Metal lattice extended

In Figure 3.7, a selection of metal clusters containing interstitial atoms is shown. Examples with interstitial H atoms as well as transition-metal atoms are also known. Addition of an interstitial metal atom is the first step towards extended metal structures. The term interstitial derives from its use in solid-state chemistry where atoms are found in the interstices of metal lattices, e.g., the tetrahedral or octahedral... 

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

The role of the metal in double layer properties can be understood in greater detail when the system is examined on the basis of the jellium model. This model was developed to describe the electron gas within sp metals. It can be used to estimate several properties of interest, including the chemical potential of an electron in the metal, the extent of electron overspill, and the work function of the metal. More recently, it has been extended to describe metal surfaces in contact with polar solvents [26]. In its simplest form, the metal atoms in the metal are modeled as a uniform positive background for the electron gas, no consideration being given to their discrete nature and position in the metal lattice. The most important property of the system is the average electron density, N ), which depends on the number of metal atoms per unit volume and the number of valence electrons per atom, n. Thus, if pjj, is the mass density of the metal, and M, its atomic mass... 

What criterion deddes whether an element is able to contribute as an interstitial atom to the stabilization of an electron defident duster Obviously, the M6 octahedron bears some relation to a microscopically small piece of metal. The vibrational frequent of the H atom in the [NbgH] unit of Nb(I,iH is nearly identical to the frequency of H in the metallic hydride NbH. [92] Since the bonding in the duster and that in the extended metal lattice are so similar, the obvious question to ask is which elements form stable compounds with the bulk metal that represents the duster atoms. The answer to this question yields a qualitative explanation for the fact that the electron defident Nb unit incorporates H, while the Mog octahedron does not. Zr forms numerous intermetallic compounds with Be, Al, and other d metals and, obviously, does not loose this ability when only six Zr atoms are joined. Inspection of the experimental data for the relevant binary systems or the use of Miedema s concept for the stability of intermetallic systems [93, 94] proves helpful in the search for possible interstitial atoms, and naturally limiting the search to atoms of appropriate size to fit into the octahedral site. Of course, if the intermetallic compounds are very stable they could also compete with the duster compound formation. 

In the next section it will be shown that these effects are much smaller on extended lattices because electrons become delocalized. The effective population of antisymmetric group orbitals present in the cluster mentioned above, becomes much smaller in an extended lattice. The n interaction is therefore smaller on the extended substrate. The reverse may also happen. In metals that have too low an electron count to populate antisymmetric combinations of s atomic orbitals in small clusters (the d orbitals will be populated instead), emvedding in an extended metallic lattice will populate such antisymmetric group orbitals to some extent. We already discussed such a situation for the Nii3 cluster. In such cases, interaction with 7c-type orbitals will be enhanced in the extended lattice as compared to the small cluster. 

The successful rationalization of these transition-metal inverse spinel structures in terms of the relative LFSE s of tetrahedral and octahedral sites is another attractive vindication of ligand-field theory as applied to structure and thermodynamic properties. Once again, however, we must be very careful not to extrapolate this success. Thus, we have a clear prediction that LSFE contributions favour tetrahedral over octahedral coordination, except for d" with n = 0, 5 or 10. We do not expect to rationalize the relative paucity of tetrahedral nickel(ii) species relative to octahedral ones on this basis, however. Many factors contribute to this, the most obvious and important one being the greater stabilization engendered by the formation of six bonds in octahedral species relative to only four bonds in tetrahedral ones. Compared with that, the differences in LSFE s is small beer. Why , one asks, was our rationalization of spinel structures so successful when we neglected to include consideration of the bond count The answer is that cancellations within the extended lattice of the spinels tend to diminish the importance of this term. 

Compounds isotypic with the k phases arc found among intcrmetallics, borides, carbides and oxides and also with silicides, germanides, arsenides, sulfides and sclcnides no nitrides, however, are found. The mode of filling the various voids in the metal host lattice of the k phases follows the schemein Ref. 4 and is presented in Table 1 for all those compounds for which the atom distribution is well known from x-ray or neutron diffraction. Accordingly, B atoms in tc-borides, Zr, Mo, W, Re)4B and Hfy(Mo, W, Re, Os)4B , occupy the trigonal prismatic interstices within the parent metal framework of a Mn, Aln,-type structure (see Table 1 see also ref. 48). Extended solid solutions are found for (Hf, Al)[Pg.140]

Although several types of lattices have been described for ionic crystals and metals, it should be remembered that no crystal is perfect. The irregularities or defects in crystal structures are of two general types. The first type consists of defects that occur at specific sites in the lattice, and they are known as point defects. The second type of defect is a more general type that affects larger regions of the crystal. These are the extended defects or dislocations. Point defects will be discussed first. 

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