Dispersion relation diatomic chain

For a chain of diatomic atoms (or ions) with different masses, the dispersion relation separates into two branches an acoustic branch and an optical branch. In the vicinity of fc = 0, the dispersion relations are given for the optical branch and the acoustic branch, respectively, by = 2/3//a 2/3(fca) /(M -I- m), where a is the spacing between the individual... 

In this chapter, we start with the classical mechanics of the linear diatomic chain with nearest-neighbour interactions. Using periodic boundary conditions the equations of motion are solved and the dispersion relations are discussed. We also discuss the transition from the diatomic chain to the monoatomic chain. 

The cosine curve duplicates the results and in its middle part represents the upper LO curve in Fig.2.7. We note that the optical vibration at q = 0 of the diatomic chain becomes an acoustic vibration at q = n/d of the mono-atomic chain. The upper part AB of the sine curve in Fig.2.7 can be obtained by folding out the LO branch AC, or equivalently, by translation of the LO branch CD through 2-n/a, that is, by a reciprocal lattice vector t = M. This is called an Umptapp process. The dispersion relation (2.43) can of course be obtained if we start directly with the Hamiltonian of the monoatomic chain which follows easily from (2.3) and then solving the resulting equations of motion by assuming a solution of the form... 

In the Einstein model, each atom is bound by an elastic spring to its equilibrium position, but there are no interactions between different atoms. The dispersion relation for this model can symbolically be represented by a horizontal line and the corresponding density of states of the diatomic chain with 2N atoms is given by... 

---