**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 2*N* ions of alternating charge ± *q* with a repulsive potential energy *A/R*^{n} between nearest neighbors.

a) Show that at the equilibrium separation *R*_{0}

b) Suppose one compresses the 1D crystal so that *R*_{0} ® *R*_{0} (1–d ). Show that—to leading order—the work [per length], U(R_{0}-R_{0}d )* *- U(R_{0}), 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 *R*_{0} = 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 *R*_{0 }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: *p _{x} = x f(r)*;

Consider the linear combination of *p* wavefunctions y *= a _{x}p_{x} + a_{y}p_{y} + a_{z}p_{z}* .

Find four sets of coefficients (*a _{x}*,

b) Consider the linear combination f *= bs + c**y , *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 *sp*^{3} hybrids) in terms of *p _{x} + p_{y} + p_{z }*, and