1. Magic Numbers of Boson Clusters
Shell Effects – Erice 1
Magic Numbers of Boson Clusters
a) He cluster mass selection via diffraction
b) The magic 4He dimer
c) Magic numbers in larger 4He clusters? The Auger evaporation picture
Giorgio Benedek with
J. Peter Toennies (MPI-DSO, Göttingen)
Oleg Kornilov (UCB, Berkeley)
Elena Spreafico (UNIMIB, Milano)

2. Can discriminate against atoms with mass spectrometer set at mass 8 and larger
from J. P. Toennies

3. from J. P. Toennies

4. At Low Source Temperatures New Diffraction Peaks Appear

5. Searching for Large 4He Clusters: 4HeN
from J. P. Toennies
He2+

6. from J. P. Toennies

7. Effective Slit Widths vs Particle Velocity He Atom versus He Dimer
500
1000
1500
2000 56 57 58 59 60 61 62 63 64 E f e c t i v S l W d h s [ n m ]
Particle Velocity v [m/s]
Effective Slit Widths vs Particle Velocity
He Atom
versus
He Dimer
Scattering length a = 2 <R> = 97 A
C =0.12 meV nm 3 He 2
Grisenti, Schöllkopf, Toennies Hegerfeldt, Köhler and Stoll
Phys. Rev. Lett (2000)
=2.5 nm
eff D
T1-Schr. o
V (particle-wall) = C X -
<R> = 52.0 + b ~ 4m
<R>
= K -3 .
0.4 A
=1.1
10-3 K
10-3 K 1
104 A
Grisenti; Schöllkopf, Toennies, Hegerfeldt, Köhler and Stoll, Phys. Rev. Lett (2000)

8. A frail GIANT! The 4He dimer: the world‘s weakest bound and
largest ground state molecule
Since <R> is much greater than Rout the dimer
is a classically forbidden molecule
<R>
A frail
GIANT!
High SR
from J. P. Toennies

9. To Further Study the Dimer it is Interesting
to Scatter from an Object Smaller than the Dimer: an Atom!
A.Kalinin, O. Kornilov, L. Rusin, J. P. Toennies, and G. Vladimirov, Phys. Rev. Lett. 93, (2004)

10. magic! The Kr atom can pass through the middle of the molecule
from J. P. Toennies
The Kr atom can pass through
the middle of the molecule
without its being affected
The dimer is nearly invisible:
magic!
trim
end of lecture 6

11. are there magic numbers or stability regions for boson clusters?
b) Magic numbers (or stability regions)
 Classical noble gas (van der Waals) clusters:
- geometrical constraints only
- magic numbers = highest point symmetry
 Quantum Bose clusters (4He)N are superfluid
- no apparent geometrical constraint
- no shell-closure argument
are there magic numbers or stability regions for boson clusters?
Shell Effects – Erice 2

12. clusters are superfluid!
4He clusters
T0= 6.7K
P0 ≥ 20bar
T= 0.37K
- formed in nozzle beam
vacuum expansion
- stabilized through
evaporative cooling
clusters are superfluid!
Shell Effects – Erice 3

13. Theory (QMC): no magic numbers predicted for 4He clusters!
- R. Melzer and J. G. Zabolitzky (1984)
- M. Barranco, R. Guardiola, S. Hernàndez, R. Mayol, J. Navarro, and M. Pi. (2006)
Binding energy per atom vs. N:
a monotonous slope, with no peaks nor regions of larger stability!
Shell Effects – Erice 4

14. More recent highly accurate diffusion Monte Carlo (T=0) calculation
rules out existence of magic numbers due to stabilities:
Cluster Number Size N
R. Guardiola,O. Kornilov, J. Navarro and J. P. Toennies, J. Chem Phys, 2006

