Dispersion curve diatomic chain

The dispersion curve for a diatomic chain is given in Figure 8.9. The curve consists of two distinct branches the acoustic and the optic. In the first of these the frequency varies from zero to a maximum cop. The second one has a maximum value of (Oq at q = 0 and decreases to co2 at q - qmax. There are no allowed frequencies in the gap between a>i and a>2. 

Figure 8.9 The dispersion curve for a one-dimensional diatomic chain of atoms. m2

Figure 4.5-1 Dispersion curve for unidimensional diatomic chain. The ratio of the masses is Mhn = 5. Note that a is the interatomic distance while the lattice periodicity is 2a.

A plot of frequency v versus k is known as a dispersion curve. The first period of this function represents the first Brillouin zone. For a linear diatomic chain in which all bonds are identical and the atoms are equally spaced, but have different masses of Mi and Mj, the result is ... 

Note that there is a gap between the two allowable frequencies (Equation 16.22) the diatomic chain can support. Also note that values of k + irfa produce the same result as k, hence we can fold the dispersion curve into a reduced zone plot. 

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... 

Fig. 2.24. Dispersion curve for the longitudinal vibrations of a diatomic chain.

The results are shown in Fig. 2.26. Thus, for the three-dimensional motion of a diatomic chain there is one pair of dispersion curves (one acoustical and one optical branch) for each direction in space. In the three-dimensional motions of a diatomic chain, the transverse directions x and y are equivalent. Consequently, only one transverse optic and acoustic dispersion curve is displayed, as they are degenerate (i.e., have the energy or vibrational motion). 

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