Herzfeld criterion

The criterion propounded by Herzfeld in 1927 is instructive in understanding the metallic state. According to the Herzfeld criterion, a material is metallic when R/V I and insulating when R/V < 1. Here R is the molar refractivity of the atomic species in the gaseous state and V is the molar volume in the condensed state. This criterion explains the metallic character of the elements in nature (Edwards Sienko, 1983). Rao Ganguly (1986) have shown that the latent heat of vaporization provides a satisfactory criterion to delineate metallic elements from nonmetallic ones. 

When the condition (RIV) = 1 is fulfilled, the refractive index/ dielectric constant goes to infinity, and we have a NM-M transition. The Herzfeld criterion when applied to metal-ammonia solutions does indeed predict (67,93) that localized, solvated electrons are set free by mutual action of neighboring electrons at metal concentrations above 4-5 MPM, and measurements (Fig. 20) similarly indicate a dielectric catastrophe in this concentration range. The simple Herzfeld picture has recently been applied by Edwards and Sienko (70) to explain the occurrence of metallic character in the Periodic Table. 

The earlier theoretical prediction of a M—NM transition is that derived from the work of Goldhammer23 and Herzfeld.14 These authors considered the effect of increasing density on the atomic polarizability and suggested that there would be a divergence in the polarizability or the dielectric constant causing the release of bound electrons. The Herzfeld criterion for dielectric catastrophe is given by. 

Fujii et al.30 have reported experimental evidence for the molecular dissociation process in Br2 near 80 GPa. This transition, which is coincident with the onset of pressure-induced metallization, was first discovered in molecular/metallic iodine.3 A diatomic molecular crystal loses its molecular character in the limit when the intermolecular distance becomes equal to the intramolecular bond length. Fujii et al.30 applied the Herzfeld criterion to I2 and Br2 and estimated that the molar reffactivity reaches the atomic limit around 20 GPa in I2 and 80 GPa in B12. In both cases, the computed pressure coincides with that for molecular dissociation accompanied by metallization. 

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. 

Fig. 3.22. DC electrical conductivity of fluid cesium (Hensel et al., 1985, 1991), rubidium (Freyland, 1981), and hydrogen (Weir et al., 1996), versus atomic density. Vertical arrows indicate predicted metallization densities according to the Goldhammer-Herzfeld criterion described in the text.

In 1927, K. F. Herzfeld published a paper (93) in which he proposed a simple criterion for determining when an element or system will ex-... 

F. O. Rice and K. F. Herzfeld, J, Phys, Colloid Chem.y 65, 975 (1951). As pointed out in this article such a criterion may be deceptive in that it may simply mean that the reaction may still be heterogeneous but that the rate is not sensitive to the surface/volume ratio. This will happen in chain reactions if chains are both initiated and destroyed at the walls. 

One more experimental method of characterizing the metallic state is to compare the volumes and refractions (R) of solids. As the refractive indices of metals are very great, the Lorentz-Lorenz function (Eq. 2.18) is close to 1 and V. According to the Goldhammer-Herzfeld [127, 128] criterion, V- R when a dielectric converts into a metal. As the measure of bond metallicity, the ratio... 

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... 

---