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Thermodynamics of Bose-Einstein CondensationDavid SchefflerInterface of Quark-Gluon Plasma and Cold Quantum Gases
from NIST/JILA/CU-Boulder
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 1 NHQ
Overview
IntroductionTheory
Partition Function of the Ideal Bose GasMean Occupation NumberBose - Einstein StatisticsTotal Particle NumberChemical PotentialCritical TemperatureMean Occupation of the lowest Energy-StateEnergyHeat Capacity
Comparison with Experimental Resultsλ-Transition of 4HeBEC of 23Na and 87RbBEC of Hydrogen
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 2 NHQ
History
1908 H. K. Onnes liquefies 4He and observes the Onnes effect (T < 2.17 K)1911 H. K. Onnes observes superconductivity in mercury at 4.2 K1924 S. N. Bose sends Einstein a paper where he derives the Planck distribution
law for photons by statistical arguments Bose statistics1924 based upon Bose’s work Einstein publishes “Quantentheorie des
einatomigen idealen Gases” where he predicts a phase transition if totalparticle number is conserved
1938 F. London proposes that superfluidity is related to BEC1941 L. D. Landau publishes a phenomenological theory of superfluidity1957 BCS theory explains superconductivity1972 Lee, Osheroff and Richardson find superfluidity in liquid 3He (T < 3 mK)
which relates superfluidity and superconductivity1995 Cornell and Wieman: first “real” BECs in 87Rb
1995-97 Ketterle at MIT: BEC of 23Na, interference, “atom laser”
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 3 NHQ
Partition Function of the Ideal Bose Gas
microstate with energy Er
all possible distributions of momenta that have a fixed energy Er
r = {np} = (np1 , np2 , np3 , ...)
Z (T , V ,µ) =X
r
exp{−β(Er − µNr )} β =1
kBT
=∞X
np1 =0
exp{−β(εp1 − µ)np1} ·∞X
np2 =0
exp{−β(εp2 − µ)np2} ...
=1
1− exp{−β(εp1 − µ)} ·1
1− exp{−β(εp2 − µ)} · ...
=Y
pi
11− exp{−β(εpi − µ)}
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 4 NHQ
Mean Occupation Number 〈npi 〉
〈npi 〉 =X
r
Pr n(r )pi
=1Z
Xr
exp{−β(Er − µNr )} n(r )pi
=1Z·∞X
np1 =0
exp{−β(εp1 − µ)np1} · ... ·∞X
npi =0
npi exp{−β(εpi − µ)npi } · ...
=∞X
npi =0
npi exp{−β(εpi − µ)npi } ·„
1− exp{−β(εpi − µ)}«
=1β
∂
∂µ
0@ ∞Xnpi =0
exp{−β(εpi − µ)npi }
1A · „1− exp{−β(εpi − µ)}«
=1β
∂
∂µ
„1
1− exp{−β(εpi − µ)}
«·„
1− exp{−β(εpi − µ)}«
=exp{−β(εpi − µ)}
1− exp{−β(εpi − µ)} =1
exp{β(εpi − µ)} − 1
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 5 NHQ
Bose - Einstein Statistics
〈npi 〉 = 1exp{β(εpi−µ)}−1
0pi
0.
〈np i〉
increasing µ
I only µ ≤ 0 is possible otherwise〈npi 〉 has a singularity
I for µ = 0 and εp0 = 0 the meanoccupation of the lowest energydiverges
I how does µ(T , x) look like?I what happens if µ = 0?
