An infinite monatomic chain has equal masses joined by identical springs of stiffness

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids

Modes, Packets and Media, Ch Normal modes of many coupled oscillators: eigenvalues, symmetry and response

Problem

An infinite monatomic chain has equal masses mm joined by identical springs of stiffness κ\kappa, with band maximum ωmax⁡=2κ/m\omega_{\max} = 2\sqrt{\kappa/m}. One mass is replaced by a lighter defect of mass m′=0.75 mm' = 0.75\,m (a fractional deficit ϵ=0.25\epsilon = 0.25), which supports a localised mode above the band with xj∝(−1)je−γ∣j∣x_j \propto (-1)^j e^{-\gamma|j|} away from the defect, where e−γ=1−ϵ1+ϵ,ωL=ωmax⁡1−ϵ2.e^{-\gamma} = \frac{1-\epsilon}{1+\epsilon}, \qquad \omega_{\rm L} = \frac{\omega_{\max}}{\sqrt{1-\epsilon^2}}. ((a)) Find e−γe^{-\gamma}, the factor by which the displacement magnitude falls from one site to the next away from the defect. ((b)) Find ωL/ωmax⁡\omega_{\rm L}/\omega_{\max}. ((c)) Given ωmax⁡=2.00×1013 rad s−1\omega_{\max} = 2.00\times10^{13}\,\mathrm{rad\,s^{-1}}, find ωL\omega_{\rm L} in rad s−1\mathrm{rad\,s^{-1}}.

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