15. Diffraction experiments with neutral (4Ne)N clusters
show instead stability regions!
Shell Effects – Erice 5

16. Magic numbers, excitation levels, and other properties of small neutral
4He clusters
Rafael Guardiola
Departamento de Física Atómica y Nuclear, Facultad de Fisica, Universidad de Valencia, Burjassot,
Spain
Oleg Kornilov
Max-Planck-Institut fur Dynamik und Selbstorganisation, Bunsenstrasse 10, Gottingen, Germany
Jesús Navarro
IFIC (CSIC-Universidad de Valencia), Apartado 22085, Valencia, Spain
J. Peter Toennies

17. R. Brühl, R. Guardiola, A. Kalinin, O. Kornilov, J. Navarro, T
R. Brühl, R. Guardiola, A. Kalinin, O. Kornilov, J. Navarro, T. Savas and J. P. Toennies,
Phys. Rev. Lett. 92, (2004)
Shell Effects – Erice 6

18. The size of 4He clusters R(N) = (1.88Å) N 1/3 + (1.13 Å) / (N 1/3  1)
QMC (V. R. Pandharipande, J.G. Zabolitzky, S. C. Pieper, R. B. Wiringa, and U. Helmbrecht,
Phys. Rev. Lett. 50, 1676 (1973)
R(N) = (1.88Å) N 1/3 + (1.13 Å) / (N 1/3  1)
Shell Effects – Erice 7

19. Single-particle excitation theory of evaporation and cluster stability
spherical box model
Magic numbers!
Shell Effects – Erice 8

20. Atomic radial distributions
4Hen
3Hen
Barranco et al (2006)

21. Fitting a spherical-box model (SBM) to QMC calculations
Condition: same number of quantum single-particle levels
this can be achieved with:
- a(N) = QMC average radius
- V0(N) = μB of bulk liquid
- a constant effective mass m*
From:
Shell Effects – Erice 12

22. this m*/m value works well for all N since
QMC (Pandharipande et al 1988)
the linear fit of QMC shell energies () for (4He)70 rescaled to the bulk liquid μB gives
m*~ 3.2 m
this m*/m value works well for all N since
Shell Effects – Erice 13

23. The Auger-evaporation mechanism
exchange-symmetric two-atom wavefunction

24. 6-12 Lennard-Jones potential
= 40 Å3
C6 = a.u.
d0 < r < R(N)
Integration volume
R(N) = cluster radius
Shell Effects – Erice 10

25. Tang-Toennies potential
Replaced by co-volume (excluded volume)
Shell Effects – Erice 11

26. - Center-of-mass reference
total L = even
μ() = 7.3 K
m* = 3.2  4 a.u.
- Auger-evaporation probability
Shell Effects – Erice 14

27. Ionisation efficiency
Shell Effects – Erice 15
- Cluster kinetics in a supersonic beam
stationary
fission and coalescence neglected:
cluster relative velocity very small
- Cluster size distribution:
- Comparison to experiment:
Jacobian factor
Gaussian spread (s  0.002)
Ionisation efficiency

28. Calculated 4He cluster size distribution at different temperatures
Shell Effects – Erice 16

29. Comparison to experiment I

30. Comparison to experiment II

31. Guardiola et al thermodynamic approach
HeN-1 + He ↔ HeN
Formation-evaporation equilibrium:
Equilibrium constant:
ZN = partition function:
Magic Numbers
at each insertion of a new bound state
Guardiola et al., JCP (2006)
SIF 2008 Genova - 14

33. In conclusion we have seen that…
 High-resolution grating diffraction experiments allow to study the
stability of 4He clusters
 Experimental evidence for the stability of the 4He dimer and the existence of magic numbers in 4He boson clusters
A kinetic theory based on the Auger evaporation mechanism for a spherical-box model qualitatively accounts for the experimental cluster size distributions
 Substantial agreement with Guardiola et al thermodynamic approach:
magic numbers related to the insertion of new bound states with increasing N

