Lattice plane index

Miller s index system is similar to the lattice plane index system but with the difference that the hkl values are the lowest possible integer values. The Miller s index notation for both the sets of planes with the lattice plane indices (010) and (020) is simply (010). Miller s indices thus provide information only about the orientation of the planes and disregard the interplanar distances involved. 

In crystals it is necessary to denote planes and direction. This can be done in the conventional way either by Miller s index or by a lattice plane index (Fig. 7.5). Directions are given as the lowest integer vector referring to the coordinate system x a), y b)... 

Figure 7.5 Definition of lattice plane index illustrated with three examples.

The most densely packed diffraction planes along the chain axis for polyethylene are denoted (002) in lattice plane index notation. The distance between these planes is thus c/2 0.127 nm. In Miller s index notation they are (001). 

The clean siuface of solids sustains not only surface relaxation but also surface reconstruction in which the displacement of surface atoms produces a two-dimensional superlattice overlapped with, but different from, the interior lattice structure. While the lattice planes in crystals are conventionally expressed in terms of Miller indices (e.g. (100) and (110) for low index planes in the face centered cubic lattice), but for the surface of solid crystals, we use an index of the form (1 X 1) to describe a two-dimensional surface lattice which is exactly the same as the interior lattice. An index (5 x 20) is used to express a surface plane in which a surface atom exactly overlaps an interior lattice atom at every five atomic distances in the x direction and at twenty atomic distances in the y direction. 

Note that the particle and lattice site indexing hidden in this notation mandates a one-one mapping between the lattice sites of the two systems we are free to choose any of the N possibilities. In the case of hep and fee lattices [66] it is natural to exploit the fact that the one lattice can be made out of the other merely by translating close-packed planes, in the fashion depicted in Fig. 6. 

This approximation is sufficient for broad lines and particle diameters <5 nm. The particle size is determined in the direction normal to the lattice plane indicated by the Miller index of the line used for profile determination. The formula featuring a constant accounting for the fact that the breadth is measured in radians (as is 20) is only valid for spherical particles. 

Immediately after the introduction of a constant refractive index Bethe developed a dispersion theory of electron diffraction which is very closely related to the Darwin-Ewald theory. In this theory the propagation of de Broglie waves through a crystal is investigated, the potential being expanded in a triple Fourier series in terms of the contributions of the individual lattice planes hkl. Thus Vq in Schrodinger s equation is replaced by a triple Fourier series with the coefficients In accordance with this assumption, the solution... 

The exceptionally high hardness of diamond, the easy cleavage along the three main lattice planes of the crystal, its high refractive index and its optical dispersion makes diamonds a valuable raw material for brilliants. The gem diamonds utilized are still exclusively natural diamonds. 

Fig. Al-5 The reciprocal lattice of a hexagonal crystal which has = 4 A. (Here the three-symbol system of plane indexing is used and 2l is the axis usually designated c.) The axes as and bs are normal to the drawing.

Diffraction can occur at different scattering angles 26 for the same crystallographic plane, giving the different orders n of diffraction. For simplicity, the number n will be incorporated in the indexing of the lattice planes, where ... 

The step-free energy k Ts(T) can be conveniently introduced by considering the interfacial tension of interfaces which are tilted through an angle 8 relative to a low-index lattice plane (fig. 65). For small 9 the angular-dependent surface tension takes the form... 

As one generally uses a vector normal to a lattice plane to specify its orientation, one can as well use a reciprocal lattice vector. This allows to define the Miller indices of a lattice plane as the coordinates of the shortest reciprocal lattice vector normal to that plane, with respect to a specified set of direct lattice vectors. These indices are integers with no common factor other than 1. A plane with Miller indices h, k, l is thus normal to the reciprocal lattice vector G = hb + kb > + lb->. and it is contained in a continuous plane G.r = constant. This plane intersects the primitive vectors a of the direct lattice at the points of coordinates xiai, X2a2 and X3a3, where the Xi must satisfy separately G.Xjai = constant. Since G.ai, G.a2 and G.as are equal to h, k and /, respectively, the Xi are inversely proportional to the Miller indices of the plane. When the plane is parallel to a given axis, the corresponding x value is taken for infinity and the corresponding Miller index taken equal to zero. 

Index the lattice planes drawn in the figure below. The c-axis in all lattices is normal to the plane of the page and hence the index l is 0 in all cases. 

The reflections of Bragg peaks can be indexed by two Miller indices, hk. Their angular positions, 29i, , yield the lattice plane spacing dfj, = 2nlq for the two-dimensional lattice structure. It is possible to have an estimate of the dimensions of the domains (D ) by analyzing the full width at half maximum (FWHM) of a GIXD peak (A p), corrected from the instrumental broadening with the Scherrer equation ... 

The cell of the allotwin lattice has a period of 6cq along c and contains six lattice planes of the IM polytype and three lattice planes of the 2M polytype. Of these, only the plane with z = 0(mod 6) has all the nodes from both polytypes overlapped by the allotwin operation, whereas in all the other lattice planes the nodes from the two polytypes are separated. Consequently, the allotwin index of IMis 6, and that of 2M is 3. 

If the sides of the crystal are kept straight during growth, the lattice planes remain intact. However, if the crystal is necked-in" or "pulled-in" during growth, line-defects appear in the lattice structure. If this is a laser-crystal that we are growing, the line defects have a localized effect upon the refractive index of the crystal, making it useless as a laser crystal. 

Fig. 4.12 (a) Definition of lattice planes in a fcc-lattice (b) hydrogen adsorbed on Ni(lll) (c) high symmetry sites of low index surfaces. 

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