Problem Set 5

Due October 18

Problem A: The standing waves of a square drumhead are written z = sin(k_xx)sin(k_yy)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_x² + k_y²)^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