Chapter I

Quantum Fluids

Chapt. I - 2

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Contents:

I.1 Foundations and General PropertiesI.1.1 Quantum FluidsI.1.2 HeliumI.1.3 Van der Waals BondingI.1.4 Zero-Point FluctuationsI.1.5 Helium under PressureI.1.6 pT-Phase Diagram of 4He and 3HeI.1.7 Characteristic Properties of 4He and 3HeI.1.8 Specific Heat of 4He and 3He

I.2 4He as an Ideal Bose GasI.2.1 Bose-Einstein CondensationI.2.2 Ideal Bose GasI.2.3 Bose-Einstein Condensation of 4He

I.3 Superfluid 4HeI.3.1 Experimental ObservationsI.3.2 Two-Fluid ModelI.3.3 Excitation Spectrum of 4He

I.4 VorticesI.4.1 Quantization of CirculationI.4.2 Experimental Study of Vortices

I.5 3HeI.5.1 normal fluid 3HeI.5.2 solid 3He and Pomeranchuk

effectI.5.3 superfluid 3He

I.6 3He / 4He mixtures

Chapter I: Quantum Fluids – Part 1

Chapt. I - 3

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1 Foundations and General PropertiesI.1.1 Quantum gases

• 1924: prediction of Bose-Einstein Condensation (BEC) by Einstein

• 1995: first observation of BEC on alkali atoms by

Eric A. Cornell Wolfgang Ketterle Carl E. Wieman

Nobel Prize in Physics 2001

"for the achievement of Bose-Einstein

condensation in dilute gases of alkali atoms,

and for early fundamental studies of

the properties of the condensates"

long period between theoretical prediction and experimental realization:high density, ultra-cold gas of noninteracting atoms required

problem: gas liquefies or solidifies

more simple to realize:interacting quantum gas, e.g. superfluid 4He

Chapt. I - 4

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

supercurrents U-tube oscillation thermomech. effects

superleaks beaker film flow critical velocitySource: D.I. Bradley, Lancaster Univ.

I.1 Foundations and General PropertiesI.1.1 Quantum Fluids

quantum fluids show fascinating properties

Chapt. I - 5

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• fluids and solids:

characteristic energies: (1) average interaction energy: V or Epot

(2) thermal (kinetic) energy: kBT or Ekin

criterion for fluid/solid: fluid: kBT >> V

solid: kBT << V

I.1.1 Quantum Fluids

• characteristic length scales in a gas:

- particle distance:

- interaction range: (scattering length a)

- de Broglie wavelength:

classical noninteracting gas: d >> r0, since atoms interact only during collisions

ℏ2𝑘T2

2𝑚≃ 𝑘B𝑇 𝑘T =

2𝜋

𝜆T

ℎ2

2𝑚𝜆T2 ≃ 𝑘B𝑇 𝜆T ≃

ℎ2

2𝑚𝜆T2

1/2

Chapt. I - 6

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.1 Quantum Fluids

• cooling down a classical gas

increases and we can distinguish three regimes

(1) classical regime: d

r0 lT

(2) quantum collisions: d

r0 lT

(3) quantum degeneracy:

d

r0 lT

collissions can no longerbe treated classically

all degrees of freedom of thegas must be treatedquantum mechanically

Chapt. I - 7

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

) (3) zero-point energy Ezp

with particle density 𝑛 = 𝑁/𝑉, 𝑑 = 𝑛−1/3

de Broglie wavelength 𝜆T = 𝑑

momentum uncertainty

zero-point energy

characteristic temperature

I.1.1 Quantum Fluids

• define new energy scale related to d = lT

for an ideal gas theenergy 𝐸 = 𝜋𝑘B𝑇 is

assumed; the factor 𝜋results from the

partition function

Chapt. I - 8

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.1 Quantum Fluids

• calculation of the average thermal wavelength of a statistical ensemble

the wave number 𝑘 is given by

𝑘T =1

ℏ

−∞

∞

d𝑝 exp −𝑝2/2𝑚

𝑘B𝑇

𝑘T =1

ℏ2𝜋𝑚𝑘B𝑇

𝑘T =2𝜋

ℎ2𝜋𝑚𝑘B𝑇 𝜆T =

2𝜋

𝑘T=

ℎ

2𝜋𝑚𝑘B𝑇

Chapt. I - 9

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.1 Quantum Fluids

• classification of fluids (interacting fluids with 𝑽 > 𝟎) :

(a) classical fluid: Tzp << Tmelting < T

(b) quantum fluid: Tmelting < T < Tzp

proportional to V or Epot

quantum parameter: quantum liquid:

L > 1

(quantum fluctuation play no role)

(quantum fluctuation dominate)

classical liquid:

L << 1

Chapt. I - 10

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.1 Quantum Fluids

• classification of fluids (non-interacting fluids with 𝑽 = 𝟎) :

(a) classical fluid: Tzp < T

(b) quantum fluid: T < Tzp

(a) classical fluid: lT ≤ n-1/3 d n ∙ lT3 ≤ 1

(a) quantum fluid: lT ≥ n-1/3 d n ∙ lT3 ≥ 1

average particle distance

particles become indistinguishable degenerate quantum fluid

e.g. electron @ 300 K: lT = 4.2 nm, d ≈ 0.2 nm for Cu (n ≈ 8.5 x 1028m-3)

𝑇cp =ℎ2𝑛2/3

2𝜋𝑚𝑘B=

ℎ2

2𝜋𝑚𝑘B𝑑2

𝑇 =ℎ2

2𝜋𝑚𝑘B𝜆T2

Chapt. I - 12

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0 10000 200000

1

2

3

40 1 2

T (K)

lT /

d

T (K)

I.1.1 Quantum Fluids

quantumfluid:

lT > dT < Tzp

classicalfluid:

lT < dT > Tzp

T = Tzp

4He: m = 6.65 x 10-27 kgn ≈ 4 x 1027 m-3

e- : m = 9.11 x 10-31 kgn ≈ 3.6 x 1027 m-3

degenerate:indistinguishable particles

non-degenerate:distinguishable particles

Chapt. I - 13

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.1 Quantum Fluids

• quantum statistics:

photons, mesons, 4He-atoms, ... integral spin S = 0, , 2, 3, …

electrons, nucleons, 3He-atoms, ... half-integral spin S = /2, 3/2, 5/2, …

indistinguishability of identical particles: symmetry of many-particle wavefunction with respect to particle exchange

• bosons (integral spin)

• fermions (semi-integral spin)

• Pauli principle: two fermions cannot occupy the same quantum state

Chapt. I - 14

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.1 Quantum Fluids

• example: two particles (1,2) in quantum states A and B:

classical: 4 possibilitiesdouble-occupancy ratio: 1/2

bosons: 3 possibilitiesdouble-occupancy ratio: 2/3

fermions: 1 possibilitydouble-occupancy ratio: 0

bosons: tendency to occupy the same quantum state

fermions: multiple occupancy of the same quantum state is forbidden

Chapt. I - 15

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0 2 4 6 8 100

1

2

3

Bose-Einstein

µ = 0

Fermi-Dirac

µ = 5kBT

Maxwell-Boltzmann

f (

)

n(

)

/ kBT

Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann distributions as a function of normalized energy. Forthe Bose-Einstein distribution we assumed μ = 0, for the Fermi-Dirac distribution μ = 5kBT (for a metal μ

is several eV corresponding to temperatures of several 10 000 K). For the Maxwell-Boltzmann distributions we also assumed μ = 0 und 5kBT so that the classical distribution coincides with the quantum

mechanical distributions for large >> kBT.

I.1.1 Quantum Fluids - Bosons and Fermions

Chapt. I - 16

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0 10 20 30 400

10

20

30

40

50

60

70

Fermi-Dirac

Maxwell-

Boltzmann

dN

/ d

/ kBT

0 2 4 6 80

10

20

30

40

50

60

70

Fermi-Dirac

Maxwell-Boltzmann

dN

/ d

/ k

BT

particle number per energy interval:

I.1.1 Quantum Fluids - Bosons and Fermions

Chapt. I - 17

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

bosons fermions

number of bosons

qu

an

tum

sta

tes

number of fermions

qu

an

tum

sta

tes

qu

an

tum

sta

tes

qu

an

tum

sta

teshigh T high T

low T low T

I.1.1 Quantum Fluids - Bosons and Fermions

Chapt. I - 18

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.1 Quantum Fluids

• well-known quantum fluids:

1. liquid helium: - small mass and d high Tzp (a few K)- small V liquid down to T = 0- 4He: spin 0 Bose liquid- 3He: spin ½ Fermi liquid

2. electrons in metals: - “non-interacting” electron gas- very small electron mass, higher density as for He

very high 𝑇zp (a few 104 K)

𝑇zp ≫ 𝑇melting

- spin ½ Fermi liquid

3. nuclear matter: - fermions in nuclei, neutron stars, …- very high density very high Tzp (a few 1012 K)

- spin ½ Fermi liquid

Chapt. I - 19

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.1 Quantum Fluids

mass (kg) n (m-3) Tzp (K) statistics

3He 5.01 x 10-27 1.63 x 1028 6.50 Fermi

4He 6.65 x 10-27 2.18 x 1028 5.93 Bose

electrons in Na 9.11 x 10-31 2.65 x 1028 4.94 x 104 Fermi

nuclear matter 1.67 x 10-27 1.95 x 1038 1 x 1012 Fermi

Chapt. I - 21

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1 Quantum Fluids – Foundations and General PropertiesI.1.2 Helium

1868: discovery of 4He in optical spectrum of the sunFrench Astronomer Jules Jansen:solar eclipse, India, yellow line in spectrum of the sun

1871: confirmation by Joseph Norman Lockyer name derived from greek word „helios“ Ἥλιος (sun)

1895: discovery on earth by William Ramsey, Per Teodor Cleve and Abraham Langlet in Norwegian rock

1907: a-particles are 4He nuclei (Rutherford, Royds)

1923: specific heat discontinuity at 2.2 K (Dana, Onnes)interpretation in terms of He-I and He-II in 1927 (Keesom, Wolfke)

1924/25: prediction of Bose condensation (Bose, Einstein)

1938: vanishing flow resistance (Allen, Misener) superfluidity (Kapitza, Nobel Prize in Physics 1978) oscillating disc experiments (Keesom, McWood) phenomenological quantum mechanical description (F. London)

• history: 4He

Chapt. I - 22

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1 Quantum Fluids – Foundations and General PropertiesI.1.2 Helium

soon after discovery on earth, race for the liquefaction of Hemain opponents: Heike Kamerlingh Onnes and James Dewar Onnes extracted 360 l of He gas out of Monazit sand from North Carolina

1908: first liquefaction: July 10, 1908

1925: first liquefaction of He in Germany (March 7, 1925: 4.2 K, 200 ml) by W. Meißner, 3rd system world-wide besides Leiden and Toronto

1932: Linde supplies first industrial He liequfier to University of Charkov, Ukraine

1947: first commercial liquefier by S.C. Collins

Heike KamerlinghOnnes* 21. 09. 1853 † 21. 02. 1926

James Dewar* 20. 09. 1842 † 27.03.1923

• history: 4Heapplication of 4He in low temperature physics

Chapt. I - 23

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

SIR JAMES DEWAR

Sir James Dewar, (1842-1923) was a chemist and physicist, best known for his work with low-temperature phenomena. Dewar was born in Kincardine, Scotland, and educated at the University of Edinburgh. He was professor of experimental natural philosophy at the University of Cambridge, England, in 1875 and professor of chemistry at the Royal Institution of Great Britain in 1877, where he was appointed director of the Davy-Faraday Research Laboratory. Dewar developed structural formulas for benzene (1867). He studied the specific heat of hydrogen and was the first person to produce hydrogen in liquid form (1898) and to solidify it (1899). He constructed a machine for producing liquid oxygen in quantity (1891). He invented the Dewar flask or thermos (1892) and co-invented cordite (1889), a smokeless gunpowder, with Sir Frederick Abel. His discovery (1905) that cooled charcoal can be used to help create high vacuums later proved useful in atomic physics. Dewar was knighted in 1904. The Dewar flask or vacuum flask/bottle is a container for storing hot or cold substances, i.e. liquid air. It consists of two flasks, one inside the other, separated by a vacuum. The vacuum greatly reduces the transfer of heat, preventing a temperature change. The walls are usually made of glass because it is a poor conductor of heat; its surfaces are usually lined with a reflective metal to reduce the transfer of heat by radiation. Dewar used silver. The whole fragile flask rests on a shock-absorbing spring within a metal or plastic container, and the air between the flask and the container provides further insulation. The common thermos bottle is an adaptation of the Dewar flask. Dewar invented the Dewar flask in 1892 to aid him in his work with liquid gases. The vacuum flask was not manufactured for commercial/home use until 1904, when two German glass blowers formed Thermos GmbH. They held a contest to rename the vacuum flask and a resident of Munich submitted "Thermos", which came from the Greek word "Therme" meaning "hot."

I.1.2 Helium

Chapt. I - 24

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.2 Helium

1933: discovery by Oliphant, Kinsey and Rutherford(they believed that 3He is unstable, disproven by Alvarez and Cornog in 1939)

1947: first experiments with gaseous 3He

1949: first condensation of 3He gas (Sydorniak and Co.)

1956: theory of Fermi liquids (L.D. Landau, Nobel Prize in Physics 1962)

1960s: first experiments with the Fermi liquid 3He (various groups)

1963: first theory of superfluid 3He (Anderson and Morel, Balian and Werthamer)

1971: discovery of superfluid phase transition at 2 mK (Lee, Osheroff, Richardson,Nobel Prize in Physics 1996)

1975: extension of BCS theory to triplet pairing (Leggett, Nobel Prize in Physics 2003)

1977: first experiments on superfluid 3He in Germany at WMI (Eska, Schuberth, Uhlig)

• history: 3He

Chapt. I - 25

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.2 Helium

• 4He: - obtained out of natural gas (e.g. sources in USA, Poland, Algeria containup to 10%)

- L = 0, S = 0, nuclear spin I = 0 boson

• 3He: - natural amount of 3He in 4He is only 0.14 ppm

- artificial generation of 3He by nuclear reaction

- 3He is very expensive, several 1000 € per liter

- L = 0, S = 0, nuclear spin I = ½ fermion

• despite their simple electronic structure, the liquids of both He isotopes show a rich variety of exotic phenomena pronounced differences due to different quantum statistics

• He isotops: 4He and 3He

(12.5 years)

Chapt. I - 26

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.3 Van der Waals Bonding

pA (t)

= e r(t)

+qB

++

++

+

++

RB A

+ +qA

e-

momentary dipole moment pA(t) induced dipole moment

interaction potential:

(a = polarizability)

Chapt. I - 27

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Johannes Diderik Van der Waals (1837 - 1923)Nobel Prize in Physics (1910)

“for his work on the equation of state for gases and liquids”

Van der Waals (right) andHeike Kamerlingh Onnes in Leiden with the helium'liquefactor' (1908)

I.1.3 Van der Waals

Chapt. I - 28

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• interaction potential with short range repulsion due to Pauli principle:

0.8 1.0 1.2 1.4 1.6 1.8-0.02

0.00

0.02

0.04

0.06

Ep

ot (

eV

)

R /

Ar = 0.0104 eV = 3.40 Å

-

R0 Lennard-Jonespotential

1K ↔ 0.1 meV

Van der Waals bondingis very weak !!

