Physics 322-2007 Solid State Physics Walter F. Smith Haverford College
Assignment 3
Due: Friday, Sept. 21 at 4 pm (Turn in to envelope outside my office.)
Reading: Livingston Ch. 7
Assigned exercises (except as noted, these are group problems, i.e. you may work on them with other students in small groups)
3A. The primitive vectors for the hexagonal lattice are
a. Show that the primitive vectors for the corresponding reciprocal lattice are
b. Explain briefly why this means that the reciprocal lattice is also hexagonal, but rotated relative to the direct lattice.
3B. Does a real lattice vector T have a corresponding unique reciprocal lattice vector G? Explain.
3C. a. Show that the reciprocal lattice vector 
is perpendicular to the direct lattice plane with Miller
indices (hkl). (See hints below.)
b. Show that the distance
between adjacent parallel planes with indices (hkl) is 
. (See hints below.)
Hint: Convince yourself that, relative to the origin, the plane hkl is
a plane defined by the points a1/h, a2/k, and a3/l. For
part a, show that G is perpendicular to two vectors that lie in
the plane (any two vectors that aren’t parallel will do). For part b, note that the distance between
planes is the same as the distance from the origin (which contains a lattice
point) to the plane described above. You
want the distance from the origin to the plane along a line perpendicular to
the plane, i.e. along a line parallel to 
. Finally, recall that
the length of any vector in the direction is given by the dot product of the vector with .
IMPORTANT: The results of this problem are quite handy in several contexts. Also, note that the result of part c is also valid for fcc and bcc lattices.
3D. (individual problem)
Show that, for a lattice with cubic symmetry, (simple cubic, fcc, bcc, diamond,
etc.), that the distance between planes (hkl)
is 
, where a is the
sidelength of the conventional unit cell.
Hint: Use the results of problem 3Cb.
3E. (individual problem) Describe and sketch the 1BZ for the hexagonal Bravais lattice.