Chapter 4 fixed copper's nearest-neighbour spring constant from its measured sound speed:

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 Thermal expansion and lattice thermal conductivity

Problem

Chapter 4 fixed copper's nearest-neighbour spring constant from its measured sound speed: K=36.6 N m−1K = 36.6\,\mathrm{N\,m^{-1}}, with bond length r0=a=2.556 A˚r_0 = a = 2.556\,\mathrm{\mathring{A}}, the same one-spring chain used again for the simple cubic crystal of Section 7.3. Section 7.2 found the cubic coefficient copper's own measured expansion needs, β=1.36×1011 J m−3\beta = 1.36\times10^{11}\,\mathrm{J\,m^{-3}}. (a) Use the simple cubic relation Bmod=K/3aB_{\rm mod} = K/3a to predict copper's bulk modulus from this model, and compare it with the measured 140 GPa140\,\mathrm{GPa}. (b) Use γG=βa/K\gamma_G = \beta a/K to find the model's own Gruneisen parameter, and compare it with the measured γG=2.0\gamma_G = 2.0 used in Section 7.3. (c) A student combines this model BmodB_{\rm mod} and γG\gamma_G with copper's measured heat capacity, CV=24.4 J mol−1K−1C_V = 24.4\,\mathrm{J\,mol^{-1}K^{-1}}, and molar volume, Vm=7.11×10−6 m3 mol−1V_m = 7.11\times10^{-6}\,\mathrm{m^3\,mol^{-1}}, in the general Gruneisen relation. Find the α\alpha this gives, and explain why it agrees with neither Section 7.3's own prediction of 1.63×10−5 K−11.63\times10^{-5}\,\mathrm{K^{-1}} nor the measured 1.65×10−5 K−11.65\times10^{-5}\,\mathrm{K^{-1}}.

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