Problem Set 5

Due October 18

Problem A:
The standing waves of a square drumhead are written z = sin(k_{x}x)sin(k_{y}y)sin(2pnt). The length of each side of the drumhead is a.

a. What are the
allowed values of k_{x} and k_{y}?

b. Mark the four lowest values of k = (k_{x}^{2}
+ k_{y}^{2})^{1/2} on the k_{x}-k_{y}
plane. Determine n for each. (Call the wave speed v.)

c. Determine the
density of states g(k)dk.

d. Determine g(n)dn.

e. Determine g(l)dl.

3-8

3-9

3-10. b stands for 6pηr.

3-42

3-57

Problem B:
Monochromatic X-rays are incident on a crystal in the geometry shown in
Figure 3-16 of the textbook. Assume the
surface of the crystal is a (100) plane.
The first order Bragg peak is observed when the Bragg angle, q, is 34°.
The crystal spacing is known to be d = 0.347 nm.

a. What is the
wavelength of the X-rays?

Now consider a set of crystal (Bragg) planes
that make an angle of 45° with the surface of the
crystal.

b. Find the angle between the incident X-rays
and the *surface* of the crystal that
produces the first order Bragg peak from this new set of crystal planes.

c. Determine the
angle between the scattered X-rays and the *surface*
of the crystal for this case.

d. Sketch a third set
of Bragg planes for this crystal.

Extra credit:
4-50