=|{\bf a}^*|]$ , $[b^*=|{\bf b}^*|]$ , $[c^*=|{\bf c}^*|]$ .

In the case of the triple intersection, (5.3.2.8) is satisfied simultaneously by three sets of diffracting planes, the Miller indices of those being [(h_ik_il_i)] , i = 1, 2, 3. From the Ewald construction, it follows that the triple point [(x^*_0,y^*_0,z^*_0)] must lie on the sphere of reflection: $[x^{*2}+y^{*2} + z^{*2} = {4 \over \lambda^2_0}. \eqno (5.3.2.8b)]$ The radius of the sphere, $[2/\lambda_0]$ , is the modulus of the double wavevector defined by Isherwood & Wallace (1971).

For cubic crystals, where $[H=(h^2+k^2+l^2)^{1/2}/a]$ , the set of equations to be solved, resulting from (5.3.2.8) and (5.3.2.8a), which relates to the triple point, takes the form $[h_i x^*_0+k_i y^*_0+l_i z^*_0 = (h^2_i + k^2_i+l^2_i)/a, \eqno (5.3.2.9)]$ where i = 1, 2, 3.

First, coordinates [x^*_0,y^*_0,z^*_0] dependent on a are determined from (5.3.2.9), and next a is calculated from (5.3.2.8b). It should be noted that the measurements performed on the film are used here for determination of the wavelength only. As shown (theoretically and experimentally) by Brühl & Rhan (1985) for cubic lattices, positions of the lines on the film that result from the multiple-diffraction phenomenon are insensitive to lattice-parameter changes (caused by thermal expansion, for example), while positions of the primary reflections depend on actual lattice-parameter values. Practical examples of photographic multiple-diffraction methods are given by Lonsdale (1947) (see also Tixier & Waché, 1970; Chang, 1984), Isherwood & Wallace (1966), Isherwood (1968), Isherwood & Wallace (1970), Spooner & Wilson (1973), Brown, Halliwell & Isherwood (1980), and Isherwood, Brown & Halliwell (1981, 1982).

The technique, in which triple intersections of Kossel lines are analysed, can be used both for back-reflection and for transmission. In the second case, the thickness t of the crystal should be such that $[\mu_Lt\simeq 1]$ [cf. equation (5.3.2.7)]. However, thicker crystals, for which $[\mu_Lt\,\gtrsim10]$ , can be examined by anomalous transmission, if the degree of crystal perfection is high (Glass & Moudy, 1974). A correction for displacement of the conics due to wafer thickness t is necessary in the case when the intersection lies along the normal to the specimen surface. One triple intersection allows the determination of the lattice parameter of a cubic crystal, but for a structure in the orthorhombic system three such intersections would be required.

Two intersecting Kossel lines sometimes form a lens configuration (Fig. 5.3.2.4a ). The use of such a figure, consisting of two lenses (Fig. 5.3.2.4b) owing to the resolved doublet of $[K\alpha_1]$ and $[K\alpha_2]$ (or Kβ) radiation, makes it possible to determine lattice parameters without a knowledge of the distance between the source and the film. Lattice parameters are then calculated from the ratio [L_1/L_2] of the distances [L_1] and [L_2] between pairs of the sections. Heise (1962) used this method for cubic crystals in the simplest case, in which the cone axes are perpendicular to the film (symmetrical method). His idea had been generalized by Gielen et al. (1965), who formed a theory of the lens in the case of arbitrarily situated diffracting planes and arbitrary wavelengths, but for cubic crystals only. Lutts (1968) derived suitable formulae for cubic, tetragonal, and hexagonal systems by combining the ratio [L_1/L_2] with interplanar spacings and lattice parameters.

Figure 5.3.2.4| top | pdf |

Lens-shaped figures formed by pairs of intersecting conics. (a) Schematic representation of the method of Heise (1962). (b) The use of the Kα_1,2 doublet for precise and accurate lattice-parameter determination.

Several features of the Kossel pattern may be jointly taken into account for its interpretation and lattice-parameter determination. Hanneman, Ogilvie & Modrzejewski (1962) used the conic sections formed by $[K\alpha_1]$ and $[K\beta]$ radiation and the lens figures.

Lang & Pang (1995) observed and analysed fine streaks in the transmitted pseudo-Kossel patterns caused by both the coherent multiple diffraction and the enhanced Borrmann (anomalous) transmission. As they have found, these fine-scale features of a few arcseconds in angular width, which add markers to the broad-line Kossel patterns, may be taken into account in accurate lattice-parameter measurements.

(iii) Determination of lattice parameters by means of techniques utilizing a highly divergent beam becomes much more complicated if there is no information about indices and the crystal system. Such a problem arises in the case when the crystal system of the specimen is unknown or when the lattice is deformed . Then, a three-dimensional array of intersecting cones with a common vertex should be taken into consideration.

