Physics 141A
Problem Set 7 Solutions
April 6, 2008
1
Germanium [111] dispersion
Solution
a) The velocity of sound is given by the slope of the tangent lines to the acoustic branches at the origin.
For the longitudinal velocity we have f /(q/qmax ) THz, an
Physics 141A
Problem Set 13 Solutions
May 14, 2008
1 Kittel 8.1 - Impurity orbits
Solution
For indium antimonide, the energy of the gap is, E g = 0.23eV, = 18, m e = 0.015m.
a) The donor ionization energy is given to be (eqn. 51)
Ed =
e 4 me
2(20 )2
= 0.3
Physics 141A Spring 2008
Final Exam, 21May08
1.
a) For beryllium c/a 1.581, somewhat lower than the ideal value 1.633 found in
Problem 2 of the rst homework.
b) The lattice is hexagonal. A possible choice of primitive translation vectors is
a1 = a (1, 0,
Physics 141A
Problem Set 4 Solutions
March 9, 2008
1
Basis of unlike atoms - Kittel problem 4.3
For the problem treated by (18) to (26), nd the amplitude ratios u/v for the two branches at Kmax = /a.
Show that at this value of K the two lattices act as if
141A Spring 2008
Homework 1 Solutions
Kevin Young and Previous (Unnamed) GSIs
1 Symmetries of the Hexagonal Lattice
Draw a two-dimensional hexagonal lattice, and indicate in your gure the positions of
1. A 3-fold rotation axis (mark it with a small triang
Physics 141A Solution Sets
February 21, 2008
1
Cuprite crystal structure
The gure shows the conventional cubic cell of a cuprite crystal, Cu2 O. The oxygen atoms are located at the
corners and at the center of the cube. The copper atoms are along the body
Physics 141A
Problem Set 6 Solutions
April 6, 2008
1
Heat Capacity of a layer lattice - Kittel 5.4
Solution
a) The slope of the acoustic phonon dispersion near q = 0 is larger when the spring constant is larger. Thus
for rigid interplanar coupling, the ve
Physics 141A
Problem Set 4 Solutions
March 3, 2008
1
Long wavelength phonons
Show that for long wavelenghts the equation of motion for a monatomic linear chain
M
d2 us
= C(us+1 + us1 2us )
dt2
reduces to the continuum elastic wave equation:
2u
2u
= v2 2 .
Physics 141A
Problem Set 8 Solutions
April 6, 2008
1
Bulk properties independence of boundary condition
Solution
a) For hard-wall boundary conditions in 2D, the solutions are of the type
(x, y) = sin(kx x) sin(ky y),
with kx = nx /Lx and ky = ny /Ly , whe
Physics 141A
Problem Set 9 Solutions
April 16, 2008
1 Free electron energies in a square lattice
Solution
2
a , and
and k c = ( a , a ) respectively.
a) The Brillouin zone of a 2D square lattice is a square with side
side face and of a
corner are k m =
Physics 141A
Problem Set 9 Solutions
April 23, 2008
1 Potential energy in the diamond structure
Solution
a) There are two atoms in the basis, atom 1 at (0, 0, 0) and atom 2 at (a/4, a/4, a/4). Then the crystal potential
may be written as
U = U 1 +U 2 = U
Physics 141A
Problem Set 11 Solutions
May 7, 2008
1 Metallic binding
Solution
a) a) First we calculate the ground state energy of the isolated atoms. This can be done by tting half a wavelength into the box width, /2 = a. With k = 2/ and E = 2 k 2 /2m e ,
Physics 141A
Problem Set 12 Solutions
May 12, 2008
1 Shape of the optical absorption edge
Solution
a) Given the dispersions for the valence band and conduction band:
E v (k) = 2 k 2 /2m h
E c (k) = E g + 2 k 2 /2m e
The energy of a direct transition as a
Physics 141A
Problem Set 3 Solutions
February 20, 2008
1
Kittel: Width of diraction maximum
We suppose that in a linear crystal there are identical point scattering centers at every lattice point, m = ma,
where m is an integer. By analogy with Eqn. (20),
Physics 141A
Ivo Souza
Reading: Chapter 8 of Kittel.
Problem Set # 13:
1. Impurity orbits.
Problem 1 of Chapter 8 of Kittel.
