Physics 114 by W7u5JOM6


									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

   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
        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
       f (T / TB ) where f is a function of T/TB. Note that your answer does not depend on the
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
 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

(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,
   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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