A chain of identical atoms

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 chain of identical atoms, each of mass MM, is joined by springs that alternate in stiffness along the chain: K1K_1, then K2K_2, then K1K_1 again, and so on, so every atom is pulled by one spring of each kind. Let xnx_n and yny_n be the displacements of the two atoms in the nnth cell, with yny_n joined to xnx_n by K1K_1 on its left and to xn+1x_{n+1} by K2K_2 on its right, and let aa be the repeat length of one K1K_1-K2K_2 pair. (a) Write the two equations of motion for xnx_n and yny_n, and use the travelling-wave trial solution of this chapter, with one amplitude for each atom, to derive ω±2(q)\omega_\pm^2(q). (b) Evaluate both branches at the zone boundary, show that a gap opens whenever K1≠K2K_1\ne K_2, and find its width, in THz, for M=1.00×10−25 kgM=1.00\times10^{-25}\ \mathrm{kg}, K1=20.0 N m−1K_1=20.0\ \mathrm{N\,m^{-1}} and K2=8.0 N m−1K_2=8.0\ \mathrm{N\,m^{-1}}.

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