2. Ionization of donors.
Problem 2 of Chapter 8 of Kittel.
3. Hall eect with two carrier types.
Problem 3 of Chapter 8 of Kittel.
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, May 2nd
Reading: Chapter 8 of Kittel.
Problem Set # 12:
1. Shape of the optical absorption edge.
Consider optical absorption by vertical transitions from the valence band with dispersion
Ev = 2 k 2 /2m to th
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, April 25th
Reading: Finish Chapter 7 of Kittel; begin Chapter 8.
Problem Set # 11:
1. Metallic binding. Binding in simple metals such as Na arises from the lowering of
kinetic energy of valence electrons whe
Physics 141A Midterm
11Mar2008
1. (25 points) Answer the following questions succintly:
a) Which stacking sequence of a planar triangular net of hard spheres results in the
Hexagonal close-packed (hcp) structure?
Face-centered cubic (fcc) structure?
b)
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, Feb. 1st
Reading: Chapter 1 of Kittel
Problem Set # 1:
1. Draw a two-dimensional hexagonal lattice, and indicate in your gure the positions of
(i) a 3-fold rotation axis (mark it with a small triangle ); (ii
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, Feb. 9th
Reading: Chapter 2 of Kittel
Problem Set # 2:
1. The gure shows the conventional cubic cell of a cuprite crystal, Cu2 O. The oxygen
atoms are located at the corners and at the center of the cube. Th
Physics 141A Spring 2008
Solution to Midterm
Instructor: Prof. Ivo Souza
1 (25 points)
Answer the following questions succintly:
a) Which stacking sequence of a planar triangular net of hard spheres results in the
Hexagonal close-packed (hcp) structure?
Final Exam, 14th December 2004
Physics 141A Fall 2004
Do 5 out of the 6 problems. Please indicate clearly in the front page which
ones should be graded.
1. (20 points) Answer the following questions succintly.
;
0 Explain why the fact that the effective
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, Feb. 29th
Reading: Finish Chapter 5 of Kittel.
Problem Set # 5:
1. Basis of two unlike atoms. Prob. 3 in Ch. 4 of Kittel.
2. Atomic vibrations in a metal. Prob. 6 in Ch. 4 of Kittel.
3. Singularity in the de
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, Feb. 15th
Reading: Chapter 4 of Kittel
Problem Set # 3:
1. Problem 4, Chapter 2 of Kittel (8th edition).
2. Let us simulate a diraction experiment on a one-dimensisonal model crystal using
Mathematica to per
Physics 141A
Ivo Souza
Spring 2008
Due: Friday 22nd of Feb.
Reading: Finish Chapter 4 of Kittel, begin Chapter 5.
Problem Set # 4:
1. Show that for long wavelenghts the equation of motion for a monatomic linear chain
M
d 2 us
= C(us+1 + us1 2us )
dt2
redu
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, March 7th
Reading: Finish Chapter 5 of Kittel.
Problem Set # 6:
1. Heat capacity of layer lattice. Prob. 4 in Ch. 5 of Kittel.
2. Alternative derivation of the 3D density of states.
Many properties of crysta
Physics 141A
Ivo Souza
Spring 2008
Due: Monday, March 31st
Reading: Chapter 6 of Kittel.
Problem Set # 8:
1. The goal of this problem is to illustrate how the results for bulk properties are insensitive
to the choice of boundary conditions. Consider ident
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, April 11th
Reading: Chapter 7 of Kittel.
Problem Set # 9:
1. Problem 1, Chapter 7 of Kittel. Square lattice, free electron energies.
2. Problem 2, Chapter 7 of Kittel. Free electron energies in reduced zone.
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, April 18th
Reading: Chapter 7 of Kittel.
Problem Set # 10:
1. Problem 4, Chapter 7 of Kittel. Potential energy in the diamond structure.
2. Problem 6, Chapter 7 of Kittel. Square lattice.
3. Magnitude of the
Physics 141A
Ivo Souza
Spring 2008
Due: Friday, March 21st
Reading: Chapter 5 of Kittel, Chapter 1 of Ashcroft and Mermin, Chapter 6 of Kittel.
Problem Set # 7:
1. The gure shows the phonon dispersion relation for germanium along [111], measured
1 1
by in