Problem 1

Consider a hydrogen molecule. The bond length is 0.076 nm. The mass of a hydrogen atom is $1.67\times 10^{-27}$ kg.

(a) The binding energy of H₂ is 4.52 eV. Sketch approximately the bond potential of H₂ assuming it can be described by a Morse potential. Label both axes.

(b) From your sketch of the bond potential, explain how you could calculate the vibrational frequency of this molecule. What is this frequency approximately?

(c) Write down the two-electron wavefunction of the first excited electron state of H₂ as a Slater determinant.

Problem 2
Sodium has a bcc crystal structure with a lattice constant of $a=$ 0.429 nm.

(a) Draw the arrangement of the atoms in the (110) plane using the Miller indices of the conventional unit cell. Label the directions in your drawing with Miller indices. Indicate the distance bewteen the atoms in the two perpendicular directions.

(b) The reciprocal lattice of bcc is fcc. The closest point on the Brilluoin zone boundary to $\Gamma$ is $N$. What is the maximum wavelength photon that can be diffracted from a sodium crystal?   $\overline{\Gamma N} = \frac{\sqrt{2}\pi}{a}$

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(c) Sketch the phonon density of states of sodium.

(d) Sodium is well described by the free electron model. Sketch the electron density of states.

Problem 3

(a ) Draw an Ewald sphere and explain what it is used for.

(b) What is the difference between a metal, an insulator, and a semiconductor?

(c) The electron density of states of a two-dimensional metal is a constant $D(E)=\alpha$ for $E > 0$ and $D(E)=0$ for $E < 0$. Give an expression for the Fermi energy in terms of $\alpha$ and the electron density $n$.

(d) How could you experimentally determine the entropy of a bar of iron?

(e) What is the Bloch theorem?