ENMA 460

Physics of Solid Materials

Homework #1

Date: September 19, 2001

Due date: September 26, 2001, in class

Please answer and turn in all the following problems. Two of them will be graded (20 points).

1. Miller indices, interplanar distance d: A sample has a cubic crystal structure with side a = 3Å.

a. What are the Miller indices for the plane with intercepts (x,y,z) = (3,6,2)?

b. Calculate the interplanar distance dhkl corresponding to the plane above.

c. If this sample is heated, the structure becomes a tetragonal crystal structure with sides a=3Å, c=4Å. Recalculate the interplanar distance dhkl.

 

2. Study example 7.1 in de Podesta's book (red book). This example makes an estimate of the atomic separation for Fe. Re-calculate the atomic separation for a-Fe, which occurs in the b. c. c. structure with side a = 2.86Å.

3. A certain application requires a high atomic density surface. The material to be used has a s. c. crystal unit cell, with a lattice spacing a = 3.5 Å. Which plane, (100) or (110) is best suited for this application.

Follow these steps.

a. Make a sketch of the planes above, indicating clearly the dimensions of the sides shown.

b. Calculate the atomic density using the information in a.

c. Choose the most appropriate plane. Justify your answer.

4. Reciprocal lattice. The primitive unit cell vectors for a hexagonal lattice are given by:

a1 = 31/2a/2 x + a/2 y; a2 = -(31/2)a/2 x + a/2 y; a3 = c z.

a. Calculate the reciprocal lattice vectors, b1, b2, b3. SHOW YOUR WORK.

b. Make a rough sketch showing the reciprocal lattice vectors b1, b2. Show that these two vectors generate a haxagonal cross section.

5. Diffraction, Bragg's law. In problem 1, we looked at a material that undergoes a phase transformation from cubic to tetragonal. The cubic phase has a constant a = 3Å, and the tetragonal phase has constants a = 3 Å, c = 4 Å.

a. Use Bragg's law to calculate the diffraction angle, 2q, for the following planes in both the cubic and tetragonal phases: (100), (001), (111), (110), (011). The incidence wavelength l = 1.54Å.

b. Which planes have the same diffraction angle in both the cubic and tetragonal phases? Which planes do not?

c. If this phase transformation occurs as a function of temperature, which diffraction peaks will you "monitor" to determine when the transformation occurs?