Reciprocal lattices vectors specifying

Fig. 7. Intensity (arbitrary units) as a function of final electron energy for emission from an s orbital adsorbed (a) in the position (0, 0, a) and (b) in the position (a/2, a/2, aj2), relative to the substrate. Single scattering (solid curves), multiple scattering (dotted curves), and no scattering (dashed curves). The arrows indicate the resonance energies specified by the reciprocal lattice vectors, and the corresponding band crossings in the free-electron band structure at the top of the figure.

Any reciprocal lattice vector, or reciprocal lattice point is uniquely specified by the set of three integers, hkl, which are the Miller indexes of the family of planes it represents in the crystal. Thus there is a one-to-one correspondence between reciprocal lattice points and families of planes in a crystal. It will be seen shortly that the reciprocal lattice is the Fourier transform of the real lattice, and vice versa. This was in fact demonstrated experimentally in Figure 1.7 of Chapter 1 by optical diffraction. As such, reciprocal space is intimately related to the distribution of diffracted rays and the positions at which they can be observed. Reciprocal space, in a sense, is the coordinate system of diffraction space. 

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. 

Figure 3.25 The direction of the reciprocal lattice vector rj in the crystallite is specified by means of angles / and defined with reference to the coordinate system o-xyz fixed to the crystallite.

The concept of the reciprocal lattice is very useful in discussing the diffraction of x-rays and neutrons from crystalline materials, especially in conjunction with the Ewald sphere construction discussed in Section 1.5.3. The regular arrangement of atoms and atomic groupings in a crystal can be described in terms of the crystal lattice, which is uniquely specified by giving the three unit cell vectors a, b9 and c. It turns out that the diffraction from a crystal is similarly associated with a lattice in reciprocal space. The reciprocal lattice is specified by means of the three unit cell vectors a, b, and c in the same way as the crystal lattice is based on a9 b, and c. In fact, the crystal lattice and the reciprocal lattice are related to each other by the Fourier transform relationship. 

The wave-like modulations are generally described with reference to the reciprocal lattice of the phase rather than the direct unit cell. The modulation wave is specified in terms of a wave vector q, which is defined in terms of the reciprocal lattice vectors a, b or c. The diffraction pattern of the phase will now show a set of reflections corresponding to the subcell plus superlattice reflections due to the modulation. In cases with a single modulation wave, the indices are described, not by conventional Miller indices but by a four-index extension, so that each reflection is indexed as ha + kb + lc +mq. The value of q is obtained directly from the diffraction pattern. When there are two distinct modulations, the diffraction pattern reflections must be indexed on a five-index system ha +kb +lc +mq +mq, and for the system with modulation in three independent directs a six index notation is needed (also see Section 3.2.2). 

In O Eq. 7.67, the summation is over aU reciprocal lattice vectors G which fulfill the condition G T = 27tM, M being an integer number. In practice, this plane-wave expansion of the Kohn-Sham orbitals is truncated such that the individual terms all yield kinetic energies lower than a specified cutoff value, Ecut,... 

Figure 5.11 The direct lattice and reciprocal lattice of a cubic crystal (a), (c) the direct lattice, specified by vectors a, b and c, with unit cell edges aq ( 0 = = cq) (b), (d) the reciprocal lattice, specified by vectors a, b and c, with unit...

Equation (3.34) specifies the possible values of q. In order to express these conditions in a simple manner, we introduce the reciprocal lattice. The primitive translation vectors of the reciprocal lattice are the three vectors 2, definded by [3.1]... 

The reciprocal lattice point defined by FB, where the components of the vector 1 are the Miller indices, corresponds to planes in direct space specified by the Miller indices. The normal to these planes is l B and hence the distance between such neighboring planes is... 

To decide which of these many vectors to use, it is usual to specify the points at which the plane intersects the three axes of the material s primitive cell or the conventional cell (either may be used). The reciprocals of these intercepts are then multiplied by a scaling factor that makes each reciprocal an integer and also makes each integer as small as possible. The resulting set of numbers is called the Miller index of the surface. For the example in Fig. 4.4, the plane intersects the z axis of the conventional cell at 1 (in units of the lattice constant) and does not intersect the x and y axes at all. The reciprocals of these intercepts are(l/oo,l/oo,l/l), and thus the surface is denoted (001). No scaling is needed for this set of indices, so the surface shown in the figure is called the (001) surface. 

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