by Giorgio Benedek,

Dipartimento Scienza dei Materiali

Università di Milano-Bicocca

1. He: a superatom

2. Superfluid helium

3. Supersonic helium

4. A surface superprobe

5. Helium clusters

6. Flying refrigerators

7. From fountains to geygers

8. Supersolid helium

Helium: a brief biography of a superatom

- 18 Aug 1868 solar eclipse : Pierre Janssen 587.49 nm: Na?

- 20 Oct 1868: Norman Lockyer same line (D3) in solar spectrum.

Proposal with Edward Frankland of a new atom: Helium!

- 26 Mar 1895: William Ramsay looks for Ar in rocks but finds

something else; Lockyer and William Crookes confirm: Helium!

William F. Hillebrand (US) found it earlier but….

- later in 1895: Teodor Cleve & Abraham Langlet (Uppsala)

determine the atomic weight of He with great accuracy

-1907: Ernest Rutherford & Thomas Royds prove that α rays

are 4He nuclei Sir Ernest Rutherford

- 1908: Heike Kamerlingh Onnes liquifies He, but attempts

to solidify He are unsuccesful

- 1926: Kamerlingh’s student Willem Hendrik Keesom

succeeds in solidifying 4He at 25 atm

- 1938: Pyotr Leonidovich Kapitsa discovers superfluidity of 4He

- 1972: Douglas D. Osheroff, D. M. Lee & R. C. Richardson

observe superfluidity in 3He as an effect of Cooper pairing

- 1969: Andreev & Lifshitz predict a superfluidity in solid 4He

( supersolid)

- 2001: C. Cohen-Tannoudji et al obtain Bose-Einstein condensation in 4He*

Helium gets condensed

4He: nuclear structure

Carbon can be produced within stars (triple-alpha process), thus making life possible

Big Bang nucleosynthesis predicts an abundance of ~23% of 4He (by mass)

This is due to:

(a) helium-4 is very stable and most neutrons combine with protons to form 4He;

(b) two 4He atoms cannot combine to form a stable atom: 8Be is unstable

Helium: a protected species

Nearly all helium on Earth from radioactive decay (~0.0034 m3/km3/year):

most helium comes from natural gas. Concentrations:

- in rocks: 8·10-9

- in seawater: 4·10-12

- in atmosphere: 5.2·10-6

In 1958 John Bardeen and other influential scientists warned the

Congress that all our helium would be gone by 1980. Congress reacted by

spending $1 billion on a separation plant in Amarillo, Texas, and began

stockpiling helium in empty gas wells.

Most helium in the Earth's atmosphere escapes into space due to its

inertness and low mass. In a part of the upper atmosphere, He and other

lighter gases are the most abundant elements.

Thanks to the conservation measures, helium supplies were not exhausted by 1980. Still worldwide consumption of helium has increased by 5 to 10 % a year in the past decade. Presently it is about 100 million cubic metres, and is predicted to rise by 4 to 5 % a year.

No one is claiming that we are in imminent danger of running out of helium--there should be at least 20 years supply left. However, new sources of the gas will have to be found to meet the ever-growing demand.

If not, God forbid, we may have to celebrate helium's 200th birthday in the year 2095--without any Mickey Mouse balloons.

4He vs 3He

Nucleus Spin Magnetic moment [μN]

proton p 1/2 2.79278

neutron n 1/2 -1.91315

deuterium d 1 0.85742

3He 1/2 -2.1276

4He 0 0

TJM

e

pN /1005078.5

227

3He nuclear magnetic resonance

3He hyperfine structure

3He spin-echo spectroscopy

3He: 0.000137% 4He: 99.999863%

He: electronic

structure I

He: electronic structure II

Atomic radius: 0.31 Å VdW radius: 1.40 Å

He*(23S)

Refractive index of

liquid He: 1.026 (!)

He: an ideal gas?

))((2

bVV

apRT

bRT

a

CP

T

pHJT

21 atm)1(K402

PRb

aTinv

Van der Waals equation of state

3663 m1071.23mPa1046.3 ba

Joule-Thomson effect

Thermal conductivity: 151.3·10-3 W / mK (300 K)

Diffusivity in solids: ~3 times that of air; ~2/3 that of H2

Solubility in water: smaller than for any other gas

Helium in an electric glow discharge can form unstable compounds and molecular

ions like HeNe, HgHe10, WHe2, He2+, He2

++, HeH+, HeD+ and even He2 …..

or form otherwise a plasma supersonic cluster beam deposition

The largest wdW cluster! 4He2 is a giant, > 50 Å wide!

He: a nobleman?

Endohedral fullerenes (by heating under a high pressure of the gas): C60@He

The neutral molecules formed are stable up to high temperatures.

