Post on 12-Jun-2020
Low Temperature Properties of theBose-Einstein and Fermi-Dirac
Equations
W. C. Troy
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.1/46
Contents
1. The Models.
2. Einstein Model For Cooling A solid.
3. Fermi-Dirac Two State Paramagnetism Model
• W. C. T., Quart. Jour. Appl. Math., AMS (2012)
• W. C. T., Physics Letters A, No. 45 (2012)
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.2/46
1. The Models
N =D
e(ǫ−µ)
kT − 1, ǫ > µ (BE)
N =D
e(ǫ−µ)
kT + 1(FD)
N = most probable number of particles with energy ǫ.
µ = chemical potential, ǫ = hν = quantum of energy,
ν = frequency, h= Planck’s constant,
T = temperature (K), k = Boltzman’s constant.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.3/46
Homework Problems
N =D
e(ǫ−µ)
kT − 1, ǫ > µ (BE)
Prove : limT→0+
N = 0. (1)
N =D
e(ǫ−µ)
kT + 1(FD)
Prove : limT→0+
N =
0, ǫ > µ
1, ǫ < µ(2)
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.4/46
Textbooks
1.) D. Halliday and R. Resnick, Fundamentals of Physics
2.) P. Tipler and R. LLewellyn, Modern Physics
3.) D. Schroeder, Thermal Physics
4.) D. Griffiths, Quantum Mechanics
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.5/46
2. Einstein Model For A Solid
• R. Feynman - 1982 - proposed that development of aquantum computer is theoretically possible.
The Question. Can a solid be cooled to a temperature
T0 > 0 where all quanta of thermal energy are drained
off, leaving the object in the ground state? If ‘yes,’
quantum effects are expected (superposition of states).
This result may lead to quantum computing devices.
• D. Powell, Moved By Light. Science News, May 7, 2011
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.6/46
1982- 2006
Laser cooling - techniques based on radiation pressure
remove energy and reduce vibrations:
2006 - vibrations in glass ’doughnuts’ were reduced bycooling to T=11 mK
2006 - wobbles in mirrors were reduced bycooling to T=10 mK
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.7/46
2006-2009
• Further refinements of laser techniques were developed
to cool sticks, sails, drums and other multi-atom objectsto low temperatures where vibrations are quelled.
• Papers ’flowed in’ as researchers competed to removeevery quantum of energy from an object, leaving it in theground state where quantum effects are expected
• D. Powell, Moved By Light. Science News, May 7, 2011
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.8/46
2009-2010
• 63 quanta left - Park and Wang, Nature Physics 2009
• 37 quanta left - Schliesser et al, Nature Physics 2009
• 30 quanta left - Groblacher et al, Nature Physics 2009
• 4 quanta left - Rocheleau et al, Nature 2010
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.9/46
2010 - 2011
• O’Connell et al (Nature, 2010) reduced a quantumdrum (one trillion atoms) to its ground state at
T0 ≈ 20mK
- Science magazine 2010 breakthrough of the year.
• Teufel et al (Nature, 2011) reduced a drum ( 10−13 kg) toits ground state where it stayed for 100 microseconds, muchlonger than the 6 nano second result of O’Connell et al.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.10/46
TheoryOur Goal. Answer the question theoretically: can allquanta of thermal energy be drained from a solid?
T0 ≡ Lowest Possible Temperature
Models.
• Einstein 1907 Model For A Solid.
• Stat. Mech. Microcanonical Mds.: T0 > 0.
Goal: show that T0 > 0.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.11/46
Einstein Theory For A Solid
Einstein (1907) developed a formula for specific heat
Cv = 1n
∂U∂T in terms of T and the quantum ǫ = hν.
A1. Each atom is a 3D quantum oscillator, which
is attached to a preferred position by a spring.
A2. q > 0 quanta of energy have been added to the solid.
A3. Each quantum has energy ǫ = νh.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.12/46
Petit-Dulong Law of Specific Heat
• P. L. Dulong (1785 - 1838) - French physicist and chemist.
• A. T. Petit (1791-1820) - French physicist.
• 1819 - Petit-Dulong Law of specific heat:
Cv =1
n
∂U
∂T= 5.94
(
cal
gm K
)
.
• A. T. Petit and P. L. Dulong, "Recherches sur quelquespoints importants de la Théorie de la Chaleur," Annales deChimie et de Physique 10 (1819)
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.13/46
Specific Heat:Cv =1n
∂U∂T
• Dulong and Petit (1819): for all solids,
Cv = 5.94
(
cal
gm K
)
.
