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)

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

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! Magic Numbers– Ratisbona 3

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

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! Magic Numbers– Ratisbona 5

Can discriminate against atoms with mass spectrometer set at mass 8 and larger from J. P. Toennies Ratisbona 7

Electron Microscope Picture of the SiNx Transmission Gratings Courtesy of Hank Smith and Tim Savas, MIT Ratisbona 7

Searching for Large 4He Clusters: 4HeN from J. P. Toennies Ratisbona 8

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, 46100 Burjassot, Spain Oleg Kornilov Max-Planck-Institut fur Dynamik und Selbstorganisation, Bunsenstrasse 10, 37073 Gottingen, Germany Jesús Navarro IFIC (CSIC-Universidad de Valencia), Apartado 22085, 46071 Valencia, Spain J. Peter Toennies Ratisbona 9

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, 185301 (2004) Ratisbona 10

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) Ratisbona 11

Single-particle excitation theory of evaporation and cluster stability spherical box model Magic numbers! Ratisbona 12

Atomic radial distributions 4Hen Barranco et al (2006) 3Hen Ratisbona 13

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: Ratisbona 14

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 Ratisbona 15

fair linearity also for 3He clusters

The Auger-evaporation mechanism exchange-symmetric two-atom wavefunction Ratisbona 16

6-12 Lennard-Jones potential = 40 Å3 C6 = 1.461 a.u. d0 < r < R(N) Integration volume R(N) = cluster radius Ratisbona 17

Tang-Toennies potential Replaced by co-volume (excluded volume) Ratisbona 18

- Center-of-mass reference total L = even μ() = 7.3 K m* = 3.2  4 a.u. - Auger-evaporation probability Ratisbona 19

Ionisation efficiency Ratisbona 20 - 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

Calculated 4He cluster size distribution at different temperatures Ratisbona 21

Comparison to experiment I Ratisbona 22

Comparison to experiment II dependence on choice of potential bottom [remember: m*(N) ~ const/V(N)] Ratisbona 23

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) Ratisbona 24

Ratisbona 25

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

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) Ratisbona 26

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 Ratisbona 27

i.e., The actual amplitudes of n*(N) oscillations are smaller than those calculated for n(N,Tf) : Ratisbona 28

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 Ratisbona 28