I.1.3 Van der Waals Bonding

Chapt. I - 29

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

determination of equilibrium distance of atoms in lattice:minimize total potential energy

summation overall pairs i,j of atoms

equilibrium distance

we rewriteR: nearest neighbor

distance

k: number of nearestneighbors

for fcc-structure

with

I.1.3 Van der Waals Bonding

hcp has lowest energy but fcc is found experimentally fornobel gases effect of zero point fluctuations

Chapt. I - 30

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• estimate of zero-point fluctuation amplitude for harmonic oscillator:

• for 𝑥 = 𝑥max:

Helium stays liquid even at T = 0 due to large zero point fluctuations caused by(i) weak Van der Waals forces (small k)(ii) small mass M

• spring constant k is given by second derivative of the interaction potential

for He: 𝑥max ≈ 0.3 − 0.4 times the lattice constant a

𝑎 ≈𝑉mol

𝑁A

1/3

𝑉mol = mole volume𝑁A = Avogadro number

(Lindeman criterion for melting of solid: xmax ≈ 0.3 ∙ a)

I.1.4 Zero-Point Fluctuations

• by equating1

2𝑘𝑥max

2 with the ground state energy1

2ℏ𝜔 of the harmonic oscillator,

we obtain

Chapt. I - 31

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• we can use uncertainty relation to estimate the kinetic energydue to zero point fluctuations:

large quantum mechanical zero-point energy due to small mass M

kinetic energy due to zero point fluctuations is comparable to potential energy due toVan der Waals interaction

due to strong influence of quantum fluctuations, helium stays liquid down totemperatures, where 𝝀𝐓 ≥ 𝒅 ≃ 𝒂

Helium becomes a quantum liquid

I.1.4 Zero-Point Fluctuations

Chapt. I - 32

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• total potential energy for solid and liquid:

solid: regular lattice

same distance between atoms Epot (R) ≈ potential Utot (R)

liquid: disordered system

varying distance between atoms Epot is smeared out due to averaging

over various distances reduced depth and increased width

of 𝐸pot 𝑅

Epot

R

Epot

⟨𝑅⟩

Why does He not become solid at low T ??

I.1.3 Van der Waals Bonding

Chapt. I - 33

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

𝑬𝒑𝒐𝒕 𝑹 ,

broadened

for liquid helium:contribution of Ezp smaller due tobroadened Epot

• without quantum fluctuations:4He would be solid at a molar volume of 10 cm³

• with quantum fluctuations:4He stays liquid with molar volume of 28 cm³

I.1.4 Van der Waals Bonding– influence of zero-point energy

0 10 20 30 40-150

-100

-50

0

50

100

ener

gy

(ca

l / m

ole

)

V (cm3 / mole)

potentialenergy

zero pointenergy

solid

liquid

12 16 20 24 28 32-20

-15

-10

-5

0

5

E po

t + E

zp (

cal /

mo

le)

V (cm3 / mole)

solid

liquid

𝑬𝒑𝒐𝒕 𝑹

Chapt. I - 34

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

quantum parameter:

liquid Xe Kr Ar Ne H24He 3He

L 0.06 0.10 0.19 0.59 1.73 2.64 3.05

large Ezp ,but also large Epot

identical Epot , but different Ezp

mass of 3He smaller than of 4He even stronger effect of quantum fluctuations larger quantum parameter larger molar volume

3He: 40 cm³/mole, 4He: 28 cm³/mole

I.1.4 Van der Waals Bonding– quantum parameter

quantum liquid:

L > 1

Chapt. I - 35

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

enthalpy: H = Etot + pVH = min ?

use of isenthalpy lines:H = const

with Etot = H – pV

straight lines with slope – p

lowest isenthalpy:

tangent with slope - p

p > 25 bar: solid

p = 25 bar: coexistence

p < 25 bar: liquid

I.1.5 Helium under Pressure

12 16 20 24 28 32-20

-15

-10

-5

0

5

E tot (

cal /

mo

le)

V (cm3 / mole)

solid liquid

0 1 2 3 4 50

10

20

30

40

pre

ssu

re (

bar

)

temperature (K)

solid 4He

superfluid 4He

normal liquid 4He

vapor

melting curve

evaporationp < 25 bar

p = 25 bar

p > 25 bar

bcc

hcp

Tc , pc

Chapt. I - 36

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Phase diagram of 4He at low temperatures. 4He remains liquid at zero temperature if the pressure is below 2.5 MPa(approximately 25 atmospheres) no triple point. The liquid has a phase transition to a superfluid phase, also known as He-II, at the temperature of 2.17 K (at vaporpressure). The solid phase has either hexagonal close packed (hcp) or body centered cubic (bcc) symmetry.

I.1.6 pT phase diagram of 4He

Clausius-Clapeyron:

@ T → 0

DSm → 0

for T → 0

(agrees with 3rd Law of thermodynamics)

0 1 2 3 4 50

10

20

30

40

pre

ssu

re (

bar

)

temperature (K)

solid 4He

bcc

hcp

superfluid 4Heor

4He-II

normal liquid 4He

vapor

melting curve

Tc , pc

TsfTb (1 bar)

pm

evaporation

Chapt. I - 40

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

log T scale !!!

Phase diagram of 3He. Note the logarithmic temperature scale. There are two superfluid phases of 3He, A and B. The linewithin the solid phase indicates a transition between spin-ordered and spin disordered structures (at low and hightemperatures, respectively).

I.1.6 pT phase diagram of 3He

vapor

10-4

10-3

10-2

10-1

100

101

0

10

20

30

40

pre

ssu

re (

bar

)

temperature (K)

solid 3He (bcc)

superfluid 3He(B-phase)

normal liquid 3He

melting curve

Tc , pc

Tsf

Tb (1 bar)

pm

evaporation

superfluid 3He(A-phase)

spin-orderedsolid 3He

29 bar

34 bar

Chapt. I - 43

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

3He 4He

boiling point Tb (K)(@ p = 1 bar)

3.19 4.21

critical temperature Tc (K) 3.32 5.20

critical pressure pc (bar) 1.16 2.29

superfluid transition temperature Ts (K) 0.0025 2.177

density r (g/cm³)(for T 0 K)

0.076 0.145

density r (g/cm³)(@ boiling point)

0.055 0.125

classical molar volume Vm (cm³ /mole)(@ saturated vapor pressure and T = 0 K)

12 12

actual molar volume Vm (cm³ /mole)(@ saturated vapor pressure and T = 0 K)

36.84 27.58

melting pressure pm (bar)(@ T = 0 K)

34.39 25.36

strong difference between superfluid properties of 4He and 3He nuclear spin different quantum statistics

I.1.7 Characteristic Properties of Helium Isotopes

Chapt. I - 44

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

4He p+↑ p+↓ n↑ n↓ e-↑ e-↓ boson3He p+↑ p+↓ n↑ e-↑ e-↓ fermion

sam

e chem

ical

pro

perties

differen

tn

uclea

rsp

in:

4He

: I= 0

3He

: I= ½

differen

t statistics3He: pairing similar to superconductivity in metals

weak attractive interaction between Fermions „Cooper pairs“4He: Bose-Einstein condensation of Bosons without „pairing interaction“

no attractive interaction required

I.1.7 Characteristic Properties of Helium Isotopes

Chapt. I - 45

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.8 Specific Heat of 4He – Lambda Point

• first measurements by Dana and Kamerlingh Onnes in 1923 - anomalous increase of Cp at about 2 K- in publication (1926) some data points were omitted,

since they were believed to result from exp. artefacts• confirmation by Clusius and Keesom in 1932 (introduced denotion He I and He II)

- He I: normal fluid, He II: superfluid

Tl = 2.1768 K

specific heat:

Buckingham, M. J. and Fairbank, W. M., "The Nature of the Lambda Transition", in Progress in Low Temperature Physics III, 1961.

specific heat diverges at Tl

1st order phase transition shape: Lambda point

free energy F: kink(F=U-TS; dF = -SdT-pdV+µdN;dU=TdS-pdV+µdN)

entropy ( -dF/dT): jumpspec. heat ( d2F/dT2):

divergence

Chapt. I - 46

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.8 Specific Heat of 4He – Lambda Point

• for ideal BEC:2nd order phase transition is expected (detailed discussion follows later)

ideal Bose gas

experimental result for 4He

CV

Chapt. I - 47

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.8 Specific Heat of 4He – Precision Measurement

• specific heat is measured with veryhigh temperature resolution (≈ nK)

• extreme purity of liquid He isimportant

model system for testingphysics of phase transitions

• experimental problem: hydrostatic pressure

space experiments

Lipa et al.

second order phase transition with logarithmic singularity

J.A. Lipa, D.R. Swanson, Phys. Rev. Lett. 51, 2291 (1983)

Chapt. I - 48

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1.8 Specific Heat of 3He

Source:Greywall, Phys. Rev. B 33, 7520 (1986) 3He specific heat and thermometry at millikelvin temperatures

CV(T) similar to a metallic superconductor

- T > Tc:linear CV(T) of Fermi liquid

- T = Tc: jump due to superfluidtransition

- T < Tc:reduction of CV due to condensation of Fermions

(Sommerfeld, 1927)

Chapt. I - 49

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.1 Summary of Experimental Observations

• specific heat of 4He is large: ≈ 1 J/g K (compared to about 10-5 J/g K of Cu, 𝑐𝑉 ∝ 𝑛 ⋅ 𝑇/𝑇F)

• latent heat of evaporation of 4He is small (20.59 J/g), but large compared to specific heat

of other materials

application as cooling liquid

• strong anomaly of specific heat at Tl, origin unknown for more than 30 years

originates from Bose-Einstein condensation (bosonic character of 4He)

• shape of anomaly resembles greek letter L Lambda point

• below Tl entropy and specific heat decrease rapidly with decreasing T

condensation in momentum space

• temperature variation of CV below Tl determined by collective excitations in superfluid

(phonons, rotons – discussion later)

Chapt. I - 50

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• for 4He phase transition at Tl Lambda point is due to bosonic character of 4He

no such transition was expected for 3He due to I = ½ (fermionic character)

• only after the development of BCS theory in 1957 (bosonic Cooper pairs of two fermions),

the search for the superfluid phase of 3He has been intensified

• 1963: Theory of superfluid 3He (Anderson and Morel, Balian and Werthamer)

• 1971: after the temperature regime below 5 mK has been accessed, the superfluid phase

of 3He has been found by Osheroff, Richardsen and Lee in NMR experiment

D.D. Osheroff, R.C. Richardsen, D.M. Lee,

Phys. Rev. Lett. 28, 885 (1972)

• there is not only a single superfluid phase,

but three, depending on temperature,

pressure and applied magnetic field

• 1977: first experiment with 3He

in Germany at WMI

(Eska, Schuberth, Uhlig, ....)

I.1 Summary of Experimental Observations

Chapt. I - 51

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Contents:

I.1 Foundations and General PropertiesI.1.1 Quantum FluidsI.1.2 HeliumI.1.3 Van der Waals BondingI.1.4 Zero-Point FluctuationsI.1.5 Helium under PressureI.1.6 pT-Phase Diagram of 4He and 3HeI.1.7 Characteristic Properties of 4He and 3HeI.1.8 Specific Heat of 4He and 3He

I.2 4He as an Ideal Bose GasI.2.1 Bose-Einstein CondensationI.2.2 Ideal Bose GasI.2.3 Bose-Einstein Condensation of 4He

I.3 Superfluid 4HeI.3.1 Experimental ObservationsI.3.2 Two-Fluid ModelI.3.3 Excitation Spectrum of 4He

I.4 VorticesI.4.1 Quantization of CirculationI.4.2 Experimental Study of

Vortices

I.5 3HeI.5.1 normal fluid 3HeI.5.2 solid 3He and Pomeranchuk

effectI.5.3 superfluid 3He

I.6 3He / 4He mixtures

Chapt. I - 52

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• high temperatures:

average distance 𝑅 of atoms in gas is large compared to de Broglie wavelength 𝝀𝐓 = 𝒉/𝒑 classical gas with Maxwell-Boltzmann statistics

with E = p² / 2m = ½ mv² = p kBT we immeadiately obtain thethermal de Broglie wave length:

• low temperatures:

- quantum mechanical description, as soon as 𝑅 = 𝜆T

- classical gas of distiguishable particles turns into quantum soup of indistinguishableidentical particles

quantum gas with Bose or Fermi statistics

Bose-Einstein condensation for Bosons

I.2 4He as Ideal Bose Gas

for an ideal gas theenergy 𝐸 = 𝜋𝑘B𝑇 is

assumed; the factor 𝜋results from the

partition function(see I.1.1)

Chapt. I - 53

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

lT

v

high temperature:thermal velocity 𝒗𝐓

density𝑵

𝑽= 𝒏 ≃

𝟏

𝑹𝟑

billiard balls

low temperature:de Broglie wavelength 𝝀𝐓

𝝀𝐓 = 𝒉/𝒎𝒗𝑻 ∝ 𝟏/ 𝑻

wave packets

T = TBEC:Bose-Einstein condensation

𝝀𝐓 ≈ 𝑹

wave packets overlap

T 0:pure Bose-Einstein condensate

„Super“-mater wave

m

I.2.1 Bose-Einstein Condensation

Chapt. I - 54

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• phase transition occurs for interactionless bosonic particles

• originates from the indistinguishability of identical particles

Albert Einstein und Satyendra Nath Bose

proposed in 1924 that below a characteristic temperature TB a macroscopic number of bosons can condense into the state oflowest energy

Eric A. Cornell Wolfgang Ketterle Carl E. Wieman

Nobel Prize in Physics 2001

"for the achievement of Bose-Einstein

condensation in dilute gases of alkali atoms,

and for early fundamental studies of

the properties of the condensates"

I.2.1 Bose-Einstein Condensation

Chapt. I - 55

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.1 Bose-Einstein Condensation

A. Einstein.

Quantentheorie des einatomigen gases.

SITZUNGSBERICHTE DER PREUSSICHEN AKADEMIE DER WISSENSCHAFTEN

PHYSIKALISCH-MATHEMATISCHE KLASSE 3, 1925.

Chapt. I - 56

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Bosons Fermions

810 nK

510 nK

240 nK

0.5 mm

Pauli exclusion principledoes not allow further

compression in momentum space

at low T

J. R. Anglin und W. Ketterle, Nature 416, 211 (2002)

I.2.1 Bose-Einstein Condensation

Chapt. I - 57

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

degenerate multiparticle system: Bosons and Fermions (cf. I.1.1)

• at given density n:particle occupies cube withd = 1 / n 1/3

• classical liquid: 𝜆T ≤ 𝑑

particles are distinguishable

• quantum liquid: 𝜆T ≥ 𝑑

particles are indistinguishable degenerate quantum liquid

• Bosons: (4He, ….)

Bose-Einstein condensation

• Fermions: (3He, e, p, n, ….)