• Weber (1875), Kopp (1904), Dewar (1904):
0 < Cv << 5.94
for many solids at low T.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.14/46
Einstein Formula (1907):
Cv = 5.94( ǫ
kT
)2 exp( ǫkT )
(
exp( ǫkT ) − 1
)2 , 0 < T < ∞.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6C
v
Figure 1. Cv vs. scaled temperature for diamond.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.15/46
θ = temperature where Cv = 12 (5.94) ≈ 2.97
Lewis and Randall: ‘Thermodynamics And The FreeEnergy Of Chemical Substances‘ (1923)
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.16/46
Behavior of Cv asT → 0+
Cv = 5.94( ǫ
kT
)2 exp( ǫkT )
(
exp( ǫkT ) − 1
)2 , 0 < T < ∞.
It is easily verified that
limT→0+
Cv = 0. (3)
Can T → 0 as the number of quanta decreases to zero?
To answer this, study the derivation of Cv.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.17/46
Micro. Canonical Derivation
N = no. of atoms, N ′ = 3N = no. of degrees of freedom.
W = no. of ways to distribute q quanta of energyover N ′ degrees of freedom:
W =(q + N ′ − 1)!
q!(N ′ − 1)!.
Entropy:
S = k ln(W ) = k ln
(
(q + N ′ − 1)!
q!(N ′ − 1)!
)
.
Simplification: de Moivre - Stirling approximation to n!
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.18/46
James Stirling 1692-1770
• Scottish mathematician.
• 1715 - Oxford - expelled from Baliol college for
correspondence supporting the 1708 Scottish
"Gathering of the Brig o’ Turk" uprising against the Stuarts.
• 1725 - Venice - feared assassination for discovering
trade secrets of Venice glassmkaers. Newton helped
him return to England.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.19/46
Stirling’s Formula
• 1733 - de Moivre, "Miscellanes Analytica,"
N ! ∼ [constant]√
N
(
N
e
)N
as N → ∞.
• 1733 - Stirling, "Methodus Differentials,"
N ! ∼√
2πN
(
N
e
)N
as N → ∞.
• Statistical Mechanics and Physics textbooks:
ln(N !) = N ln(N) − N, N >> 1.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.20/46
Derivation of Cv
Apply ln(M !) = M ln(M) − M when M >> 1.
S = k(
ln((q + N ′ − 1)!) − ln(q!) − ln((N ′ − 1)!))
becomes
S = k(
(q + N ′ − 1) ln(q + N ′ − 1) − q ln(q) − (N ′ − 1) ln(N ′ − 1))
Therefore,
dS
dq= k ln
(
1 +N ′ − 1
q
)
.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.21/46
Internal Energy:
U = qǫ + N ′ǫ
2and
dU
dq= ǫ.
Temperature:
1
T=
∂S
∂U=
dSdq
dUdq
=k
ǫln
(
1 +N ′ − 1
q
)
.
Conclusion I:
T0 = limq→0+
T = 0.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.22/46
Invert Temperature Equation:
q =N ′ − 1
eǫ
kT − 1, N ′ = 3nNA.
U = qǫ + N ′ǫ
2=
(N ′ − 1)ǫ
eǫ
kT − 1+ N ′
ǫ
2.
Specific Heat: Cv = 1n
∂U∂T .
Cv = 5.94( ǫ
kT
)2 exp( ǫkT )
(
exp( ǫkT ) − 1
)2 , 0 < T < ∞.
Conclusion II:
Cv exists for all T > 0 and limT→0+
Cv = 0.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.23/46
Widely quoted:
limT→0+
Cv = 0. (4)
Property (4) is mathematically questionable becauseits derivation is based on
ln(q!) = q ln(q) − q,
which loses accuracy as q → 0+.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.24/46
The Error In Stirling’s Approximation At Low q:
ln(q!) = q ln(q) − q
When q=10 Relative Error = 13 Percent.
When q=2 Relative Error = 188 Percent.
When q=2 Stirling’s Approximation Gives
.69 = −.61
When q=1 Stirling’s Approximation Gives
0 = −1
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.25/46
New Derivation
Replace Stirling ’s approximation
ln(M !) = M ln(M) − M
with the exact formula
ln(M !) = ln(Γ(M + 1)),
where the Gamma function Γ(z) is defined by
Γ(z) =
∫
∞
0tz−1e−tdt, Re(z) > 0.
