Primitive lattice wave numbers

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

In all properties studied with pseudopotenlial theory, the first step is the evaluation of the structure factors. For simplicity, let us consider a metallic crystal with a single ion per primitive cell -either a body-centered or face-centered cubic structure. We must specify the ion positions in the presence of a lattice vibration, as we did in Section 9-D for covalent solids. There, however, we were able to work with the linear force equations and could give displacements in complex form. Here the energy must be computed, and that requires terms quadratic in the displacements. It is easier to keep everything straight if we specify displacements as real. Fora lattice vibration of wave number k, we write the displacement of the ion with equilibrium position r, as... 

Equation (8.4.2) suggests that a wavefunction uk(r) needs to be found by standard quantum-chemical means for only the atoms or molecules in the one direct-lattice primitive unit cell. For each of the Avogadro s number s worth of fermions in a solid, the factor exp(ik R) in Eq. (8.4.2) provides a new quantum "number," the wavevector k, that guarantees the fermion requirement of a unique set of quantum numbers. The Bloch waves were conceived to explain the behavior of conduction electrons in a metal. 

The Bom-von Karman contour condition demonstrates that the Bloch wave vector of free electrons in a cubic lattice is, according to Sommerfeld, constituted only by real components. The number of k values (k = p/h) admitted in a primitive cell of a reciprocal lattice is equal to the number of sites in the crystal. The linear momenta operator, p, is... 

The wavevector mismatch is given by Ak it is set experimentally to approximately zero (hence, S(Ak) l), Finally, At is the temporal delay between the pulses of the probe laser, p, and those of the lasers Z and s m is the reduced lattice mass, N is the number of primitive cells per cm, and A = 2iru L /ce. In the expression for A, c is the speed of light, L is the effective length within the medium over which spatial overlap of the three-wave mixing exists, and Eg is the dielectric constant of the medium at Wj,. 

This form shows explicitly that the elements of the dynamical matrix D(q) depend only on the difference L = - 5, and are therefore independent of The dimension of the dynamical matrix is 3n, where n is the number of atoms in the primitive unit cell (a, 3 = x, y, z k, k = l...n). We have therefore reduced the infinite set of equations of motion (3,16) to the -problem (3,21) which represents a set of 3n lineccr homogeneous equations in the 3n unknown amplitudes e (K q) for each wave vector q. This reduction is a direct consequence of the periodicity of the lattice as expressed by (3.19), Equation (3.21) can be rewritten as... 

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