1. Magic Numbers– Ratisbona 1
Magic Numbers of Boson 4He Clusters:
the Auger evaporation mechanism
Giorgio Benedek with
Elena Spreafico (UNIMIB, Milano)
J. Peter Toennies (MPI-DS, Göttingen)
Oleg Kornilov (MBI-NOK, Berlin)

2. are there magic numbers or stability regions for boson clusters?
 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?
Magic Numbers– Ratisbona 2

3. 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!
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4. Magic Numbers– Ratisbona 4
More recent highly accurate diffusion Monte Carlo (T=0) calculation
rules out existence of magic numbers due to stabilities:
R. Guardiola,O. Kornilov,
J. Navarro and J. P. Toennies,
J. Chem Phys, 2006
Cluster Number Size N
Magic Numbers– Ratisbona 4

5. 4He clusters - formed in nozzle beam vacuum expansion
T0= 6.7K
P0 ≥ 20bar
T= 0.37K
- formed in nozzle beam
vacuum expansion
- stabilized through
evaporative cooling
clusters are superfluid!
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6. Can discriminate against atoms with mass spectrometer set at mass 8 and larger
from J. P. Toennies
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7. Electron Microscope Picture of the SiNx Transmission Gratings
Courtesy of Hank Smith and Tim Savas, MIT
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8. Searching for Large 4He Clusters: 4HeN
from J. P. Toennies
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9. Diffraction experiments with (4He)N clusters show stability regions!
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
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10. 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)
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11. 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)
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12. Single-particle excitation theory of evaporation and cluster stability
spherical box model
Magic numbers!
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13. Atomic radial distributions
4Hen
Barranco et al (2006)
3Hen
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14. 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:
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15. 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
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16. fair linearity also
for 3He clusters

17. The Auger-evaporation mechanism
exchange-symmetric two-atom wavefunction
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18. 6-12 Lennard-Jones potential
= 40 Å3
C6 = a.u.
d0 < r < R(N)
Integration volume
R(N) = cluster radius
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19. Tang-Toennies potential
Replaced by co-volume (excluded volume)
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20. - Center-of-mass reference
total L = even
μ() = 7.3 K
m* = 3.2  4 a.u.
- Auger-evaporation probability
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21. Ionisation efficiency
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- 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

22. Calculated 4He cluster size distribution at different temperatures
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23. Comparison to experiment I
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24. Comparison to experiment II
dependence on choice
of potential bottom
[remember: m*(N) ~ const/V(N)]
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25. 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)
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26. Ratisbona 25

27. Do magic numbers persist in doped clusters?
larger stability
= larger density
= larger renormalization of the dopant
effective momentum of inertia I*
same oscillations?

28. For each N there is a distribution n(N,T) derived from P(N,T)
Auger evaporation lowers T,
and P = P(N,T) changes with N and T
T = 0.37 K
For each N there is a distribution n(N,T) derived from P(N,T)
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29. For each evaporation process N+1 → N :
The cooling for the evaporation sequence N’→N is then
e.g., for a cooling T0 = 6.7 K → Tf = 0.37K
and |V0|= 7.3 K : N = N’/2.38
in principle the distribution n*(N)
contains clusters of all temperatures < T0
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30. i.e.,
The actual amplitudes of n*(N) oscillations are smaller than those calculated for n(N,Tf) :
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31. 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
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