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Physical Review Letters

Diffraction of Neutral Helium Clusters: Evidence for “Magic Numbers”

Rüdiger Brühl, R. Guardiola, Anton Kalinin, Oleg Kornilov, Jesús Navarro, Tim A. Savas, Jan Peter Toennies

Published May 6, 2004• 65 citations
Publisher

Abstract

The size distributions of neutral 4He clusters in cryogenic jet beams, analyzed by diffraction from a 100 nm period transmission grating, reveal magic numbers at N=10-11, 14, 22, 26-27, and 44 atoms. Whereas magic numbers in nuclei and clusters are attributed to enhanced stabilities, this is not expected for quantum fluid He clusters on the basis of numerous calculations. These magic numbers occur at threshold sizes for which the quantized excitations calculated with the diffusion Monte Carlo method are stabilized, thereby providing the first experimental confirmation for the energy levels of 4He clusters.

DOI
10.1103/physrevlett.92.185301

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The size distributions of neutral 4He clusters in cryogenic jet beams, analyzed by diffraction from a 100 nm period transmission grating, reveal magic numbers at N=10-11, 14, 22, 26-27, and 44 atoms. Whereas magic numbers in nuclei and clusters are attributed to enhanced stabilities, this is not expected for quantum fluid He clusters on the basis of numerous calculations. These magic numbers occur at threshold sizes for which the quantized excitations calculated with the diffusion Monte Carlo method are stabilized, thereby providing the first experimental confirmation for the energy levels of 4He clusters.
He clusters predicted by all theories since 1984 has been challenged by high-resolution matter-wave diffraction experiments. The observed magic numbers were explained in terms of enhanced growth rates of specific cluster sizes for which an additional excitation level calculated by diffusion Monte Carlo is stabilized. The present theoretical study provides an alternative explanation based on a simple independent particle model of the He clusters. Collisions between cluster atoms in excited states within the cluster lead to selective evaporation via an Auger process. The calculated magic numbers as well as the shape of the number distributions are in quite reasonable agreement with the experiments.
Small weakly bound boson-fermion 4Hem3Hen^{4}\mathrm{He}_{m}^{3}\mathrm{He}_{n}4Hem3​Hen​ clusters formed in a free jet expansion are identified using nondestructive transmission grating diffraction. The observations confirm the existence of more than 11 very tenuous complexes including the three-body halo molecule 4He23He^{4}\mathrm{He}_{2}^{3}\mathrm{He}4He23​He and the pseudo-Borromean complex 4He23He2^{4}\mathrm{He}_{2}^{3}\mathrm{He}_{2}4He23​He2​. Effective cluster formation temperatures, extracted from a sudden freeze model for cluster growth using theoretical binding energies, increase smoothly with cluster size, thereby confirming the calculations with the possible exception of 4He23He2^{4}\mathrm{He}_{2}^{3}\mathrm{He}_{2}4He23​He2​.
Clusters of He atoms formed in a supersonic free jet expansion have been studied by electron-bombardment ionization and mass spectrometry. Distinct magic-number enhancements of ion intensity are observed for clusters of 7, 10, 14, and 30 atoms for both helium isotopes, and for 23 atoms of 4He^{4}\mathrm{He}4He. Dramatic differences in the onset of cluster formation between the two isotopes may be related to the stability of 4He^{4}\mathrm{He}4He dimers.
Diffuse clusters of arbitrary numbers of helium atoms are treated in a mean-field theory of the many-body system. The nonlinear Hartree self-consistent-field equations are solved numerically to give their energy eigenvalues and eigenfunctions. Both isotopes 4He and 3He are treated according to their bosonic and fermionic statistics and symmetry requirements. Clusters of all sizes of 4He atoms are stable, but there is a minimum threshold of ∼70 atoms for the formation of 3He clusters. All the excited states in addition to the ground state were found. The wave functions are seen to have interesting differences from electronic wave functions in atoms.
The size of the helium trimer is determined by diffracting a beam of 4He clusters from a 100 nm period grating inclined by 21 degrees. Because of the bar thickness the projected slit width is roughly halved to 27 nm, increasing the sensitivity to the trimer size. The peak intensities measured out to the eighth order are evaluated via a few-body scattering theory. The trimer pair distance is found to be (r) = 1.1(+0.4)(-0.5) nm in agreement with predictions for the ground state. No evidence for a significant amount of Efimov trimers is found. Their concentration is estimated to be under 6%, less than expected.
A stability chart of mixed 4He{}^{4}\mathrm{He}4He and 3He{}^{3}\mathrm{He}3He clusters has been obtained by means of the diffusion Monte Carlo method, using both the Aziz HFD-B and the Tang-Toennies-Yiu atom-atom interaction. The investigated clusters contain up to eight 4He{}^{4}\mathrm{He}4He atoms and up to 20 3He{}^{3}\mathrm{He}3He atoms. One single 4He{}^{4}\mathrm{He}4He binds 20 3He{}^{3}\mathrm{He}3He atoms, and two 4He{}^{4}\mathrm{He}4He bind 1, 2, 8, and more than 14 3He{}^{3}\mathrm{He}3He atoms. All clusters with three or more 4He{}^{4}\mathrm{He}4He atoms are bound, although the combinations 4He33He9,10,11{}^{4}{\mathrm{He}}_{3}^{3}{\mathrm{He}}_{9,10,11}4He33​He9,10,11​ and 4He43He9{}^{4}{\mathrm{He}}_{4}^{3}{\mathrm{He}}_{9}4He43​He9​ are metastable. Clusters with 2, 8, and 20 3He{}^{3}\mathrm{He}3He atoms are particularly stable and define magic 3He{}^{3}\mathrm{He}3He numbers.