Lattice plane oriented perpendicular to a direction (lamellar texture)

Zvyagin, B. B.

doi:10.1107/97809553602060000593

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 412-413

Section 4.3.5.2. Lattice plane oriented perpendicular to a direction (lamellar texture)

B. B. Zvyaginⁿ ^‡

4.3.5.2. Lattice plane oriented perpendicular to a direction (lamellar texture)

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If in the plane of orientation (the texture basis ) the crystal has a two-dimensional cell a, b, γ, the c* axis of the reciprocal cell will be the texture axis . Reciprocal-lattice rods parallel to c* intersect the plane normal to them (the ab plane of the direct lattice) in the positions hk of a two-dimensional net that has periods $[1/a\sin\gamma]$ and $[1/b\sin \gamma]$ with an angle γ′ = π − γ between them, whatever the direction of the c axis in the direct lattice. The latter is defined by the absolute value c and the normal projection [c_n] on the ab plane, with components [x_n] , [y_n] along the axes a, b. In the triclinic case, $[{x_n=(c/a)(\cos\beta-\cos\alpha\cos\gamma)/\sin^2\gamma} \eqno (4.3.5.1)]$ $[{y_n=(c/b)(\cos\alpha-\cos\beta\cos\gamma)/\sin^2\gamma} \eqno (4.3.5.2)]$ (Zvyagin et al., 1979). The lattice points of each rod with constant hk and integer l are at intervals of $[c^*=1/d_{001}]$ , but their real positions, described by their distances $[D_{hkl}]$ from the plane ab, depend on the projections of the axes a* and b* on c* (see Fig. 4.3.5.1 ), the equations $[{x_n=-a^*\cos\beta^*/c^*} \eqno (4.3.5.3)]$ $[{y_n=-b^*\cos\alpha^*/c^*} \eqno (4.3.5.4)]$ being satisfied.

Figure 4.3.5.1| top | pdf |

The relative orientations of the direct and the reciprocal axes and their projections on the plane ab, with indication of the distances B_hk and D_hkl that define the positions of reflections in lamellar texture patterns.

The reciprocal-space representation of a lamellar texture is formed by the rotation of the reciprocal lattice of a single crystal about the c* axis. The rods hk become cylinders and the lattice points become circles lying on the cylinders. In the case of high-energy electron diffraction (HEED ), the wavelength of the electrons is very short, and the Ewald sphere, of radius 1/λ, is so great that it may be approximated by a plane passing through the origin of reciprocal space and normal to the incident beam. The patterns differ in their geometry, depending on the angle $[\varphi]$ through which the specimen is tilted from perpendicularity to the primary beam. At $[\varphi=0]$ , the pattern consists of hk rings. When $[\varphi\neq0]$ it contains a two-dimensional set of reflections hkl falling on hk ellipses formed by oblique sections of the hk cylinders. In the limiting case of $[\varphi=\pi/2]$ , the ellipses degenerate into pairs of parallel straight lines theoretically containing the maximum numbers of reflections. The reflection positions are defined by two kinds of distances: (1) between the straight lines hk (length of the short axes of the ellipses hk): $[B_{hk}=(1/\sin\gamma)(h^2/a^2+k^2/b^2 - 2hk\cos\gamma/ab)^{1/2} \eqno (4.3.5.5)]$ and (2) from the reflection hkl to the line of the short axes: $[\eqalignno{D_{hkl} &= (ha^*\cos\beta^*/c^*+kb^*\cos\alpha^*/c^*+l)c^* &(4.3.5.6)\cr &=(-hx_n-ky_n+l)/d_{001}. &(4.3.5.7)}%fd4.3.5.7]$ In patterns obtained under real conditions $[(0\lt\varphi\lt\pi/2]$ , accelerating voltage V proportional to $[\lambda^{-2}]$ , distance L between the specimen and the screen), these values are presented in the scale of $[L\lambda]$ , $[D_{hkl}]$ also being proportional to $[1/\sin\varphi]$ with maximum value $[D_{\rm max}=B_{hk}\tan\varphi]$ for the registrable reflections. The values of $[B_{hk}]$ and $[D_{hkl}]$ , determined by the unit cells and the indices hkl, are the objects of the geometrical analysis of the OT patterns . When the symmetry is higher than triclinic, the expression for $[B_{hk}]$ and $[D_{hkl}]$ are much simpler.

Such OT patterns are very informative, because the regular two-dimensional distribution of the hkl reflections permits definite indexing, cell determination, and intensity measurements. For low-symmetry and fine-grained substances, they present unique advantages for phase identification, polytypism studies , and structure analysis .

In the X-ray study of textures , it is impossible to neglect the curvature of the Ewald sphere and the number of reflections recorded is restricted to larger d values. However, there are advantages in that thicker specimens can be used and reflections with small values of $[B_{hk}]$ , especially the 00l reflections, can be recorded. Such patterns are obtained in usual powder cameras with the incident beam parallel to the platelets of the oriented aggregate and are recorded on photographic film in the form of hkl reflection sequences along hk lines, as was demonstrated by Mamy & Gaultier (1976). The hk lines are no longer straight, but have the shapes described by Bernal (1926) for rotation photographs. It is difficult, however, to prepare good specimens. Other arrangements have been developed recently with advantages for precise intensity measurements. The reflections are recorded consecutively by means of a powder diffractometer fitted with a goniometer head. The relation between the angle of tilt $[\varphi]$ and the angle of diffraction (twice the Bragg angle) $[2\theta]$ depends on the reciprocal-lattice point to be recorded. If the latter is defined by a vector of length $[H=(2\sin\theta)/\lambda]$ and by the angle $[\omega]$ between the vector and the plane of orientation (texture basis), the relation $[\varphi=\theta-\omega]$ permits scanning of reciprocal space along any trajectory by proper choice of consecutive values of $[\omega]$ or $[\theta]$ . In particular, if $[\omega]$ is constant, the trajectory is a straight line passing through the origin at an angle $[\omega]$ to the plane of orientation (Krinary, 1975). Using additional conditions $[[\omega=\arctan(D/B)]$ , $[H=(B^2+D^2)^{1/2}]]$ , Plançon et al. (1982) realized the recording and the measurement of intensities along the cylinder-generating hk rods for different shapes of the misorientation function N(α).

In the course of development of electron diffractometry, a deflecting system has been developed that permits scanning the electron diffraction pattern across the fixed detector along any direction over any interval (Fig. 4.3.5.2 ). The intensities are measured point by point in steps of variable length. This system is applicable to any kind of two-dimensional intensity pattern, and in particular to texture patterns (Zvyagin, Zhukhlistov & Plotnikov, 1996). Electron diffractometry provides very precise intensity measurements and very reliable structural data (Zhukhlistov et al., 1997).

Figure 4.3.5.2| top | pdf |

(a) Part of the OTED pattern of the clay mineral kaolinite and (b) the intensity profile of a characteristic quadruplet of reflections recorded with the electron diffractometry system. The scanning direction is indicated in (a).

If the effective thickness of the lamellae is very small, of the order of the lattice parameter c, the diffraction pattern generates into a combination of broad but recognizably distinct 00l reflections and broad asymmetrical hk bands (Warren, 1941). The classical treatments of the shape of the bands were given by Méring (1949) and Wilson (1949) [for an elementary introduction see Wilson (1962)].

References

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, pp. 412-413