Fermi sea

0 10000 20000

0 1 2

0

1

2

3

4

T (K)

lT /

d

T (K)

quantum classical

4He: n = 4 ∙ 1021 cm-3,

e- : n = 3.6 ∙ 1021 cm-3

I.2.1 Bose-Einstein Condensation

Chapt. I - 58

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

temperature density (1/cm³)phase space

density

atomic oven 500 K 10-1 10-13

Laser cooling 50 µK 1011 10-6

evaporation cooling 500 nK 1014 2.612

Bose-Einstein condensation 1015 107

I.2.1 Bose-Einstein Condensation

𝜆T/𝑑• cooling down atoms

Chapt. I - 59

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• Laser cooling: techniques in which atomic and molecular samples are cooled down to near absolute zero through the interaction with one or more laser fields

I.2.1.1 Laser Cooling

• Laser cooling techniques

Doppler cooling (Schawlow, Hänsch)T.W. Hänsch, and A.L. Schawlow, Cooling of gases by laser radiation, Optics Communications, 13, 68 (1975)

other techniques• Sisyphus cooling• Resolved sideband cooling• Velocity selective coherent population trapping (VSCPT)• Anti-Stokes inelastic light scattering (typically in the form of fluorescence or

Raman scattering)• Cavity mediated cooling• Sympathetic cooling

• Nobel Price in Physics 1997:

„for the development of methods to cool and trap atoms with laser light”

for Steven Chu, Claude Cohen-Tannoudji und William D. Phillips

Chapt. I - 60

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.1.1 Laser Cooling – Doppler cooling

• in Doppler cooling, the frequency of light is tuned slightly below an electronic transition in the atom

• because the light is detuned to the "red" (i.e., at lower frequency) of the transition, the atoms will absorb more photons if they move towards the light source, due to the Doppler effect

a stationary atom sees the laser neither red- nor blue-shifted and does not absorb the photon

an atom moving away from the laser sees it red-shifted and does not absorb the photon

an atom moving towards the laser sees it blue-shifted and absorbs the photon, slowing the atom

the photon excites the atom, moving an electron to a higher quantum state.

the atom re-emits a photon,as its direction is random, there is no net change in momentum over many absorption-emission cycles

Chapt. I - 61

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.1.1 Laser Cooling – Doppler cooling

• Doppler shift (how does the moving atom see the laser frequency)

atom = observer (O)fixed position

laser = signal source (S) with frequency 𝑓𝑆moving at volocity 𝑣𝑠 towards atom

𝐯𝑠

𝐞𝑆𝑂

𝒇𝑶 =𝒇𝑺

𝟏 −𝐯𝑺 ⋅ 𝐞𝑺𝑶

𝒄

laser frequency seen by the atom isblue shifted, if laser and atom move towards each other

red detuned laser light becomes resonant if atom moves towards the laser

• momentum/velocity change of atom due to photon absorption

𝚫𝒑

𝒑=ℏ𝒌𝒑𝒉

𝒎𝒗=𝚫𝒗

𝒗⇒ 𝚫𝒗 =

ℏ𝒌𝒑𝒉

𝒎

Chapt. I - 62

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.1.1 Laser Cooling – Doppler cooling

• atom-laser detuning with atom at rest:

𝛿0 = 𝜔𝑎𝑡𝑜𝑚 − 𝜔𝑙𝑎𝑠𝑒𝑟

• atom-laser detuning with atom moving at velocity v:

𝛿Doppler = 2𝜋 𝑓𝑂 − 𝑓𝑆 = 2𝜋𝑓𝑂𝐯 ⋅ 𝐞𝑆𝑂𝑐

= 2𝜋𝑓𝑂ℏ𝐤 ⋅ 𝐯

ℏ𝜔𝑂= 𝐤 ⋅ 𝐯

we use:

𝑝 =𝐸

𝑐

𝑐 =𝐸

𝑝=ℏ𝜔

𝑚𝑣=ℏ𝜔

ℏ𝑘

effective detuning between atom and laser

𝛿eff = 𝛿0 − 𝛿Doppler = 𝛿0 − 𝐤 ⋅ 𝐯

• force acting on the atom:

𝐅 = ℏ𝐤 𝛾 = ℏ𝐤Γ

2

𝐼/𝐼0

1 +𝐼𝐼0+

2𝛿effΓ

2

𝛾 = absorption rateΓ = line width of atom transition,

spontaneous emisson rate𝐼0 = saturation intensity

large intensity 𝐼 ≫ 𝐼0:

𝐅 → ℏ𝐤Γ

2

Chapt. I - 63

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.1.1 Laser Cooling – Doppler cooling

• velocity distribution of an atom beam before and after (dashed line) theinteraction with a laser beam of fixedfrequency

• velocity distribution of an atom beam before and after (dashed line) theinteraction with a laser beam ofcontinuously decreasing frequency

Chapt. I - 64

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.1.1 Laser Cooling – Doppler cooling

• achievable minimum temperature in Doppler cooling:

𝒌𝐁𝑻𝐃𝐨𝐩𝐩𝐥𝐞𝐫 =ℏ𝜞

𝟐

- spontaneous absorption and emission are random processes

the average force acting on the atoms is statistically fluctuating Brownian motion in phase space

- minimum temperature is given by the compensation of heating effect due to therandom absorption and emission processes and maximum cooling rate

- e.g. 𝑇Doppler = 240 𝜇K for Na corresponding to average velocity of 1 km/h

𝑇Doppler = 140 𝜇K for Rb

Chapt. I - 65

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• optical molasses (Steven Chu, 1985)

• an optical molasses consists of 3 pairs of counter-propagating circularly polarized laser beams intersecting in the region where the atoms are present

• while for simple Doppler cooling the minimum temperature for Na atoms is about 250 µK, optical molasses can cool the atoms down to 40 μK, i.e. an order of magnitude colder.

I.2.1.1 Laser Cooling – Optical Molasses

Chapt. I - 66

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.1.1 Laser Cooling – Optical Molasses

• force acting by two laser with opposite directions

𝐅 = +ℏ𝐤Γ

2

𝐼/𝐼0

1 +𝐼𝐼0+

2 𝛿0 − 𝐤 ⋅ 𝐯Γ

2 − ℏ𝐤Γ

2

𝐼/𝐼0

1 +𝐼𝐼0+

2 𝛿0 + 𝐤 ⋅ 𝐯Γ

2

total force

Chapt. I - 67

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• eigen energies: with 𝑛2 = 𝑛𝑥2 + 𝑛𝑦

2 + 𝑛𝑧2

consider a gas of N (≈ 1022) bosons of mass m in a cube of volume V = L³ (≈ 1 cm3) canonical ensemble (given 𝑻, 𝑽,𝑵)

µ = chemical potential

(note that always 𝜇 < 휀0 to avoid negative occupation probabilities)

• energy and particlenumber:

• occupation probability: (Bose-Einstein distribution)

• Δ휀/𝑘𝐵𝑇 ≤ 10−14 level spacing small compared to thermal energy(e.g. He @ 10 K)

no influence on experimental result

𝑈 𝑇, 𝑉, 𝜇 =

𝑘

휀𝑘 𝑛𝑘

휀𝑘 =ℏ2𝑘2

2𝑚=

ℏ2

2𝑚

𝜋

𝐿

2

𝑛2

𝑁 𝑇, 𝑉, 𝜇 =

𝑘

𝑛𝑘

𝑛𝑘 =1

exp휀𝑘 − 𝜇𝑘𝐵𝑇

− 1

I.2.2 Ideal Bose Gas

Chapt. I - 68

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

how to get rid of the summation over k ?

𝑘

… ⇒ 𝑍 𝑘 𝑑3𝑘 =𝑉

2𝜋 3 𝑑3𝑘

𝑘

… ⇒ 𝐷 휀 𝑑휀 =𝑉

4𝜋22𝑚

ℏ2

3/2

휀 𝑑휀

𝑍 𝑘 =𝑉

2𝜋 3

𝐷 휀 =𝑉

4𝜋22𝑚

ℏ2

3/2

휀

for free bosons with dispersion

휀𝑘 =ℏ2𝑘2

2𝑚:

𝑈 𝑇, 𝑉, 𝜇 =

𝑘

휀𝑘 𝑛𝑘 ⇒𝑉

4𝜋22𝑚

ℏ2

3/2

휀3/2

exp휀 − 𝜇𝑘B𝑇

− 1𝑑휀

𝑁 𝑇, 𝑉, 𝜇 =

𝑘

𝑛𝑘 ⇒𝑉

4𝜋2

2𝑚

ℏ2

3/2

휀1/2

exp휀 − 𝜇𝑘B𝑇

− 1𝑑휀

Chapt. I - 69

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

• particle number: 𝑁 =𝑉

2𝜋 3 𝑛𝑘 𝑑

3𝑘 =𝑉

2𝜋 3

1

exp휀𝑘 − 𝜇

𝑘B𝑇− 1

𝑑3𝑘

• problem:

the contribution of the lowest energy state 휀𝑘=0 = 휀0 is lost when we replace the summation by an integration:

for 𝜇 → 0, 𝑘 → 0:

on the other hand:

𝑛𝑘 =1

exp휀𝑘 − 𝜇𝑘B𝑇

− 1≃

𝑘𝐵𝑇

휀𝑘 − 𝜇∝

1

𝑘2⇒ 𝑛0 → ∞

𝑁 =𝑉

2𝜋 3 𝑛𝑘 𝑑3𝑘 ∝

1

𝑘24𝜋𝑘2𝑑𝑘 ∝ 𝑑𝑘

since the contribution of 𝑛0 is not contained in the integral when we replace thesummation by an integration, we have to take it into account separately

𝑁 = 𝑛𝑜 +

𝑘>0

𝑛𝑘 = 𝑁𝑜 + 𝑁𝑒𝑥

the lowest states yields a contribution ∝ 𝒅𝒌 in the integration, i.e., only a vanishing amount in an infinitesimal region around 𝒌 = 𝟎

Chapt. I - 70

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

• particle number: 𝑁 =1

exp휀0 − 𝜇𝑘𝐵𝑇

− 1+

𝑘>0

𝑔𝑘 𝑛𝑘

𝑁 = 𝑁0 +

0

∞

𝐷 휀 𝑛 휀 𝑑휀

particles inground state

particles inexcited states

for free bosons with dispersion:

density of states (3D):

• we obtain for free bosons:

𝑁0 =1

exp휀0 − 𝜇𝑘B𝑇

− 1=

1

𝛾−1 exp휀0𝑘B𝑇

− 1with fugacity 𝛾 = exp

𝜇

𝑘B𝑇

𝑁𝑒𝑥 =𝑉

4𝜋22𝑚

ℏ2

3/2

0

∞휀

exp휀 − 𝜇𝑘B𝑇

− 1𝑑휀

degeneracy: 𝑔𝑘we assume 𝑔𝑘 = 1

Chapt. I - 71

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

𝝀𝐓−𝟑 𝑽 ⋅ 𝜻𝟑/𝟐 𝜸

-2 -1 00

1

2

𝝁/𝒌𝐁𝑻

3

𝜻𝟑/𝟐 𝜸

𝜻𝟑/𝟐 𝜸 = 𝟏

= 𝟐. 𝟔𝟏𝟐𝟑𝟕…

thermal de Broglie wavelength

(Riemann zeta function)

I.2.2 Ideal Bose Gas

• at given 𝑛 = 𝑁/𝑉 ≃ 𝑛𝑒𝑥 and 𝑇: above equation determines 𝜇(𝑇, 𝑛)

• for 𝑻 → ∞: 𝜆𝑇 → 0 and therefore 휁3/2 𝛾 → 0, 𝜇 → −∞

• for 𝑻 → 𝟎: 𝜆𝑇 → ∞ and therefore 휁3/2 𝛾 → ∞ impossible !!

Chapt. I - 72

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

• what happens with decreasing 𝑻 ?

− 𝜆𝑇 increases 휁3/2 𝛾 has to increase to keep 𝑛𝑒𝑥 ≃ 𝑛 = 𝑁/𝑉 constant (𝑁0 still negligible)

• problem:

휁3/2 𝛾 has maximum value for 𝛾 = 1, resp. 𝜇 = 0

maximum value of 휁3/2 𝛾 for 𝛾 = 1, resp. 𝜇 = 0,

is related to maximum value 𝜆𝑇,BEC of the de Broglie wavelength

𝜆𝑇,BEC3 ⋅

𝑁

𝑉= 𝜆𝑇,BEC

3 ⋅ 𝑛 = 휁3/2 1 = 2.61237…

• with 𝜆𝑇,BEC = ℎ/ 2𝜋𝑚𝑘B𝑇BEC, we obtain:

𝑛 = 2.612𝑚𝑘B𝑇BEC2𝜋ℏ2

3/2

𝑇BEC =2𝜋ℏ2

𝑚𝑘B

𝑛

2.612

2/3

down to 𝑻 = 𝑻𝐁𝐄𝐂 we still can accommodate all particles in the excited states: 𝑵 ≃ 𝑵𝒆𝒙, 𝑵𝟎 ≃ 𝟎

Chapt. I - 73

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

• 𝑻 = 𝑻𝐁𝐄𝐂:

what happens at the characteristic temperature 𝑻𝐁𝐄𝐂 =𝟐𝝅ℏ𝟐

𝒎𝒌𝐁

𝒏

𝟐.𝟔𝟏𝟐

𝟐/𝟑

at 𝑇 = 𝑇BEC: 𝜆𝑇,BEC3 ⋅ 𝑛 =

𝜆𝑇,BEC3

𝑑3= 2.612 ≃ 1, 𝜇 → 0

de Broglie wavelength 𝝀𝑻,𝐁𝐄𝐂 ≃ average particle distance 𝒅

for 𝜇 ≥ 0, there is no solution of 𝑁 = 𝑁𝑒𝑥 =𝑉

4𝜋22𝑚

ℏ2

3/2

0∞ 𝜀

exp𝜀−𝜇

𝑘B𝑇−1

𝑑휀

restricted temperature range 𝑻 ≥ 𝑻𝐁𝐄𝐂 𝝁 ≤ 𝟎

𝜆𝑇,BEC3 ⋅ 𝑛 ≃ ℴ 1 criterion for the importance of quantum effects or degeneracy

• we have defined TBEC as the temperature, where 𝑁 = 𝑁𝑒𝑥 or 𝑛 = 𝑛𝑒𝑥

1 =2.612

𝜆𝑇,BEC3 ⋅ 𝑛

𝜆𝑇,BEC3 = 2.612 ⋅ 𝑑3 =

ℎ

2𝜋𝑚𝑘B𝑇BEC

3

or

Chapt. I - 74

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

• 𝑻 ≤ 𝑻𝐁𝐄𝐂:

how are the 𝑵 particles distributed among the available states ?

