Homework 2
Due: 10/21/03 before class
I am giving quite a bit of extra time for this homework because it may require a lot of
work and reading to tackle the concepts. This homework will lead up to the final
question which is question one in chapter six of Ashcroft and Mermin. By the time you
get to the final question, you should have the knowledge and numbers to get a correct
answer.
1a.
For a S.C., F.C.C, and B.C.C structure, each with a lattice constant a, calculate the
reciprocal lattice vectors and draw as best you can the reciprocal lattice for each structure
and determine the length of the each reciprocal lattice unit cell.
Note: You have some freedom of choice as to what you take as the basis vectors when
you are calculating the reciprocal lattice vectors. For instance, you do not have to choose
primitive lattice vectors but you can use the conventional lattice vectors for the
calculations. However, it is very important that you remember that your basis will no
longer be one atom but will be two atoms for the B.C.C. structure and four atoms for the
F.C.C structure. Hence you will have to calculate the structure factor for these multiatom basis and explain how it affects the reciprocal lattice points and x-ray diffraction.
Be sure to do this before you answer the following three questions.
1b.
1c.
1d.
The S.C. lattice becomes what well known bravais lattice in reciprocal space?
The F.C.C. lattice becomes what well known bravais lattice in reciprocal space?
The B.C.C. lattice becomes what well known bravais lattice in reciprocal space?
2.
For the S.C. structure the ratio between the magnitudes of the reciprocal lattice
vectors are given in the table below:
Ratio
Table


Ko Ko


K1 K o


K2 Ko


K3 Ko


K4 Ko


K5 Ko


K6 Ko


K7 Ko
S.C. Reciprocal
Lattice
1
2
3
2
5
6
2 2
3
B.C.C. Reciprocal
Lattice
F.C.C. Reciprocal
Lattice
Now I want you to calculate similar data for the F.C.C. and B.C.C. structures. Calculate
the eight smallest ratios (the ratio is defined to be with respect to the reciprocal lattice
vector with the smallest magnitude for that particular reciprocal lattice) of the reciprocal
lattice vectors for the F.C.C. and B.C.C. reciprocal lattice points. Remember that two
reciprocal lattice vectors that have the same magnitude only count as one! If you used
multi-atom basis to calculate the reciprocal lattices, be sure to disregard any reciprocal
lattice point that has a structure factor of zero. (Show your work)
3.
As I have said before, the diamond lattice can be viewed as either two
interpenetrating F.C.C. lattices or also as an F.C.C. lattice with a basis of two
atoms. Use this second view to help you in calculating the structure factor for
a diamond lattice. Indicate how it affects the x-ray intensity peaks of the
standard F.C.C. structure.
Note: Again, there are several ways to do this problem. One way that may be
easier than other ways is to consider the diamond structure as a simple cubic
crystal structure with a basis of eight atoms. You may find the reference in
the following webpage useful:
http://www.stoner.leeds.ac.uk/teaching/CM2/CM6/sld001.htm
4.
Now armed with solutions to problems 1-3, onto the problem in Ashcroft and
Mermin. Identify the F.C.C. and B.C.C. structure and calculate the lattice
constant. (Extra credit) Identify the diamond structure and calculate the
lattice constant.
You may find the formula for the magnitude of the reciprocal lattice vector
useful:
 
K  2k sin  
2
Hint: Compute the magnitudes of K, divide these values by the magnitude of the K that
has the smallest magnitude. This will get you the ratios which should match your results
for problem 2.