N =
� ∞
0
< n(E, μ(T ), T ) > D(E) dE
This expression implicitly determines μ = μ(T ).
U =
� ∞
0
< n(E, μ(T ), T ) > E D(E) dE
To determine CV from this expression one must
take into account the temperature dependence of
μ in addition to the explicit dependence of < n >
on T .
8.044 L19B15
���
�
��
�
ε−μ
8.044 L22B1
�ε)
D(
0
ε=µ(T)
ε
8.044 L22B2
0
ε=μ (T)
ε
8.044 L22B3
ε=μ (T)
0
ε
8.044 L22B4
ε=μ
(T)
0
ε
8.044 L22B5
μ (T �
�
0
ε
8.044 L22B6
N =
=
Z�
∞
0
dE
< n > D(E)
Z�
∞
1
⎡
(2S + 1)V
0 eE/kB T − 1
4π 2
(2S + 1)V
=
4π 2
(2S + 1)V
=
4π 2
⎤
2m
⎦ dE
E
n2
√
Z
�
3/2
∞
E
2m
dE
2
E/k
T
0
n
e B −1
√
3/2
Z
�
∞
2mkB T
x
dx
2
x
0
n
e -− 1 �
'
⎣
⎛
= (2S +
3/2 √
1)ζ(3/2) ⎝
⎞
√
( π/2)ζ(3/2)
V ⎠
λ3(T )
8.044 L22B7
ζ(3/2) = 2.612 . . .
√
and λ(T ) = h/ 2πmkB T .
For a fixed number density n = N/V the critical
temperature for S = 0 Bosons is given by
h2
⎛
⎞2/3
n ⎠
Tc =
2πmkB ζ(3/2)
⎝
∝ n2/3
At a fixed temperature the critical number density
is given by
nc = ζ(3/2) λ−3(T ) ∝ T 3/2
8.044 L22B8
For a fixed number of atoms N , once T falls below
Tc those atoms that can not be accommodated
in states with finite begin accumulating in the
ground state where = 0.
N0 = N −
Z ∞
|0
< n > D() d = N (1 − (T /Tc)3/2)
{z
∝T 3/2
}
8.044 L22B9
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
BOSE-EINSTEIN CONDENSATION
IS A QUANUM MECHANICAL EFFECT
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1980
1960
2000
300 K
CLASSICAL MODEL
QUANTUM REALITY
λ
30 K
3K
300 mK
30 mK
3 mK
300
30
POINT-LIKE
PARTICLES
FOLLOWING
TRAJECTORIES
WAVES
PROPAGATING
THROUGH
SPACE
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
λ∝
1
m ×V
30 K
3K
300 mK
FO
FOR
OR ATOMS MOVI
MOVING AT THERMAL VELOC
VELOCIITY
T
AT R
ROOM
OOM TEMPERATURE (300K
300K)),
λ<T
THE
HE
HEIIR PHYSI
PHYSICAL SI
SIZE.
ZE
30 mK
3 mK
300
OR
R THE ELECTRONS MOVI
MOVING AROUND THE NUCLE
NUCLEII
FO
IN THOSE
THOSE ATOMS,
ATOMS, λ ≈ 1 ANGSTROM
ANGSTROM.
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
THE WAVE NATURE OF THE ELECTRONS
STABILIZES THEM AGAINST LOSING ENERGY AND
FALLING INTO THE NUCLEUS.
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
THE WAVE NATURE OF PROTONS ALLOWS THEM
TO GET CLOSE ENOUGH DURING COLLISIONS IN THE
SUN TO INITIATE FUSION.
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
QM ALLOWS MOLECULES TO HAVE A STATISTICAL
CHANCE OF ADSORBING ON A SURFACE INSTEAD OF
REMAINING IN THE BULK GAS
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
IN 1924 AND 1925 SATYENDRA BOSE AND ALBERT
EINSTEIN INVESTIGATED THE INFLUENCE OF QM ON
THE COLLECTIVE BEHAVIOR OF PARTICLES.
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
1921
"for his services to Theoretical Physics, and especially
for his discovery of the law of the photoelectric effect"
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
TH
THEY
HEY APPLI
APPLIED THE
THEIIR THEORY TO THE S
SIIMPLEST
MPLES
POSS
OSSIIBLE CASE:
CASE: A GAS OF NONNON-IINTERACT
NTERACTIIN
NG ATOM
ATOMS
TOMS
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
AS
S THE ATOMS GET COLDER THEI
THEIR VELOC
VELOCIITY
T
TY
DI
DIM
MIINI
NISHES AND THE
THEIIR WAVELENGTH GROWS
GROWS.
GROWS.
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
WH
WHEN
HEN THE WAVELENGTH BECOMES COMPARABLE TO
T
THE
TH
HE SEPARAT
SEPARATIION
ON,, A PHASE TRANSI
TRANSIT
TIION OCCURS.