𝑁 = 𝑁𝑒𝑥 =𝑉

4𝜋22𝑚

ℏ2

3/2

0∞ 𝜀

𝑒

𝜀−𝜇𝑘B𝑇−1

𝑑휀 yields 𝜇 → 0− for 𝑇 → 𝑇BEC+

for 𝜇 → 0− (𝑇 → 𝑇BEC+ ), we obtain 𝑁0 =

1

exp𝜀0−𝜇

𝑘𝐵𝑇−1

≃𝑘𝐵𝑇

𝜀0→ ∞

below TBEC: Nex < N or nex < nmacroscopic occupation 𝑵𝟎 ≫ 𝟏 of the ground state

• key result:

at 𝑇 ≤ 𝑇BEC a finite fraction of all particles occupies the ground state

this phenomenon is called Bose-Einstein condensation

the name „condensation“ is used since in analogy to the gaseous-liqud transitionthe condensed particles do no longer contribute to the pressure

휀0 =ℏ2

2𝑚

𝜋

𝐿

2∝

1

𝑉2/3∝

1

𝑁2/3 → 0 for 𝑁 → ∞ (thermodynamic limit)

Chapt. I - 75

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

temperature dependence of 𝑵𝟎 and 𝑵𝒆𝒙

𝑁 = 𝑁0 𝑇 + 𝑁𝑒𝑥 𝑇 = const• we have constant particle number:

𝑻 > 𝑻𝐁𝐄𝐂: 𝑁0 = ℴ(1) can be neglected

𝑁 = 𝑁𝑒𝑥 =𝑉

4𝜋22𝑚

ℏ2

3/2

0

∞휀

exp휀 − 𝜇𝑘B𝑇

− 1𝑑휀 is valid

𝑻 = 𝑻𝐁𝐄𝐂: 𝜇 → 0, 1 =2.612

𝜆𝑇,BEC3 ⋅ 𝑛

𝑁 = 𝑁𝑒𝑥 = 2.612 ⋅𝑉

𝜆𝑇,BEC3

𝑻 ≤ 𝑻𝐁𝐄𝐂: 𝑁 = 𝑁0 + 𝑁𝑒𝑥 = 𝑁0 + 2.612 ⋅𝑉

𝜆𝑇3 = 𝑁0 + 2.612 ⋅

𝑉

𝜆𝑇3 ⋅

𝜆𝑇,BEC3

𝜆𝑇,BEC3

𝜇 = 0,

𝑁 = 𝑁0 + 2.612 ⋅𝑉

𝜆𝑇,BEC3 ⋅

𝑇

𝑇BEC

3/2

= 𝑁0 + 𝑁 ⋅𝑇

𝑇BEC

3/2

𝑵𝟎

𝑵= 𝟏 −

𝑻

𝑻𝐁𝐄𝐂

𝟑/𝟐

Chapt. I - 76

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

T > TBEC T < TBEC T << TBEC

macroscopic occupationof ground state

0

1

n n n

µ < 0

µ = 0 µ = 0

0

1

0

1

• calculation of 𝑁0(𝑇) = 𝑁 휀0, 𝑇 for (휀0−𝜇) → 0:

𝑁0 𝑇, 𝜇 =1

exp휀0 − 𝜇𝑘𝐵𝑇

− 1≃

1

1 +휀0 − 𝜇𝑘𝐵𝑇

− 1=

𝑘𝐵𝑇

휀0 − 𝜇

𝑵𝟎 𝑻, 𝝁 grows dramtically for (𝜺𝟎−𝝁) → 𝟎

I.2.2 Ideal Bose Gas

Chapt. I - 77

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0.0 0.5 1.0 1.5 2.00

1

2

3

nk

k / k

BT

g = 1, µ = 0

g = 0.5, µ = - 0.69 kBT

g = 0.25, µ = - 1.39 kBT

I.2.2 Ideal Bose Gas

𝑁0

𝑁0

Chapt. I - 78

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

𝑻

𝝁

𝟎𝑻𝐁𝐄𝐂

𝑻

𝑵𝟎/𝑵

𝟎𝑻𝐁𝐄𝐂

𝟏

𝜺𝒌

𝒏𝒌

𝟎

𝟎

𝑵𝟎

𝜺𝟎𝑵𝟎

𝑵= 𝟏 −

𝑻

𝑻𝐁𝐄𝐂

𝟑/𝟐

𝒏𝒌 =𝟏

𝐞𝐱𝐩𝜺𝒌 − 𝝁𝒌𝑩𝑻

− 𝟏

Chapt. I - 79

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• occupation of the first excited state for µ → 0 and T 1K:

level spacing for 1022 atoms in a box with L = 1 cm: Δ휀/𝑘𝐵𝑇 = 휀1/𝑘𝐵𝑇 ≤ 10−14

• more detailed calculation: 𝑛1/𝑁0 ∝ 𝑁−1/3, i.e. 𝑛1/𝑁0 → 0 for 𝑁 → ∞

(justifies why only the ground state was singled out in the calculation of TBEC)

𝑛1 𝑇 =1

1 + exp𝜖1 − 𝜇𝑘𝐵𝑇

− 1≃

1

1 +휀1 − 𝜇𝑘𝐵𝑇

− 1≃

𝑘𝐵𝑇

휀1 − 𝜇≈ 1014

𝑛1 and all 𝑛𝑘>1 are small compared to the total particle number N = 1022

𝑛1𝑁0

=𝑘𝐵𝑇

휀1 − 𝜇⋅휀0 − 𝜇

𝑘𝐵𝑇= −

𝜇

휀1 − 𝜇≃ −

𝜇

휀1≃𝑘𝐵𝑇

𝑁0휀1𝑁0 𝑇, 𝜇 ≃

𝑘𝐵𝑇

휀0 − 𝜇=𝑘𝐵𝑇

−𝜇we use

for 𝑇 < 𝑇BEC/2 we can use the crude approximations 𝑁0 ≃ 𝑁, 𝑇 ≃ 𝑇BEC ∝ 𝑁2/3:

𝑛1𝑁0

∝1

𝑁1/3 already at 𝑇 ≃ 𝑇𝐵𝐸𝐶/2, we have 𝑁0 ≫ 𝑛1, 𝑛𝑘>1

I.2.2 Ideal Bose Gas

Chapt. I - 81

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0 2 4 6 8 100

1

2

3

4

5

D(

) n

()

/ D

• we defined the Bose-Einstein temperature TBEC as the temperature, where nex /n = 1

𝜺𝟎 = 𝟎𝝁 = −𝟎. 𝟏 𝚫𝜺

𝚫𝜺/𝒌𝑩𝑻 = 𝟎. 𝟏

𝚫𝜺/𝒌𝑩𝑻 = 𝟎. 𝟐

𝚫𝜺/𝒌𝑩𝑻 = 𝟏

𝜺𝟎 = 𝟎𝚫𝜺/𝒌𝑩𝑻 = 𝟎. 𝟏

0 1 2 3 40

1

2

3

4

D(

) n

()

/ D

𝝁 = −𝟎. 𝟏𝚫𝜺

𝝁 = −𝟎. 𝟎𝟏𝚫𝜺

𝝁 = −𝚫𝜺

decrease T at µ = const. decrease µ at T = const.

I.2.2 Ideal Bose Gas

Chapt. I - 82

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• Bose-Einstein condensation can be viewed as condensation of gasin momentum space

all atoms have the same momentum

0

0

T = 0

only ground stateis occupied

0 < T = TBEC

still macroscopicoccupation of the

ground state

• Bose-Einstein condensation corresponds to order-disorder phase transition

I.2.2 Ideal Bose Gas

Chapt. I - 83

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

T / TBEC

• excitation density (T < TBEC)

• condensate density nex(T)/n

n0(T)/n

I.2.2 Ideal Bose Gas

Chapt. I - 84

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

T / TBEC

𝝆𝒏 𝑻 /𝝆

for non-interacting Bosons:

(not valid in general)

• normal fluid density 𝜌𝑛 𝑇

I.2.2 Ideal Bose Gas

Chapt. I - 85

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

for free bosons with dispersion:

density of states (3D):

• we obtain:

calculation of total energy

with

I.2.2 Ideal Bose Gas

Chapt. I - 86

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

with

Gamma function

Riemann zeta

function

I.2.2 Ideal Bose Gas

Chapt. I - 87

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

• heat capacity: 𝐶𝑉 𝑇 =𝜕𝑈

𝜕𝑇𝑉,𝑁

𝑻 ≤ 𝑻𝐁𝐄𝐂: 𝜇 = 0, 𝛾 = 1

𝐶𝑉 𝑇 =15

4𝑁𝑘B

휁5/2 1

휁3/2 1

𝑇

𝑇𝐵𝐸𝐶

3/2

= 1.925 ⋅ 𝑁𝑘B𝑇

𝑇𝐵𝐸𝐶

3/2

𝑁𝑇

𝑇𝐵𝐸𝐶

3/2non-condensed particles are contribution ℴ 𝑘𝐵 to the specific heat

𝑻 ≥ 𝑻𝐁𝐄𝐂: 𝜇 < 0, 𝛾 < 1 are functions of 𝑇 and 𝑛 = 𝑁/𝑉

𝐶𝑉 𝑇 =15

4𝑁𝑘B

휁5/2 𝛾

휁3/2 1

𝑇

𝑇𝐵𝐸𝐶

3/2

+3

2𝑁𝑘B

𝑇

휁3/2 1

𝑇

𝑇𝐵𝐸𝐶

3/2

휁5/2′ 𝛾

𝜕𝛾

𝜕𝑇𝑛

Chapt. I - 88

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

𝑻 ≥ 𝑻𝐁𝐄𝐂: 𝜇 < 0, 𝛾 < 1 are functions of 𝑇 and 𝑛 = 𝑁/𝑉

the chemical potential 𝜇 (fugacity 𝛾) is obtained from

𝜆𝑇3 ⋅

𝑁

𝑉= 𝜆𝑇

3 ⋅ 𝑛 = 휁3/2 𝛾

with 𝜆𝑇3 ∝ 𝑇−3/2 temperature derivative yields

1 =휁3/2 𝛾

𝜆𝑇3 ⋅ 𝑛

=휁3/2 𝛾

휁3/2 1

𝑇

𝑇BEC

3/2

0 =3

2𝑇

휁3/2 𝛾

휁3/2 1

𝑇

𝑇BEC

3/2

+휁3/2′ 𝛾

휁3/2 1

𝑇

𝑇BEC

3/2𝜕𝛾

𝜕𝑇𝑛

=3

2𝑇휁3/2 𝛾 + 휁3/2

′ 𝛾𝜕𝛾

𝜕𝑇𝑛

𝜕𝛾

𝜕𝑇𝑛

= −3

2𝑇

휁3/2 𝛾

휁3/2′ 𝛾

𝐶𝑉 𝑇 =15

4𝑁𝑘B

휁5/2 𝛾

휁3/2 1

𝑇

𝑇BEC

3/2

+3

2𝑁𝑘B

𝑇

휁3/2 1

𝑇

𝑇BEC

3/2

휁5/2′ 𝛾

𝜕𝛾

𝜕𝑇𝑛

insert into

𝐶𝑉 𝑇 = 𝑁𝑘B휁5/2 𝛾

휁3/2 1

𝑇

𝑇BEC

3/215

4−9

4

휁5/2′ 𝛾

휁3/2′ 𝛾

휁3/2 𝛾

휁5/2 𝛾

𝐶𝑉 𝑇 = 𝑁𝑘B휁5/2 𝛾

휁3/2 1

𝑇

𝑇BEC

3/215

4−9

4

휁3/2 𝛾

휁1/2 𝛾

휁3/2 𝛾

휁5/2 1=15

4

휁5/2 𝛾

휁3/2 𝛾−9

4

휁3/2 𝛾

휁1/2 𝛾

𝛾휁𝑙′ 𝛾 = 휁𝑙−1

Chapt. I - 89

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• heat capacity (T < TBEC)

heat capacity is continuous at 𝑻𝑩𝑬𝑪 2nd order phase transition

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2.5

CV /

nk B

T / TC

I.2.2 Ideal Bose Gas

• heat capacity (T > TBEC)

𝑐𝑉 𝑇 =𝐶𝑉 𝑇

𝑉= 1.925 ⋅ 𝑛 𝑘B

𝑇

𝑇𝐵𝐸𝐶

3/2

1.925

𝑐𝑉 𝑇 = 𝑛 𝑘B15

4

휁5/2 𝛾

휁3/2 𝛾−9

4

휁3/2 𝛾

휁1/2 𝛾

• heat capacity: 𝐶𝑉 𝑇 =𝜕𝑈

𝜕𝑇𝑉,𝑁

Chapt. I - 90

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.2 Ideal Bose Gas

• equation of state:

𝑝 𝑇, 𝑉, 𝑁 = −𝜕𝐸𝑡𝑜𝑡𝜕𝑉

𝑁

=2

3

𝐸𝑡𝑜𝑡 𝑇, 𝑉, 𝑁

𝑉

𝑻 ≤ 𝑻𝐁𝐄𝐂: 𝜇 = 0, 𝛾 = 1

𝑈 𝑇, 𝑉, 𝑁 =3

2𝑛𝑉𝑘𝐵𝑇

휁5/2 1

휁3/2 1

𝑇

𝑇𝐵𝐸𝐶

3/2

=3

2𝑁𝑘𝐵𝑇

휁5/2 1

𝜆𝑇3

𝑝 𝑇, 𝑉, 𝑁 = 𝑘𝐵𝑇휁5/2 1

𝜆𝑇3 = 1.341

𝑘𝐵𝑇

𝜆𝑇3

pressure does not depend on volume: 𝒅𝒑

𝒅𝑽= 𝟎 !!

system does not have to provide work on reducing volume:only more particles are shifted into the ground state

more detailed treatment has to take into account zero point fluctuation(smaller volume results in stronger zero point fluctuations)

Chapt. I - 91

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• comparison of TBEC and TF (Fermi temperature):

both temperatures are similar for similar m and n = N/V

quantum behavior in Bose and Fermi systems sets in below the degeneracytemperature T* defined by n · l³T ≈ O(1) bosons: T* ≈ TBEC, fermions: T* ≈ TF

physical properties of Fermi and Bose liquids below TBEC and TF are completelydifferent due to Pauli exclusion principle

I.2.2 Ideal Bose Gas

Chapt. I - 92

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

comparison of systems with fixed and varying particle number

• phonons: N ≠ const, µ = 0

𝑛𝑘 → 0 for 𝑇 → 0 𝑁 = ∑𝑛𝑘 → 0

phonons „freeze out“

T

N

µ

N

T = const• He gas: N = const, µ = µ(N,T) ≠ 0

𝝁 < 𝟎 is determined by N and T:

0

0

I.2.2 Ideal Bose Gas

Chapt. I - 93

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• temperature dependence of µ (qualitative discussion)

T

Nges

𝜇 = 𝜇 𝑇

µ = const

TBEC

N

• T > TBEC

• T < TBEC

µ(T) < µ(TBEC)=0

µ(T) > µ(TBEC)=0, but always µ(T) < 0

I.2.2 Ideal Bose Gas

µ(T)

T

TBEC

0

µ(TBEC)

𝜇 𝑇

Chapt. I - 94

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

variation of T at fixed µ variation of µ at fixed T

𝐷 휀 𝑛 휀 𝑑휀 decreases with T 𝐷 휀 𝑛 휀 𝑑휀 increases with 𝜇 → 휀0

0 2 4 6 8 100

1

2

3

4

5

D(

) n

()

/ D

𝜺𝟎 = 𝟎𝝁 = −𝟎. 𝟏 𝚫𝜺

𝚫𝜺/𝒌𝑩𝑻 = 𝟎. 𝟏

𝚫𝜺/𝒌𝑩𝑻 = 𝟎. 𝟐

𝚫𝜺/𝒌𝑩𝑻 = 𝟏

𝜺𝟎 = 𝟎𝚫𝜺/𝒌𝑩𝑻 = 𝟎. 𝟏

0 1 2 3 40

1

2

3

4

D(

) n

()

/ D

𝝁 = −𝟎. 𝟏𝚫𝜺

𝝁 = −𝟎. 𝟎𝟏𝚫𝜺

𝝁 = −𝚫𝜺

I.2.2 Ideal Bose Gas

Chapt. I - 95

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• does

describe the condensation temperature of 4He ???

