Soft phonon mode consider a line of ions of equal mass but alternating in charge, with e_p = e(?1)^p as the charge on the pth ion the inter atomic potential is the sum of two contributions (1) a short-range interaction of force constant C_1R = ? that acts between nearest neighbors only, and (2) a coulomb interaction between all ions.?

(a) Show that the contribution of the coulomb interaction to the atomic force constants is C_pC?= 2(?1)^p e²/p^3?a³, where a is the equilibrium nearest-neighbor distance.?

(b) From (16a) show that the dispersion relation may be written as where w²₀ ? 4?/M and ? = e²/?a³.

(c) Show that w² is negative (unstable mode) at the zone boundary Ka = ? if ? > 0.475 or 4/7? (3), where ? is a Riemann zeta function. Show further that the speed of sound at small K_a is imaginary if ? > (2 In 2)^?1?= 0.721. Thus w² goes to zero and the lattice is unstable for some value of Ka in the interval (0, ?) if 0,475

Transcribed Image Text:

w²/w-sin Ka+o(-1) (1 - cos pKa)p-3 + o2 (–19² (1 P-1