Question

Soft phonon modes. Consider a line of ions of equal mass but alternating in charge, with $e_{p}=e(-1)^{p}$ as the charge on the $p$ th ion. The interatomic potential is the sum of two contributions: (1) a short-range interaction of force constant $\mathrm{C}_{1 R}=\gamma$ 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 $C_{p C}=2(-1)^{p} e^{2} / p^{3} a^{3}$, where $a$ is the equilibrium nearest-neighbor distance. (b) From (16a) show that the dispersion relation may be written as $$ \omega^{2} / \omega_{0}^{2}-\sin ^{2} \frac{1}{2} K a+\sigma \sum_{p-1}^{\infty}(-1)^{p}(1-\cos p K a) p^{-3} $$ where $\omega_{0}^{2} \equiv 4 \gamma / M$ and $\sigma=e^{2} / \gamma a^{3} .$ (c) Show that $\omega^{2}$ is negative (unstable mode) at the zone boundary $K a=\pi$ if $\sigma>0.475$ or $4 / 7 \zeta(3)$, where $\zeta$ is a Riemann zeta function. Show further that the speed of sound at small $K a$ is imaginary if $\sigma>(2 \ln 2)^{-1}$ $=0.721$. Thus $\omega^{2}$ goes to zero and the lattice is unstable for some value of $K a$ in the interval $(0, \pi)$ if $0.475<\sigma<0.721 .$ Notice that the phonon spectrum is not that of a diatomic lattice because the interaction of any ion with its neighbors is the same as that of any other ion.

          Soft phonon modes. Consider a line of ions of equal mass but alternating in charge, with $e_{p}=e(-1)^{p}$ as the charge on the $p$ th ion. The interatomic potential is
the sum of two contributions: (1) a short-range interaction of force constant $\mathrm{C}_{1 R}=\gamma$ 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 $C_{p C}=2(-1)^{p} e^{2} / p^{3} a^{3}$, where $a$ is the equilibrium nearest-neighbor distance. (b) From (16a) show that the dispersion relation may be written as
$$
\omega^{2} / \omega_{0}^{2}-\sin ^{2} \frac{1}{2} K a+\sigma \sum_{p-1}^{\infty}(-1)^{p}(1-\cos p K a) p^{-3}
$$
where $\omega_{0}^{2} \equiv 4 \gamma / M$ and $\sigma=e^{2} / \gamma a^{3} .$ (c) Show that $\omega^{2}$ is negative (unstable mode) at the zone boundary $K a=\pi$ if $\sigma>0.475$ or $4 / 7 \zeta(3)$, where $\zeta$ is a Riemann zeta function. Show further that the speed of sound at small $K a$ is imaginary if $\sigma>(2 \ln 2)^{-1}$ $=0.721$. Thus $\omega^{2}$ goes to zero and the lattice is unstable for some value of $K a$ in the interval $(0, \pi)$ if $0.475<\sigma<0.721 .$ Notice that the phonon spectrum is not that of a diatomic lattice because the interaction of any ion with its neighbors is the same as that of any other ion.

Added by Heidi R.

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