Basic Property: M ! = Γ(M +1) when M is a positive integer.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.26/46
Entropy S = k ln(
(q+N ′−1)!
q!(N ′−1)!
)
becomes
S = k(
ln(Γ(q + N ′)) − ln(Γ(q + 1)) − ln(Γ(N ′)))
.
U = qǫ + N ′ǫ
2.
1
T=
dSdq
dUdq
1
T=
k
ǫ
(
Γ′(q + N ′)
Γ(q + N ′)− Γ′(q + 1)
Γ(q + 1)
)
∀q ≥ 0.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.27/46
Temperature:
T =ǫ
k
(
Γ′(q + N ′)
Γ(q + N ′)− Γ′(q + 1)
Γ(q + 1)
)
−1
> 0 ∀q ≥ 0.
Specific heat:
CV =1
n
dU
dT=
1
n
dUdq
dTdq
=
−k
n
(
Γ′(q + N ′)
Γ(q + N ′)− Γ′(q + 1)
Γ(q + 1)
)2 [d
dq
(
Γ′(q + N ′)
Γ(q + N ′)− Γ′(q + 1)
Γ(q + 1)
)]
−1
> 0 ∀q ≥ 0.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.28/46
Lowest Temperature:
T0 = limq→0+
T =νh
k
(
Γ′(N ′)
Γ(N ′)+ γ
)
−1
> 0.
γ = Euler’s constant. T0 is large when ν is large!
Lowest Specific heat:
limq→0+
Cv =
−k
n
(
Γ′(N ′)
Γ(N ′)− Γ′(1)
Γ(1)
)2[
d
dq
(
Γ′(q + N ′)
Γ(q + N ′)− Γ′(q + 1)
Γ(q + 1)
∣
∣
∣
∣
q=0
)]
−1
> 0.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.29/46
Derivative Free Method
Goal: Derive temperature formula without differentaition.
S = k ln
(
(q + N ′ − 1)!
q!(N ′ − 1)!
)
.
U = qνh + N ′νh
2.
1
T=
∆S
∆U=
S(q + 1) − S(q)
U(q + 1) − U(q)=
k
νhln
(
q + N ′
q + 1
)
.
T 0 = limq→0
T =νh
k ln(N ′)> T0 =
νh
k
(
Γ′(N ′)
Γ(N ′)+ γ
)
−1
> 0.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.30/46
Comparison With Experiment
O’Connell et al (Nature, 2010): ν = 6 × 109Hz, one trillion atoms.
Ground state at T0 = 20mK
New Temperature Formula:
Ground state at q = 0, T0 = 9.8mK
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.31/46
Diamond
T0 = limq→0+
T = 23.205 (K) and limq→0+
Cv = .63 × 10−20
250 500 750 1000 12500
1
2
3
4
5
6C
v Diamond
T(Kelvin)
New Cv
23.1 23.205 23.3
0.3
0.63
0.9
x 10−20
Cv
Blowup
Einstein Cv
New Cv
C0
T0 T
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.32/46
Diamond
T0 = limq→0+
T = 23.205 (K) and limq→0+
Cv = .63 × 10−20
250 500 750 1000 12500
1
2
3
4
5
6C
v Diamond
T(Kelvin)
New Cv
23.1 23.205 23.3
0.3
0.63
0.9
x 10−20
Cv
Blowup
Einstein Cv
New Cv
C0
T0 T
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.33/46
When N ! = Γ(N + 1), the new derivation gives
limq→0+
T = T0 > 0 and limq→0+
U = N ′ǫ
2= Ground State.
Does this happen experimentally?
Le et al, Science (2011), show how two diamonds can
exhibit quantum entanglement at room temperature.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.34/46
3. Fermi-Dirac Model
N =D
e(ǫ−µ)
kT + 1(FD)
limT→0+
N =
0, ǫ > µ
1, ǫ < µ(5)
Two-state paramagnetism model (Schroeder, ThermalPhysics, p. 98):
Paramagnetic materials ( e.g. magnesium, molybdenum),
are attrated to a magnetic field, B̄. The material does
not retain magnetic properties when B̄ is removed.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.35/46
Two State Model
Assumptions: a paramagnetic system has N >> 1
electrons attracted to external field B̄ = B0k̂, B0 > 0.