OCCURS
SO
SOME
OME OF THE ATOMS LOSE THE
THEIIR IDENTI
DENTITY AND BECOM
BECOME
PART
PA
ART OF A SI
SINGLE WAVE SPANN
SPANNIING THE CONTA
CONTAIINER.
NER
NER.
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
HALF THE ATOMS IN THE WORLD FOLLOW THE RULES
OF BOSE AND EINSTEIN AND ARE CALLED "BOSONS".
THE OTHER HALF FOLLOW RULES SET OUT BY ENRICO
FERMI AND PAUL DIRAC AND ARE CALLED "FERMIONS".
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
Enrico Fermi 1938
"for his demonstrations of the existence
of new radioactive elements produced by
neutron irradiation, and for his related
discovery of nuclear reactions brought
about by slow neutrons"
P.A.M. Dirac 1933 (with Erwin Schrodinger)
"for the discovery of new productive forms
of atomic theory"
μK
μK
μK
300 nK
30 nK
1900
1920
1940
2000
1980
1960
TE
ET
T
FI
FINIT
T=0
<n>
<n>
1
300 K
~ kT
30 K
1
T <<
FE
ER
R
RM
MI
εF
ε
εF
ε
εF/kB
3K
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
2000
1980
1960
TE
ET
T
FI
FINIT
T=0
<n>
<n>
1
1
300 K
~ kT
30 K
T <<
FE
ER
R
RM
MI
ε
εF
<n>
BOSE
OS
N δ (ε )
εF
ε
εF/kB
3K
300 mK
30 mK
<n>
3 mK
~ kT
(GOOD GUESS)
ε
ε
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
2000
1980
1960
TE
ET
T
FI
FINIT
T=0
<n>
<n>
1
1
300 K
~ kT
30 K
T <<
FE
ER
R
RM
MI
ε
εF
<n>
BOSE
OS
N δ (ε )
εF
εF/kB
ε
3K
300 mK
30 mK
<n>
3 mK
~ kT
(GOOD GUESS)
300
ε
ε
30
<n>
BOSE
OS
<n>
N δ (ε )
N0(T)δ(ε)
(ACTUAL)
ε
1
Tc T
ε
μK
N0/N
3
continuous
part
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
O2 LIQUEFIES AT 90K
O2 FREEZES AT 50K
H2 LIQUEFIES AT 20K
H2 FREEZES AT 14K
He LIQUEFIES AT 4K
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
REAL ATOMS OR MOLECULES DO INTERACT WITH
EACH OTHER AND UNDERGO LIQUEFICATION AND
FREEZING DUE TO THESE INTERACTIONS.
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
TE
TEMPERATURES
EMPERATURES BELOW 4.2 K CAN BE ACHI
ACHIEVED IN
N
4H e B
BY
Y PU
PUMP
PUMPI
MPING ON THE
THE VAPOR
VAP
VAPOR
OR ABOVE
ABOV
AB
OVE
E THE
THE L
LIIQU
QUIID
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
SUPERCONDUCTIVITY WAS DISCOVERED BY
KAMERLINGH ONNES IN 1911.
300 K
30 K
RESISTANCE
3K
300 mK
30 mK
3 mK
300
4K
μK
TEMPERATURE
30
1913
"for his investigations on the properties of matter at
low temperatures which led, inter alia, to the production
of liquid helium"
3
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
The Nobel Prize in Physics 1972
30 K
3K
The Nobel Prize in Physics 1972 was awarded jointly to John Bardeen, Leon Neil Cooper
and John Robert Schrieffer "for their jointly developed theory of superconductivity,
usually called the BCS-theory".
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
LIQUID HELIUM-4 FLOWS
INSIDE THE TUBE
30 K
3K
300 mK
30 mK
3 mK
SOLID TUBE
300
FINE POWDER
30
3
SUPERFLUIDITY WAS DISCOVERED IN HELIUM-4 IN
THE 1930s AT A TEMPERATURE OF 2 KELVIN.
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
DOUG
OUG OSHEROFF SEES STRANGE GL
GLIITCHE
TCHES
ON THE
THE MELTI
MELTING CURVE OF
O 3He
He
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
WH
HICH TURNED OUT TO BE TWO DI
DIFFEREN
FFERENT
FFERENT
SUP
UP
PERFLU
ERFLU
ERFLUIID PHASES OF
O 3He
He
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
The Nobel Prize in Physics 1996
300 K
30 K
3K
The Nobel Prize in Physics 1996 was awarded jointly to David M. Lee, Douglas D. Osheroff
and Robert C. Richardson "for their discovery of superfluidity in helium-3".
