Laser Cooling/Trapping of atoms We will discuss this in more detail toward the end of the semester, but it is possible to slow-down (cool) atoms by passing them through a region with counter-oriented laser beams tuned to just below an optical transition [“optical molassses”; so that atoms moving toward the laser will see photons Doppler shifted onto the resonance and absorb the photon (along with its momentum), and other atoms will interact minimally with the photons]. Using this technique, along with simple evaporative cooling, you can get VERY COLD gases of atoms in an optical/magnetic trap

U.C. Boulder group’s BEC This is a map of the momentum distribution in the gas cloud (measured by looking at the gas after the trap has been turned off) for various temperatures (yes, it is O(10 -7 K)). The sharp white peak in the middle is the BEC showing up at ~ 200 nK

Bose-Einstein Condensation There is a great applet at this web sit: You can slowly cool a model gas down and get BEC, or cool it down too quickly and just have all the gas leave your trap (you can even stop the gas in some state and then have it reestablish equilibrium in another smaller trap). The site shows the temperature of the gas, you can watch it cool as some molecules (the most energetic) leave the trap, and it shows the temperature at which BEC would be expected for the density in the trap. See also the movie at:

Another interesting applet (more for chapter 14 when we talk about cooling) Is at the related website:

Intended CALM Question In dealing with systems in which the atoms or molecules possess finite magnetic moments, it is possible to do work on a system by changing the applied magnetic field. The infinitesimal work performed on a substance with magnetization M could then be written (dbar)W=MdH. Should M or H be the “natural variable for the internal energy in such a system? Show how one might define a generalization of the Helmholtz free energy to have the other of this pair (M,H) as its natural variable.

Key Definitions: E=E(S,V,N) Internal energy (fundamental relation) H=H(S, P, N) = E + PV (Enthalpy) F=F(T, V, N) = E - TS (Helmholtz Free Energy) G=G(T, P, N) = E + PV –TS (Gibbs Free Energy) For hydro-static systems (volume the only external parameter). dE = TdS – PdV +  dN dH = TdS +VdP +  dN dF = -SdT –PdV +  dN dG = -SdT + VdP +  dN

Examples: Baeirlein DVB I: Show that you can compute changes in entropy for any combination of isothermal and isobaric processes in a hydrostatic thermodynamic system if you know the specific heat capacity at constant pressure and the equation of state. DVB II: Suppose you take an ideal gas from (100kPa,300K) to (100kPa, 450K) and then to (400kPa, 450K). What is the net amount of heat that must be transferred to the gas to accomplish this?

Jacobians From vol. 5 of the “Course in Theoretical Physics” by Landau and Lifshitz ; L. M. Lifshitz and L. P. Pitaevskii “Statistical Physics 3 rd Ed. Part 1 page 53) Jacobians can be extremely useful in developing non-obvious relationships among functions of several interdependent variables (as we are often wanting to do in thermodynamics). For more details and examples see extract from L&L as a pdf file on the course web site under lecture presentations.

CALM: C P or C V ? 7 of 11 respondents knew that C P was easier to measure and C V easier to compute. Most could give a good explanation for the former, nobody was convincing for the latter

Examples: DVB Show that C P and C V may be related to each other through quantities that may be determined from the equation of state (i.e. by knowing V as a function of P, T, and N). Does this formula work for the case we already know (ideal gas)? Examples: Reif