Graduate Lecture Series29 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 EinsteinJune 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

L

L

L

Standing Wave Eigenfunction

At T > 0

Probability of Occupancy

Fermion

Boson

Probability of Occupancy

Classical limit

Large E

Bose-Einstein Distribution

for spin-0 bosons

or

fugacity

Positive octant

E=0

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 critical 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

400 nK

200 nK

50 nK

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 appleton

Magnetic Evaporative Cooling

Go tohttp://www.colorado.edu/physics/2000/atomic_lab.html

For a visual appleton

Laser Cooling of Atoms

Go tohttp://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

Douglas Osheroff

1945 -

Transition Temperature

Nobel Prize in Physics 1997

Photon Gas2 Polarization states

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 pressure-energy relation for the photon gas is

Or

Phonon Gas

! (k) = ! (k+2¼n=a)http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-730Physics-for-Solid-State-ApplicationsSpring2003/8A2B76D2-7D99-445B-B511-EDFA06C0482B/0/lecture12c.pdf

Speed of sound in solid

Peter Debye

1884 – 1966

Verh. Deut. Phys. Ges. 15, 678-689 (1913)

Low Temperature Limit

Specific Heat of Solid at low temperature

Low Temperature Limit

High Temperature Limit

Dulong Petit Law

http://web.mit.edu/6.730/www/ST04/Lectures/Lecture12.pdf