Physics 322-2007 Solid State Physics Walter F. Smith Haverford College
Assignment 2 (Revised)
Due: Friday, Sept. 14 at 4 pm (Turn in to envelope outside my office.)
Reading: sections 1-7 of the online "Fundamentals of Crystallography" here:
http://www.chemsoc.org/exemplarchem/entries/2003/bristol_cook/home.htm
and/or read Kittel pp. 24-38.
Assigned exercises (except as noted, these are group problems, i.e. you may work on them with other students in small groups)
2A A “mole” of a substance contains Avogadro’s number of atoms or molecules of the substance. The “atomic weight” or “molecular weight” is conveniently defined so that, if you measure an amount in grams equal to the atomic or molecular weight of the substance, you get exactly one mole.
Aluminum (Al) has an fcc structure.
a. An x-ray diffraction experiment is carried out, using x-rays with a wavelength of 1.54 Angstroms. It is found that the Bragg angle for first order “reflection” from the (111) planes in Al is 19.2o. Compute the interplanar distance for these planes.
b. The density of Al is 2.7 g/cm3 and its atomic weight is 27.0 u. Use this information to calculate Avogadro’s number, and compare your result with the known value.
2B. You can do diffraction
experiments with x-rays, electrons, or neutrons. Why is the energy of a neutron that would
typically be used for crystal diffraction so much smaller than that of an
electron that would typically be used?
2C. Miller indices for hexagonal
lattices.
a. For planes that intercept the
axes at negative values (relative to the origin), one uses a bar over the
number in the Miller index corresponding to the negative intercept. For example, the
intercepts the a1 axis at 2a1,. the a2 axis at –a2,
and the a3 axis at
infinity. Explain why this notation is
never really needed for cubic, tetragonal, or orthorhombic lattices, when the
basis has only one atom.
b.
For the hexagonal lattice, there is a peculiar and redundant convention
for specifying the Miller indices.
First, one determines the intercepts of the plane in question along four
axes, as defined by the vectors shown in the figure. These are placed in the order:
intercept along a1 axis, along a2 axis, along d axis, along a3 axis.
The rest of the procedure is the same as for other lattices, i.e. we take the inverse of each number, then find the smallest set of integers with the same ratio. This results in Miller index of the form (hkil), where the numbers h, k, and l are exactly the same as for the regular procedure. One can show that
i = -h – k.
Sketch the conventional unit cell
for the hexagonal lattice, and show how the 
plane cuts through it,
assuming the lattice point in the center of the bottom face is the origin.
2D. (individual problem) The sines and cosines version of the Fourier expansion is given by:
The complex exponential version of the Fourier expansion is given by:
Show that, for all p
> 0, we have 
and 
, assuming that S0
= 0.
Hint: start by
re-expressing the above sum into 
plus a sum running
from p =1 to infinity.