next stepderive an expression for µ(T , V , N) from N(T , V ,µ)
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 6 NHQ
Total Particle Number N(T , V ,µ)
N(T , V ,µ) =X
pi
〈npi 〉 =X
pi
exp{−β(εpi − µ)}1− exp{−β(εpi − µ)}
=X
pi
exp{−β(εpi − µ)} ·∞Xj=0
exp{−β(εpi − µ)}j
=X
pi
∞Xj=1
exp{−β(εpi − µ)j} εp =p2
2m
=∞Xj=1
exp{βµj} V(2π~)3
Zd3p exp{−β p2j
2m} p2 =
p′2
j
=∞Xj=1
exp{βµj} V(2π~)3
Zd3p′
j3/2exp{−β p′2
2m} dp =
dp′√j
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 7 NHQ
Total Particle Number N(T , V ,µ)
N(T , V ,µ) =∞∑j=1
exp{βµj} V(2π~)3 4π
∫dp′
j3/2p′2 exp{−β p′2
2m} �→©
=∞∑j=1
exp{βµj} V(2π~)3
1j3/2
(2mπβ
)3/2
=∞∑j=1
exp{βµj} Vj3/2
(√2mπkBT
2π~
)3
=V
λ(T )3
∞∑j=1
exp{βµj}j3/2
(1)
thermal wavelengthλ(T ) = 2π~√
2πmkBTfrom E = p2
2m = πkBT and p = ~ 2πλ
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 8 NHQ
Intermediate Recapitulation
derived the partition function for the ideal Bose Gas
Z (T , V ,µ) =∏pi
11− exp{−β(εpi − µ)}⇓
calculated the Bose-statistics and found µ ≤ 0
〈npi 〉 =1
exp{β(εpi − µ)} − 1
⇓
asked whether µ(T , V , N) can be zero
N(T , V ,µ) =V
λ(T )3
∞∑j=1
exp{βµj}j3/2
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 9 NHQ
Chemical Potential µ(T , V , N)
fixed number of particlesN(T , V ,µ) = N v = V/N
1 =v
λ(T )3
∞∑j=1
exp{ µjkBT }
j3/2(2)
I T →∞ µ→ −∞I µ = 0 at T = Tc
I no solution for T < Tc
µ(T , V , N)
-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1
0 0.1
0 0.5 1 1.5 2 2.5µ
T/Tc
µ can indeed be zero at a critical temperature Tc
particles start to ”condense” into the lowest energy state⇒BEC2009/10/29 | TU Darmstadt - IKP | David Scheffler | 10 NHQ
Critical Temperature Tc of the Ideal Bose Gas
1 =v
λ(T )3
∞∑j=1
exp{βµj}j3/2
µ=0−→ vλ(Tc)3
∞∑j=1
1j3/2
=v
λ(Tc)3 ζ(3/2) (3)
Riemann’s Zeta Function
ζ(x) =∞∑j=1
1jx
ζ(3/2) ≈ 2.612
critical temperature
Tc =2π~2
ζ(3/2)2/3mv2/3kB
constraint for BECparticlesVolume
· λ(T )3 ≥ 2.612
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 11 NHQ
What happens below Tc ? – eq. (2)
for µ = 0 and p2 → 0, 〈np〉 = 1
exp{β( p22m−µ)}−1
∝ 1/p2
N =X
p
〈np〉 = 〈n0〉 +Xp 6=0
〈np〉
N = ∆
Zp〈np〉p2dp = ∆
Z δp
0dp| {z }
=0 δp→0
+ ∆
Z ∞δp〈np〉p2dp
in contradiction to diverging 〈n0〉 for µ = 0 !
treat 〈n0〉 separately below Tc∑p
〈np〉 = 〈n0〉 +∑p 6=0
〈np〉 → 〈n0〉 +4πV
(2π~)3
∫p〈np〉p2dp (4)
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 12 NHQ
Mean Occupation of the lowest Energy-State
N(T , V ,µ) =V
λ(T )3
∞∑j=1
exp{βµj}j3/2
for T > Tc eq. (1)
N(T , V ,µ) = 〈n0〉 +V
λ(T )3 ζ(3/2) for T ≤ Tc eq. (4)
T > Tc
〈n0〉N
= O(1/N) ≈ 0
T ≤ Tc with eq. (3)Vλ3 ζ(3/2) = Vλ3
cvλ3 = N
“TTc
”3/2
〈n0〉N
= 1− VNλ(T )3
ζ(3/2) = 1−„
TTc
«3/2
〈n0(T )〉
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
<n 0
/N>
T/Tc
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 13 NHQ
E(T , V ,µ) of non-condensed particles
similar to N(T , V ,µ)
E(T , V ,µ) =∑
p
εp〈np〉 = ... =3kBTV2λ(T )3
∞∑j=1
exp{βµj}j5/2
T ≤ Tc, µ = 0 with eq. (3) V = Nλ3c
ζ(3/2)
E(T , V , N) =3kBTV2λ(T )3
∞∑j=1
1j5/2
=3ζ(5/2)2ζ(3/2)
NkBT(
TTc
)3/2
T > Tc
E(T , V ,µ) =3kBTV2λ(T )3
∞∑j=1
exp{βµj}j5/2
