**Department of Physics, University of Maryland, College Park, MD 20742-4111**

**Physics 731 HOMEWORK ASSIGNMENT #2 ** **Due: Sept. 19, 2000**

During the next lecture following the **due date**, solutions will be distributed. After solutions are passed out, late submissions of homework will not be accepted. (Translation: There is no penalty for turning in homework one lecture after the due date, but it is risky to plan on delaying till that deadline, since if some complication prevents you from doing the homework then, you will be out of luck.) Homework should not be viewed as a take-home test: you are invited, even encouraged, to talk with others. On the other hand, each student must write up answers in his/her own words; students submitting identical papers will be penalized.

Read Ashcroft & Mermin (A&M), chaps. 5-6.

1. A&M 5-3

2. A&M 6-2

3a) Show as a corollary to problem 2 [A&M 6-2] that the {111} planes of a simple cubic crystal are triangular lattices. (So are the {111} planes of the bcc crystal.) What is the interplanar spacing? [Hint: problem 1, A&M 5-3, may be helpful.]

b) For an fcc crystal, viewed from the [111] direction as a sequence of stacked close-packed planes, write down a third primitive vector **a****3**, given that the first two are in a close-packed plane [e.g. **a****1**= a **x**; **a****2**= (a/2) (**x** + **yÖ
**3)]. Then show explicitly how the ABCABC stacking sequence is realized, i.e. that after 3 translations by **a****3 **the lattice points coincide with those in the original plane, translated perpendicular to this plane by 3 times the interplanar spacing *d*. This problem explicates assertions made in class.

4. A&M 6-3

5. A&M 6-5

6. Consider the reciprocal lattice of a two-dimensional (2D) lattice. Write **k **= **k‘
** + kz.

a) Show that **K**3D** **= **K**2D** **+ Kz, Kz arbitrary, so that the reciprocal lattice can be represented by a net of rods. For elastic scattering, **k ®
kŒ
**, write the relation between **k‘
** and **k‘
Œ
**. What added constraint comes from energy conservation?

b) Generalize Fig. 6.7 to show the Ewald construction for diffraction from a 2D lattice. Note that one observes a diffraction pattern of electrons from a surface for all values and orientations of the incident wavevector k above a critical value.

c) Show that for electrons incident perpendicularly on a {100} surface of a copper crystal, the critical *energy *at which the first diffracted beam appears (as incicent energy is raised) is about 22 eV.

Read and think about (but do not turn in) A&M 5-4, A&M 6-4.