2026 MATSE 413 Homework 7

Please clearly indicate your answer to aid in grading. You are welcome to use any math software

you like (or none at all). If you use software and do not show your work, you cannot receive

partial credit. If you show your work either by writing down your equations or by taking a

screenshot of your program, you will be eligible to receive partial credit. If you use Python or

similar, please comment your code if you wish to receive partial credit. Remember: if your

answer does not have SI-based units, you will receive zero points! Note: in this class, I generally

expect that you can look up constants like the mass of the electron or the speed of light.

1 √3

1) The hexagonal crystal structure has the primitive vectors ⃗⃗⃗⃗

1 = ̂, ⃗⃗⃗⃗

2 = (− 2 ̂ + − 2 ̂),

3 = ̂ . Find the reciprocal lattice vectors. Draw at least one unit cell of the reciprocal

⃗⃗⃗⃗

lattice. Make sure to indicate the relative length of each vector (e.g. if ⃗⃗⃗

1 > ⃗⃗⃗⃗

2 clearly

indicate that somewhere in your drawing). Your drawing does not have to be to scale. Which

Bravais lattice is this? (6 points)

2) InSb has a zincblende crystal structure. We are interested in calculating the structure factor

for various reflections. For mathematical simplicity, you can consider this structure to be

simple cubic with an eight-atom basis.

a. Write down the positions of the eight atoms in the basis in terms of the simple cubic

lattice constant . The simple cubic lattice constant is the distance from one corner to

the other corner of the face centered cubic unit cell. (3 points) Hint: Remember that

the zincblende structure is two interpenetrating FCC lattices. You can use the FCC

atomic positions from the in-class activity as a starting point.

b. Calculate the general expression for the structure factor, simplifying as much as

possible. You can assume that In and Sb have the same atomic form factor, . This is

a reasonably good assumption since In and Sb have similar numbers of electrons and

the InSb bond has a relatively small ionic character. (3 points)

c. List the possible values for the structure factor for InSb in terms of , the atomic form

factor. (3 points)

d. Now consider the case of InP, which also has a zincblende structure but whose atoms

have very different form factors. How does this change your results from parts b and

c? (3 points)

1 3

3) Consider a 2D lattice with the following primitive lattice vectors: ⃗⃗⃗⃗

1 = ̂, ⃗⃗⃗⃗

2 = − 2 ̂ + 4 ̂.

a. Draw the lattice. (2 points)

b. Draw the Wigner-Seitz cell. What type of 2D Bravais lattice is that? (2 points)

c. Calculate the reciprocal lattice vectors ⃗⃗⃗

1 and ⃗⃗⃗⃗

2 from the primitive lattice vectors ⃗⃗⃗⃗

1

and ⃗⃗⃗⃗

2 . Remember that ∙ = 2 { = 0 ≠ , = 1 = } (3 points)

d. Draw the reciprocal lattice. (2 points)

e. Indicate a primitive unit cell of the reciprocal lattice and construct the 1st Brillouin

zone. (2 points)

f. Compare the real and reciprocal lattice. (2 points)

4) Shown below is real x-ray diffraction data for a film grown on a substrate. The film is an

alloy. In each of the five scans, the film composition and therefore the film lattice constant is

changed while the substrate composition and lattice constant remains the same.

a. Identify which peaks come from the film and which come from the substrate. Put a

circle or oval around the peaks coming from the film and put a square or rectangle

around the peaks coming from the substrate. You do not need to identify the very

small peaks near 40° and 45°. This is a 2 − scan, meaning that we are only

measuring the out-of-plane (c-axis) lattice constant. Hint: The substrate peaks should

remain the same in all five scans, while the film peaks should change as the lattice

constant changes. The film peaks should have roughly similar widths corresponding

to the quality of the film, as should the substrate peaks. (4 points)

b. Assume the red scan on the bottom has an alloy content of = 0% and the black

scan on the top has an alloy content of = 100%. Given this data, does the c-axis

lattice constant increase or decrease as increases? How do you know? (3 points)

The following question is extra credit. You are not obligated to do it. If you do the problem

correctly, you get extra credit points. If you do the problem incorrectly or do not do the problem

at all, you are not penalized.

5) An X-ray experiment is performed with the source and detector located in the x-z plane.

Therefore ⃗⃗⃗⃗⃗

and ⃗⃗⃗⃗⃗⃗⃗⃗

are in the same plane as well. The wavelength of the Al Kα line

λ=1.54 Å is being used to perform the diffraction experiment. The X-ray source is positioned

such that X-rays are incident on the crystal top surface [i.e. the (001) plane] at an angle of in

= 55.933°. A strong diffraction peak is measured when the detector is at an angle of out =

19.06°. Construct the Ewald circle and give indicate the set of lattice planes that gives rise to

constructive interference using the (ℎ) notation. (6 points)