PROBLEMS -
1: Draw the following planes and directions in a tetragonal unit cell: (001), (011), (113), [110], [201], [101]. Show cell axes.
2: Show by means of a (110) sectional drawing that [1 11] is perpendicular to (111) e cubic system, but not, in general, in the tetragonal system.
3: In a drawing of a hexagonal prism, indicate the following planes and directions (1210), (1012), (1011), [110], [111], [021]. Show cell axes.
4: Show that the planes (110), (121), and (312) belong to the zone [111].
5: Do the following planes all belong to the same zone: (110), (311), (132)? If so, what is the zone axis? Give the indices of any other plane belonging to this zone.
6: Prepare a cross-sectional drawing of an HCP structure which will show that all atoms do not have identical surroundings and therefore do not lie on a point lattice.
7: Show that c/a for hexagonal close packing of spheres is 1.633.
8: Show that the HCP structure (with c/a = 1.633) and the FCC structure are equally close-packed, and that the BCC structure is less closely packed than either of the former.
9: The unit cells of several orthorhombic crystals are described below. What is the Bravais lattice of each and how do you know? Do not change axes. (In solving this kind of problem, examining the given atom positions for the existence or nonexistence of centering translations is generally more helpful than making a drawing of the structure.)
a) Two atoms of the same kind per unit cell located at 0 ½ 0, ½ 0 ½.
b) Four atoms of the same kind per unit cell located at 0 0 z, 0 ½ z, 0 ½ (½ +z), 0 0 (½ + z)
c) Four atoms of the same kind per unit cell located at x y z, x- y- z, (½ + x) (½ - y)z-, (½ - x) (½ + y) z-.
d) Two atoms of one kind A located at ½ 0 0, 0 ½ ½; and two atoms of another kind B located at 0 0 ½, ½ ½ 0.
10: Construct a Wulff net, 18 cm in diameter and graduated at 30o intervals, by the use of compass, dividers, and straightedge only. Show all construction lines.
In some of the following problems, the coordinates of a point on a stereographic projection are given in terms of its latitude and longitude, measured from the center of the projection. Thus, the N pole is 90o N, 0o E; the E pole is 00 N, 90o E; etc.
11: Plane A is represented on a stereographic projection by a great circle passing through the N and S poles and the point 0°N, 70°W. The pole of plane B is located at 30°N, 50°W.
a) Find the angle between the two planes by measuring the angle between the poles of A and B.
b) Draw the great circle of plane B and demonstrate that the stereographic projection is angle-true by measuring with a protractor the angle between the great circles of A and B.
12: Pole A, whose coordinates are 20°N, 50°E, is to be rotated about the axes described below. In each case, find the coordinates of the final position of pole A and show the path traced out during its rotation.
a) 100° rotation about the NS axis, counterclockwise looking from N to S.
b) 60° rotation about an axis normal to the plane of projection, clockwise to the observer.
c) 60° rotation about an inclined axis B, whose coordinates are 10°S, 30°W, clockwise to the observer.
13: Draw a standard (111) projection of a cubic crystal, showing all poles of the form {100}, {110}, {111} and the important zone circles between them.
14: Draw a standard (001) projection of white tin (tetragonal, c/a = 0.545), showing all poles of the form {001}, {100}, {110}, {011}, {111} and the important zone circles between them.
15: Draw a standard (0001) projection of beryllium (hexagonal, c/a = 1.57), showing all poles of the form {2110}, {1010}, {2111}, {1011} and the important zone circles between them.
16: Plot the great-circle route from Washington, D.C. (39°N, 77°W) to Moscow (56°N, 38°E).
a) What is the distance between the two cities? (Radius of the earth = 6360 km.)
b) What is the true bearing of an airplane flying from Washington to Moscow at the beginning, midpoint, and end of the trip? (The bearing is the angle measured clockwise from north to the flight direction. Thus east is 90° and west is 270°.)
17: Cellulose (C6H10O5)x crystallizes as monoclinic crystals with lattice parameters a= 7.87 Å, b = 10.31 Å, c = 10.13 Å, and β = 122°.
a) Plot the lattice points for (h0l), i.e., in direct space. Superimpose the lattice points of the adjacent (hol) on the first plot.
b) Plot the hol net of the reciprocal lattice (i.e., the reciprocal lattice plane containing reciprocal lattice points of the form hol). Superimpose the points of the (h1l) reciprocal lattice net onto the first plot.
18: Lutetium has a hexagonal structure with lattice parameters a = 3.516 Å and 570 Å. Plot the h0l plane of the reciprocal lattice of this material.
19: Aluminum silicate (mullite) Al6Si2O13 has an orthorhombic Bravais lattice and lattice parameters a = 7.5456 Å, b = 7.6898 Å and c = 2.8842 Å. Assuming that the Bravais lattice is simple orthorhombic, in one diagram plot the h0l net of the reciprocal lattice, and in a second diagram plot the 0kl net of the reciprocal lattice.