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Graduate Lecture Series
29 June – 3 July, 2009
Prof Ngee-Pong Chang
Lecture 3
Bose Gas
&
Bose-Einstein Condensate
Bose – Einstein Statistics
1924
1894 - 1974
1879 - 1955
Bose letter to Einstein
June 4, 1924
SN Bose, Zeit f Phys v26, 178 (1924);
v27, 384 (1924)
The Historical Development of Quantum Theory, Jagdish Mehra and Helmut
Rechenberg, p569 (2001); Writings on physics and philosophy by W Pauli,
Charles Paul Enz, K. vMeyenn, R. Schlapp, p. 94 (citation of Bose papers)
Inclusive vs Exclusive
Bosons in a Box
Standing Wave Eigenfunction
L
L
L
At T > 0
Probability of
Occupancy
Probability of Occupancy
Fermion
Classical
limit
Boson
Large E
or
Bose-Einstein Distribution
for spin-0 bosons
fugacity
E=0
Positive
octant
Bose-Einstein
Condensate
Bose-Einstein Distribution
for spin-0 bosons
or
fugacity
Riemann Zeta function
varies monotically from z = 0 to z = 1
For small z
At high temperatures
At crit ical Tc
Critical
Temperature
Crude estimate of Tc
for He4
You can check that this gives a Tc value of 3.17 K,
to be compared with the experimentally observed
value of 2.18 K
J.C. Davies group (Cornell)
http://www.iranika.ir/articles%20page/mohandesi/mechanic.superfluids1.htm
Bose-Einstein Condensate Term
The number of zero energy
bosons per unit volume
University of Stuttgart measurement of
BEC of Chromium atoms
http://www.pi5.uni-stuttgart.de/news/050303/news050303.html
Bose-Einstein Condensate
Nobel Prize in Physics 2001
Cornell – Wieman experiment:
Cooling two thousand Rubidium-87 atoms to
below 170 nK using combination of laser
cooling and magnetic evaporative cooling.
Ketterle (MIT) experiment
Cooled some hundred times more atoms
(Na23) , and was able to demonstrate
quantum mechanical interference between
two BEC.
Magnetic Evaporative Cooling
For a visual applet
on
Magnetic Evaporative Cooling
Go to
http://www.colorado.edu/physics/2000/atomic_lab.html
For a visual applet
on
Laser Cooling of Atoms
Go to
http://www.colorado.edu/physics/2000/atomic_lab.html
Ketterle
experiment
http://online.itp.ucsb.edu/plecture/ketterle/oh/45.html
Nobel Prize in Physics 1997
Transition Temperature
Douglas Osheroff
1945 -
Nobel Prize in Physics 1997
2 Polarization states
Photon Gas
Photon energy
N is a function of temperature
At equilibrium, fugacity, z, rises to maximum value of 1
Pressure
By comparing with the equation for U, we find that the pressureenergy relation for the photon gas is
Or
Phonon Gas
! (k) = ! (k + 2¼n=a)
http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6730Physics-for-Solid-State-ApplicationsSpring2003/8A2B76D2-7D99-445BB511-EDFA06C0482B/0/lecture12c.pdf
Speed of sound in solid
Peter Debye
1884 – 1966
Verh. Deut. Phys. Ges. 15,
678-689 (1913)
Low Temperature Limit
Low Temperature Limit
Specific Heat of Solid at
low temperature
High Temperature Limit
Dulong Petit Law
http://web.mit.edu/6.730/www/ST04/Lectures/Lecture12.pdf