TBEC ≈ 0.5 K by using data of gaseous 4He (is below condensation temperature) TBEC ≈ 3.1 K by using data of liquid 4He, agrees only qualitatively with measured Tl = 2.17 K

I.2.3 Bose-Einstein Condensation of 4He

0 1 2 3 4 5 6 70.0

0.2

0.4

0.6

0.8

1.0

1.2

g(r

)

r (Å)

radialdistributionfunction

withoutinteraction

4He is not an ideal Bose gas: interaction of He atoms hard core repulsion admixture of higher k-

states (is equivalent to larger k-uncertainty)

reduction of particlenumber in ground state

Chapt. I - 96

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.2.3 Bose-Einstein Condensation of 4He

• superfluid fraction of 4He is only about 14% for T 0

neutron scattering

Chapt. I - 98

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Contents:

I.1 Foundations and General PropertiesI.1.1 Quantum FluidsI.1.2 HeliumI.1.3 Van der Waals BondingI.1.4 Zero-Point FluctuationsI.1.5 Helium under PressureI.1.6 pT-Phase Diagram of 4He and 3HeI.1.7 Characteristic Properties of 4He and 3HeI.1.8 Specific Heat of 4He and 3He

I.2 4He as an Ideal Bose GasI.2.1 Bose-Einstein CondensationI.2.2 Ideal Bose GasI.2.3 Bose-Einstein Condensation of 4He

I.3 Superfluid 4HeI.3.1 Experimental ObservationsI.3.2 Two-Fluid ModelI.3.3 Excitation Spectrum of 4He

I.4 VorticesI.4.1 Quantization of CirculationI.4.2 Experimental Study of

Vortices

I.5 3HeI.5.1 normal fluid 3HeI.5.2 solid 3He and Pomeranchuk

effectI.5.3 superfluid 3He

I.6 3He / 4He mixtures

Chapt. I - 99

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• LHe can flow through tiny capillaries without any friction superfluidity

• superfluidity discovered in 1938 by two groups:(i) P. Kapitza, Nature 141, 74 (1938)

(ii) J.F. Allen, A.D. Misener, Nature 141, 75 (1938)

• theory of superfluidity:

(i) F. London (1938): superfluidity caused by ordered state in momentum spaceF. London, Nature 141, 643 (1938); Phys. Rev. 54, 947 (1938).

(ii) L. Tisza (1938): two-fluid model: prediction of second soundL. Tisza, Nature 141, 913 (1938).

(iii) L.D. Landau (1941-47): two-fluid hydrodynamics of superfluid 4He superfluidity as a consequence of the excitation spectrumL.D. Landau, J. Phys. USSR 5, 71 (1941); ibid. 8, 1 (1944); ibid. 11, 91 (1947).

(iv) R.P. Feynman (1953): excitation spectrum postulated by Landau follows fromquantum mechanicsR.P. Feynman, Phys. Rev. 90, 1116 (1953).

I.3 Superfluid 4He

Chapt. I - 100

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• T > Tl: - liquid He is boiling- latent heat of evaporation is removed from bath- temperature is decreasing

• T ≈ Tl: - temperature is decreasing only slowly large specific heat- temperature does not stay constant no latent heat, 2. order transition

- boiling suddenly stops !!!

superfluid helium:

- very high thermal conductivity- low T not only at surface, but

also within liquid- no vapor bubbles in liquid

evaporation only fromsurface, no boiling

boiling of „normal“ liquid:

vapor bubbles in liquid

lower T at surface: vapor pressure decreases

higher T within liquid: vapor pressure > hydrostatic pressure generation of vapor

bubbles

evaporation

I.3.1 Experimental Observations:Boiling of Liquid Helium

Chapt. I - 101

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

two different measuring methods for viscosity h of superfluid 4He

flow through narrow capillary

h < 10-11 poise

• flow without measurable friction• viscosity h ≈ 0 superfluid component flows out

torsional pendulum

h ≈ 10-5 poise

• damping of oscillation: h > 0• scattering of phonons and rotons normal component picks up

momentum

capillary

(1 Poise = 1 g/cm s = 0.1 Pa ∙ s)

I.3.1 Experimental Observations:Viscosity

(cf. water: 8.90 × 10−5 poise @ 25°C)

Chapt. I - 102

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• flow through narrow capillary

• flow velocity about independent of pressure• flow velocity increases with decreasing

diameter a of capillary

experimental result depends on measuring method !! explanation by two-fluid model

I.3.1 Experimental Observations:Viscosity

0.8 1.2 1.6 2.010

1

102

visc

osity

hn (

10-7

Pa

· s)

temperature (K)

0 4 8 12 16

4

8

12

mea

n ve

loci

ty (

cm/s

)

pressure head (N / a2)

1.2 x 10-7 m

7.9 x 10-7 m

3.9 x 10-6 m

Tl

• torsional pendulum

Chapt. I - 103

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• helium film covers container via absorption of atomsfrom the vapor phase (van der Waals forces)

• superfluidity of He: „mobile“ film

containerfills up

containeris draining

out

containerdripsempty

overshoot during fill up and draining out oscillation of filling height

I.3.1 Experimental Observations:Superfluid Film Flow

Chapt. I - 104

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• estimate of superfluid film thickness:

in equilibrium chemical potentials are identical: µfilm = µvap = µ0 (for film in saturated vapor)

bulk liquid gravitation van der Waals potential

(n ≈ 3 for d < 5 nm, a = Hamaker constant, a isdetermined by dielectric properties of wall and He atoms)

µfilm = µ0 + mgh – a /dn

µvap = µ0 + mgh

h

µ = µ0

bulk liquid

film

vapor

d

equilibrium condition: µfilm = µ0

typical values:

d (nm) ≈ 30 / [h (cm)]1/3

h = 5 cm d = 20 nm

normal fluid blocked in thin film

superleak

• for vapor: µvap = µ0 + mgh

• for film: µfilm = µ0 + mgh – a /dn

I.3.1 Experimental Observations:Superfluid Film Flow

= 0

Chapt. I - 105

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• estimate of the flow of liquid for a critical superfluid velocity ofvs = 30 cm/s:

• superfluid flow acts as heat leak:

helium pump

Hebath1.2 K

vacu

um

vacu

um

• film creeps up along the container walls and pumping line• film creeps up to the position, where T = 2.18 K• evaporation from film dominates, since vapor pressure

at height h is much lower• estimate of film thickness for „unsaturated film“ d depends on gas pressure p (kinetic gas theory)

Tmin ≈ 1 K

I.3.1 Experimental Observations:Superfluid Film Flow

• min. pressure achieved by vacuum pump with

Chapt. I - 106

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

T-DT

A B

Dp

capillary or tubefilled with powder

• starting point: same filling level and temperature T of container A and B

• apply Dp liquid flows through capillary from A to B

• temperature decreases in B and increases in A

• remove Dp original state is recovered

• „temperature flow“ is opposite to mass flow

T+DT

only superfluid flows from A to B through narrow capillaryincrease/decrease of normal fluid concentration in A/B

similar experiment:

• only superfluid flows through powder

• increase of normal fluid concentration

• increase of T in container

thermometer

powder

vacuum isolation

I.3.1 Experimental Observations:Thermomechanical Effects

Chapt. I - 107

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• inverse thermomechanical effect: generate Dp by applying DT

capillary

heater

T < Tl

• heating generates increase of normal fluid concentration

• superfluid components flows through capillary to establish balance

• analoguous to osmotic pressure

• superfluid flows very rapidly through capillary so that a fountain pressure can be generated

fountain effect

I.3.1 Experimental Observations:Thermomechanical Effects

Chapt. I - 108

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• light is absorbed by emery powder heating

• lower opening is closed by cotton wool only superfluid can enter

• after switching on the light, a He fountain is generated by the thermomechanical effect

I.3.1 Experimental Observations:Fountain Effect

Chapt. I - 109

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• thermal conductivity of superfluid 4He is by more than five orders of magnitude larger than that of normal liquid 4He

• explains disappearance of boiling below 𝑇𝜆

• heat conduction through narrow capillaries has maximum at ≈ 1.8 K

• heat current density varies as 𝑱𝑸 ≃ −𝝀 𝛁𝑻𝟏/𝟑 (not as 𝑱𝑸 ≃ −𝝀 𝛁𝑻)

no U processes possible in superfluid 4He (no periodic lattice)

excitations (phonons, rotons) move ballistically, only surface scattering

I.3.1 Experimental Observations:Heat Conduction

Chapt. I - 110

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

-thermal conductivity k is proportional to

- mean velocity of the particles- mean free path- density of the particles- specific heat of the particles

compare to solid at low T:

ballistic phonons N processes do not cause

momentum relaxation only surface scattering

kinetic gas theory:

I.3.1 Experimental Observations:Heat Conduction

function of 𝑑(e.g. Hagen-Poiseuille)

𝑛 ⋅ 𝑐𝑉⋆

Chapt. I - 111

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• anomalous properties of superfluid 4He can be well described by two-fluid model(l. Tisza, J. de Phys. et Radium 1, 164 (1938))

• formal description of superfluid 4He as the sum of a normal and a superfluid component

• T = 0:

• T = Tl:

• superfluid component: no entropy: Ss = 0, zero viscosity: hs = 0

• normal component: carries total entropy: Sn = S, finite viscosity: hn = h

I.3.2 Two-Fluid Model

• microscopic description of superfluids is very difficult phenomenological treatment using hydrodynamics

Chapt. I - 112

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• determination of 𝝆𝒏 (E.L. Andronikashvili, Zh. Eksperim. i. Teor. Fiz. 18, 424 (1948))

• torsional pendulum experiment: 50 Al disks, thickness: 13 µm, diameter: 3.5 cm,disk separation: 210 µm

• only normal component is moved during rotation

• skin depth of viscous wave: 𝟐𝜼𝒏/𝝆𝒏𝝎

large compared to disk separation below Tl

complete normal component is moved

• moved mass of normal component results in change of moment of inertia change of oscillation frequency: w ~ 1/mass

empirical relation:

I.3.2 Two-Fluid Model:Experiment of Andronikashvili

0.0 0.4 0.8 1.2 1.6 2.0 2.40.0

0.2

0.4

0.6

0.8

1.0

rs /

r , r

n /

r

temperature (K)

cf. normal fluid density of ideal BEC:

Chapt. I - 113

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

for details of hydrodynamic description see appendix 1

I.3.2 Two-Fluid Model: Hydrodynamics

• starting point for hydrodynamic description are Navier-Stokes equations:

- normal fluid:

inertia pressure gradient

temperaturegradient

viscosityadditional termdue to compressibility

- with the definition of the substantive derivative Dv/Dt:

and

the tensor derivative of the velocity vectorcurl of the velocity (called vorticity)

disappears for irrotational flow(no vortices)

entropy per mass

Chapt. I - 114

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

- superfluid fluid: no viscous friction

I.3.2 Two-Fluid Model: Hydrodynamics

inertia pressure gradient

temperaturegradient

viscosity:hs = 0

additional termdue to compressibility

Euler type equation for superfluid (if there are no vortices)

detailed treatment including vortices: Gross-Pitaevskii equation

Chapt. I - 115

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• first sound:normal and superfluid component move in-phase:

• second sound:normal and superfluid component move out-of-phase:(Peshkov 1944)

m/s 1373

1

2

vv

v1 = 238 m/s for T → 0

for T → 0

• third sound: waves propagating in thin He films

• fourth sound: compression wave propagating in a super leak

(super leak: very small opening or capillary, in which normal fluidcannot move because of its finite viscosity)

I.3.2 Two-Fluid Model: Hydrodynamics

Chapt. I - 124

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

SC closeto Tc

measurement of R(T)

diffusion

resistiveheater

heatpulse

substrate

I.3.2 Two-Fluid Model: Second Sound - Experiment

1st sound

2nd sound

Chapt. I - 127

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

strong

interactionrotons

phonons

T → Tl:

collision probability increases picture of elementary excitations no longer valid vortex rings

pulse experiment

D

Tkrotonn

B

rot

,- exp r

4

, Tphononn r

I.3.2 Two-Fluid Model: Second Sound

Chapt. I - 129

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

T-

DT

A B

Dp

capillary or tubefilled with powder

T+

DT

I.3.2 Two-Fluid Model – explanation of thermomechanical effect

since = 0 for superfluid and

Dp causes DT and vice versa

Dp → 0 for rn → 0

• work required for mass current from A to B:

• energy gained by heat flow from B to A:

• energy balance yields:

same result follows from (see appendix)

with dvs /dt = 0 in equilibrium

Chapt. I - 131

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• heater generates normal fluid: phonons and rotons• radiation to the left• recoil to the right• measurement of displacement of the glas plate via Newton‘s rings between

glas plate and lens

heater

glas plate Newton‘s rings

lens (fixed position)

recoil

wire support

4He

I.3.2 Two-Fluid Model – momentum transfer due to heat flow

• momentum flow per volume:given by r v ∙ v

• resulting pressure on heat source:

• with heat flow per area:

• we obtain with(no mass transport) pressure

Chapt. I - 132

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.3.3 Excitation Spectrum of Superfluid 4He

Question: Does the formation of a condensate automatically result in superfluidity ?

Answer: Structure of the excitation spectrum is crucial for observabilityof superfluidity (Lev Landau, 1947)

• what happens, if an object of mass m moves at velocity vi through the superfluid ? at T = 0, deceleration only due to generation of excitations at which velocity does this motion cause an excitation of energy Ep and

momentum p ?

• with velocity difference vi - vf of object and superfluid we obtain (energy and momentum conservation):

• elimination of vf yields

Landau critical velocity: vL = 0 for free bosons !!

(mass of object can be large)

smallest vi for momentum p of excitation parallel to vi

Chapt. I - 133

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

) • deceleration of moving particle by generation of excitations with energy Ep

• we therefore obtain

• Landau critical velocity:

no deceleration of particle, if vi < vL

Ep

= ħ

w

p = ħk

dispersion offree particle

vL = 0 for free particles !!

I.3.3 Excitation Spectrum of Superfluid 4He

• motivation for Landau critical velocity

• how to obtain blue dispersion curve ??

Chapt. I - 134

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

dra

g(1

0-1

4N

)

<v> (m/s)

0.35 K

4.0 K

vL ≈ 45 m/s

measurement of dragon ions in He

no generation of excitationsfor v < vL

no friction

I.3.3 Excitation Spectrum of Superfluid 4He – measurement of critical velocity

• experiment:

There is a finite Landau critical velocity !

Chapt. I - 135

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Lev Landau (1947) and Richard Feynman (1953): discussion of situation for superfluid 4He (stronger interaction) postulation of the following excitation spectrum (phonons and rotons):

(i) small p:

(ii) higher p:

• free bosons: vL = 0, since Ep = p²/2m and arbitrarily small momentum transfer is possible

phonon-likecollective excitations

rotons(finite energy Drot forexcitation)

I.3.3 Excitation Spectrum of Superfluid 4He

Nikolai Bogoliubov (1947): interaction effects drastically change the nature of elementary excitations

collective excitations (many particles move at the same time)

even for weak repulsive interaction: Ep = c |p| (phonon-like)

expression „roton“ used by Feynman due to analogy with „smoke ring“,since it is connected with a forward motion of a particle accompanied

by a ring of back-flowing particles however: detailed nature of rotons still not clarified

Chapt. I - 136

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.3.3 Excitation Spectrum of Superfluid 4He

Richard P. Feynman:(from: „The beat of a different drum“Jagdish Mehra, 1994)

„I cannot remember, how it happened.I was walking along the street... andzing! I understood it!“ ...

