Reciprocal lattice angles

A standard set of reference axes and equations to describe spontaneous strains is now well established (Schlenker et al. 1978, Redfern and Salje 1987, Carpenter et al. 1998a). The orthogonal reference axes, X, Y and Z, are selected so that Y is parallel to the crystallographic y-axis, Z is parallel to the normal to the (001) plane (i.e. parallel to c ) and X is perpendicular to both. The +X direction is chosen to conform to a right-handed coordinate system. Strain is a second rank tensor three linear components, cn, 622 and 33 are tensile strain parallel to X, Y and Z respectively and co, 623, eu are shear strains in the XZ, YZ and XY planes, respectively. The general equations of Schlenker et al. (1978) define the strains in terms of the lattice parameters of a crystal (a, b, c, a, P, y, where P is the reciprocal lattice angle) with respect to the reference state for the crystal ( , bo, Co, cto, Po,Yo) ... 

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

Electrons having energies and incident angles typical of RHEED can be treated as nearly nonpenetrating. As a result, atoms in the outermost plane are responsible for most of the scattering, and the resulting reciprocal lattice will be an array of rods perpendicular to the surfrce plane. 

Fig. 2.13. Two-dimensional Bravais lattice with the basis vectors a)s a2, and the reciprocal lattice vectors bi, b2. The solid and dashed arrows at angles A and 0A give the ferroelectric (k = 0) and antiferroelectric (k = bi/2) configurations of dipoles in the ground state.

From a comparison of various spot electron diffraction patterns of a given crystal, a three-dimensional system of axis in the reeiproeal lattice may be established. The reeiproeal unit cell may be eompletely determined, if all the photographs indexed. For this it is sufficient to have two electron diffraction patterns and to know the angle between the seetions of the reeiproeal lattice represented by them, or to have three patterns which do not all have a particular row of points in common (Fig.5). Crystals of any compound usually grow with a particular face parallel to the surface of the specimen support. Various sections of the reciprocal lattice may, in this case, be obtained by the rotation method (Fig.5). 

Fig. 4.3. Experimental intensity vs. voltage (energy) curves for electron diffraction from at Pt(l 11) surface. Beams are identified by different labels (h,k) representing reciprocal lattice vectors parallel to the surface. An incidence angle of 4° from the surface normal is used...

Triclinic crystals. None of the angles of a triclinic cell are right angles in consequence, none of the axes of the reciprocal lattice are... 

These formulae are so unwieldy that it is better to derive the reciprocal lattice elements directly from the spacings and angles of the planes ... 

All reciprocal lattice levels and all angles of oscillation can be dealt with in this way, care being taken always to use the correct radius for the circle of contact. In the same way, if for any purpose it is desired to know at what angle any plane reflects, if is only neeessary to draw... 

If, as in Fig. 108 a, the c axis of the crystal is displaced from the axis of rotation in the plane normal to the beam (for the mean position of the crystal), the zero layer (hk0) of the reciprocal lattice is tilted in this same direction, and its plane cuts the sphere of reflection in the circle AD. During the 15° oscillation a number of hkO points pass through the surface of the sphere, and thus X-rays reflected by these hkO planes of the crystal strike the film at corresponding points on the flattened-out film (Fig. 108 b) the spots fall on a curve BAD, whose distance from the equator is a maximum at a Bragg angle 0 = 45° and zero at 6 = 90°. If, on the other hand, the displacement of the c axis is in the plane containing the beam (Fig. 108 c), the spots on the film fall on a curve whose maximum distance from the equator is at 6 = 90° (Fig. 108 d). When the displacement of the c axis has components in both directions,... 

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