Magical helium clusters

Physicists are well accustomed to the concept of magic numbers in size distributions. They crop up in many contexts, ranging from condensed matter physics to nuclear physics. Usually, such magic numbers correspond to states of enhanced ground state stability, and this is often also true for small atomic clusters, e.g. at the completion of structural shells in Van der Waals clusters, or when valence electrons fill closed shells in metal clusters. There are explicit theoretical predictions, however, that magic numbers of this kind should not occur for He clusters because the ground state properties are calculated to change smoothly with the number of atoms in the cluster. So it seems astonishing at first sight that Peter Toennies and colleagues in Göttingen, Burjassot, Valencia and MIT should have recently reported the observation of magic numbers for helium clusters (Physical Review Letters 92, 185301). Their experiment involved the creation helium clusters by free expansion of fluid helium at cryogenic temperatures, using the apparatus shown schematically in the figure. The collimated beam of clusters thus created was diffracted from a nanostructured transmission grating with a period of 100 nm, treating the clusters as matter waves. Because the de Broglie wavelength is inversely proportional to momentum – or, in this case also to mass because all the clusters were moving at the same velocity – the different cluster sizes diffracted at different angles enabling them to be distinguished. The cluster detector was a mass spectrometer. By plotting the mass spectrometer signal as a function of diffraction angle, the size distributions of the clusters could be plotted out with excellent resolution as shown in the lower part of the figure. It is immediately evident that, contrary to expectation, certain cluster sizes appear to be favoured. The authors carried out several tests to check these results and to confirm that peaks were genuine, i.e. independent of conditions in the source region (where the authors varied the temperature and pressure) and in the detector (where they varied the ionizing electron impact energies). They also varied the angle of the diffraction grating relative to the incident beam of clusters. They conluded that there were definitely magic numbers corresponding to cluster sizes of 10/11, 14, 22, 26/27 and 44 atoms. Why should this be? Preference for particular magic cluster sizes must presumably reflect events taking place during cluster creation in the source region. Usually, one would assume that these preferred sizes were more stable, and therefore energetically favoured during the creation processes. But this seems to be ruled out by the ground state energy calculations. However, there is no reason why stabilty considerations should be applied only to the ground state. Given the “heat” and turmoil leading to cluster production it seems highly plausible that they will often be created in excited states. Thus the stability of the latter may also be relevant. So the authors calculated the energies of excited states of the cluster corresponding to radial modes and finite angular momentum. Their calculation was non-trivial, but seems to have produced some very interesting results. They compared the excitation energies with the chemical potential, i.e. with the binding energy of a single additional atom to