Graduate Lecture Series 29 June – 3 July, 2009 Prof Ngee-Pong Chang Lecture 3 Bose Gas & Bose-Einstein Condensate

Bose – Einstein Statistics

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 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

A t cr i t i ca l T c Critical Temperature

Crude estimate of T c for He 4 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)

Bose-Einstein Condensate Term The number of zero energy bosons per unit volume

University of Stuttgart measurement of BEC of Chromium atoms

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 (Na 23 ), and was able to demonstrate quantum mechanical interference between two BEC.

Magnetic Evaporative Cooling

For a visual applet on Magnetic Evaporative Cooling Go to

For a visual applet on Laser Cooling of Atoms Go to

Ketterle experiment

Nobel Prize in Physics 1997

Douglas Osheroff Transition Temperature

Nobel Prize in Physics 1997

Photon Gas 2 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 + 2 ¼n = a ) 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, (1913)

Low Temperature Limit

Specific Heat of Solid at low temperature Low Temperature Limit

High Temperature Limit Dulong Petit Law