If 3He is used, it can be readily observed by helium NMR spectroscopy:

Fullerenes compounds, nanotubes, supramolecular compounds can be studied

in this way. 3He sneaks into everywhere and tells about the electronic

environment (a nobleman?)

End of lecture 1

Helium:

the only substance which doesn’t freeze at absolute zero and normal pressure

ordinary substances

P. Kapitza (1938)

Lee, Osheroff, Richardson

from D. Vollhardt & P Wölfle 1990

log scale!

λ – line specific heat

W. H. Keesom et al (1932)

VSdTdPm //

VPSTU mVT

T

P

dT

dPU mm

)(, III lslsls VVUUUU

4He solid vs. liquid II

kinmEpp

rr

22

0

4He: a quantum solid

fighting against Heisenberg’s indetermination principle

in solid Helium unfavorable conditions:

- attractive forces (Epot) are weak

- both m and r0 are small

r0

21pr

20

2

8mrEE kinpot

pressure needed!

Classical (Maxwell-Boltzmann) statistics

A B

B

B

A

A

Quantum Bose-Einstein statistics

Quantum Fermi-Dirac statistics

BA PP

BA PP 5

0AP

4He

3He

Fermions (3He) also fight against Pauli’s esclusion principle!

“Keesom and van den Ende (1930) observed quite accidentally that

liquid helium II passed with very annoying ease through certain

extremely small leaks which at a higher temperature were perfectly tight

for liquid helium I and even for gaseous helium. ...

… This observation seemes to indicate an enormous drop of viscosity

when liquid helium passes the λ-point.”

Fritz London, Superfluids, Vol. II (John Wiley & Sons, New York 1954)

the supersurface film

H. Kamerlingh Onnes (1922)

a Helium fountain

“At any rate the fountain effect

experiments show that, in liquid

helium, heat transfer and matter

transfer are inseparably

interconnected”.

F. London, ibidem

Two-fluid model of the superfluid state (L. Tisza)

a normal (viscous) component with atoms having different excited-state velocities

a superfluid component with all atoms having the same ground state velocity (BEC no dissipation zero viscosity)

0/)( 1

)( d

e

gN

kTB

2/12/33

)2(4

)( mh

Vg

01)( / B

kTBegN

)(2

1,)1(

2

)(

2/3

2/3

2

/

02/3

12/3

2

0

/)(/

0

/

Agh

mkTV

eAA

h

mkTV

eegdeN

kT

kTkTkT

B

BB

simple ideas about Bose–Einstein condensation (BEC)

)()()1(

)(/)(1

Riemanng

Bosexxg

V,N

density of states

)(2

)1(1

2

22/3

2

12/3

2/3

2

Aoh

mkTA

A

h

mkTA

V

Nn

)(2

2

3

1

)(2/5

2/3

20

/)(Ag

h

mkTVkTd

e

gE

kTB

])(2

1[2

3

)(

)(

2

3 22/5

2/3

2/5 AoA

kTAg

AgkT

N

EE

kTAomkT

hnkTE

2

3)(

161

2

3 22/32

total energy

average energy per atom

atom density

condensation on the

ground state

3/1

32/3

23

993.1

32

3

)(

11

adB

dBdBa

v

mkT

hv

nBEC

3/223/2

2/5

2

978.122

0 nm

n

m

hTkE c

3/22

3/22

21

4n

mn

m

hTkEE cclassic

3/223/2

23

2

311.3)(2

: nm

n

m

hTkBEC c

BEC critical temperature & conditions

de Broglie wavelength 2 x interatomic distance

Micromégas, bien meilleur observateur que son nain, vit clairement que les atomes se parlaient …

(Voltaire, Micromégas)

atoms “talk” each other if their average mean distance is smaller than their de Broglie wavelength

but the boson attitudes are totally different from fermion attitudes….

de Broglie

(even L)(singlet) =

(odd L)(triplet)

L = 0 S = 0 s-wave Cooper pair: unfavoured

L = 1 S = 1 p-wave Cooper pair: favoured

J = L + S J = 0 (3P0), J = 1 (3P1), J = 2 (3P2)

but spin-orbit interaction is below

the mK range and can be neglected:

9-fold ~ degeneracy mixing of Sz = 1, 0,-1 states

(like the ground state of ortho-H2 !)

L = 2 S = 0 d-wave Cooper pair: less favoured

3He condensation

B-phase: Balia-Werthamer state (isotropic gap (T))

= |> + 2-1/2[ |> + |>] + |>

A-phase: Anderson-Brinkham-Morel “axial” state

= |> + |> anisotropic gap (k,T) = 0(T)[1 – (k·L)2]1/2 ^ ^

the superfluid phases of 3He

a third phase (A1) is induced by a

magnetic field with spin wavefunction

= |>

a magnetic superfluid!

End of lecture 2