MIDTERM EXAM, PHYSICS 5305, Fall, 2011, Dr. Charles W. Myles
In Class Exam, Tuesday, October 25
NOTE: This Exam begins at 6:00pm.
Lets make every effort to finish it by 8:30pm!! Thank you!!
INSTRUCTIONS:
PLEASE read all of these before doing anything else!!! Failure to follow them may lower your
grade!!
1. PLEASE write on ONE SIDE of the paper only!! This wastes paper, but it makes my grading
easier!
2. PLEASE do not write on the exam sheets, there will not be room! Use other paper!!
As you can see, many of the questions on this exam are Qualitative!!!
3. PLEASE answer the qualitative Questions briefly, including the “How” or the “Why” of
each, not just the “What”. Keep your answers to these short & aimed at the main ideas. A
sketch with appropriate labels or an equation with a brief comment is often the easiest way to
make a point. NOTE: The words “Discuss” , “Define”, and “Explain” below mean to write
complete, grammatically correct, English sentences. I don’t want to see a lot of equations! I
want to see WORDS describing the physics!
4. PLEASE write neatly. If I cannot read or find your answer, you can't expect me to give it the
credit it deserves and you are apt to lose credit.
5. PLEASE put the Questions in order and the pages in order before turning in this exam!
PLEASE FOLLOW THESE SIMPLE DIRECTIONS!!!! THANK YOU!!!
NOTE: Most questions are about solids with crystal structures based on ideal 3-dimensional
lattices. There are 4 qualitative questions & 4 quantitative problems. Answer any two (2)
qualitative questions plus any three (3) quantitative problems for a total of five (5) questions
required. Each is equally weighted & worth 20 points for a total of 100 points on this exam!
Qualitative Questions! Answer ANY TWO (2) of these 4 questions!
Answer briefly in a few English Sentences, with as few math symbols as possible!
1. Crystal Structure #1: Briefly Define or Explain:
a.
b.
c.
d.
Lattice
Basis
Primitive Unit Cell
Conventional Unit Cell
e.
f.
g.
h.
Primitive Lattice Vectors
Packing Fraction
Coordination Number
Miller Indices
2. Crystal Binding #1:
a. Explain what is meant by the cohesive energy of a solid. Discuss the nature of the
interactions that contribute to the cohesive energy of solids.
b. Define the Madelung Energy in ionic solids. Briefly Discuss the reasons that calculating
this contribution to cohesive energies must be done carefully to avoid obtaining divergent
results.
Crystalline solids are often considered in terms of the 4 idealized categories listed in parts c
to f. Yet, for each of these 4 kinds of bonding, it is the electrostatic Coulomb interaction that
provides the attractive force. For each of the 4 types of idealized crystals listed in parts c to f,
and by considering only the outer valence electrons, Discuss how the Coulomb force is
operating in each case. That is, Discuss the Physical Mechanisms which are primarily
responsible for the bonding energy each of these kinds of solids.
c. Ionic Bonding
e. Metallic Bonding
d. Covalent Bonding
f. Van der Waals Bonding
Qualitative Questions (continued)! Answer ANY TWO (2) of these 4 questions!
3. Reciprocal Lattice & Wave Diffraction #1: Briefly Define or Explain:
a. Reciprocal Lattice
e. Atomic Form Factor
b. Reciprocal Lattice Vector
f. Bragg’s Law of Diffraction
c. Brillouin Zone
g. Laue Condition (Laue Equations)
d. Structure Factor
h. Ewald Construction.
4. Lattice Vibrations & Lattice Dynamics #1: In parts a. -f. briefly Define or Explain:
a. Lattice Dynamics
e. Longitudinal Modes
(or Longitudinal Polarization)
b. Harmonic Approximation
f.
Transverse
Modes
c. Normal Modes
(or Transverse Polarization)
d. Dispersion Relations
g. Discuss the qualitative differences in the behavior (as a function of wave vector) of
acoustic and optic phonon modes.
h. What characteristics must the crystal structure of a solid have in order for optic modes to
exist? Is it possible for the phonon dispersion relations of a solid to contain acoustic
modes only? If so, what characteristics are required for the crystal structure?
(Hopefully) Short Quantitative Problems! Answer ANY THREE (3) of these 4 problems!
If any of these seems to be mathematically messy, you’ve probably done something wrong!
5. Crystal Structure #2: The vectors of the primitive bcc cell connect atoms at the vertices
of the bcc cube with an atom in the cube center.
a. Write these vectors in standard Miller notation & calculate the angle between any two of
them.
b. Prove that the volume of the bcc primitive cell is half of the volume of the conventional
cell.
6.
Crystal Binding #2: In our discussion of crystal binding from Ch. 3 of Kittel’s book, we
talked about the Lennard-Jones potential, which is sometimes used to model the nearestneighbor interatomic potential in some solids. This potential has the form:
x
y
where ε & σ are adjustable constants and r = nearest-neighbor distance.
a. Briefly discuss the physical origin of the r-6 term and of the r-12 term.
Express your answers to the following in terms of the constants ε & σ.
b. Sketch this potential as a function of r. Note: If you don’t do this, you will lose points!
c. Calculate the equilibrium nearest-neighbor distance r0.
d. Consider a solid for which the interatomic potential U(r) given above is a good
approximation. In this solid, the atoms are undergoing small displacements (vibrations)
about the equilibrium nearest-neighbor distance r0. In the spirit of the harmonic
approximation for the phonons, assume that the displacements r – r0 are small enough that
U(r) can be expanded in a Taylor’s series in the displacements r - r0 keeping only up
through the quadratic term. By this means, find an expression for the “effective spring
constant k” for this Lennard-Jones system.
Quantitative Problems (continued)!
Answer ANY THREE (3) of these 4 problems!
7.
Reciprocal Lattice & Wave Diffraction #2
a. Derive the reciprocal lattice vectors for the conventional & the primitive unit cells of the
fcc lattice.
b. Calculate the length of the [211] vector, where the Miller indices are referred to the fcc
primitive reciprocal lattice. Use this result to obtain the separation of the (211) planes in
the primitive lattice.
8.
Lattice Vibrations & Lattice Dynamics #2:
Consider the model one-dimensional monatomic chain of N atoms, equally spaced with
separation a, and each with the same mass m. The force constant coupling each atom to its
nearest-neighbors is K. We discussed in class that the normal mode vibrational
frequency ω(k) of a mode with wavevector k for this model is:
[-(π/a) ≤ k ≤ (π/a)]
a. Derive an expression for the group velocity vg as a function of k.
b. Using the results of part a, evaluate vg at very small values of k (k → 0). Briefly Discuss
the physical significance of this low k group velocity.
c. Using the results of part a, evaluate vg for k at the Brillouin Zone boundary [k = (π/a)],
Briefly Discuss the physical significance of this Brillouin Zone boundary group velocity.
Specifically, what we say about propagation of longitudinal waves in this lattice at
frequency ω(k = π/a)?