(a) At the zone boundary, , . Substitute this directly into the two linear equations…

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids · Lattice dynamics and thermal properties of solids

Condensed Matter Physics, Ch Two atoms in the cell: acoustic and optical branches

Problem

(a) At the zone boundary,

q=\pi/a

e^{\pm iqa}=e^{\pm i\pi}=-1

. Substitute this directly into the two linear equations

\begin{aligned}(2K-M_1\omega^2)U &=K(1+e^{-iqa})W,\\(2K-M_2\omega^2)W &=K(1+e^{iqa})U\end{aligned}

of Section 5.2, and show that the two equations decouple completely. Use this, without appealing to the general formula for

\omega_\pm^2

or to the difference-of-squares identity, to derive

\omega_{\rm opt}(\pi/a)=\sqrt{2K/M_1}

(with

U\ne0

W=0

) and

\omega_{\rm ac}(\pi/a)=\sqrt{2K/M_2}

(with

W\ne0

U=0

). (b) Magnesium oxide has the rock-salt structure, with

M_{\rm Mg}=24.305\ \mathrm{u}

M_{\rm O}=15.999\ \mathrm{u}

and a quoted force constant

K=88.0\ \mathrm{N\,m^{-1}}

. Find the width of the forbidden gap at the zone boundary, in THz.

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