It is difficult to dispense with the data concerning the camera geometry. However, the distance of X-ray source from the film center may be eliminated in calculations when the multiple-exposure technique is used. This technique, introduced by Ellis et al. (1964) for back-reflection patterns, depends on recording the Kossel lines at variable but controlled distances from the focus to the film (Fig. 5.3.2.5 ), so that three or more positions of a cone generator can be established and, as a consequence, the cone axis and the semivertical angle are determined. The interpretation of the multiple-exposure pictures is based, in principle, on the coordinates of general points of lines rather than on their special properties.

Figure 5.3.2.5| top | pdf |
Schematic representation of the multiple-exposure technique (after Fischer & Harris, 1970).

The basic formula valid for all the methods applying the Kossel idea, $[{\bf P}\cdot{\bf N}=\cos\alpha, \eqno (5.3.2.10)]$ where P is the unit vector defining the cone generator and N is the axial direction of a cone, can now be fully utilized, since multiple-exposure techniques make possible accurate calculations of direction cosines. Lengthy and complicated calculations resulting from measurements performed on the film may be realized by means of a computer. A suitable program is given by Fischer & Harris (1970). This technique has also been applied and developed by Slade, Weissmann, Nakajima & Hirabayshi (1964), Shrier, Kalman & Weissmann (1966), Newman & Weissmann (1968), Schneider & Weik (1968), Fischer & Harris (1970), Newman & Shrier (1970), Aristov, Shekhtman & Shmytko (1973), and Soares & Pimentel (1983) for both the back-reflection and the transmission region.

As mentioned above, the Kossel lines occurring in the back-reflection region are similar to ellipses; they can be described using an equation of the fourth degree (Newman, 1970). In general, the major axes of such ellipse-shaped figures have been taken into account in lattice-parameter determination. A novelty introduced by Lider & Rozhansky (1967) was to also use the minor axes in the calculations. The essential feature of their method is the location of the X-ray source in the plane of a flat film.
(iv) The other possibility for gathering the necessary information for the recorded picture is a more detailed study of the form of the Kossel lines. Morris (1968) proposed a method based on the mathematical analysis of a cone, which makes possible the determination of lattice parameters in any crystal system, with a relative accuracy as high as 10 parts in 10⁶. The necessary calculations can be made by a computer program. A conic section can usually be expressed by a general equation of the second degree (Bevis, Fearon & Rowlands, 1970; Harris & Kirkham, 1971; Morris, 1968): $[x^2+Ay^2+Bxy+Cx+Dy+E=0, \eqno (5.3.2.11)]$ which results from a combination and a transformation of (5.3.2.6b) and (5.3.2.6c). The coefficients A, B, C, D, and E, being functions of the direction cosines and of the ratio 2d/nλ, can be found by the method of least squares. Methods based on Kossel-line fitting can be realized both in the single-exposure (Harris & Kirkham, 1971) and in the multiple-exposure technique (Aristov, Shekhtman & Shmytko, 1973; Aristov & Shmytko, 1978).
(v) From theoretical considerations based on the shape of pseudo-Kossel lines (Harris & Kirkham, 1971), it is possible to eliminate the need for information concerning camera geometry if the source and the pattern centre are accurately located. Lattice parameters of an unknown or deformed crystal can thus be determined with no information other than measurements on the film and a knowledge of the wavelength.

A general method for locating the X-ray source and the centre of the pattern – which permits the realization of the above idea – has been developed by Biggin & Dingley (1977). Its characteristic feature is the introduction of steel balls between the specimen and the film; these cast sharp shadows on the film by blocking the diffuse radiation. Coordinates of points along the Kossel lines as well as the shadow ellipses recorded on the film are taken into account in calculations.

5.3.2.4.3. Accuracy and precision

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Although the precision theoretically obtainable by means of the Lonsdale (1947) method is of the order of 1 part in 10⁶, this limit is unattainable in practice. The reported values are in the range of about 10⁻⁴–10⁻⁵ Å, depending not only on the method but also on the crystal – its symmetry and perfection. The highest accuracy known by the author was achieved by Lonsdale [(1947), ±5 × 10⁻⁵ Å, for diamond], Morris [(1968), 2 parts in 10⁵] and Aristov & Shmytko [(1978), |δd|/d ∼ 3 × 10⁻⁵, 1–5 × 10⁻⁵ rad for angles between crystallographic directions].

Systematic errors due to the methods in which a divergent beam is applied have been discussed by Hanneman, Ogilvie & Modrzejewski (1962), Gielen, Yakowitz, Ganow & Ogilvie (1965), Beu (1967), Lutts (1968), and Aristov & Shmytko (1978). The main sources of systematic error are:

(i) those common to all X-ray methods, resulting from a finite depth of X-ray penetration, wavelength dispersion, refraction (Isherwood & Wallace, 1966; Isherwood, 1968), and from the real structure (substructures and mosaic blocks) and
(ii) those common to methods with photographic recording , resulting from film shrinkage and inaccurate determination of camera dimensions and distances on the film.

The errors of the second group may be to some extent removed if small differences of the length resulting from the resolved $[K\alpha_{1,2}]$ doublet are measured on the film rather than distances due to only one wavelength, and/or if the camera dimensions can be eliminated from the equations used in the calculations of lattice parameters (see §5.3.2.4.2). A relative misorientation between the specimen and the flat film has been analysed by Lutts (1973).