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
TH
THE
HE SEARCH BEG
BEGIINS FOR BOSE-E
BOSE-EIINSTE
NSTEIIN
N
CONDENSATI
ONDENSATION IN A REAL GAS
GA
CONDENSAT
2000
300 K
30 K
3K
300 mK
See the article Possible "New" Quantum Systems
In Physics Review Letters, Volume 36, Number 15 pages 910–913 (1976)
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
ATOMIC HYDROGEN WILL REMAIN A GAS DOWN TO
ABSOLUTE ZERO IF ITS MAGNETIC MOMENTS ARE
ALIGNED BY A MAGNETIC FIELD.
μK
μK
μK
300 nK
30 nK
1900
1920
1940
2000
1980
1960
300 K
HARVARD
CORNELL
MIT
AMSTERDAM
1980
TURKU
30 K
MOSCOW
3K
300 mK
KYOTO
30 mK
3 mK
UNIV. BRITISH COLUMBIA
300
30
3
GROUPS WORKING ON SPIN-POLARIZED ATOMIC HYDROGEN
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
CONFINEMENT BY WALLS
TRAPPING BY A
MAGNETIC FIELD
300
30
WALLS CAUSE THE MOMENTS TO FLIP; THEN THE ATOMS
RECOMBINE INTO MOLECULES AND FREEZE OUT.
3
μK
μK
μK
300 nK
A MAGNETIC TRAP KEEPS THE ATOMS OFF THE WALLS.
30 nK
1900
1920
1940
1960
1980
2000
300 K
EVAPORATION
PRODUCES
COOLING
30 K
3K
HOTTEST ATOMS
300 mK
30 mK
3 mK
300
COLDEST ATOMS
30
IN 1986 HARALD HESS, A POSTDOCTORAL FELLOW IN
MIT'S HYDROGEN GROUP, PROPOSES MAGNETIC TRAPPING
AND EVAPORATIVE COOLING.
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
BEGINNING IN THE 1980s, METHODS WERE DEVELOPED
TO COOL ATOMS INTO THE MICROKELVIN REGION OF
TEMPERATURES USING LASERS.
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
The Nobel Prize in Physics 1997
30 K
3K
The Nobel Prize in Physics 1997 was awarded jointly to Steven Chu, Claude
Cohen-Tannoudji and William D. Phillips "for development of methods to
cool and trap atoms with laser light".
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
LASER COOLING WORKS BEST WITH CERTAIN ATOMS
SUCH AS LITHIUM (Li), SODIUM (Na), and RUBIDIUM (Rb).
BUT LASER COOLING ALONE CAN NOT GET THESE
ATOMS COLD ENOUGH TO ACHIEVE BEC.
30 K
3K
300 mK
30 mK
3 mK
FOR THE FINAL STAGE OF COOLING ONE MUST TURN
TO EVAPORATIVE COOLING.
300
30
THEN THE RACE BEGAN: LOWER THE TEMPERATURE
WHILE INCREASING THE DENSITY.
3
μK
μK
μK
300 nK
30 nK
1900
1920
.
.
.
1940
1960
1980
2000
300 K
ERIC CORNELL & CARL WIEMAN
JILA (NIST AND UNIV. OF COLORADO)
30 K
Rb
3K
300 mK
RANDALL HULET
RICE UNIVERSITY
30 mK
Li
3 mK
300
WOLFGANG KETTERLE
MIT
Na
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
Race to BEC
[or normalized density of atoms]
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
2000
1980
Bose-Einstein Condensation in a Parabolic Trap
300 K
30 K
Density
Energy
3K
∝√h
300 mK
∝√T
r
condensate
normal component
Momentum
3 mK
300
30
∝√h
r
30 mK
3
∝√T
μK
μK
μK
300 nK
30 nK
r
1900
1920
1940
1960
1980
2000
HOW DID THEY KNOW THEY HAD
BOSE-EINSTEIN CONDENSATION?
300 K
30 K
3K
300 mK
30 mK
3 mK
300
IN THE TRAP, ATOMS IN THE
CONDENSATE ARE ALMOST AT
REST, THE REMAINDER HAVE
THERMAL SPEEDS.
30
3
WHEN THE TRAP IS TURNED OFF
THE THERMAL ATOMS SPEED AWAY,
BUT THE CONDENSATE ATOMS
REMAIN NEAR THE ORIGIN.
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
SUCCESSIVE REAL SPACE IMAGES OF A SODIUM
CONDENSATE FORMING IN A KETTERLE TRAP
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
The Nobel Prize in Physics 2001
300 K
30 K
3K
The Nobel Prize in Physics 2001 was awarded jointly to Eric A. Cornell, Wolfgang Ketterle and
Carl E. Wieman "for the achievement of Bose-Einstein condensation in dilute gases of alkali
atoms, and for early fundamental studies of the properties of the condensates".
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
300 nK
30 nK
1900
1920
1940
1960
1980
2000
300 K
30 K
3K
300 mK
30 mK
3 mK
300
30
3
μK
μK
μK
INTERFERENCE OF MATTER WAVES
300 nK
30 nK
MIT OpenCourseWare
http://ocw.mit.edu
8.044 Statistical Physics I
Spring 2013
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