Give each of the twelve nearest-neighbour bonds of an fcc crystal the general…

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids

Solid State Physics, Ch Lattice dynamics in three dimensions

Problem

Give each of the twelve nearest-neighbour bonds of an fcc crystal the general stiffness matrix C=−ΦC=-\Phi allowed by symmetry, so that the dynamical matrix is D(q)=1M∑RC(R) [1−cos⁡(q⋅R)]D(\mathbf q)=\frac{1}{M}\sum_{\mathbf R}C(\mathbf R)\,[1-\cos(\mathbf q\cdot\mathbf R)], summed over the twelve neighbours at R\mathbf R. For the bond (a/2)(1,1,0)(a/2)(1,1,0), CC has α\alpha in its xxxx and yyyy entries, β\beta in xyxy and yxyx, and γ\gamma in zzzz. The other bonds follow by relabelling axes: (a/2)(1,0,1)(a/2)(1,0,1) has α\alpha in xxxx and zzzz, β\beta in xzxz and zxzx, γ\gamma in yyyy, and a bond such as (a/2)(1,1ˉ,0)(a/2)(1,\bar1,0) has −β-\beta off the diagonal. (a) Expand D(q)D(\mathbf q) to order q2q^2 along [100][100] and along [110][110], and show that c11=4α/ac_{11}=4\alpha/a, c44=2(α+γ)/ac_{44}=2(\alpha+\gamma)/a and c12=2(2β−α−γ)/ac_{12}=2(2\beta-\alpha-\gamma)/a, with ρm=4M/a3\rho_m=4M/a^3. (b) Copper has a=3.615 A˚a=3.615\ \text{\AA}, c11=168.4c_{11}=168.4, c12=121.4c_{12}=121.4 and c44=75.4 GPac_{44}=75.4\ \mathrm{GPa}. Find α\alpha, β\beta and γ\gamma in N m−1\mathrm{N\,m^{-1}}. (c) With M=63.55 uM=63.55\ \mathrm{u}, predict the longitudinal and transverse frequencies at XX, q=(2π/a)(1,0,0)\mathbf q=(2\pi/a)(1,0,0), in THz. (d) How many of these constants do the three elastic constants determine, and what could test the fit? Could elastic data also fix a second shell of neighbours?

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