Physics 322-2007 Solid State Physics Walter F. Smith Haverford College

Assignment 1

Due: Friday, Sept. 7 at 4 pm (Turn in to envelope outside my office.)

Reading: van Zeghbroeck, 2.1 and 2.2 (This is available at http://ece‑www.colorado.edu/~bart/book/fundamen.htm )

(Optional reserve reading: Kittel Chapter 1 plus pages 23-38)

Assigned exercises (except as noted, these are group problems, i.e. you may work on them with other students in small groups)

1A. a) Explain why the “side-centered cubic” crystal structure (simple cubic with additional points in the centers of the vertical faces of the cubic cell) is not, as is, a Bravais lattice.

b) Describe this structure in terms of a Bravais lattice with a basis. (i.e. What is the Bravais lattice underlying the structure? What is the basis?)

The face-centered cubic (fcc), body-centered cubic (bcc), and diamond structure are the three most important crystal structures for materials typically used by physicists and engineers. In the following exercises, you’ll explore these in more detail. (The simple cubic structure is a bit rare, but is included as a “warm up”.) There is no nasty three-dimensional geometry or trigonometry required for any of these problems. (Only a bit of 2D geometry.)

1B. The face-centered cubic is the most dense and the simple cubic is the least dense of the three cubic Bravais lattices. The diamond structure is less dense than any of these. One measure of this is the “coordination number” (number of nearest neighbors).

a) Draw diagrams for each of the following, highlighting the nearest neighbors of one atom. As an aid, I have given you the coordination number for each case – make sure your diagram shows this number of highlighted nearest neighbors.

i) simple cubic (coordination number 6) ii) face-centered cubic (coordination number 12)

iii) body-centered cubic (coordination number 8) iv) diamond (coordination number 4)

1C. For the following, let the side of the cubic unit cell have length 1. In terms of this measure, calculate a) the nearest neighbor distance r₁ and b) the next-nearest neighbor distance r₂ for each of the following lattices. Show your work! Arrange all your results in a nice table.

i) simple cubic (answers: r₁ = 1, r₂ = 1.41) ii) face-centered cubic (answers: r₁ = 0.707, r₂ = 1)

iii) body-centered cubic iv) diamond

1D. Suppose identical solid spheres are distributed through space in such a way that their centers lie on the points of each of the following four structures, and spheres on neighboring points just touch, without overlapping. (Such an arrangement of spheres is called a close-packing arrangement.) Show that the “packing fraction” (i.e. the fraction of the unit cell volume occupied by spheres) is:

i) simple cubic 0.52 ii) face-centered cubic 0.74

iii) body-centered cubic 0.68 iv) diamond 0.34

(Note that, not coincidentally, this follows the same pattern as the coordination number from 1B.)