General Instructions
1. Chapter 2&3: Phys 224 &
Macrostates and Microstates.
Due Monday, April 3
2. Chapter 4: Entropy,
Energy and Temperature: Due Monday
April 10
3. Chapter 5: Boltzman
Distribution and Partition Function. Due Monday April 17
4. Chapter 6: Density of States
Due Monday April 25
5. Chapter 7: Chemical Potential.
Due Monday
May 1
6. Chapter 8&9: Perfect and Ideal
Gases.
Due Monday May 8
7. Chapter 10&Supplmental: Chemical Equilibrium and
Thermal Radiation. Due Wednesday May 17 (due to midterm)
8. Chapter 11&12: Bose and Fermi Gases.
Due Wednesday
May 24 (due to midterm)
9. Chapter 13&??. Semiconductors
& Special Topics.
Due Thursday June 2 (so
solutions may be posted before the final)
Back to Physics 328 Home Page.
Comments on Reading
- Read through Chapter 2 to review the first and second law from Physics 224. Reassure yourself that if you spent the time you could solve most of the problems at the end of the chapter. In particular, check out the qualitative question on reversibility (2.2), surface tension (2.5), and one of the engine cycle problems (2.8 or 2.9).
- In Chapter 3, refer to the appendices as needed to refresh your math. Most of you stated on the enrollment survey that you were unfamiliar with the random walk -- see example 3.1 for an explanation. Problem 3.1 is worth thinking about, though not necessarily in complete sentences.
- This week's assignment is short since we'll only start covering the material on Wednesday. Expect 5 problems in a more typical week.
Assignment
Comments on Reading
- We are skipping for now the bulk thermodynamic aspects of chapter 4. If we choose to do cryogenics in June, we'll come back to Joule-Thompson expansion and cooling. We will also return to entropy of mixing and solution when we discuss the ideal gas.
Assignment
Comments on Reading: Chapter 5 establishes the Canonical Distribution, characteristic of systems in thermal equilibrium with constant volume and number of microsystems. In class we will concentrate on the harmonic oscillator examples; you should read the other examples on your own. The heat capacity of a collection of oscillators is important in solid state physics. The homework addresses spin systems, ideal gases, and excited atoms. Chapter 5 also gives us the tools to go beyond thermodynamics into fluctuations.
Assignment
- 5.3 Fluctuating magnetic moments
- 5.4 Magnetic Cooling -- useful for cryogenics.
- 5.6 Stellar temperature -- an example of how to include degeneracy
- 5.7 Ruby decay rates -- how to get real numbers from experiments
- 5.13 Rayleigh scattering -- why the sky is blue
Comments on Reading:
The concept of Density of States is ubiquitous throughout physics -- how to count states when they are so close together in energy that it pays to use an integral. This is where statistical mechanics started, with densities of states in phase space. With the benefit of knowing quantum mechanics, we can avoid the problem of the minimum cell size in phase space by starting with quantum states. Equipartition is also a classical phenomenon -- it only works when the states have low enough energy to be classically excited.
The assignment is short this week since you'll be busy studying for midterms and we'll be busy grading them. Read over the other problems at the end of the chapter, and you'll be amazed at the wide variety of situations to which these simple principles may be applied.
Assignment
6.3 DOS for relativistic particle. This is relevant for neutron stars.
6.5 2D partition function for particle in a box. This is relevant for quantum well lasers.
6.8 Energy fluctuations in a cubic millimeter of silicon. Hint: use the heat capacity approach to fluctuations.
Comments on Reading.
This week we relax the assumption of constant N and allow diffusive equilibrium to be established between two systems. Now not only is entropy maximized with respect to energy transfer, but also with respect to particle transfer. In the same way that two temperatures are equal in thermal equilibrium, two chemical potentials are equal in diffusive equilibrium. The Grand Partition Function becomes our new normalization factor for probabilities, and systems that have their state and number described by the grand partition function are called a grand canonical ensemble.
Assignment
- 6.9 Brownian motion and equipartition.
