Magic number and quasi-melting temperature discovered in crystal growth and surface phase transitions

For crystal growth and surface phase transitions, we discovered a simple rule in the relationship between surface structure and temperature. This rule states that changes in crystal growth modes and those in surface structures occur at temperatures that are simple fractions, such as 12, 23, and 34, of the melting temperature, Tm, of the material. Here, we first show our experimental results of transition temperatures for Si-MBE (molecular beam epitaxy) on Si(111) and Si(001) substrates and for structure changes on Si(111) surface. We then discuss them for other materials from the literature, having fcc (face centered cubic) or diamond structures, which are similar crystal structures to Si. We found that the transition temperatures were the same, regardless of material, depending on crystal surface orientation. That is, transitions take place at the temperatures of 26, 36, 46, 56 Tm for (111) surfaces, and 14, 24, 34 Tm for (001) surfaces, when transition temperatures are normalized to the melting temperature of each material. We discuss the origin of these fraction temperatures by consideration of surface atom energy based on the geometry of atom arrangement on a crystal surface. We think that the reason why transition temperatures were same is due to the same coordination numbers of the respective surface atoms if material has similar crystal structure even if materials differ. We consider these fractions are one kind of magic numbers, and also call these temperatures as quasi-melting temperatures, from the analogy of the melting for bulk crystal. For example, at 12 Tm, atoms bound at kink sites become free, i.e. kink melting occurs, since kink-site atoms have half the coordination number of bulk atoms. Finally, we point out that it is important for understanding various surface phenomena to consider the role of surface-defects such as terraces, ledges, and kinks.