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Physical Review Letters

Determination of the Bond Length and Binding Energy of the Helium Dimer by Diffraction from a Transmission Grating

R. E. Grisenti, Wieland Schöllkopf, Jan Peter Toennies, Gerhard C. Hegerfeldt, Thorsten Köhler, M. Stoll

Published Sep 11, 2000• 326 citations
Publisher

Abstract

A molecular beam consisting of small helium clusters is diffracted from a 100 nm period transmission grating. The relative dimer intensities have been measured out to the 7th order and are used to determine the reduction of the effective slit width resulting from the finite size of the dimer. From a theoretical analysis of the data which also takes into account the van der Waals interaction with the grating bars, the bond length (mean internuclear distance) and the binding energy are found to be $〈r〉\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}52\ifmmode\pm\else\textpm\fi{}4\AA{}$ and $|{E}_{b}|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.1+0.3/\ensuremath{-}0.2$ mK.

DOI
10.1103/physrevlett.85.2284

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A molecular beam consisting of small helium clusters is diffracted from a 100 nm period transmission grating. The relative dimer intensities have been measured out to the 7th order and are used to determine the reduction of the effective slit width resulting from the finite size of the dimer. From a theoretical analysis of the data which also takes into account the van der Waals interaction with the grating bars, the bond length (mean internuclear distance) and the binding energy are found to be 〈r〉=52\ifmmode±\else\textpm\fi4A˚〈r〉\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}52\ifmmode\pm\else\textpm\fi{}4\AA{}〈r〉=52\ifmmode±\else\textpm\fi4A˚ and ∣Eb∣=1.1+0.3/\ensuremath−0.2|{E}_{b}|\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.1+0.3/\ensuremath{-}0.2∣Eb​∣=1.1+0.3/\ensuremath−0.2 mK.
The relative transmissions of helium dimer and helium atom beams through a set of nanoscale sieves were measured as a function of hole size in the range from 98–410 nm. From the relative transmission coefficients, the mean internuclear distance of helium dimer was determined to be 〈r〉=62±10 Å. This enormous bond length—by far the largest known—confirms recent theoretical estimates of the potential energy well depth ε and the extremely small binding energy Eb of helium dimer. The ranges of these parameters corresponding to the experimental uncertainty in 〈r〉 are ε/k=10.88–10.98 K and Eb/k=0.65–1.30 mK, where k is the Boltzmann constant.
Molecular beams of rare gas atoms and D2{\mathrm{D}}_{2}D2​ have been diffracted from 100-nm-period SiNx{\mathrm{SiN}}_{x}SiNx​ transmission gratings. The relative intensities of the diffraction peaks out to the eighth order depend on the diffracting particle and are interpreted in terms of effective slit widths. These differences have been analyzed by a new theory which accounts for the long-range van der Waals ${\ensuremath{-}C}_{3}/{l}^{3}$ interaction of the particles with the walls of the grating bars. The values of the C3{C}_{3}C3​ constant for two different gratings are in good agreement and the results exhibit the expected linear dependence on the dipole polarizability.
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.
Van der Waals attraction between helium atoms can lead to the formation of weakly-bound He 2 diatomic molecules stable at temperatures in the millikelvin range. Helium dimers were first detected experimentally in 1993 and 1994. Sophisticated quantum-mechanical computations confirm that the He–He bond is the weakest known, with a binding energy corresponding to 1.176 mK, about 10 -4 eV. The average internuclear distance is computed to be in excess of 50 Å, which makes He 2 the largest known diatomic molecule as well. The authors have introduced an idealized deltafunction model for the helium dimer that succinctly represents its remarkable properties. Despite the weakness of each individual He–He bond, an intriguing mechanism exists for helium to form trimers and larger clusters with much greater binding energies.
The advance of matter optics, thus the coherent scattering of an atom or molecule beam from a nanostructured grating, recently gave conclusive evidence for the extremely weakly bound diatomic and triatomic van der Waals clusters of helium in a precision experiment. A quantum mechanical multi-channel many-body scattering theory, as it is presented and extended in this work, is essential for the quantitative description of such an experiment. In order to enhance the resolution of the most recent measurements the grating was tilted with respect to the incident beam. At first, the asymmetry of the diffraction pattern caused by this was theoretically studied for the case of atom diffraction. Subsequently, the ground state of the helium trimer was investigated and its bond length determined to be about 1 nm. Another part of this work considers inelastic diffraction and shows, using small hydrogen dimers as an example, that matter optics allows, in principle, for the spectroscopy of weakly bound clusters.
In atom optics, optical wave theory has been very successful in describing the diffraction of atoms by microfabricated structures. For a molecule, however, with a binding energy smaller than the kinetic energy and with a non-negligible diameter, wave optics may no longer be adequate due to the effects of finite size and breakups. To study the role of the two effects we consider the diffraction of weakly bound diatomic systems such as the helium dimer. We find striking deviations from classical optics, in particular from Babinet's principle, and use the results to clarify the role of finite size and breakups, respectively, in the diffraction from strips, slits, and transmission gratings. As an application the determination of the He2{\mathrm{He}}_{2}He2​ size by transmission through holes and by diffraction experiments is investigated.