PES 446 / PHYS 546 Solid State Physics

Spring 1997
Tentative Assignments


PROBLEM SET 1 DUE Monday Feb. 10

1. Kittel Chapter 1 # 3 2. Calculate the density of atoms (atoms/cm^2) for the (100), (110) and (111) planes of a Cu crystal (Cu has fcc structure). 3. a) The Bragg scattering angle for n=1 from the (110) planes in iron (bcc) is 22 degrees for a x-ray wavelength of 1.54 Angstroms. Determine the length of the edge of the cube for iron. b) What is the Bragg scattering angle from the (111) planes of iron ? 4. Kittel Chapter 2 # 1 (assume n = 1) 5. Kittel Chapter 2 # 5 6. PHYS 546 Students only: (based on problem from Ashcroft and Mermin Ch. 5 #2) The simple hexagonal Bravais lattice is shown in the Figure below. It is formed by stacking two-dimensional triangular nets directly above each other. The direction of stacking ( a3 in Figure 4.19 in Ashcroft and Mermin) is known as the c-axis. Three primitive vectors are: a1 = a x a2 = (a/2)x + (SQRT(3) a / 2)y a3 = c z The first two of these primitive vectors generate a triangular lattice in the x-y plane. The third primitive vector stacks the planes a distance c above one another. Using these primitive vectors and Equation 13 from Chapter 2 in Kittel, show that the reciprocal of the simple hexagonal Bravais lattice is also simple hexagonal, with lattice constants 2(pi) / c and 4 (pi) / SQRT(3) a, rotated 30 degrees about the c-axis with respect to the direct (real space) lattice.

PROBLEM SET 2 DUE

1. Kittel Chapter 3 # 2 To get the answer in the back of the book, use theoretical values for Ro rather than the Neon values in the text. Which configuration (bcc or fcc) is more stable ? 2. Kittel Chapter 3 # 5 NOTE: The expression in the book for C is wrong. The denominator should be Ro squared instead of just Ro. 3. For an inert gas crystal with fcc structure, show that the equilibrium separation between atoms is Ro = 1.09 sigma. 4. Kittel Chapter 3 # 6 (Use iteration on your calculator, graphical analysis, or numerical math packages on computers to solve) See inside back cover of Kittel for cgs units and constants

PROBLEM SET 3 DUE: March 5, 1997

1. Kittel Chapter 4 # 2 (6th edition # 3) You may want to consider the fact that "s" is a large number and expand u(s+p) in a Taylor's series expansion about the point s. 2. Kittel Chapter 4 # 4 (6th edition # 5) 3. Kittel Chapter 4 # 5 (6th edition # 6) 4. (based on Ashcroft and Mermin Chapter 22 # 1) Let us reexamine the theory of the linear chain of atoms (one dimensional crystal), but now expand our model by removing the assumption that only nearest neighbors interact. We can now replace equation (1) from Kittel Chapter 4 with: Fs,p = Cp (us+p - us) + Cp (us-p - us) which give us the force on the s plane from atoms p planes away. a) Show that the dispersion relation (equation (9) in Kittel) must now be generalized to: omega = 2 SQRT{ SUM [ Cp [sin^2 (pKa/2)] / M]} PHYS 546 only: b) Show that the long-wavelength limit of the dispersion relation (equation (15) in Kittel) must be generalized to: w = a [ SUM p^2 Cp / M ]^1/2 |K| assuming that SUM p^2 Cp converges. c) Show that if Cp = 1 / p^n (where 1 < n < 3) , so that the sum does NOT converge, then in the long-wavelength limit w is proportional to K^(n-1)/2 . Hint: It is no longer permissible to use the small K expansion of the sine in part a), but one can replace the sum by an integral in the limit of small K.

PROBLEM SET 4 DUE March 12, 1997

1. Kittel Chapter 5 # 1 2. We examined the Debye model in three dimensions. Now consider the Debye approximation for the lattice energy and specific heat of a one-dimensional monatomic lattice with atomic spacing a and a speed of sound v given by the long wavelength limit of equation 15 in Chapter 4: v = SQRT [ C a a / M ] We can define the Debye frequency and temperature slightly differently from what we did in the book: Debye frequency = (pi) v / a Debye temperature = (h-bar) (Debye frequency) / (Boltz. const.) a. Derive integral expressions for the lattice energy and specific heat using these definitions for one polarization state. b. Examine the low and high temperature limits. Show that at low temperatures Cv = (pi)(pi) L (Boltz. const.) T / 3 a (Debye temp) and at high temperatures, Cv approaches the classical limit of L (Boltzconst)/ a. 3. Consider a three dimensional solid which has a density of states in agreement with the Debye approximation. Show that at temperatures lower than the Debye temperature the most probable phonon energy is (h-bar) (plasma frequency) where exp[(h-bar)(plasmafreq)/(Boltzconst)T)][1-((h-bar)(plasmafreq)/2(Boltzconst)T)] = 1

PROBLEM SET 5 DUE April 14, 1997

1. Kittel Chapter 6 # 1 2. Kittel Chapter 6 # 2 HINT: p = - dU/dV 3. Kittel Chapter 6 # 6 HINT: let E = E' exp(-wt) and v = v'exp(-wt) 4. Kittel Chapter 6 # 10 HINT: assume lamda e = d approx. PHYS 546 only: 5. We can set up charge density waves in our free electron gas and then quantize them to form "plasmons". These waves can be understood by considering a simple model. Imagine displacing the entire electron gas, as a whole, through a distance d with respect to the fixed positive background of the ions. The resulting "surface" charge on both ends gives rise to an electric field which can act as a restoring force and produce oscillations. a. Let n = electron number density, m = mass of an electron, e = charge on an electron and u = electron displacement. Show that the equation of motion for the electrons is d2u/dt2 = (plasma frequency)**2 u = 0 where plasma frequency = SQRT (4(pi)n e e / m) in CGS units or = SQRT(n e e / m epsilon nought) in MKS units b. Now let us change the geometry to a uniform spherical region instead of a slab. Again assume that the electrons are moved collectively in some direction relative to the uniform positive background. Determine an expression for the plasma frequency, wp , in this geometry. [Hint: GaussUs Law may help. This does NOT involve much math ! Your answer should not look all that different from part a - except for some numerical factor. ]

PROBLEM SET 6 DUE April 21, 1997

1. Kittel Chapter 7 # 1 in part b: compare the cube corner with the center of a face of the cube part c is extra credit for undergraduates, required for graduate students 2. (20 points) We qualitatively discussed the idea that electron waves of certain wavelengths (or k values) could not propagate through a period structure. We suggested that this would lead to energy bands and gaps. Let us return to this model in a more quantitative manner. We anticipate that at k = n(pi)/a, wave functions will contain both forward and backward travelling components: Psi(+/-) = (1/SQRT(2)) (exp(ikx) +/- exp(-ikx)) where the prefactor gives the correct normalization. We could also write these solutions as: PSI(+) = SQRT(2) cos(kx) PSI(-) = SQRT(2) i sin(kx) We are interested in the average potential energy of electrons in these two states. These electrons are not uniformly distributed in space. They have a probability distribution determined from PSI* PSI. V(+/-) = (1/a) Integral (0 ->a) PSI*PSI V(x) dx where V(x) is similar to the Kronig-Penney model which we discussed. a. Find the potential energy of the two states at the zone boundaries. b. What is the total energy of the electrons in the two states at the zone boundaries ? c. What is the width of the first energy gap ? d. How does the width of the band depend on the height (n value) of the band ? -------------------------------------- last updated: April 28, 1997