Name___________________________

Grade_____________

MatSc 530
First homework
The reciprocal lattice and the Ewald construction

1. A number of materials have structures for which only planes in their hexagonal lattices whose Miller indices obey the rule - h + k + l = 3n (n = 0,1,2....) can actually be observed in the diffraction pattern. This implies that these are the only sets of planes which actually exist in the lattice.

a. Construct below limited regions (include negative indices) of the l = 0,1,2,3 layers of the reciprocal lattice for such a hexagonal lattice for which a = 3, c = 8 Å. Index each allowed reciprocal lattice point

b. From these layers, construct a 3D-like drawing of the hexagonal unit cell for the reciprocal lattice. Show only the allowed reciprocal lattice points in the cell. Identify the cell axes.

c. Locate another cell in the reciprocal lattice each corner of which is occupied by an allowed reciprocal lattice point. Identify these cell axes. What is the shape of this cell?

d. From the cell in part c, derive the cell in real space and determine its cell constants.

2. An almost infinite number of perfectly cubic-shaped grains of salt, about 1 micron in size, are affixed to a glass plate so that they are extremely densely packed. For NaCl, a = 5.64 Å.

a. Show below the representation, to scale, of the reciprocal lattice for this specimen, and explain.

b. The specimen is placed in a diffractometer and two theta/two-theta scans are made with CuKalpha radiation. For one scan, the specimen is rotated so the normal to the glass plate is 45° away from the normal position in the incident beam/diffracted beam plane. For the other scan, the specimen is moved so that the normal to the glass plate is 45° away from the normal position along the plane perpendicular to the incident beam/diffracted beam plane. Use the Ewald construction to determine if any reflections will be observed in either scan, and, if so, calculate the two-theta angle for several of them. If no reflections are observed, what needs to be done to get some peaks on the pattern (show Ewald construction)?

3. The reciprocal lattice "points" for a certain single crystal specimen are long and rod-shaped. Give a series of drawings of the appropriate Ewald constructions below to show how to use omega (specimen only) and two-theta scans to determine the length and diameter of any reflection from the single crystal. Why do you think the reciprocal lattice "points" are rod-shaped?