Extend the chain of Section 4.1 by adding a second spring between next-nearest neighbours

PHYSICS · Multi-part · 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 Elastic waves and the vibrating chain

Problem

Extend the chain of Section 4.1 by adding a second spring K2K_2 between next-nearest neighbours, so the total force on atom nn becomes Fn=K(un+1+un−1−2un)+K2(un+2+un−2−2un).F_n=K(u_{n+1}+u_{n-1}-2u_n)+K_2(u_{n+2}+u_{n-2}-2u_n). ((a)) Using the same travelling-wave trial solution as Section 4.2, show that ω2=2KM(1−cos⁡qa)+2K2M(1−cos⁡2qa).\omega^2=\frac{2K}{M}(1-\cos qa)+\frac{2K_2}{M}(1-\cos2qa). ((b)) Show that the long-wavelength sound speed depends on both springs, vs=a(K+4K2)/Mv_s=a\sqrt{(K+4K_2)/M}, while the zone-boundary frequency depends only on KK, ω(π/a)2=4K/M\omega(\pi/a)^2=4K/M, reproducing Section 4.6's result.

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