Girvin, Yang Modern Condensed Matter Physics 2019 SSP · Problem 5.7

Dynamics of Lattice Vibrations

Physics — Condensed Matter · 2d-square-lattice-phonons

Problem

Consider a 2D square lattice made of atoms with mass

M

and interaction energy

U = \frac{1}{2a^2} \sum_{(ij)} \lambda_{(ij)}[(\vec{R}_i - \vec{R}_j) \cdot (\vec{u}_i - \vec{u}_j)]^2,

where

a

is the lattice constant,

(ij)

represent pairs of atoms, and

\lambda_{(ij)}

equals

\lambda_1

and

\lambda_2

for nearest- and next-nearest neighbors, respectively, and is zero otherwise. Find the frequency and polarization vectors of the normal modes along the directions of nearest-neighbor bonds.

Related theory

Lattice Vibrations And Phonons From Atoms To Quanta

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