An error typical for methods realized by means of an electron microscope or an electron-beam probe may result from the thermal effects of the electron beam generating a divergent X-ray beam at the crystal surface. Uncontrolled thermal effects may also occur in the case of the Kossel method, since the sample is situated inside the X-ray tube. In the latter method, the wavelength of the radiation emitted depends on the chemical composition of the sample, since the sample plays the role of the anode of the X-ray tube.

The reported precision of the methods, limited by the finite width of the lines on the photograph, and depending also on the geometrical features taken into account, is 1 part in 10³ to 1 part in 10⁵. The highest $[[\sigma(d)/d=10^{-5}]]$ is reported by Hanneman, Ogilvie & Modrzejewski (1962), Gielen, Yakowitz, Ganow & Ogilvie (1965), and Lider & Rozhansky (1967). On the other hand, the lowest (1 part in 10³), obtained by Harris & Kirkham (1971), is attributed to the method in which neither the indexing of the lines nor a knowledge of the crystallographic system or camera geometry is required.

For precision determination of lattice-parameter differences, a `point' source (i.e. as small as possible) is required and the high-order Kossel lines should be used to obtain both well resolved $[K\alpha_{1,2}]$ doublets and `thin' figures. The near-intersections of conic sections, applied in Lonsdale's (1947) method, the major axes of lens-shaped figures, used in Heise's (1962) method, and the small spherical polygons formed by several Kossel cones are very sensitive to lattice-parameter changes, so that these figures can be taken into account in the precise measurements reported in §5.3.2.4.4.

5.3.2.4.4. Applications

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As was mentioned in §5.3.2.4.3, the methods in which a highly divergent beam is used are applied both to the accurate determination of the unit cell and to the precision detection of lattice-parameter changes or differences. It should be added that the Kossel method is especially suitable for small single crystals or fine-grained polycrystals , whereas the other divergent-beam techniques need larger specimens (Lutts, 1968).

Since all the methods are relatively simple (stationary specimen, stationary film, simple construction of the camera) and, on the other hand, are applicable mainly for highly symmetric systems, they proved to be particularly useful in studies of metals and semiconductors. Various applications of the Kossel method and other divergent-beam techniques for this task have been discussed by Ullrich (1967), Ullrich & Schulze (1972), and Geist & Ascheron (1984). The latter paper relates especially to semiconductors.

A task that arises both in metallurgy and in the semiconductor industry is the examination of the real structure – in particular, measurements of strains introduced by variation in temperature, pressure, mechanical stress (elastic strains) or by point defects, deviation from exact stoichiometry, irradiation damage, and phase changes (permanent strains).

Measurements of small changes in interplanar spacings of independent sets of crystal planes enable a stress–strain analysis to be made (Imura, Weissmann & Slade, 1962; Ellis et al., 1964; Slade et al., 1964; Newman & Weissmann, 1968; Berg & Hall, 1975). A special case of strains is an extensional deformation of the lattice in the direction of crystal growth (Isherwood, 1968).

A typical metallurgical problem is the effect of heat treatment on the microstructure of alloys. An example of the application of the Kossel method to the task is given by Shinoda, Isokawa & Umeno (1969), who reported a study of precipitation of α from β in copper–zinc alloys. The lattice parameters and thermal expansion of α-iron and its alloys were examined by Lutts & Gielen (1971). Structure defects resulting from over-pressure experiments and annealing were investigated by Potts & Pearson (1966). Irradiation effects caused by neutrons were the subject of papers of Hanneman, Ogilvie & Modrzejewski (1962), Yakowitz (1972), and Spooner & Wilson (1973); those caused by electron bombardment were reported by Ullrich (1967).

Divergent-beam techniques are considered to be a suitable tool for studying strains in epitaxic layers (Hart, 1981), since corresponding lines of the layer and substrate, observed on one photograph, can be readily identified. Relevant examples are given by Brühl (1978), Chang, Patel, Nannichi & de Prince (1979), and Chang (1979), who examined lattice mismatch in LPE heterojunction systems, and by Brown, Halliwell & Isherwood (1980), and Isherwood, Brown & Halliwell (1981, 1982), who reported characterization of distortions in heteroepitaxic structures together with a theoretical basis (multiple diffraction) for the method.

Another task of real-structure examination is the determination of angles between crystal blocks. A method has been worked out by Aristov, Shmytko & Shulakov (1974a,b).

Divergent-beam techniques can also be used in X-ray topographic studies, realized either by means of Kossel-line scanning (Rozhansky, Lider & Lyutzau, 1966) or by line-profile analysis (Glass & Weissmann, 1969).

Schetelich & Geist (1993) used the Kossel method for lattice-parameter determination and a qualitative estimation of the crystal perfection of quasicrystals and showed that the fine structure of Kossel lines of quasicrystals is the same as observed for conventional crystals.

Mendelssohn & Milledge (1999) used a Dingley–Kossel camera for quick and simple computer-aided measurements of cell parameters of isotopically distinct samples of LiF over a wide temperature range of 15–375 K.

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