, 1 =V
Nλ(T )3
∞∑j=1
1j3/2
exp{βµj}
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 14 NHQ
Expansion of E(T , V ,µ) for T > Tc
E(T , V ,µ) =3kBTV2λ(T )3
∞Xj=1
exp{βµj}j5/2
, 1 =V
Nλ(T )3
∞Xj=1
1j3/2
exp{βµj}
j = 1 E(T , V ,µ) =3kBTV2λ(T )3
exp{βµ}, 1 =V
Nλ(T )3exp{βµ}
E1(T , V , N) =32
kB N T
j = 2E(T , V ,µ) =
3kBTV2λ(T )3
exp{βµ2}25/2
, (exp{βµ})2 =„
Nλ(T )3
V
«2
E2(T , V , N) =32
kB N Tλ(T )3N25/2V
E(T , V , N) ≈ E1 + E2 =32
kBNT
"1 +
ζ(3/2)25/2
„Tc
T
«3/2#
withVN
=λ3
c
ζ(3/2)eq. (2)
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 15 NHQ
Heat Capacity CV (T )
CV (T ) =(∂Q∂T
)NV
=(∂E∂T
)NV
with dW = 0
T ≤ Tc
CV (T , N) =15ζ(5/2)4ζ(3/2)
NkB
„TTc
«3/2
T > Tc
CV (T , N) ≈ 32
kBN
"1 +
ζ(3/2)27/2
„Tc
T
«3/2#
cV (T )
Tc T
3/2
.
c V[k
B]
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 16 NHQ
Summary
calculated the critical temperature
Tc =2π~2
ζ(3/2)2/3mv2/3kB⇓
observed an inconsistency in our equation for µ for T < Tc
1 =v
λ(T )3
∞∑j=1
exp{ µjkBT }
j3/2
and fixed it
⇓
calculated N(T , V , N) −→ E(T , V , N) −→ cV (T , N)
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 17 NHQ
λ-Transition of 4He
I in 1908 Heike Kamerlingh Onnes liquefiedHelium
I in 1913 Onnes recieved the Nobel prize“for his investigations on the properties of matterat low temperatures which led, inter alia, to theproduction of liquid helium”
I λ-transition at T = 2.17 KI in 1962 Landau recieved the Nobel prize
“for his pioneering theories for condensedmatter, especially liquid helium”
(from http://ltl.tkk.fi/research/theory/helium.html)Ideal Bose Gas
Tc =2π~2
ζ(3/2)2/3mv2/3kBwith v ≈ 46 Å
3, m = 4u ⇒ Tc = 3.11K
well, not bad for an ideal gas model
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 18 NHQ
BEC of dilute 23Na and 87Rb gases
I first observation of real BEC in1995 by Cornell and Wieman (Rb,CU-Boulder), Ketterle (Na, MIT)
I Nobel prize in 2001“for the achievement of Bose-Einsteincondensation in dilute gases of alkali atoms, andfor early fundamental studies of the properties ofthe condensates”
I 23Na: T exp ≈ 2µK Tc = 2.2µK
I 87Rb: T exp ≈ 170 nK Tc = 34 nK
upper picture time of flight imaging of a BEC from Ketterle et al., Phys. Rev. Lett. 75, 3969 (1995)lower picture from NIST/JILA/CU-Boulder
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 19 NHQ
BEC of Hydrogen
Ideal Bose gasv =
1n
m = 1u ⇒ Tc = 51µK
v1/3/rB = ∆r/rB ≈ 3300 λ/rB ≈ 4600 rB : Bohr radius
in molecular H2 ∆r/rB ≈ 1.4→ BEC is driven by quantum statistics and not by interactions!
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 20 NHQ
Atom Laser & Interference of BECs
I interference fringes between separatedparts of a BEC cloud ”macroscopicmatter wave” by MIT group
I coherent emission of condensed atoms”pulsed atom laser” by MIT group
more about this inI Fermion-Fermion and Boson-Boson
Interaction at low TI Experimental production of BECs
upper picture from Ketterle et al., Science 275, 637 (1997)lower picture from http://cua.mit.edu/ketterle_group/Nice_pics.htm
2009/10/29 | TU Darmstadt - IKP | David Scheffler | 23 NHQ