... „But of course, that is just what liquidsmust do. If you measure X-ray diffraction,then because of the spatial structure ofthe liquid, which is almost like a solid, there will be a maximum correspondingto the first diffraction ring of the X-raypattern! That was a terriffic moment!“

Chapt. I - 137

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.3.3 Excitation Spectrum of Superfluid 4He

• interaction effects in Bose systems finite stiffness solid like excitations

weak interactions:

phonons[Bogoliubov, 1947]

strong interactions:

phonons and rotons[Landau, 1947, Feynman & Cohen, 1957]

Chapt. I - 138

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.3.3 Excitation Spectrum of Superfluid 4He

• physical interpretation of the photon and roton part of the excitation spectrum

phonon roton

• de Broglie wavelength 𝜆 = ℎ/𝑝 >> atomic distance

• coupled motion of group of atoms(like for acoustic phonon in solid)

• de Broglie wavelength 𝜆 = ℎ/𝑝 ≃ interatomic distance

• central atom moves forward while closely packed neighbors move out of the way in circular motion (like smoke ring)

Chapt. I - 139

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.3.3 Excitation Spectrum of Superfluid 4He

• experimental study:

inelastic neutron scattering(coupling to density)

• experimental parameter:

Drot / kB = 8.67 K

p0 / ħ = 1.94 Å-1

mrot = 0.15 mHe

Drot

p0

phonons

rotonsE p/

k B(K

)

p / ħ (Å-1)

Chapt. I - 141

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Quasiparticle breakdown in a quantum spin liquidMatthew B. Stone, Igor A. Zaliznyak, Tao Hong, Collin L. Broholm and Daniel H. ReichNature 440, 187-190 (9 March 2006)

Liquid helium excitation spectrum S(Q, w) from inelastic neutron scattering measurements. Main panel, excitation spectrum in 4He for 1.5 < T < 1.8 K. Data for wavevector Q ≥ 2.3 Å-1 are reproduced from ref. 13,

data at smaller Q are from C.L.B. and S.-H. Lee, unpublished results. Solid black line, dispersion from ref. 13; red circle

with cross, spectrum termination point at Q = Qc and w 2D. White line, Feynman–Cohen bare dispersion in absence of

decays; horizontal red line at w = 2D, onset of two-roton states for w ≥ 2D. Inset, excitations near termination point, at Q

= 2.6 Å-1 ≈ Qc, for several temperatures.

I.3.3 Excitation Spectrum of Superfluid 4He – roton minimum in dispersion relation of liquid

Chapt. I - 142

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

a = 3.6 Å 2p / a ≈ 1.8 Å-1

(reasonable agreement)

periodic crystal:

k and k+G are equivalent

liquid:only short distance ordering a becomes smeared out also G = 2p/a is smeared out

I.3.3 Excitation Spectrum of Superfluid 4He – comparison to linear chain of atoms

• excitation spectrum:qualitative explanation

0.0 0.8 1.6 2.4 3.20

4

8

12

16

E p /

kB (

K)

p / ћ (Å-1

)

linearchain

p / a

G

experimental data

Chapt. I - 143

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

averagesolid

• smearing of G = 2p / a

modelling by average solid +continuum

• other liquids: rotons strongly damped

• superfluid He: no damping, since there are no

single particle excitations

• critical velocities:

vL = 238 m/s for phonons

vL ≈ 60 m/s for rotons

measured vL usually smaller

excitation of vortices

I.3.3 Excitation Spectrum of Superfluid 4He – roton minimum in dispersion relation of liquid

aver. solid

Chapt. I - 144

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• average distance abecomes smaller

• He becomes harder phonon frequency increases

• ordering increases roton frequency decreases

I.3.3 Excitation Spectrum of Superfluid 4He – pressure dependence of dispersion

Chapt. I - 145

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• cusp-like specific heat

• no jump of specific heat atphase transition as expectedfor 2nd order phase transition

J.A. Lipa, D.R. Swanson, Phys. Rev. Lett. 51, 2291 (1983)

log

l

l

T

TTC

V

experiment

I.3.3 Excitation Spectrum of Superfluid 4He – specific heat of an ideal Bose gas

ideal Bose gas

experimental result for 4He

CV

/ n

k B

deviations from ideal Bose gas expectationdue to finite interactions collective excitations: phonons, rotons

CV(T) determined by freeze out of phonons androtons at T < Tl

Chapt. I - 146

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

) (i) T < 0.6 K:in this temperature regime the dominant excitations arelong wave length phonons

(ii) 0.6 K < T < 1.2 K:in this temperature regime the dominant excitations are rotons,number NR of rotons increases with T thermal activation

(iii) 1.2 K < T < Tl:- lifetime broadening of roton states- additional excitations- more complicated behavior

(corresponds to Debye model

in solid state physics)

I.3.3 Excitation Spectrum of Superfluid 4He – specific heat below Tl

• below Tl: three different temperature regimes

Chapt. I - 147

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

D.S. Greywall, Phys. Rev. B18, 2127 (1978);Phys. Rev. B21, 1329 (1979)

T³

I.3.3 Excitation Spectrum of Superfluid 4He – specific heat below Tl

Chapt. I - 151

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Contents:

I.1 Foundations and General PropertiesI.1.1 Quantum FluidsI.1.2 HeliumI.1.3 Van der Waals BondingI.1.4 Zero-Point FluctuationsI.1.5 Helium under PressureI.1.6 pT-Phase Diagram of 4He and 3HeI.1.7 Characteristic Properties of 4He and 3HeI.1.8 Specific Heat of 4He and 3He

I.2 4He as an Ideal Bose GasI.2.1 Bose-Einstein CondensationI.2.2 Ideal Bose GasI.2.3 Bose-Einstein Condensation of 4He

I.3 Superfluid 4HeI.3.1 Experimental ObservationsI.3.2 Two-Fluid ModelI.3.3 Excitation Spectrum of 4He

I.4 VorticesI.4.1 Quantization of CirculationI.4.2 Experimental Study of

Vortices

I.5 3HeI.5.1 normal fluid 3HeI.5.2 solid 3He and Pomeranchuk

effectI.5.3 superfluid 3He

I.6 3He / 4He mixtures

Chapt. I - 152

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.4 Vortices

Chapt. I - 153

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• in the discussion of the two-fluid model (see Appendix) we assumed the absence of turbulence in 𝜌𝑠 𝛻 × 𝑣𝑠 = 0

• 1941: Landau proposed to check this assumption in experiments withsuperfluid He in rotating containers

• 1949: Onsager predicted the appearance of vortices in rotating superfluid He

• 1953: Feynman noted that circulation in superfluid He should be quantized

• 1961: first experimental proof of quantization by Vinen

quantized vortices in superfluid He as in superconductors

difference: He is not charged, neutral circulation

I.4 Vortices

Chapt. I - 154

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• ideal vortex in z-direction at r = 0:

r

v

infinite range

vortex core

x

y

zA

I.4.1 Quantization of Circulation– Vortices in Hydrodynamics

superfluid He: G is quantized

• if A is a circle, then

and

Stokes Thomson

• circulation G:

𝛻 × 𝒗 =

𝜕𝑤

𝜕𝑦−𝜕𝑣

𝜕𝑧𝜕𝑢

𝜕𝑧−𝜕𝑤

𝜕𝑥𝜕𝑣

𝜕𝑥−𝜕𝑢

𝜕𝑦

Chapt. I - 156

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• below Tl: wave functions of He-atoms overlap formation of coherent ground state

• phase is related to velocity of superfluid:

I.4.1 Quantization of Circulation– coherent ground state of superfluid He

quantization of circulation in superfluid He analogous to flux quantization in superconductors

• description of ground state by macroscopic wave function (as for a superconductor)

• since 𝑁 ≫ 1 in ground state: 1/𝑁 → 0 and hence Δ𝑁/𝑁 and Δ𝜑 are arbitrarily small

• uncertainty relation between particle number and phase: DN · Dj ≥ 1 or

𝒗𝒔 =𝟏

𝒎𝒔ℏ𝛁𝝋(𝒓)

Chapt. I - 157

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• for superconductor (London theory):

I.4.1 Quantization of Circulation

• superfluid He is uncharged: qs = 0

2𝜋𝑛

• circulation

𝜿

….. is quantized in units of 𝒉/𝒎𝐇𝐞 = 𝜿 (vorticity)

• generation of vortices by rotation: angular frequency W magnetic field B

Coriolis force Lorentz force

Chapt. I - 158

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.4.1 Quantization of Circulation

an uncharged superfluid cannot rotate homogeneously

• superfluid velocity

• curl of the gradient of a scalar function vanishes

rotation takes place via vortex lines

Chapt. I - 159

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• excitation energy ≈ Ekin should be as small as possible:

minimum vs n = 1minimum length vortex rings smallest ring atomic dimension

I.4.1 Quantization of Circulation– are there vortices in thermal equilibrium ?

• vortex still macroscopic object

E >> kBT except for T ≈ Tl

momentum is large vL = E / p is small for vortex rings

• critical velocity vL is determined by excitation of vortex rings

Chapt. I - 160

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• cool down a rotating 4He container below Tl

• above Tl : rigid rotation

weak circulation strong circulation

flat surface

screening of circulation„Meißner State“

penetration of circulation„Mixed State“

• below Tl :

I.4.2 Experimental Study of Vortices(first experiments by Vinen in 1961)

Chapt. I - 161

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.4.2 Experimental Study of Vortices

• what is the density of vortices inside the container ?

- circulation at the edge of the container with radius R:

- must be equal to number of vortices 𝑛 × vorticity 𝜅 × area 𝜋𝑅2

Γ = 𝑛 𝜅 𝜋𝑅2 = 2𝜋𝑅2Ω

≈ 20 /mm² × Ω [Hz]

Chapt. I - 162

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.4.2 Experimental Study of Vortices

http://ltl.tkk.fi/research/applied/rotating3he.html

• rotating cryostat at Helsiki university

Chapt. I - 163

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• imaging of vortices: - He is covered by electrons- electrons are trapped in vortex cores- on applying an electric field, electrons are extracted to

fluorescent screen

T = 0.1 K

pattern and camera are rotating

pinning of vorticesat bottom ofcontainer

E.J. Yarmchuk, M.J.V. Gordon, R.E. Packard, Phys. Rev. Lett. 43, 214 (1979)E.J. Yarmchuk, R.E. Packard, J. Low Temp. Phys. 46, 479 (1982)

I.4.2 Experimental Study of Vortices

Chapt. I - 165

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

a still image captures the vortex lattice along the axis of rotation in a vat of superfluid helium. A perfect lattice is not observed due to waves and other boundary effects

I.4.2 Experimental Study of Vortices

http://www.aps.org/units/dfd/pressroom/papers/gaff09.cfm

Chapt. I - 166

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Contents:

I.1 Foundations and General PropertiesI.1.1 Quantum FluidsI.1.2 HeliumI.1.3 Van der Waals BondingI.1.4 Zero-Point FluctuationsI.1.5 Helium under PressureI.1.6 pT-Phase Diagram of 4He and 3HeI.1.7 Characteristic Properties of 4He and 3HeI.1.8 Specific Heat of 4He and 3He

I.2 4He as an Ideal Bose GasI.2.1 Bose-Einstein CondensationI.2.2 Bosons and FermionsI.2.3 Bose-Einstein Condensation of 4He

I.3 Superfluid 4HeI.3.1 Experimental ObservationsI.3.2 Two-Fluid ModelI.3.3 Excitation Spectrum of 4He

I.4 VorticesI.4.1 Quantization of CirculationI.4.2 Experimental Study of

Vortices

I.5 3He as a model Fermi gasI.5.1 normal fluid 3HeI.5.2 solid 3He and Pomeranchuk

effectI.5.3 superfluid 3He

I.6 3He / 4He mixtures

Chapt. I - 167

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

4He p+↑ p+↓ n↑ n↓ e-↑ e-↓ boson3He p+↑ p+↓ n↑ e-↑ e-↓ fermion

• 3He has spin ½ Fermion

• fluid 3He: real Fermi gas Fermi liquid (ideal non-interacting Fermi gas with finiteinteraction)

theoretical description by Landau (1956-1958)L.D. Landau, Soviet. Phys. JETP 3, 920 (1957)

L.D. Landau, Soviet. Phys. JETP 5, 101 (1957)

• solid 3He: nuclear magnetism

due to different quantum statistics:liquid 3He has completely different properties than liquid 4He

I.5 3Helium

Chapt. I - 168

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0.4 0.8 1.2 1.6 2.00.0

0.2

0.4

0.6

0.8

1.0

0.5

1

2

5

EF/k

BT = 200

f (E

)

E / EF

10

• first approximation: description by an ideal Fermi gas (non-interacting Fermions)

𝝁 𝑻

T

TF

EF

0

0.9887·TF

degenerate

Boltzmann

I.5.1 Normal Fluid 3Helium

• Fermi-Dirac distribution:

Chapt. I - 169

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0 10 20 30 400

10

20

30

40

50

60

70

Fermi-Dirac

Maxwell-

Boltzmann

dN

/ d

E

E / kBT

0 2 4 6 80

10

20

30

40

50

60

70

Fermi-Dirac

Maxwell-Boltzmann

dN

/ d

E

E / kBT

I.5.1 Normal Fluid 3Helium

0.00 0.02 0.04 0.06 0.08 0.100.98

0.99

1.00

1.01

/

EF

T / TF

for metals:T / TF ≈ 0.01 at room temperature µ ≈ EF

Chapt. I - 170

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Fermi energy

Fermi temperature

Fermi wavelength

Fermi velocity

m = mHe is large expected Fermi temperature is small: 𝑻𝑭 ≃ 𝟒. 𝟐 K

comparison to electron gas in metal:

electrons in metals: n ≈ 1023 cm-3 is large, m = me is small TF ≈ 105 K is large

3He: n ≈ 1022 cm-3 is smaller, m = mHe is much larger TF ≈ few K is small

I.5.1 Normal Fluid 3Helium: Fermi Temperature

N = particle number n = particle density

[1/m³]

Chapt. I - 171

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• classical:

• only small fraction ≃ 𝑇/𝑇𝐹 of particles in energyinterval 𝑘𝐵𝑇 around 𝐸𝐹 contributes to specific heat

• energy change per particle ≈ 𝑘𝐵𝑇

• total energy change ≈ 𝑘𝐵𝑇 ⋅ 𝑁𝑘𝐵𝑇

𝑇𝐹∝ 𝐷 𝐸𝐹 𝑇2

• 𝐶𝑉 ∝ 𝐷 𝐸𝐹 𝑇

I.5.1 Normal Fluid 3Helium: specific heat

(Dulong-Petit)

Sommerfeld expansion

• quantum mechanical:

spin degreesof freedom

contribution perdegree of freedom

(equipartition theorem)

