Problem Set 5

Due October 18

Problem A:  The standing waves of a square drumhead are written z = sin(kxx)sin(kyy)sin(2pnt).  The length of each side of the drumhead is a.

a.  What are the allowed values of kx and ky?

b.  Mark the four lowest values of k = (kx2 + ky2)1/2 on the kx-ky 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