Dispersion curve monoatomic chain

Figure 8.5 The dispersion curve for a one-dimensional monoatomic chain of atoms.

In Fig. 21.1 this condition is visualized as the intersection of the straight line cq with the dispersion curve (Om q) = 2cm m ql2). Going back to the original units, cOm can be identified as the dispersion of a monoatomic chain of masses M = + M2 with a linear in-... 

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.4.17. Dispersion curves for a monoatomic linear chain with adiabatically moving bond charges, calculated according (4.125)...

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