Assuming a spherical mean field for electrons in metal clusters, single-particle level densities and electronic binding energies are calculated for clusters with up to 4000 valence electrons. Two phenomenological mean-field potentials, simulating microscopically calculated ones, are used. A global beating pattern, which envelopes individual shell oscillations, emerge from the calculations. The semiclassical interpretation of such a supershell structure, as proposed by Balian and Bloch in terms of interference of amplitudes associated with classical closed orbits, is found to be valid in the present case. Thermal effects, which tend to smear out shell and, therefore, supershell structures, are investigated qualitatively. Consequences of the shell structure are not obscured for cluster sizes up to several thousand atoms under the experimentally accessible temperature of 100–1000 K.

Statistical fragmentation can give valuable information on the internal structure of clusters due to the strong dependence of the evaporation rate on the activation energy for the process. In particular, the abundances in an ensemble of hot clusters will reflect the relative stabilities of the individual species. We derive the relation between evaporative activation energies and relative abundances for large clusters. Quantitative criteria for the applicability of the theory are derived and the analytical results are compared with numerical simulations.

Experimental mass abundance spectra are used to extract evaporative activation energies (dissociation energies) for protonated water clusters, ${\left({\text{H}}_2\text{O}\right)}_N{\text{H}}^{+}$ , and deprotonated water clusters, ${\left({\text{H}}_2\text{O}\right)}_N{\text{OH}}^{-}$ , in the size range up to hundred molecules. The inversion is achieved by application of the shell correction method adapted from nuclear physics to the abundance spectra. The well known abundance anomaly for protonated clusters which occurs for $N=20–22$ is found to have the characteristic behavior of a shell closing, whereas other apparent magic numbers are only prominent peaks in the abundance spectra because of the instability of the evaporative precursor. For the deprotonated clusters, we find a similar shell closing for $N=55–56$ .