Sommerfeld coefficient

Chapt. I - 172

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• interacting Fermi gas Fermi liquid (Landau)

• increased effective mass 𝑚⋆

smaller Fermi temperature𝑇𝐹 ∝ 1/𝑚⋆

𝑚⋆/𝑚He ≃ 2.8 (@ p =1 bar)≃ 5.5 (@ p = 30 bar)

• additional contribution due to spin fluctuations (paramagnons) strong repulsive interaction at short distance antisymmetric orbital wavefunction symmetric spin wavefunction parallel orientation of neighboring spins

0.0 0.4 0.8 1.2 1.6 2.0 2.40.0

0.2

0.4

0.6

0.8

1.0

1.2

CV

,mo

l / R

T (K)

TF ≈ 0.6K

measured

3He

measured Fermi temperature is significantly smaller than expected

I.5.1 Normal Fluid 3Helium: specific heatC

V,m

ol

T

3NAkB = 3R

TF ≈ 4 K

expected

Chapt. I - 174

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• intuitive argument for increased effective mass:

interaction causes „backflow“ of particles increased effective mass m*

reduced Fermi temperature TF

I.5.1 Normal Fluid 3Helium: effective mass

Chapt. I - 175

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• T < TF: degenerate Fermi gastemperature independent Pauli spin susceptibility (only fraction T/TF contributes)

for electron liquid:

• nuclear spin I = ½ magnetic properties

• T > TF: non-degenerate Fermi gasmagnetic susceptibility follows Curie law: 𝝌 ∝ 𝑪/𝑻

𝜇eff = 𝑔𝑛𝜇𝑛

Landé factor magnetic moment of 3He

𝑀 =𝜇effΔN

V= 𝜇effΔn

Δ𝑛 ≈ 𝜇eff𝐵𝑒𝑥𝑡 ⋅ 𝐷 𝐸𝐹 /𝑉

𝜒 = 𝜇0𝜇eff2 𝐷 𝐸𝐹 /𝑉

𝐷 𝐸𝐹 ∝ 𝑚⋆

I.5.1 Normal Fluid 3Helium: magnetic susceptibility

for 3He liquid:

Chapt. I - 176

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

T

T (K)

𝜒𝑇 ∝ 𝑇 ⇒ 𝜒 = 𝑐𝑜𝑛𝑠𝑡.(Pauli susceptibility)

𝜒𝑇 ≈ 𝑐𝑜𝑛𝑠𝑡. ⇒ 𝜒 = 𝐶/𝑇(Curie law)

TF ≈ 0.6K

degenerate non-degenerate

3He

I.5.1 Normal Fluid 3Helium: magnetic susceptibility

cannot be measured for anelectron gas system in a metalsince TF >> melting temperature

Chapt. I - 178

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• description by kinetic gas theory:

viscosity: (ℓ = mean fee path)

diffusion coefficient: (self-diffusion)

thermal conductivity:

• classical gas: 𝑣th = thermal velocity (Maxwell-Boltzmann statistics)

• Fermi gas: 𝑣F = Fermi velocity (Fermi statistics)

𝝉 ∝𝑻

𝑻𝑭

𝟐

particle-particle scattering

• Fermi gas, with ℓ = 𝑣F𝜏

I.5.1 Normal Fluid 3Helium: transport properties

Chapt. I - 179

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

) 3He Fermi Gas ratio

CV / NAkBT 2.78 1.0 2.78

vF (m/s) 56 157 2.80

/ µ0 µ²eff (Jm³)-1 3.3·1051 3.6·1050 9.2

deviations due to finite interactions Landau Fermi liquid effective mass

I.5.1 Normal Fluid 3Helium: Comparison to Ideal Fermi Gas

mole volume: 36.84 cm³ @ p = 0 bar, T = 0 K n = 1.63 x 1028 m-3

mass: m3He = 3.016 amu = 5.006 x 10-27 kg

Chapt. I - 181

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

comparison of phase diagrams of3He and 4He

• 4He mostly hcp, small region with bcc,at high T: fcc

• 3He shows bcc-hcp transitionat 100 bar,at high T: fcc

• 3He shows minimum in meltingcurve at T = 0.32 K

Isaak Jakowlewitsch Pomerantschuk

(Исаак Яковлевич Померанчук;

geb: 20. Mai 1913 in Warschau;

gest: 14. Juli 1966 in Moskau)

russischer Physiker.

can be used for cooling of ³HePomerantchuk effect

I.5.2 Solid 3He and Pomerantchuk Effect

0.1 1 1010

1

102

103

104

p (

bar

)

T (K)

phase diagram of4He and 3He

3He

4Hehcp

bcc

liquid

hcp

liquid

bcc

fcc

Chapt. I - 183

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• explanation: solid phase: atoms are ordered,

spins are disordered and determine entropy: Ssolid = R ln2at low T: antiparallel ordering of spins, S decreases towards zero

liquid phase: atoms are spatially disordered, but ordering in k-space (Fermi liquid)

entropy of Fermi liquid:

• Clausius-Clapeyron equation:

• always: Vsol < Vliq

• for T < 0.32 K: disorder larger in solid than in liquid phase ??

S

lnT

R ln2

0

liquid

solid

320 mK

I.5.2 Solid 3He: Melting Curve and Entropy

drop due to interactions

Chapt. I - 184

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

stamp stainless steelbellow

liquid

solid pressure cell

• precooling to T < Tmin

• adiabatic compression solidification and cooling

• lowest T: ≈ 1.5 mK limitation due to antiparallel

spin ordering in solid 3He

I.5.2 Solid 3He: Pomerantchuk Cooling

10-3

10-2

10-1

100

2.8

2.9

3.0

3.1

3.2

3.3

3.4

p (

MP

a)

10-3

10-2

10-1

100

0.0

0.2

0.4

0.6

0.8

S /

R

T (K)

liquid

solid

liquid

solid

Tmin

ln 2

Chapt. I - 186

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Richardson, Nobel Prize Lecture (1996)

I.5.2 Solid 3He: Pomerantchuk Cooling

Pomeranchuk's suggestion for cooling a melting mixture of 3He:The solid phase has a higher entropy than the liquid at low temperatures. As the liquid-solid mixture is compressed, heat is removed from the liquid phase as solid crystallites form. The fractional change of volume required to completely convert liquid into solid is approximately 5%. Unlike melting water, the solid phase forms at the hottest part of the container

Chapt. I - 187

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.5.2 Solid 3He: Single Crystals

Sequence of interferograms of a growing 3He crystal at T = 0.55 mK. The dashed white lines outline the identified facets marked with Miller indices.H. Alles et al., PNAS 99, 1796–1800 (2002)

Computer-generated shape of a bcc crystal presented together with an elementary patch of a crystal habit where the facets are labeled with Miller indices.

Chapt. I - 188

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• initial believe: no superfluid 3He due to Pauli‘s exclusion principle for fermions no Bose-Einstein condensation of 3He

• after development of BCS theory: pairing of 3He atoms to Cooper pairs ?? problem: liquid 3He is system of strongly interacting particles formation of Cooper pairs out of „naked“ 3He atoms not possible Landau: weakly interacting quasiparticles in Fermi liquid

• 1963: critical pairing temperature of about 100 mK was expected

• 1971: discovery of superfluid 3He by Osheroff, Richardson and Lee (used Pomerantschuk cooling)

(Nobel Prize in Physics 1996 „for their discovery of superfluid 3He“)

D.D. Osheroff, R.C. Richardson, D.M. Lee, Phys. Rev. Lett. 28, 855 (1972),

D.D. Osheroff, W.J. Gully, R.C. Richardson, D.M. Lee, Phys. Rev. Lett. 29, 920 (1972)

• initially: two superfluid phases A and Blater: three phases (in magnetic field additional A1 phase)

• experimental result: pairs have spin S = 1, spin triplet pairs pairs with S = 1 must have L = 1 (see below) occupation probability for same position is negligible

(reduction of short length repulsion)

I.5.3 Superfluid 3He

Chapt. I - 190

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Douglas D. Osheroff, Stanford University,

Stanford, California, USA

David M. Lee, Cornell University, Ithaca,

New York, USA

Robert C. Richardson, Cornell University, Ithaca,

New York, USA

The Nobel Prize in Physics 1996

"for their discovery of superfluidity in helium-3"

I.5.3 Superfluid 3He

Chapt. I - 191

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Source: Nobel Prize Lecture Lee/Osheroff (1996)

- cool down of 3He via Pomeranchuk effect- observation of melting curve anomalies give hint to phase transitions

I.5.3 Superfluid 3He: discovery

Chapt. I - 192

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

log T scale !!!

Phase diagram of 3He. Note the logarithmic temperature scale. There are two superfluid phases of 3He, A and B. The linewithin the solid phase indicates a transition between spin-ordered and spin disordered structures (at low and hightemperatures, respectively).

vapor

10-4

10-3

10-2

10-1

100

101

0

10

20

30

40

pre

ssu

re (

bar

)

temperature (K)

solid 3He (bcc)

superfluid 3He(B-phase)

normal liquid 3He

melting curve

Tc , pc

Tsf

Tb (1 bar)

pm

evaporation

superfluid 3He(A-phase)

spin-orderedsolid 3He

29 bar

34 bar

I.5.3 Superfluid 3He: Phase Diagram for B = 0

Chapt. I - 193

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

0.8 1.2 1.6 2.0 2.4 2.80

10

20

30

40

p (

bar

)

T (mK)

polycriticalpoint (PCP)21.5 bar, 2.56 mK

p = 34.4 bar

1. order transition

I.5.3 Superfluid 3He: Phase Diagram for B = 0

superfluid 3He-B

sf 3He-A

normal liquid 3He

paramagnetic bbc solidafm bbc solid

pressure increases pairing interaction

linear T scale !!!

Chapt. I - 195

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• TC: 2. order phase transition jump of Cp as for superconductor• TAB: 1. order phase transition small change of temperature dependence of Cp

hysteresis, latent heat• PCP: first and second order phase transition merge

Δ𝐶𝑝/𝐶𝑝 ≃ 2@ p = 30 bar

Δ𝐶𝑝/𝐶𝑝 ≃ 1.4@ p = 1 bar

T (mK)

10³

Cp

/ R

measurement under saturated

vapor pressure

p = 28.7 bar

I.5.3 Superfluid 3He: specific heat

T. A. Alvesalo et al., JLTP 45, 373 (1981)

Δ𝐶𝑝

𝐶𝑝≃ 1.9

weak couplingBCS theory:Δ𝐶𝑝/𝐶𝑝 = 1.43

Chapt. I - 196

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

viscosity changesby 5 orders ofmagnitude

normal fluidfreezes out

• other experiments: persistent flow experiments: - viscosity of B phase changes at least by

12 orders of magnitude- persistent flow in A phase decays slowly

(appearence of textures)- small critical velocity: 1 to 100 mm/s

• use of vibrating wire, measurement of quality factor n / Dn

I.5.3 Superfluid 3He: Proof of SuperfluidityPersistent-Current Experiments on Superfluid He 3 -B and He 3 -A

J. P. Pekola, J. T. Simola, K. K. Nummila, O. V. Lounasmaa, and R. E. Packard

Phys. Rev. Lett. 53, 70

Persistent-Current Experiments on Superfluid He 3 -B and He 3 -A J. P. Pekola, J. T. Simola, K. K. Nummila, O. V. Lounasmaa, R. E. Packard, Phys. Rev. Lett. 53, 70 (1984)

Chapt. I - 197

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• 3He is strongly interacting Fermi liquid (different to electrons in metals) exchange interaction favors parallel spin alignment Fermi statistics favors anti-parallel spin orientation

• 3He forms Cooper pairs with spin S = 1 (spin triplet pairs) in contrast to electrons inmetals: S = 0 (spin singlet pairs pairs)

• note the general requirement: antisymmetric wave function for fermions

symmetric spin wave function antisymmetric spatial wave function (L = 1,3,...)antisymmetric spin wave function symmetric spatial wave function (L = 0,2,...)

examples: (i) Cooper pairs in metallic superconductors (L = 0, S = 0) s-wave, spin-singlet pairs

(ii) Cooper pairs in high temperature superconductors (L = 2, S = 0) d-wave, spin-singlet pairs

(iii) Cooper pairs in superfluid 3He (L = 1, S = 1) p-wave, spin-triplet pairs

+p-type spatial wave functionminimizes hard core repulsion

complex behaviordue to 3 x 3 combi-nations for Lz /Sz

I.5.3 Superfluid 3He: Pairing

Chapt. I - 198

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

spin triplet pairs have magnetic moment phase diagram depends on magnetic field

new phase A1

phase A gains,phase B looses

phase B disappears

I.5.3 Superfluid 3He in an applied B-field

~20µK

Chapt. I - 199

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

3He

B = 0.88 Tp = pmelting

T / Tc

Cp

(10

-4J /

mK

mole

)

clear demonstration of existence of A1 phase !!

I.5.3 Superfluid 3He in an applied B-field:specific heat

W.P. Halperin et al., Phys. Rev. B13, 2124 (1974)

Chapt. I - 200

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

source: http://ltl.tkk.fi/research/theory/helium.html

I.5.3 Superfluid 3He in an applied B-field

Chapt. I - 201

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• starting point: BCS Hamiltonian

I.5.3 Superfluid 3He: Quantum States

𝜎1, 𝜎2, 𝜎3, 𝜎4: all possible spin configurations of the two 3He atoms

𝑐k𝜎† , 𝑐k𝜎 creates/annihilates fermion in the quantum state 𝐤, 𝜎

describes the scattering of a particle from state 𝐤′, 𝜎3 to 𝐤, 𝜎1and a second particle from −𝐤′, 𝜎4 to −𝐤, 𝜎2

(particle number operator)

Chapt. I - 202

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• pair consists of two Fermions total wave function must be antisymmetric

center of massmotion

we assume K = 0

orbitalpart

spinpart

• spin wave function:

antisymmetric spin wavefunction symmetric orbital function:

L = 0, 2, … (s, d, ….)

symmetric spin wavefunction antisymmetric orbital function:

L = 1, 3, … (p, f, ….)

