The Mott-Hubbard Metal-Insulator Transition

A concept related to the localization vs. itineracy problem of electron states, and which has been very useful in providing a frame for the understanding of the actinide metallic bond, is the Mott-Hubbard transition. By this name one calls the transition from an itinerant, electrically conducting, metallic state to a localized, insulator s state in solids, under the effect of external, thermodynamic variables, such as temperature or pressure, the effect of which is to change the interatomic distances in the lattice. 

This transition has been emphasized by Mott for the case of localized impurity states in a semiconductor, forming a metallic band at some concentration of impurities (i.e. at some average distance between the impurities). It is referred to very often as the Mott (or Mott-Hubbard) transition. 

The reason why it is a useful concept in actinide solid state physics (as in d-transition metals) is that it describes the effect of overlapping in forming a band therefore, in d or f unfilled bands, a critical value of the inter-atomic distance should determine whether the electron states will be acting as localized (insulator s) states, or as an unfilled (metallic) band. If we refer to the concept of f-f-overlapping, it is thinkable that across a series such as the actinides the f-f-overlapping might come to a critical value distinguishing between band and localized behaviour. 

In order to understand the concept, it is customary to present a sort of paradox arising from the solution of the Hamiltonian (11) in the tight-binding approximation (Eq. (12)). 

Suppose we approach M atoms of an element having an unfilled outer shell, disposed in a lattice. The point may be made clear if we suppose to approach hydrogen atoms (outer shell 1 s ). Equations (11) and (12) would predict the broadening of the electron state in a half-filled s-band, which should therefore allow metallic behaviour. Apparently, this would happen for any inter-atomic distance a and, therefore also at infinite distance. What would change is, of course, the bandwidth, which is determined by matrix elements dependent on the interatomic distance a but the metallic behaviour, depending essentially on the fact that the electrons have available energy states within the band, should occur also at distances where the atoms may well be supposed to be isolated. 

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