N1 >> 1 electrons are in the up-state,
N2 >> 1 electrons are in the down-state
N = N1 + N2
Energies: U1 = −µBB0, U2 = µBB0
µB = eh4πm = 9.27 × 10−24 Joules
Tesla
Total Energy: U = µBB0 (N2 − N1) = µBB0 (N − 2N1)
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.36/46
Entropy: S = k ln(
N !N1!N2!
)
= k ln(
N !N1!(N−N1)!
)
N >> 1, N1 >> 1, N − N1 >> 1
S ≈ k (N ln(N) − N1 ln (N1) − (N − N1) ln (N − N1))
Total Energy: U = µBB0 (N − 2N1)
1T = ∂S
∂U = dS/dN1
dU/dN1
Solve for N1 : N1 = N
exp“
−2µBB0
kT
”
+1
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.37/46
N1 =N
exp(
−2µBB0
kT
)
+ 1
limT→0+
N1 = N
Solve for T: T = − 2µBB0
k ln“
NN1
−1”
limN1→N−
T = 0
These limits are both invalid because the Stirling
approximation requires N − N1 >> 1
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.38/46
Entropy: S = k ln(
N !N1!(N−N1)!
)
= k ln(
Γ(N+1)Γ(N1+1)Γ(N−N1+1)
)
Total Energy: U = µBB0 (N − 2N1) = heB0
4πm (N − 2N1)
1
T=
∂S
∂U=
dS/dN1
dU/dN1
T =heB0
2πmk(
Γ′(N1+1)Γ(N1+1) − Γ′(N−N1+1)
Γ(N−N1+1)
)
limN1→N−
T =heB0
2πmk(
Γ′(N+1)Γ(N+1) − Γ′(1)
Γ(1)
) > 0
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.39/46
Application
BPPH (2,2-diphenyl-1-picrylhydrazyl) - paramagnetic powder
(1) Grobet et al, J. Chem Phys. (1978), p. 5225
(2) Schroeder, Thermal Physics, pp.106-108
N = 2.3 × 1023, B0 = 2.06, k = 1.38 × 10−23, µB = 9.27 × 10−24
limN1→N−
T = T1 =heB0
2πmk(
Γ′(N+1)Γ(N+1) − Γ′(1)
) = .05K > 0
Experiment: > 99 percent magnetization at T ≈ 2K
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.40/46
Acknowledgement
• Stefanos Folias• S. J. Anderson• Richard Field• Toby Chapman• Stuart Hastings
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.41/46
Debye Formula
Einstein Specific Heat (1907):
Cv = 5.94( ǫ
kT
)2 exp( ǫkT )
(
exp( ǫkT ) − 1
)2 , 0 < T < ∞.
Debye Specific Heat (1913):
Cv = 9Nk
(
T
TD
)3 ∫ TDT
0
x4ex
(ex − 1)2dx, 0 < T < ∞.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.42/46
Teufel et al (Nature, 2011) single atom analysis relieson predictions of the Bose-Einstein equation
< N > =1
exp( ǫkT ) − 1
, 0 < T < ∞.
P1: Thermal energy is present at every positive T > 0.
P2: T → 0 as < N > → 0. i.e. T0 = 0.
P2: Quantum effects become important when < N > < 1.
< N > < 1 when T <ǫ
k ln(2).
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.43/46
One Atom Model
Current theory (e.g. Teufel et al, Nature, 2011) - theBose-Einstein equation
< N >=1
exp( ǫkT ) − 1
, 0 < T < ∞,
for a single atom represents the entire solid:
< N >= ave. no. of quanta in 1 particle state with energy ǫ.
ǫ = hν = quantum of energy,
ν = frequency, h= Planck’s constant,
T = temperature (K), k = Boltzman’s constant.
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.44/46
Teufel et al (Nature, 2011) single atom analysis relieson predictions of the Bose-Einstein equation
< N > =1
exp( ǫkT ) − 1
, 0 < T < ∞.
P1: Thermal energy is present at every positive T > 0.
P2: T → 0 as < N > → 0. i.e. T0 = 0.
P2: Quantum effects become important when < N > < 1.
< N > < 1 when T <ǫ
k ln(2).
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.45/46
Comparison With Experiment
O’Connell et al experiment: ground state achieved when
< N > = .07, ν = 6GHz and T = 20mK
The one atom model
< N > =< N/D >=1
exp( ǫkT ) − 1
predicts
Ground State At T = 106mK
Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations – p.46/46