Physics 573: Problem set 2

1. A certain one dimensional crystal has ions with alternating charge ±q and a repulsive potential energy term U₀/rⁿ.
   (a) Calculate the equilibrium separation r₀, which is the lattice spacing, and the value of the potential energy at equilbrium.
   (b) What is the effective "spring constant" between the ions? (Ans: k_eff = [2(ln 2)(n-1)q²]/r₀³).

2. Find the speed of a longitudinal phonon wave that propagates along the [111] direction in a cubic crystal, which implies that the amplitudes along crystal lattice vectors (cube edges) are all equal. A good starting point is to assume a wave of the form q = q₀exp{i[ks(x+y+z) - ωt]}, where s= 1/ √ 3  .

3. Calculate the effect spring constant for shear in a cubic crystal. This can be done by setting the up a strain e = 2e_xx = - 2e_yy and searching for excess potential energy of the usual form U = (1/2)k_effe².

4. A lattice is stable if when displaced from quilibrium, atoms return toward the equilibrium position. Alternatively, if there is strain along any possible direction, the lattice supplies a restoring force (towards minimum strain energy). This set of criteria can be used to derive conditions on the amplitudes of the cubic crystal elastic constants c₁₁, c₁₂, and c₄₄. For example there are obvious conditions: c₁₁ > 0, and not so obvious conditions: c₁₁ + 2c₁₂ > 0. Obtain all of these conditions by writing the elastic tensor and invoking the condition that all sub-determinants must be positive.

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