Physics 114 Spring 2002 Seminar 10: Quantum Gases II Assigned Reading: Baierlein chapter 9, sections 1, 4-6; Reif chapter 9, sections 16-17 E-mail assignment: Thursday (Friday) people need to send their questions to me by Tuesday (Wednesday) at 5:00 pm. Remarks: Now that we are familiar with the BE/FD distribution functions, we can discuss some of the ramifications. In particular, we will focus on what happens at low temperatures – Fermions and Bosons behave quite differently in this regime. Note: Friday’s group will have to finish the problems and presentations from last week before going on to new stuff. Seminar Break: Melaku on Thursday; Robin on Friday Presentations: 1. Friday only: Joanna’s presentation from last week + Robin’s problem 9.13 2. Justify the ideal gas approximation for conduction electrons. Explain the Fermi energy and its relation to chemical potential. Finally, describe how we can calculate the heat capacity at low temperatures for metals. Be sure you refer both to the Baierlein and the Reif reading. (Lisa on Thursday; Stephanie on Friday) 3. Explain how the theoretical description of an ideal boson gas leads to a prediction of Bose-Einstein condensation. Describe both early experimental evidence of BEC (with 4 He) and the more modern experiments using laser cooling of atoms. Try to provide us with an intuitive explanation as to why BEC occurs. (Mike on Thursday; Mark on Friday) Discussion Problems Thursday Friday 8.16 Abram A1 from last week volunteer 9.17 Dan Matt 9.23 Melaku Robin A1 Andrew Joanna A2 (parts a – d) Abram Dave Hand-in Problems: 9.17, 9.18, 9.23, A2, A3 See following pages for additional problems A1 – A3 Additional problems A1. (Baierlein’s problem 9.8) A total of N fermions (spin 1/2, mass m) are restricted to motion in two dimensions on a plane of area A. There are no mutual interactions. (a) For a temperature T satisfying 0 < T << TF, calculate the average energy per particle E / N , CA (the heat capacity at constant area), and the total entropy S. (b) Calculate the same three quantities when the temperature is sufficiently high that the fermions behave like a semi-classical two-dimensional gas. A2. The Grand Potential is a Legendre transform of the energy: = E - TS - N. It can be shown that kT*lnZ where Z is the Grand Canonical Ensemble partition function. (a) Use this information to obtain expressions for p, the mean pressure, and N, the mean number of particles, for a condensed ideal boson gas with spin degeneracy gs = 2s+1 (take 0 at T < TB). You will need to consult Appendix A of Baierlein to deal with the integral you must evaluate. Note that your expression for N can be used to determine TB. (b) From your answer to (a), construct an equation of state for the gas in the form pV f (T / TB ) where f is a function of T/TB. Note that your answer does not depend on the NkTB spin of the particle. (c) For T > TB it is possible to show that one can obtain an approximate expression for the equation of state that is of the form 2 pV 1 3 N 3 N 1 c where is Baierlein’s thermal de Broglie wavelength and c is a NkT 4 2 gsV gsV constant (we will just accept this for now!). Rewrite this expression in the form of part (b) (i.e., pV/NkTB as a function of T/TB) and determine the constant c by requiring that these two expressions match at T = TB. Verify that your expression gives the classical result in the high temperature limit. (d) Plot pV/NkTB versus T/TB for the ideal boson gas and for the classical ideal gas, for 0<T/TB<3. (e) Using the above two expressions for p and the fact that p = 2E/3V, determine the specific heat at constant volume for the ideal boson gas for both T<TB and T>TB. (f) Plot CV/Nk versus T/TB over the same temperature range as your plot for part (d). ************One more problem on next page!************* A3. (Baierlein’s problem 9.20) The second experiment to produce BEC in a dilute gas used sodium atoms. The number density was N/V = 1020 atoms/m3. The mass of a sodium atom is 3.82 10-26 kg. As with the rubidium experiment, only one state of intrinsic angular momentum was populated. (a) Approximate the trap that confined the atoms by a box with rigid walls. At what temperature would you expect BEC to set in (as you lowered the temperature)? (b) How low a temperature would be required for 90% of the atoms to be in the single-particle ground state? (c) The common stable isotope of sodium is 23Na, with 12 neutrons. The unstable isotope, 21 Na, has 10 neutrons, the same nuclear spin, and a half-life of 23 seconds. Consider a box of volume 1 cm3 containing 1014 Na atoms at temperature T = 1.3 10-6 K. The atoms form a dilute gas, with one state of intrinsic angular momentum populated. Determine whether CV is an increasing or decreasing function of temperature if i) all atoms are 23Na atoms; ii) half are 23Na and half are 21Na.
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