- 7.2 Partial pressures of ideal gases
- Note: Equation 7.14 has a minus sign mistake that propagates into both equations 7.16 and 7.17. The Gibbs-Duhem relation 7.14 should read (sum over i) (Nidzi) = -SdT + Vdp. At constant temperature and constant chemical potential for all other constituents than i, this becomes (7.17 with a minus sign correction) [dzi/dp]T = V/Ni = 1/ni (where z = zeta, the chemical potential, and the derivative is at constant temperature)
- 7.3 Rising tree sap from humidity gradient
- 7.5 Alternate approach to the Rayleigh scattering problem
- 7.8 and 7.9 Ionization of Hydrogen. This is a general problem relevant for hopping conductivity in a solid, where an electron initially hops from a neutral atom to neighbor, creating a (+) (–) pair that may move through the crystal by further hopping. If the repulsion energy between the two electrons on the negative ion is larger than the energy difference, you can end up with an insulator even though in the single electron picture there should be conduction. If the repulsion energy is smaller, you can end up with charges 'self-trapping'.
Comments on Reading
Chapter 8 has the basics of perfect gases and is all relevant.
Chapter 9 includes a lot of material covered in Physics 224, including adiabatic expansion of a gas (PVγ=constant). Many of the end-of-chapter problems require only thermodynamics to solve, and we'll skip those. We will concentrate on the parts less well covered there - chemical potential, internal partition function, and ideal solutions. We will supplement chapter 9 with some material on chemical reactions, deriving the equilibrium constants from the partition function.
Assignment
- 8.1 Fermi-Dirac distribution at kT<<chemical potential.
- 8.2 Chemical potential of 2D gas. Problem asks for just Fermi level, but it is also possible to solve analytically for chemical potential as a function of T.
- 8.4 2D to 3D transition. Remember you found a constant density of states in 2D back in chapter 6
- 9.11 Star formation (note that A < 0 for a gravitational potential)
- 9.12 Sap rising under osmotic pressure
Comments on Reading
Chemical equilibrium constants are covered in Kittel and Kroemer, Chapter 9, and the first HW problem is taken from that book. Black body radiation really launched the quantum era, and is thus of great importance for both current physics and for understanding scientific history. It is used throughout all branches of physics, from cosmologists estimating the structure of the early universe, to condensed matter physicists measuring the temperature of their samples, to geopoliticians trying to curb global warming.
For the last problem on the homework, you may choose either one depending on your interest level in solids or stars (otherwise the assignment was too long).
I did not assign a problem on proving the formula J = (nc)/4 since it is worked out in appendix H. Read that appendix if you are curious.
Prof. Olmstead will be out of town on May 15, so the actual topic of that lecture may vary depending on who teaches it.
Assignment
- Extra 7.1 (Kittel and Kroemer 9.5): particle-antiparticle equilibrium
- 10.1 Planetary thermal balance
- Extra 7.2. Expanding universe.
- 10.6 Heat shields
- Measuring temperature. CHOOSE ONE: (9 is more relevant to solid state experiments; 11 to astronomers)
- 10.9 Semiconductor temperature measurement
- 10.11 Stellar temperature measurement
Comments on Reading
Quantum fluids are major research areas that have led to several Nobel prizes (superfluid He4, superfluid He3, bose condensation of gases, quantum Hall effect, fractional quantum Hall effect, stellar evolution, etc.) The basics can be understood in a farily straightforward bit of statistical physics.
Assignment
- 11.3 Fountain effect
- 11.4 2D Bose gas (hint: the potential in part (b) is that of a 2D harmonic oscillator)
- 12.2 2D Fermi Gas
- 12.8 Relativistic Fermi Gas
- 12.10 Neutron Star
Comments on Reading
The semiconductor chapter is straightforward, and we won't have time to discuss all the examples in class. One HW problem below is on the thermoelectric effect covered in the book but not in class.
For cryogenics, read chapter 12 of Kittel and Kroemer.
Assignment
- 13.3 Conduction band occupation from donors. HINT: remember the relationship between energy and chemical potential when a state has 50% probability of being occupied)
- 13.4 Semi-insulating GaAs (this can happen by putting excess As into the crystal)
- 13.A Thermopower
- K&K 12.5 Adiabatic Demagnetisation