Physics 861 - Fall 00

Sample questions from earlier years

NOTE: Elasticity and p-n junctions were not covered

1. Identical point particles of mass m are placed at the sites of a very long one-dimensional lattice of period a. The corresponding mass density $\rho (x)$ is a periodic function that can be expanded in the Fourier series

$$\rho (x)=\sum_{n=-\infty }^{\infty }\rho _{n}\,e^{iG_{n}x}$$ (1)

• What are the values of G_n and $\rho _{n}$?
• If the atoms are replaced by diatomic molecules (if each point mass is replaced by two point masses, m and M, separated by a distance $b\leq a/2$ along x), what are then the values of G_n and $\rho _{n}$?

  
  

2. For a very long one-dimensional lattice of period a consisting of atoms of mass m connected by springs between nearest neighbors having spring constant K,

• Compute the frequency spectrum, $\omega (k)$. What is the maximum value of $\omega$ and for what k does it occur?
• Find the sound velocity c and the bulk modulus B. How is B related to c² and to the average mass density m/a?
• If there is also a spring constant K₂ connecting second nearest neighbors, what is then $\omega (k)$? What is c²?
• Show that if K is negative (while K₂ is positive) the system is unstable. What phonon modes are unstable for small, negative K? What is the physical interpretation of this instability?

  
  

3. The attached sheet shows phonon dispersion curves for CuCl, CuBr, and Pb. For each figure

• Identify (if they exist) the Longitudinal Acoustic, Transverse Acoustic, Longitudinal Optic, Transverse Optic branches. If some of these branches do not exist, explain why.
• What do the labels such as $\Delta _{1}$ or X₃ mean? A qualitative explanation is sufficient.
• How were the experimental points obtained? Make a guess.

  
  

4. Verify that the positions of the electron resonances in Figure 28.9b of Ashcroft and Mermin are consistent with the electron effective masses given for silicon on page 569 and the formulas (28.6) and (28.8).

What is the expected position of the electron cyclotron resonances in silicon if the magnetic field is in the (100) direction?

5. Estimate $\left\langle u^{2}\right\rangle$ and $\left\langle \exp (-iqu)\right\rangle$ for the one-dimensional monatomic chain of problem 2, at reasonably high temperatures. Why do you need to assume that the chain has finite length L to get a finite answer? Explain why your formula is valid at temperatures that are neither too low nor too high.

6. Solve the Schrödinger equation for an electron in crossed electric and magnetic fields. Take the magnetic field along z and the electric field along x or y. Use the Landau gauge.