Mott criterion

In their model, N is the local electron density measured over an appropriate volume, which they argue is given by the Mott criterion discussed in Sect. 3.4.1. Thus, with the Mott radius of 20 A for uniform spheres, these spheres will constitute the appropriate volume over which to measure the local electron density N. 

Mott first gave an estimate (125) of this critical density at the NM-M transition in terms of the screening properties of an itinerant electron gas. The "Mott criterion is... 

The Mott Criterion for Metal Solutions and Expanded Fluid Metals"... 

Among the phase transitions where electronic factors play a major role, the most well-known are the metal-insulator transitions exhibited by transition-metal oxides, sulfides, and so on. This subject has been discussed at length.2,23,24 A recent observation26 of some interest is that the metal-nonmetal transition occurs at a critical electron concentration as given by the particular form of the Mott criterion, = 0.26 ... 

There appears to be little doubt that the Mott criterion given by eq 3 is an effective indicator of the critical condition at the M-NM transition itself. At the least, this simple criterion provides a numerical prediction for the metal-nonmetal transition in many situations. Figure 10 summarizes some of the experimental data.34-36 Interestingly, besides doped semiconductors, metal—ammonia and metal—noble gas systems and superconducting cuprates all follow the linear relation given by eq 3. This is truly remarkable. 

The electronic bands of an infinite crystal can cross as a function of some parameter (pressure, concentration etc.). If one treats the e /r,2 term of the electron repulsion correctly, one sees that the crossing transition of the two bands is a first-order phase transition, between the metallic and insulating states. This transition was predicted by Mott in 1946 and has carried his name ever since. In fact, the original Mott criterion does not predict such a transition for Hg, but the criterion was derived for monovalent atoms. For divalent mercury it should not be applicable. Also the semiempirical Herzfeld criterion, which was very successful in predicting the insulator to metal transition in compressed xenon, predicts bulk Hg to be non-metallic. All this seems to imply that Hg is a rather special case. 

It is reasonable to speculate that the differences in elemental densities at the MNM transition are related to characteristic atomic properties. One such property, for example, is the radius of the principal maximum in the charge density of the ns valence orbital, a which enters into the Mott criterion (Section 2.3.4). A related property is the static polarizability a of the isolated atom. The polarizability formed the basis of very early discussions of the MNM transition by Goldhammer (1913) and Herzfeld (1927). They pointed out that electrons localized around atomic nuclei constitute polarizable objects and their internal dynamics in dense assemblies leads to local corrections to the polarizing tendency of any external field impressed on the system. For an isotropic material, the correction factor has the form [1 — (4Tr/3)lVa] where N is the number of atoms per unit volume. If a is taken to remain roughly... 

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