Transcript Diapositiva 1

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
by Giorgio Benedek,
Dipartimento Scienza dei Materiali
Università di Milano-Bicocca
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
Helium gets condensed
- 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
- 1969: Andreev & Lifshitz predict a superfluidity in solid 4He
( supersolid)
- 1972: Douglas D. Osheroff, D. M. Lee & R. C. Richardson
observe superfluidity in 3He as an effect of Cooper pairing
- 2001: C. Cohen-Tannoudji et al obtain Bose-Einstein condensation in 4He*
4He:
nuclear structure
 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
 Carbon can be produced within stars (triple-alpha process), thus making life
possible
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
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.
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.
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
3He:
4He:
0.000137%
99.999863%
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
N 
e
 5.050781027 J / T
2M p
 3He hyperfine structure
 3He nuclear magnetic resonance
 3He spin-echo spectroscopy
He: electronic
structure I
He: electronic structure II
He*(23S)
Refractive index of
liquid He:
1.026 (!)
Atomic radius: 0.31 Å
VdW radius:
1.40 Å
He: an ideal gas?
 Van der Waals equation of state
RT  ( p 
a
V
2
)(V  b)
a  3.46103 Pa m6 b  23.71106 m3
 Joule-Thomson effect
1  2a
 T 



b



 P  H C p  RT

 JT  
Tinv 
2a
 40 K ( P  1 atm)
Rb
 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
He: a nobleman?
 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
 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?)
 The largest wdW cluster! 4He2 is a giant, > 50 Å wide!
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
log scale!
from D. Vollhardt & P Wölfle 1990
λ – line specific heat
W. H. Keesom et al (1932)
dPm / dT  S / V
U  TS  PmV
 dP P 
U   m  m  TV
T 
 dT
U s  Ul II , U s  Ul I
(Vs  Vl )
4He
solid vs. liquid II
4He:
a quantum solid
fighting against
Heisenberg’s
indetermination principle
rp  12 
r0
r  r0
p   p 2   2m Ekin
E pot  Ekin
2


8m r02
pressure needed!
in solid Helium unfavorable conditions:
- attractive forces (Epot) are weak
- both m and r0 are small
Classical (MaxwellBoltzmann) statistics
A
B
PA  PB
Quantum BoseEinstein statistics
A
B
4He
PA  5PB
Quantum Fermi-Dirac
statistics
A
B
3He
PA  0
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. ...
the supersurface film
H. Kamerlingh Onnes
(1922)
… 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)
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)
simple ideas about Bose–Einstein condensation (BEC)

N
0e
g ( )
(  B ) / kT
N  g ( ) 
N e
 B / kT
1
d
g ( ) 
4 V
h
3
m3 / 2 (2 )1/ 2
density of states
 e B / kT  1   B  0

 d g ( ) e
 / kT
0
 2 m kT 
V

2
 h

3/ 2 

 0

e
 ( B  ) / kT
 0
A1
(  1)
3/ 2
,
Ae
 B / kT
V,N
1

 2 m kT 
V

2
 h

g ( x)   x  / 
3/ 2
g3 / 2 ( A)
( Bose)
1
g (1)   ( ) ( Riem ann)

E
0e
E
 g ( )
(  B ) / kT
3
 2 m kT 
d  kT V 

2
2
1
 h

3/ 2
g ( A) 3
E 3
A
 kT 5 / 2
 kT [ 1  5 / 2  o( A2 ) ]
N 2
g3 / 2 ( A) 2
2
atom density
N
 2 m kT 
n   A

2
V
 h

total energy
g5 / 2 ( A)
3/ 2
 2 m kT 
 A

2
 h

average energy per atom
 
A 
1  
3/ 2 
  1 (  1) 
3/ 2
3/ 2

 h2  
3
n
   o( A 2 ) 
E  kT 1  
 16   m kT  
2


 o( A 2 )
3
kT
2
condensation on the
ground state
BEC critical temperature & conditions
E 0
h2  n 
k Tc 


2 m  25 / 2 
E  Eclassic
1
2
2/3
2 2 / 3
 1.978
n
m
h2 2 / 3
2 2 / 3
kTc 
n 
n
4 m
m
2/3
BEC :
h  n 
k Tc 
2 m   ( 32 ) 
BEC
1
1  3 
 va  3  
n
 ( 2 )  2 
2
dB  1.993v1a/ 3
2 2 / 3
 3.311 n
m
3/ 2
3dB
dB 
h
3m kT
de Broglie wavelength  2 x interatomic distance
Micromégas, bien meilleur observateur
que son nain, vit clairement que les
atomes se parlaient …
(Voltaire, Micromégas)
de Broglie
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….
3He
condensation
 =
(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

the superfluid
phases of 3He
B-phase: Balia-Werthamer state (isotropic gap (T))
 = |> + 2-1/2[ |> + |>] + |>
A-phase: Anderson-Brinkham-Morel “axial” state
 = |> + |>
^ ^ 2]1/2
anisotropic gap (k,T) = 0(T)[1 – (k·L)
a third phase (A1) is induced by a
magnetic field with spin wavefunction
 = |>
a magnetic superfluid!
End of lecture 2