Inert-Gas Solids

and the elements directly below it in the periodic table or the Solid State Table, form the simplest closed-shell systems. The electronic structure of the inert-gas solid, which is face-ccntercd cubic, is essentially that of the isolated atoms, and the interactions between atoms are well described by an overlap interaction that includes a correlation energy contribution (frequently described as a Van der Waals interaction). The total interaction, which can be conveniently fitted by a two-parameter Lennard-Jones potential, describes the behavior of both the gas and the solid. Electronic excitations to higher atomic states become excitons in the solid, and the atomic ionization energy becomes the band gap. Surprisingly, as noted by Pantelides, the gap varies with equilibrium nearest-neighbor distance, d, as d 

We may think of the inert gases as neutral atoms with their electronic structure 

Their weak interatomic interaction is responsible for the condensation of the normal inert gases into solids. Atoms of normal inert gas are brought together until the repulsive terms in the overlap interaction prevent further contraction. The attraction favors a close-packed structure, and all of the normal inert gases form face-centered cubic lattices. These two contributions to the total interaction will remain almost the same in the ionic crystals, but with added Coulomb interactions, so it is desirable to understand all of these contributions with some care. 

We wish to discuss interatomic interactions at two levels first, from the detailed though approximate quantum-mechanical calculation by Gordon and Kim second, in terms of the parameterized model of Lennard-Jones, which will be useful for approximate calculations in the ionic solids as well as the inert-gas solids. 

The interatomic interaction for neon. The. solid line is the overlap interaction calculated by Gordon and Kim (1972). The dashed line is the Lennard-Joncs interaction with parameters determined by Bernardes (1958). 

---

Extensive computer simulations have been caiTied out on the near-surface and surface behaviour of materials having a simple cubic lattice structure. The interaction potential between pairs of atoms which has frequently been used for inert gas solids, such as solid argon, takes die Lennard-Jones form where d is the inter-nuclear distance, is the potential interaction energy at the minimum conesponding to the point of... 

In Chapter 7.4, empty reactor volume determination of a perfect CSTR is described by following the discharge concentration from the sudden step-change injection of a non-adsorbing inert gas (solid line in the picture.) Next the same experiment is discussed if made with a chemisorbing gas and shown on the previous picture with a dotted line. In this second case, the reactor... 

The systems we have in mind here are pure monatomic solids such as metals and inert gas solids which contain only one kind of defect in a single electronic state. We consider only vacancies explicitly, but rather similar expressions hold for systems containing only interstitials. From Eqs. (74a) and (75) we find that at equilibrium... 

A special case of closed-shell configurations are the ions of zero charge, the inert gases themselves. The inert-gas solids may not be of great intrinsic interest, but they provide the best starting point for the understanding of ionic solids, so we discuss them first. We shall then see how the ionic solids can be understood in terms of transfer of protons between nuclei, much as we understood the polar covalent solids in terms of transfer of protons between the nuclei of the homopolar solids. 

To the extent that the electronic structure is describable in terms of independent atoms, the properties of inert-gas solids are easily understandable and not so interesting. There are, however, one or two points that should be made. The optical absorption spectra of isolated atoms consists of sharp lines that correspond to transitions of the atom to excited slates, and to a continuous spectrum of absorption beginning at the ionization energy and continuing to higher energy. The experimental absorption spectra of inert-gas solids (Baldini, 1962) also show fairly sharp lines corresponding to transitions from the valence p states to excited s... 

The spectra may also be described in the language of solid state theory. The atomic excited states are the same as the excitons that were described, for semiconductors, at the close of Chapter 6. They are electrons in the conduction band that are bound to the valence-band hole thus they form an excitation that cannot carry current. The difference between atomic excited states and excitons is merely that of different extremes the weakly bound exciton found in the semiconductor is frequently called a Mott-Wannier exciton-, the tightly bound cxciton found in the inert-gas solid is called a Frenkel exciton. The important point is that thecxcitonic absorption that is so prominent in the spectra for inert-gas solids does not produce free carriers and therefore it docs not give a measure of the band gap but of a smaller energy. Values for the exciton energy are given in Table 12-4. 

Pantclides (1975c) also discussed the valence bands for the inert-gas solids, indicating that they consist of a narrow p band and an s band, which may be taken as completely sharp. He gave a universal width for the p band, of fi / md), with f/v = 4.2. (Again, his numerical value was different because of a different definition of d.) Presumably the conduction bands, corresponding to electrons added to the crystal, would be quite like free-electron bands. 

The diamagnetic susceptibility of the inert-gas solids is dominated by the Lan-gevin term (see Section 5-E), is readily calculated for the atom, and can be multiplied by the density of atoms to obtain values in good agreement with experiment. (Sec, for example, Kubo and Nagamiya, 1969, p. 438.)... 

Estimate the total contribution to the cohesion from the second-neighbor interactions in a face-centered cubic inert-gas solid. If you set the spacing d at the minimum of the Lennard-Jones interaction, that is, d = 2 l a, the result can be written as a fraetion of the contribution from nearest neighbors. 

We think of the ionic solids as made up of closed-shell ions, so that to a first approximation, the electronic structure is like that of the inert-gas solids. Two important differences arise, however, from the transfer of protons to make ions first, the electronic states on different ions are not the same and, second, the spacing is sufficiently reduced that there are important effects from the matrix elements between states on adjacent ions. It will be best to begin with the simplest description, as we have in other systems, and introduce complications as we go one reason for this is that many properties are understandable without the full complexity of the true electronic structure. 

This was defensible in the inert-gas solids (though we noted that the gap was slightly reduced in those solids), but in the ionic crystal the nonmctallic ion electronic levels are greatly raised and the important excited levels (for exciton levels as well as for lower conduction-band levels) are dominated by the states on metallic ions see Fig. 14-1. Pantelides noted in fact that a critical study of the analysis of experiments in terms of the independent-ion model did not support the model. The model appeared to work for the alkali halides, but this was by fitting 16 experimental numbers with 8 adjustable parameters and the systematic variation made this fitting possible. Little success was had with other compounds. 

The identification of the band gap in ionic crystals with pscudopotentials suggests one other property that may be attributable to ionic crystals. The general insensitivity of to material or structure gives a rationalization to the observation that band gaps in ionic solids and even inert-gas solids vary as cl. However, the point cannot be made as strongly for ionic solids as it can for covalent solids. 

---