Department of Physics, University of Maryland, College Park, MD 20742-4111
Physics 731 HOMEWORK ASSIGNMENT #3 Due: Sept. 28, 1999
Finish reading Ashcroft & Mermin (AM), chaps. 19–20.
1. AM 19–1 (cf. Ibach & Lüth, 1–11 for an easier version of this problem).
2. AM 19–2
3. AM 20–1, parts a and b only.
4. AM 20–4, parts a and b only.
5. Consider a line of 2N ions of alternating charge
± q with a repulsive potential energy A/Rn between nearest neighbors.a) Show that at the equilibrium separation R0
b) Suppose one compresses the 1D crystal so that R0
® R0 (1–d ). Show that—to leading order—the work [per length], U(R0-R0d ) - U(R0), can be written (1/2)Cd 2, where
6. Barium oxide has the NaCl structure. Estimate the cohesive energies per molecule of the hypothetical crystals Ba
+O- and Ba++O- - (relative to separated neutral atoms). The observed nearest-neighbor distance is R0 = 2.76 Å; the first and second ionization potentials of Ba are 5.19 and 9.96 eV; the electron affinities of the first and second electrons added to the neutral oxygen atom are 1.5 and -9.0 eV. Which valence state (singly or doubly ionized) do you predict will occur? [Assume R0 is the same for both forms.]7. a) A set of normalized and mutually orthogonal p-state wavefunctions for an atom can be written in the form: px = x f(r); py = y f(r); pz = z f(r).
Consider the linear combination of p wavefunctions
y = axpx + aypy + azpz .Find four sets of coefficients (ax, ay, az) that give normalized p-state wavefunctions with positive lobes pointing towards the corners of a regular tetrahedron (i.e. 4 alternating corners of a cube).
b) Consider the linear combination
f = bs + cy , where y is any one of the four wavefunctions calculated above, and s is an s-state wavefunction, normalized and orthogonal to the p's. Find values of b and c which make the four resulting f wavefunctions orthogonal to each other and normalized, and write out the resulting four f wavefunctions (the sp3 hybrids) in terms of px + py + pz , and s. (Cf. Ibach & Lüth, 1–9.)