"With free nanoparticles the melting point may be as little as half that of the bulk material," says Ulrich Dahmen, head of the National Center for Electron Microscopy (NCEM), who discovered the "magic size" phenomenon while working with collaborators from the University of Copenhagen, "while a crystal embedded in a matrix of a different solid may need to be superheated to melt. This is like an ice cube refusing to melt in boiling water."

The researchers used NCEM's million-volt Atomic Resolution Microscope to make micrographs that showed numerous nanometer-sized islands of lead in a shallow aluminum sea—islands that assumed a few different shapes and a range of discrete sizes.

"What surprised us was that some sizes were preferred, while others were avoided," says Dahmen. "At first it was puzzling, but on reflection it made perfect sense."

Free crystals take shapes that minimize their surface energy; embedded crystals have to conform to their neighbors in the solid matrix. Both internal elastic energy and interface energy must be minimized, and because a lead atom is about 20 percent larger than an aluminum atom, compromise is a must.

"Putting lead atoms into aluminum has been likened to shoving grapefruits among oranges," Dahmen says. "If you try to replace them three for three, say, you expend a lot of energy squashing the grapefruits. But replace five oranges with four grapefruits and they fit reasonably well. With lead and aluminum, we found that the most preferred fit was nine for eleven." Magic sizes correspond to regularly repeating values of minimal strain energy.

The equilibrium shape of a large inclusion is a symmetrical crystal, but many lead inclusions, especially the smaller ones, are asymmetrical. By adopting asymmetrical shapes, small inclusions can maintain perfectly flat interfaces with the host matrix. This in turn suggests why smaller inclusions have to be superheated to melt: more heat energy is needed to overcome the constraint at the perfect interface of an asymmetrical, magic-size inclusion than at the imperfect interface of a symmetrical one.

This dependence of melting temperature on size did not hold, however, when Dahmen, joined by colleagues from Denmark and France, turned his attention to inclusions of lead and cadmium—a "eutectic" alloy, one that melts at a conveniently low temperature.

Neither lead not cadmium is soluble in aluminum, and they refuse to mix as solids; when melted together and recooled, they separate again, much like metal in a solder joint. As inclusions, each assumes its own characteristic crystal structure, a disk of hexagonal cadmium neatly joining a cuboctahedral chunk of lead along parallel close-packed planes.

These tiny "ingots" in their aluminum "crucible," as big as 50 nanometers across to as small as a single nanometer, all melt at the same temperature. Melting always starts at the triple junction where the crystal planes of the aluminum, lead, and cadmium meet, and always proceeds down the plane of the lead-cadmium interface.

The nano-inclusions melt at about the same temperature as the bulk alloy, and have never been observed to require superheating, no matter how small. Why not?

The melting starts at the interface of the three metals—a one-dimensional line. Thus the planar interfaces of the inclusions with their matrix play a negligible role in determining the energy needed for melting—although they must be cooled an extra 50 to 65 degrees Kelvin to resolidify.

"We can cycle the melting and solidification back and forth as we watch through the electron microscope, observing how the transformation works at the near-atomic scale," says Dahmen. "Few experiments in metallurgy are so reproducible."