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Question - Soft phonon mode consider a line of ions of equal mass but alternating in charge, with ep = 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 C1R = ? 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 CpC = 2(–
1)p e2/p3 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 s = e2/?a3.
(c) Show that w2 is negative (unstable mode) at the zone boundary Ka = p if s > 0.475 or 4/7? (3),
where ? is a Riemann zeta ...Read More function. Show further that the speed of sound at small Ka is imaginary if s
> (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, p) if 0,475 < s=""> ...Read Less

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