Physics 5337 - Homework #4

  1. Consider a two-dimensional crystal of close-packed pennies.
    1. What is the coordination number?
    2. How many nearest neighbors does a given penny have?
    3. Call the nearest neighbor distance d, one penny diameter between the centers.
    4. How many next-to-nearest neighbors does a given penny have?
    5. What is their distance in terms of d?
    6. How many next-to-next-to-nearest neighbors does a given penny have?
    7. What is their distance in terms of d?
    8. Compute the Lennard-Jones sums Σ pij-6 and Σ pij-12

  2. The Madelung constant for the infinite line of ions of alternating sign (page 64) is a conditionally (not absolutely) convergent series. Show by explicit calculation (a computer program would be handy since you need a lot of terms) that
    1. (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + ... converges to ln(2)
    2. but (1 - 1/2 -1/4) + (1/3 - 1/6 - 1/8) + (1/5 - 1/10 - 1/12) + ... converges to ln(2)/2
    The terms in the two series are identical, but the grouping is different. In fact, one can get any answer at all from a conditionally convergent series simply by grouping the terms differently. One must be very careful when computing the Madelung constant for a three-dimensional ionic crystal to group nearby positive and negative contributions together like the first series above.

  3. Consider a bcc lattice in which the cubic (non-primitive) cell dimension is "a". What is the volume of the truncated octahedron Wigner-Seitz primitive cell?