MIDTERM EXAM, PHYSICS 5305, Fall, 2009, Dr. Charles W. Myles
Take Home Exam: Distributed, Monday, November 2
DUE AT 5PM, MONDAY, NOVEMBER 9!! NO EXCEPTIONS!
Bring it to my office or put it in my mailbox. (I prefer it in my mailbox! Put it in a sealed envelope!)
RULE: You may use almost any resources (library, internet, etc.) to answer the questions. EXCEPTION: You MAY NOT COLLABORATE WITH ANY OTHER PERSON! If you have questions/difficulties, consult with me, not with other students (whether or not they are in this class!), with people who had this course earlier, with other faculty, with post-docs, or with anyone else I may have forgotten to list here. You are bound by the TTU Code of Student Conduct not to violate this! Anyone caught violating this will, at a minimum, receive an “F” on this exam!
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, most 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: “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: All questions are about solids with crystal structures which are based on ideal 3-dimensional lattices. (NOTE: All 5 Questions are required!)
Please sign the statement below and turn it in with your exam:
I have neither given nor received help on this exam
Signature ______
1. Crystal Structure
a. Briefly (in a few English sentences) Define or Explain the following terms: 1) Bravais
Lattice, 2) Basis, 3) Crystal Structure, 4) Primitive Unit Cell, 5) Conventional Unit Cell,
6) Wigner-Seitz Cell, 7) Primitive Lattice Vectors, 8) Packing Fraction, 9) Coordination
Number.
b. Briefly Discuss the reasons that a periodic lattice with a five-fold symmetry axis cannot exist.
c. Define the Miller indices of a lattice plane. Explain how they can uniquely define a set of
crystal planes.
Short Quantitative Questions
The vectors of the primitive fcc cell connect atoms at the vertices of the fcc cube with the face centered atoms.
d. Write these vectors in the standard Miller notation and calculate the angle between any
two of them.
e. Prove that the volume of the fcc primitive cell is one quarter of the volume of the
conventional cell.
2. Crystal Binding.
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
mathematically divergent results.
Briefly Explain the types of bonding (or binding) between atoms in crystals which are listed in parts b to e. For each, in a few sentences, Discuss the Physical Mechanisms which are primarily responsible for the binding energy in solids where that mechanism is the dominant one. Also, state for which categories of solids each interaction is the primary bonding mechanism.
c. Ionic Bonding
d. Covalent Bonding
e. Metallic Bonding
f. Van der Waals Bonding
3. Reciprocal Lattice and Wave Diffraction:
a. Briefly (in a few English sentences) Define or Explain the following terms: 1) Reciprocal Lattice, 2) Reciprocal Lattice Vector, 3) Brillouin Zone, 4) Structure Factor, 5) Atomic Form Factor, 6) Bragg’s Law of Diffraction, 7) Laue Condition (Laue Equations), 8) Ewald Construction.
b. Briefly Discuss the Physics underlying Bragg’s Law of Diffraction. Briefly Explain the
reasons that this law is equivalent to the Laue Condition (or the Laue Equations).
c. Name the three most important kinds of probes used in diffraction experiments on crystals.
(Hint: See p. 24 of the 8th Edition of Kittel’s book!). Discuss the essential condition that
the wavelength of each probe must satisfy if it is to be useful in understanding crystal
structure.
Short Quantitative Questions
d. Derive the reciprocal lattice vectors for both the conventional and the primitive unit cells
of the fcc lattice. Using these, show that the volume of the reciprocal cell of the primitive
lattice is four times that of the conventional lattice.
e. 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.
4. Lattice Vibrations & Lattice Dynamics:
a. Briefly (in a few English sentences) Define or Explain the following terms: 1) Lattice Dynamics, 2) Harmonic Approximation, 3) Normal Modes, 4) Longitudinal Modes, 5) Transverse Modes, 6) Dispersion Relations, 7) Phonon, 8) Elastic Constants, 9) Anharmonic Effects.
b. Discuss the qualitative differences in the behavior (as a function of wave vector) of acoustic and optic phonon modes.
c. 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?
d. Sketch the phonon dispersion curves for an elemental solid such as Si. (Si has the diamond crystal structure).
e. Discuss how your sketch in part b would be different for a compound solid such as GaAs.
(GaAs has the zincblende crystal structure). Discuss the reasons these curves are so
different, or so similar for Si & GaAs.
5. Lattice Vibrations & Thermal Properties:
a. Briefly (in a few English sentences) Define or Explain the following terms: 1) Planck (Bose-Einstein) Distribution, 2) Phonon (Vibrational) Density of States (Density of Modes), 3) Phonon (Lattice) Heat Capacity, 4) Einstein Frequency (& Einstein Temperature), 5) Debye Frequency (& Debye Temperature), 6) Thermal Expansion Coefficient, 7) Heat Current, 8) Thermal Conductivity, 9) Phonon “Gas”, 10) Umklapp Processes.
Historically, two early models of the phonon density of states used to try to explain the observed low temperature behavior of the lattice heat capacity CV(T) were the Einstein Model and the Debye Model. It is interesting that, even though modern computers allow the (in principle) exact numerical calculation of CV(T), both models are still sometimes useful today to help to understand the Physics behind this low temperature behavior of CV(T).
b. Briefly Discuss the Physics underlying the Einstein Model. What is the primary assumption that this model makes about the behavior of the phonon frequencies as a function of wave vector? Do you think that this assumption would ever be a reasonable first approximation (at least qualitatively) for any of the phonon modes in a solid? If so, for which kinds of phonon modes might it be useful?
c. Briefly Discuss the Physics underlying the Debye Model. What is the primary assumption that this model makes about the behavior of the phonon frequencies as a function of wave vector? Briefly contrast this with the assumption made in the Einstein Model. Do you think that this assumption would ever be a reasonable first approximation (at least qualitatively) for any of the phonon modes in a solid? If so, for which kinds of phonon modes might it be useful?
In Ch. 5, it was stated that the harmonic approximation to lattice dynamics cannot be used to calculate either the thermal expansion coefficient or the thermal conductivity, but that anharmonic effects are needed to calculate either of these properties.
d. Briefly Discuss the Physics that underlies the statement that the harmonic approximation cannot be used to calculate these properties. If the harmonic approximation is used to calculate them, why is it obvious that the results of this calculation aren’t reasonable? The calculation of the thermal conductivity may be viewed, as in Ch. 5, in terms of processes involving phonon-phonon scattering. Is such scattering even possible in the harmonic approximation? Why or why not?