Department of Physics University of Maryland College Park, MD 20742-4111
Physics 731 HOMEWORK ASSIGNMENT #1
Fall 19981. What is the Bravais lattice formed by all points with Cartesian coordinates (n1, n2, n3) if: a) the ni are all even or all odd?
b) the sum of the ni is even?
2. Show that the c/a ratio for an ideal hexagonal close-packed structure is Ö(8/3) = 1.633...
3. Show that only 1-, 2-, 3-, 4-, and 6-fold rotation axes are permitted in the case of a Bravais lattice. (Hint: Use the following argument of Buergers: Consider points A and A' of a Bravais lattice a repeat distance a apart. If an n-fold axis passes through A and A', the points B and B' are obtained by rotating through f = 2p/n clockwise and counterclockwise, respectively. If B and B' are Bravais lattice points, what do you know about b/a? Now express b in terms of a and f, derive the allowed values of f.)
4. a) How many atoms are there in the primitive cell of diamond?
b) What is the length in angstroms of a primitive translation vector?
c) Show that the angle between the tetrahedral bonds of diamond is cos-1(-1/3) = 109_28'
d) How many atoms are there in the conventional cubic unit cell?
5. Show that the packing fraction (the fraction of volume filled by hard spheres that just touch) have the following values for the common lattice structures: simple cubic, 0.52; body-centered cubic, 0.68; face-centered cubic, 0.74; diamond, 0.34. Without doing any calculation, give the packing fraction of a hexagonal close-packed crystal. (Explain.)
6. Show that a bcc lattice may be decomposed into 2 sc lattices A, B, with the property that none of the nearest-neighbors lattice points of a lattice point on A lie on A, and similarly for B.at to obtain the same property, a sc lattice is composed into 2 fcc lattices, and a fcc latice into 4 sc lattices. (These results are important for antiferromagnetism!)
7. Consider the CsCl structure, assuming a Cs + ion is at position 000 of an sc lattice.
a) Give the number of first, second, and third nearest-neighbor ions. Which are Cs+? Cl-?
b) Give the atomic coordinates of these neighbors, and thereby confirm your answer to a).