Copper's interatomic bond is fitted to a Morse potential with dissociation energy

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Modes, Packets and Media, Ch The anharmonic oscillator: frequency, harmonics and drift

Problem

Copper's interatomic bond is fitted to a Morse potential V(r)=D(1−e−a(r−r0))2V(r)=D\left(1-e^{-a(r-r_0)}\right)^2 with dissociation energy D=0.343 eVD=0.343\,\mathrm{eV}, range parameter a=1.36×1010 m−1a=1.36\times10^{10}\,\mathrm{m^{-1}} and equilibrium length r0=2.87×10−10 mr_0=2.87\times10^{-10}\,\mathrm{m}. Model the solid as a set of independent bonds, each vibrating as this one-dimensional anharmonic oscillator, with the following idealisations: treat the vibration classically, so that by equipartition the mean total vibrational energy of a bond in thermal equilibrium at temperature TT is ⟨E⟩=kBT\langle E\rangle = k_{\rm B}T (kBk_{\rm B} Boltzmann's constant); take the fundamental amplitude AA of the near-harmonic motion to satisfy the small-oscillation energy relation ⟨E⟩=12mω02A2\langle E\rangle = \tfrac12m\omega_0^2A^2; and use the Morse mean-position drift ⟨x⟩=34aA2\langle x\rangle = \tfrac34aA^2. ((a)) Show that the reduced mass mm cancels between these relations, and that the fractional bond expansion is ⟨x⟩/r0=3kBT/(4Dar0)\langle x\rangle/r_0 = 3k_{\rm B}T/(4Dar_0). ((b)) Evaluate the fractional bond expansion at T=300 KT=300\,\mathrm{K}. ((c)) Hence estimate the solid's linear thermal expansion coefficient αL=d(⟨x⟩/r0)/dT\alpha_L = \mathrm{d}(\langle x\rangle/r_0)/\mathrm{d}T in this one-bond model, and compare it with copper's measured value αL=1.65×10−5 K−1\alpha_L=1.65\times10^{-5}\,\mathrm{K^{-1}}.

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