Real space lattice vector

The sums in Eqs. (1) and (2) run, respectively, over the reciprocal space lattice vectors, g, and the real space lattice vectors, r and Vc= a is the unit cell volume. The value of the parameter 11 affects the convergence of both the series (1) and (2). Roughly speaking, increasing ii makes slower the convergence of Eq. (1) and faster that of Eq. (2), and vice versa. Our purpose, here, is to find out, for an arbitrary lattice and a given accuracy, the optimal choice, iiopt > tbal minimises the CPU time needed for the evaluation of the KKR structure constants. This choice turns out to depend on the Bravais lattice and the lattice parameters expressed in dimensionless units, on the... 

Let a , a2, a3 be the real space lattice vectors and b, b2, b3 be the reciprocal space lattice vectors. We have then the following relations ... 

A simple example of this calculation is the simple cubic lattice we discussed in Chapter 2. In that case, the natural choice for the real space lattice vectors has a, = a for all i. You should verify that this means that the reciprocal lattice vectors satisfy b, = 2rr/a for all i. For this system, the lattice vectors and the reciprocal lattice vectors both define cubes, the former with a side length of a and the latter with a side length of 2tt/g. 

Again, notice that the length of the reciprocal lattice vectors are inversely related to the reciprocal of the length of the real space lattice vectors. Views of these lattice vectors in real space and reciprocal space are shown in Fig. 3.1. 

Why is it useful to introduce such a complicated set of vectors This becomes obvious when we look at the scalar product between a real space lattice vector R and a reciprocal lattice vector q. Expressing these vectors by the corresponding primitive vectors we can write ... 

We need to introduce some change in notation here [15], which will be used frequently in the following calculations. Let the simulation box dimensions be A.X X A , X z- Then the real-space lattice vectors are... 

Now we can use the -function relations that result from summing the complex exponential exp( ik R) over real-space lattice vectors or reciprocal-space vectors within the BZ, which are proven in Appendix G, to simplify Eq. (3.84). With these relations, / takes the form... 

The prindple of a LEED experiment is shown schematically in Fig. 4.26. The primary electron beam impinges on a crystal with a unit cell described by vectors ai and Uj. The (00) beam is reflected direcdy back into the electron gun and can not be observed unless the crystal is tilted. The LEED image is congruent with the reciprocal lattice described by two vectors, and 02". The kinematic theory of scattering relates the redprocal lattice vectors to the real-space lattice through the following relations... 

A reciprocal lattice vector is constracted for each plane of the real-space lattice as follows ... 

Figure 3.1 View of the real space and reciprocal space lattice vectors for the fee primitive cell. In the real space picture, circles represent atoms. In the reciprocal space picture, the basis vectors are shown inside a cube with side length At fa centered at the origin.

The last condition is identical to Eq. (A.10). Therefore, g must be a reciprocal lattice vector. This shows us another important property of the reciprocal lattice it is the Fourier transform of the real space lattice. 

A crystal is a physical object - it can be touched. However, an abstract construction in Euclidean space may be envisioned, known as a direct space lattice (also referred to as the real space lattice, space lattice, or just lattice for short), which is comprised of equidistant lattice points representing the geometric centers of the structural motifs. Any two of these lattice points are connected by a primitive translation vector, r, given by ... 

It has just been stated that a band stracture diagram is a plot of the energies of the various bands in a periodic solid versus the value of the reciprocal-space wave vector k. It is now necessary to discuss the concept of the reciprocal-space lattice and its relation to the real-space lattice. The crystal structure of a solid is ordinarily presented in terms of the real-space lattice comprised of lattice points, which have an associated atom or group of atoms whose positions can be referred to them. Two real-space lattice points are connected by a primitive translation vector, R ... 

Taking the primitive translation vectors for one of the real-space cubic lattices from Table 4.1, Eqs. 4.25-4.27 can be used to obtain the primitive translation vectors for the corresponding reciprocal lattice, which are given in Table 4.2. By comparing Tables 4.1 and 4.2, it is seen that the primitive vectors of the reciprocal lattice for the real-space FCC lattice, for example, are the primitive vectors for a BCC lattice. In other words, the ECC real-space lattice has a BCC reciprocal lattice. 

BCC real-space lattices are completely determined by the condition that each inner vector, k, go over into another by all the symmetry operations. This is not the case for the tmncated octahedron. The surface of the Wigner-Seitz cell is only fixed at the truncating planes, not the octahedral planes. Nonetheless, the volume enclosed by the truncated octahedron is taken to be the first BZ for the FCC real-space lattice (Bouckaert et ak, 1936). The special high-symmetry points are shown in Table 4.5. 

What is the relationship between the lattice of the crystal and the lattice on which the diffraction intensities fall, the lattice of the transform The relationship is that between the real space lattice of the crystal and the reciprocal lattice. The point where a wave diffracted by a particular family of planes hkl appears in the diffraction pattern is related to the origin of the diffraction pattern by the reciprocal lattice vector h = hkl. The direction of the reciprocal lattice vector h is normal to the family of planes, and its length is 1 /reciprocal lattice vector defines a permissible point in diffraction space where a diffraction wave may be observed. That wave, having both amplitude and phase, is the Fourier transform of that particular family of planes hkl. 

The vector Ahki points in a direction perpendicular to a real space lattice plane. We would like to express this vector in terms of reciprocal space basis vectors a, b, c. ... 

Figure 1.7 Geometrical description of a lattice plane in terms of real space basis vectors.

In solving three-dimensional triperiodic diffraction problems the concept of the reciprocal lattice (128) helps greatly. Reciprocal space constructions are useful for diperiodic structures also. In the simplest case of a strictly two-dimensional single layer grating, the reciprocal space construction is an array of parallel rods normal to the plane of the grating. These rods cut the plane at the points of a reciprocal net generated by translations of unit reciprocal vectors a and b having properties defined below in terms of the real-space imit vectors d and b (Section IIB). 

Many of the structural and physical properties of single-walled nanotubes can easily be understood based on a the picture of a strip of a two-dimensional graphite sheet rolled into a cylindrical form. Adapted to this symmetry, we will use the real space unit vectors ai, 2 of a hexagonal lattice in x,y coordinates ... 

The conditions described here also define the conditions for diffraction of electron waves at the Brillouin zone boimdaries. Likewise the Brillouin zones described in Chapter 2 are reciprocal-space objects with the symmetry of the reciprocal lattice rather than the real-space lattice. The reciprocal lattice points in Figure 2.5, for example, are located at points hbj, kb2, and lb3. The reciprocal lattice for a simple cubic system with basis vectors ai, a 2, and as has reciprocal lattice vectors parallel to the real space vectors. However, larger distances in real space correspond to shorter distances in reciprocal space. Thus, planes that are widely spaced in real space have closely spaced reciprocal lattice points and vice versa. One may determine by examination of Figure 4.2 that the (100) planes are V3 times farther apart than are the (111) planes. In general, the distance, d, between (hkl) lattice planes in a cubic system may be shown to be ... 

A reciprocal lattice may be defined based on Equation 4.2 and the basis vectors for the real-space lattice. 

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