34. Electron Microscope Picture of the SiNx Transmission Gratings
Courtesy of Prof. H. Smith and Dr. Tim Savas, M. I. T.

35. Lecture 2: Helium Droplets
Grebenev, Toennies & Vilesov
Science 279, 2083 (1998)

36. Helium Droplets T0 ≤ 35 K P0 ≥ 20 bar Droplets are cooled
by evaporation to
=0.37 K (4He),
=0.15 K (3He)
Brink and Stringari,
Z. Phys. D 15, 257 (1990)

37. Some Microscopic Manifestations of Superfluidity
Free Rotations of Molecules
The Roton Gap (Phonon Wing)
Anomalously Small Moments of Inertia
How many atoms are needed for superfluidity?
How will this number depend on the observed
property?

38. Laser Depletion Spectroscopy

39. OCS
Sharp spectral features indicate
that the molecule rotates without
friction
The closer spacing of the lines
indicates a factor 2.7 larger
moment of inertia
Is this a new microscopic manifestation of superfluidity?

40. 2.Evidence for Superfluidity in Pure 4He Droplets:
Near UV Spectrum of the S S0 Transition of Glyoxal
Since IR absorption lines are so sharp, what about electronic transitions?

41. signature of superfluidity
The experimental sideband reflects the DOS of Elementary Excitations
Roton gap:
signature of superfluidity
rotational lines

42. Magic number in fermionic 3He clusters
(Barranco et al, 2006)
(p + 1)(p + 2)(p + 3)/3
= 2, 8, 20, 40, 70, 112, 168, 240, 330, ...
stable for N > 30

43. Mixed 4He/3He Droplets: Two Production Methods
Small 4He Clusters: N< 100
Large 4He Clusters: 100< N< 5000

44. 4He / 3He phase separation
Barranco et al (2006)
4HeN3He

45. Stable 4He + 3He mixed clusters
Barranco et al (2006) 4 3 2 1 1 3

46. Aggregation of 4He Atoms Around an OCS Molecule
Inside a 3He Droplet
3He
OCS surrounded by a cage of 4He

47. IR Spectra of OCS in 3He Droplets with Increasing Numbers of 4He Atoms
~ 60 He atoms are needed to restore free rotations:
Number needed for superfluidity?
Grebenev Toennies and Vilesov Science, 279, 2083 (1998)

48. According to this criterium 90 4He Atoms are needed for Superfluidity!
The Appearance of a Phonon Wing Heralds the Opening up of the Roton Gap
roton
maxon
Relative Depletion [%]
Wavenumber [cm-1]
According to this criterium 90 4He Atoms are needed for Superfluidity!
Pörtner, Toennies and Vilesov, in preparation

49. maxons: in both 4He and 3He
rotons: in 4He only

50. Space localization  spectral localization!
Localized phonon in 3He at the impurity molecule
Space localization  spectral localization!

51. The localized phonon (LP) is much sharper than the bulk phonon width!

52. electron – collective excitation coupling
molecule
He atoms
spatial decay of molecule electronic wavefunctions

53.  E = E(q) Inelastic part of dipolar matrix element:
Sideband absorption coefficient:
Dynamic form factor:
interatomic potential
Response function:
non-interacting atoms
 E = E(q)
Collective excitations:

54. “Shell” model for dynamics
Barranco et al
“Shell” model for dynamics n
n +1

55. particle-hole excitation spectrum
collective excitation (phonon) spectrum

57. Para-Hydrogen Has Long Been A Candidate for Superfluidity

58. Non-condensed
Bose condensed

59. The reduced coordination In small droplets favors superfluid response
para-Hydrogen
Decrease in the moment
of inertia indicates
superfluidity
The reduced coordination
In small droplets favors
superfluid response
cartoon H2 on OCS

60. Aggregation of p-H2 molecules around an OCS molecule inside a mixed 4He/3He droplet

62. (5-6 H2)
(5-6 H2)
(3-4 H2)
(3 H2)

63. Average Moments of Inertia
Ia Ib Ic
This is the first evidence
for superfluidity of p-H2