Question

Vibrations of square lattice. We consider transverse vibrations of a planar square attice of rows and columns of identical atoms, and let u, m denote the displacement normal to the plane of the lattice of the atom in the lth column and mth row (Fig. 13). The mass of each atom is M. Assume force constants such that the equation of motion is Figure 13 Square array of lattice constant a. The displacements considered are normal to the plane of the lattice. as problem 1(a) in earlier printings of this book, can result only from an unrealistic assumption about the interatomic forces It was discovered that this equation, given Continue, however, with the problein as restated here. (a) Assume solutions of the form where a is the spacing between nearest-neighbor atoms. Show that the equation of motion is satisfied if ofM-2C(2-cos Kra-cos Kya) This is the dispersion relation for the problem. (b) Show that the region of K space for which independent solutions exist may be taken as a square of side 2mla. This is the first Brillouin zone of the square lattice. Sketch ω versus K for Kx = K, with Ky =0,

Answer 1

The force constant of a crystal lattice can be calculated using the equation k = (mω²)/Δx, where k is the force constant, m is the mass of the atom, ω is the angular frequency, and Δx is the displacement of the atom.

Step-by-step explanation:

The question is asking about the force constant of a crystal lattice. The force constant is a measure of the stiffness of the lattice and can be calculated using the equation

k = (mω²)/Δx

where k is the force constant, m is the mass of the atom, ω is the angular frequency, and Δx is the displacement of the atom. To find the force constant, we need to know the mass of the atom, the angular frequency, and the displacement.