I.5.3 Superfluid 3He: Quantum States

singlet:

triplet:

symmetry of the pair wave function

Chapt. I - 203

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.5.3 Superfluid 3He: Quantum States

• pair formation in Fermi liquids due to finite pair attraction near FS

- spontaneous pair formation in k-space Gorkov pairing amplitude

- consequence:matrix momentum distribution(Nambu, 1962)

pairs holes

particles pairs

statistical average

• alternatively one can define the pairing strength:

four component function due to two values of 1 and 2

orbital structure spin structure

𝑐k𝜎† , 𝑐k𝜎 creates/annihilates fermion in the quantum state 𝐤, 𝜎

Chapt. I - 204

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.5.3 Superfluid 3He: Quantum States

• spin structure of the pairing strength (pair potential)

- two fermions with s = ½ and 𝜎1, 𝜎2 = 𝑚𝑠 = ↑ , ↓

- Pauli principle: Δ𝑘𝜎1𝜎2 = −Δ𝑘𝜎2𝜎1

singlet

triplet

- singlet (S=0): even parity orbital function

- triplet (S=1): odd parity orbital function

• orbital structure of the pairing strength (pair potential)

k

k

D

D

Chapt. I - 205

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• spin dependent pairing potential in BCS theory:(4 components covering all spin orientations before and after interaction)

probability to find electron pairs

I.5.3 Superfluid 3He: Quantum States

• reformulation:

• spin singlet:

• spin triplet:

scalar gap function d-vectorPauli spin matrix

Pauli vector:

𝜎𝑥 =0 11 0

𝜎𝑦 =0 −𝑖𝑖 0

𝜎𝑧 =1 00 −1

Chapt. I - 206

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• order parameter must describe pair density and spin wave function use of d vector

I.5.3 Superfluid 3He: Quantum States

• for all pairs same state for S and L (S = 1, L = 1)• orbital state determined by spin state spin-

orbit coupling

direction ofspin wave function

superfluiddensity

phasefactor

amplitude of d (d is vector with complex components) = pair density

direction of d = spatial orientation of spin wave function

• spin wave function is different for the three phases of superfluid 3He 0ˆ Sd

• theory: - dx, dy, dz can be chosen real, - d transforms as vector under rotations

Chapt. I - 207

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.5.3 Superfluid 3He: Quantum States

• spin triplet state, S = 1

• spin wave function

basisstates

Chapt. I - 208

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.5.3 Superfluid 3He: Quantum States

• spin triplet state, S = 1

other possible basis states: Sz = -1, 0, +1

• for B = 0: Sz states are degenerate• degeneracy is lifted by interaction of pairs: magnetic dipole-dipole interaction

more appropriate basis states, which are adopted to the perturbation due to interaction

probabilityamplitude forspin withcomponentsSx, Sy, Sz

Chapt. I - 209

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.5.3 Superfluid 3He: Quantum States

• spin wave function expressed in different basis states

Chapt. I - 210

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.5.3 Superfluid 3He: Quantum States

• different phases of 3He representation of order parameter in terms of Ylm

orbital wave functions (L = 1)

spin wave functions (S = 1)9 coefficients (according values of Lz, Sz)

Aij(r) is the wave function for the center of mass of a Cooper pair

A phase: B phase:

Chapt. I - 211

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• A phase: Anderson-Brinkman-Morel (ABM) state linear combination of 𝑆𝑧 = ±1 𝑐1 ≠ 0, 𝑐2 = 0, 𝑐3 ≠ 0

(i) 𝐵 = 0: 𝑐1 = |𝑐3| 𝑆𝑧 = 0(ii) 𝐵 ≠ 0: 𝑐1 ≠ |𝑐3| 𝑆𝑧 ≠ 0 (anisotropic, paramagnetic)

I.5.3 Superfluid 3He: Quantum States

• general spin wave function:

• A1 phase: 𝑐1 = 0, 𝑐2 = 0, 𝑐3 ≠ 0 𝑆𝑧 = 1 (“ferro”magnetic)

exists only for 𝐵 ≠ 0, reversal of magnetic field: 𝑐1 ≠ 0, 𝑐2 = 0, 𝑐3 = 0

• B phase: Balian-Werthamer (BW) state |𝑐1 = 𝑐2 = 𝑐3| ≠ 0 𝑆𝑧 = 0 (isotropic, non-magnetic)

𝐵 ≠ 0: Sz = 0 component is non-magnetic

𝜒𝐵 =2

3𝜒𝐴 B phase energetically less favorable

in applied magnetic field !!

Chapt. I - 212

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.5.3 Superfluid 3He: d vectors of different phases

• general spin wave function:

• A phase: 𝑐1 ≠ 0, 𝑐2 = 0, 𝑐3 ≠ 0 d vector in xy-plane

• order parameter:

orbital wave function spin wave functionno component of orbital

momentum along ℓ (L in xy-plane)

(L=1, S=1)x y

z z

spin component is

zero along 𝑑

Chapt. I - 213

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• unpaired atoms moving in L direction do not perturb no energy gap in L-direction

B S

L

x

z S

S

Fermisphere

anisotropicenergy

gap

kx

kz

nodes low lying

qp excitations

L

d

note that L || S would result in repulsive spin-orbit interaction

I.5.3 Superfluid 3He: d vectors of different phases

• A phase: • both spins are always parallel

• all paired atoms are moving in plane perpendicular to L

Chapt. I - 214

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

B S

d

L

x

z

L kz

kx

I.5.3 Superfluid 3He:order parameter of the A phase

Chapt. I - 215

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

B S

d

L

x

z

B S

d

x

z

N

dd

d

104°

equatorial planeof Fermi spherein k-space

A phase B phase

(S, L, N = expectation values for spin, orbital momentum and texture)

• B phase: |c1| = |c2| = |c3| complex behavior, direction of d is

k-dependent, amplitude is constant

I.5.3 Superfluid 3He: d vectors of different phases

• order parameter:

Chapt. I - 216

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Fermisphere

isotropicenergy

gap

kx

kz

no nodes !

B phase

I.5.3 Superfluid 3He:order parameter of the B phase

• spherically symmetric state- characterized by d-vector with constant

magnitude all over the Fermi surface

Chapt. I - 217

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• A1 phase: c1 = 0, c2 = 0, c3 ≠ 0 eigenstate of Sz

d vector not required

xd2

2

I.5.3 Superfluid 3He: d vectors of different phases

Chapt. I - 219

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• Andronikashvili type experiment:

• effect of wall:

torsionalresonator

B║

B┴

1-(T / Tc)

rs

/ r

left: small momentum transfer

right: large momentum transfer

I.5.3 Superfluid 3He: normal fluid density

Chapt. I - 220

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• Sr2RuO4

• organic superconductors such as (TMTSF)2PF6

• heavy Fermion superconductors UPt3, UBe13

• neutron stars ??

Tetramethyltetraselenafulvalen

I.5.3 Superfluid 3He: other spin-triplet systems

Chapt. I - 221

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Contents:

I.1 Foundations and General PropertiesI.1.1 Quantum FluidsI.1.2 HeliumI.1.3 Van der Waals BondingI.1.4 Zero-Point FluctuationsI.1.5 Helium under PressureI.1.6 pT-Phase Diagram of 4He and 3HeI.1.7 Characteristic Properties of 4He and 3HeI.1.8 Specific Heat of 4He and 3He

I.2 4He as an Ideal Bose GasI.2.1 Bose-Einstein CondensationI.2.2 Ideal Bose GasI.2.3 Bose-Einstein Condensation of 4He

I.3 Superfluid 4HeI.3.1 Experimental ObservationsI.3.2 Two-Fluid ModelI.3.3 Excitation Spectrum of 4He

I.4 VorticesI.4.1 Quantization of CirculationI.4.2 Experimental Study of

Vortices

I.5 3HeI.5.1 normal fluid 3HeI.5.2 solid 3He and Pomeranchuk

effectI.5.3 superfluid 3He

I.6 3He / 4He mixtures

Chapt. I - 222

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

I.6 3He/4He mixtures

• why interesting:

model system for testing Fermi liquid theories(3He is fermion, can be diluted in passive 4He background change of interaction)

technical application in cooling machinesH.E.Hall, P.J.Ford and K.Thomson, Cryogenic 6 (1966) 80B.S.Neganov, N.S.Borisov and M.Yu.Liburg, Zh.Exp. Teor. Fiz. 50 (1966) 1445.

• experimental observations:

specific heat curve of 4He is shifted to lower T on adding 3He, but doesnot change its shape

phase separation of 4He and 3He at low T

Chapt. I - 223

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

3He4He

miscibility

gap

• high T: 3He reduces 𝑇𝜆 ∝ 𝑛.4He

• tricritical point Tk = 0.87 K

• T < Tk

phase separation

3He rich phase: lighter3He poor phase: heavier

• T 0 incomplete phase separation

6.5% of 3He still dissolved in 4He

• 𝑥3 =𝑛3

𝑛3+𝑛4; 𝑥4 =

𝑛4

𝑛3+𝑛4

limiting concentrations:

𝑥4 = 𝑎𝑇3/2𝑒−𝑏/𝑇 (𝑎 = 0.85 K−3/2, 𝑏 = 0.56 K−1)

𝑥3 = 0.0648 1 + 𝑐𝑇2 (𝑐 = 10 K−2)

I.6 3He/4He mixtures: Phase diagram

𝑇𝜆 𝑥4

Chapt. I - 224

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• at T < 0.5 K < 𝑇𝜆:

4He forms superfluid with negligible amount of excitations (phonons, rotons)

viscosity, entropy and specific heat go to zero: inert superfluid background

𝑚3∗ : effective mass of 3He dissolved in 4He

𝑛3 = 𝑥3𝑁3/𝑉: density of 3He atoms

• 3He obeys Fermi statistics

Fermi temperature is about 1 K for 3He dissolving 3He in 4He reduces the Fermi temperature

I.6 3He/4He mixture as a Fermi Liquid

Chapt. I - 225

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

x3 = 0.013

x3 = 0.05

⅓ TF

TF

⅓ TF

classical gas(Dulong-Petit)

• Fermi gas (𝑇 ≪ 𝑇𝐹)

• classical gas (𝑇 > 𝑇𝐹)

• 𝑇𝐹 increases with 𝑥3

x3 (%)

𝑚3∗ = 2.4 𝑚3𝐻𝑒

interaction with 4He

I.6 3He/4He mixture as FL: specific heat

Chapt. I - 226

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• bonding:

• 𝑉𝐴𝐵 >1

2𝑉𝐴𝐴 + 𝑉𝐵𝐵 complete miscibility (e.g. water and alcohol)

•1

2𝑉𝐴𝐴 + 𝑉𝐵𝐵 > 𝑉𝐴𝐵 phase separation (e.g. water and petrol)

A A A B B B

VAA VAB VBB

critical point

miscibility gap

increasing mixing with increasing T:

F = U – TS min

bindingenergy

thermalmotion

optimization of binding energycomplete phase separation @ T = 0

minimization of free energy

I.6 3He/4He mixtures:miscibility gap in binary systems

Chapt. I - 227

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• binding energy of 3He in 3He (V33) and 4He (V34):

• 𝜖3,𝑑 0 > 𝜖3,𝑐 or vice versa ?

3He has smaller mass larger zero point fluctuations occupies larger volume binding of 3He is larger in 4He than in 3He due to larger density of 4He

|𝜖3,𝑑 0 | > |𝜖3,𝑐|

corresponds to case 𝑉𝐴𝐵 >1

2𝑉𝐴𝐴 + 𝑉𝐵𝐵 complete miscibility expected,

why miscibility only up to 6.5% ??

I.6 3He/4He mixtures:finite solubility of 3He in 4He

(ii) single 3He atom in liquid 4He:

binding energy for single 3He atom: 𝜖3,𝑑(𝑥3 → 0) =𝜇3,𝑑 𝑥3→0

𝑁𝐴𝜇3,𝑑 = chemical potential of dilute phase

(i) 3He in 3He: binding energy is given by the latent heat of evaporation L3:

binding energy for single 3He atom: 𝜖3,𝑐 = −𝐿3

𝑁𝐴=

𝜇3,𝑐

𝑁𝐴

𝜇3,𝑐 = chemical potential of pure

(concentrated) liquid

Chapt. I - 228

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

Questions:

• why can‘t we dissolve more than 6.5% of 3He in 4He at T = 0 ?

two effects:

(i) 3He forms degenerate Fermi liquid 𝑇𝐹 increases with 𝑥3 for 𝑥3 > 6.5%, the Fermi energy exceeds the gain in binding energy

(ii) 𝜖3,𝑑 𝑥3 > 𝜖3,𝑑(𝑥3 = 0) effective attraction between two ³He atoms (magnetic and volume effect)

I.6 3He/4He mixtures:finite solubility of 3He in 4He

• why don‘t we have a complete phase separation into 3He and 4He at T = 0 ?

results in finite disorder, violation of 3rd law of thermodynamic ? no: we have degenerate Fermi gas, ordering in k-space

Chapt. I - 229

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

x3

0.065

−𝜖3,𝑑(0)

−𝜖3,𝑐

𝑬𝐩𝐨𝐭 𝒙𝟑 + 𝒌𝐁𝑻𝐅 𝒙𝟑

0gaseous ³He

pure liquid ³He

³He diluted in 4He

𝑬𝐩𝐨𝐭 𝒙𝟑 = 𝟏 + 𝒌𝐁𝑻𝐅 𝒙𝟑 = 𝟏

for 𝑥3 > 6.5%: 𝑬𝐩𝐨𝐭 𝒙𝟑 + 𝒌𝐁𝑻𝐅 𝒙𝟑 > 𝝐𝟑,𝒄 = 𝑬𝐩𝐨𝐭 𝒙𝟑 = 𝟏 + 𝒌𝑩𝑻𝐅 𝒙𝟑 = 𝟏

separation of pure ³He

energy

𝑬𝐩𝐨𝐭 𝒙𝟑

I.6 3He/4He mixtures:finite solubility of 3He in 4He

Chapt. I - 230

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• for Fermi liquid: Cconcentrated < Cdiluted (x3 = 0.065) (𝐶𝑉 ∝ 𝑇/𝑇𝐹, 𝑇𝐹 ∝ 𝑛2/3)

• with we therefore obtain (𝐶𝑉 = 𝛾𝑇, 𝛾 = Sommerfeld coefficient):

Uconcentrated (T) < Udiluted (T)

• operation principle:

remove ³He atoms from the dilute phase below Tk = 0.87 K transport of ³He atoms across phase boundary

to maintain equilibrium concentration corresponds to evaporation of ³He from concentrated phase cools as the latent heat of evaporation is removed

concentrated(lighter)

diluted,6.5%(heavier)

³He

³He

I.6 3He/4He mixtures:cooling effect

U

T

Uconcentrated

Udiluted

Δ𝑄

• on transition across phase boundary:

𝑑𝐺 = 0 = 𝑑𝑈 − 𝑇𝑑𝑆 ⇒ 𝑑𝑈 = 𝑇𝑑𝑆 = 𝑑𝑄

removal of heat𝛥𝑄 = 𝑇𝛥𝑆 = 𝑇 𝑆dil 𝑇 − 𝑆con 𝑇 cooling effect

³He/4He dilution refrigerator

Chapt. I - 231

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

• assumption: one mole of ³He crosses boundary

• since there is no volume change

(standard expressions for Fermi liquid)

I.6 3He/4He mixtures:cooling effect (more detailed)

𝛥𝑄 = 𝑇𝛥𝑆 = 𝑇 𝑆dil 𝑇 − 𝑆con 𝑇

• removed heat:

• cooling power Δ 𝑄 = Δ𝑄/𝑛3 ⋅ 𝑛3

Δ 𝑄 = 𝑛3𝑇 [𝑠dil 𝑇 − 𝑠con 𝑇 ]

with and

we obtain the entropy

Chapt. I - 232

R. G

ross

and

A. M

arx

, © W

alth

er-

Me

ißn

er-

Inst

itu

t (2

00

4 -

20

15

)

3He, concentrated

3He, diluted

Fermisphere

large 3He densitylarge Fermi spherehigh TF

small 3He densitysmall Fermi spherelow TF

fraction of thermallyexcited 3He atomsincreases (∝ 𝑻/𝑻𝑭)

entropy increases goingfrom concentrated todiluted phase

removed heat: dQ = TdS

thermallyexcited 3He atoms

plausibility considerationkBT

kBT

Fermisphere

I.6 3He/4He mixtures:cooling effect (more detailed)