Vacant site

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

If the molecule under consideration were being placed on an empty lattice, the second segment could go into any one of the z sites adjacent to the first. However, ni of the sites are already filled, so there is a chance that one of the z sites in the coordination sphere of the first segment is already occupied. To deal with this possibility, we assume that the fraction of vacant sites on the lattice as a whole also applies in the immediate vicinity of the segment positioned above. This fraction is (N - ni)/N, so the number of possible locations for segment 2 of the (i + l)th molecule is z(N - ni)/N. 

The vacant sites will be distributed among the N lattice sites, and the interstitial defects on the N interstitial sites in the lattice, leaving a conesponding number of vacancies on die N lattice sites. In the case of ionic species, it is necessaty to differentiate between cationic sites and anionic sites, because in any particular substance tire defects will occur mainly on one of the sublattices that are formed by each of these species. In the case of vacant-site point defects in a metal, Schottky defects, if the number of these is n, tire random distribution of the n vacancies on the N lattice sites cair be achieved in... 

B Surface vacant site C Single vacancy kink D Adatom E Kinked ledge F Terrace... 

Figure 4.2 Terraces, ledges and kinks on a solid surface, together with an emerging screw dislocation, a vacant site, and an adatom...

Figure 6.2 The increase in electrical conductivity when a metal sample is heated to a high temperature and then quenched to room temperature, arising from the introduction of vacant sites at high temperature...

Typical values of the energy to form vacancies are for silver, lOSkJmol and for aluminium, 65.5kJmol These values should be compared with the values for the activation enthalpy for diffusion which are given in Table 6.2. It can also be seen from the Table 6.2 that die activation enthalpy for selfdiffusion which is related to the energy to break metal-metal bonds and form a vacant site is related semi-quantitatively to the energy of sublimation of the metal, in which process all of the metal atom bonds are broken. 

The process requires the interchange of atoms on the atomic lattice from a state where all sites of one type, e.g. the face centres, are occupied by one species, and the cube corner sites by the other species in a face-centred lattice. Since atomic re-aiTangement cannot occur by dhect place-exchange, vacant sites must play a role in the re-distribution, and die rate of the process is controlled by the self-diffusion coefficients. Experimental measurements of the... 

In all of these oxide phases it is possible that departures from the simple stoichiometric composition occur dirough variation of the charges of some of the cationic species. Furthermore, if a cation is raised to a higher oxidation state, by the addition of oxygen to tire lattice, a conesponding number of vacant cation sites must be formed to compensate tire structure. Thus in nickel oxide NiO, which at stoichiomen ic composition has only Ni + cations, oxidation leads to Ni + ion formation to counterbalance the addition of extra oxide ions. At the same time vacant sites must be added to the cation lattice to retain dre NaCl sUmcture. This balanced process can be described by a normal chemical equation thus... 

By analogy with similar materials in which free elecU ons and electron holes are formed, NiO is called a p-type compound having vacant site Schottky defects, and ZnO is an n-type compound having interstitial Frenkel defects. The concentrations of these defects and their relation to the oxygen pressure in the suiTounding atmosphere can be calculated, for a dilute solution of defects by the application of a mass action equation. The two reactions shown above are represented by the equations... 

It is not necessary for a compound to depart from stoichiometry in order to contain point defects such as vacant sites on the cation sub-lattice. All compounds contain such iirndirsic defects even at the precisely stoichiometric ratio. The Schottky defects, in which an equal number of vacant sites are present on both cation and anion sub-lattices, may occur at a given tempe-ramre in such a large concentration drat die effects of small departures from stoichiometry are masked. Thus, in MnOi+ it is thought that the intrinsic concentration of defects (Mn + ions) is so large that when there are only small departures from stoichiometry, the additional concentration of Mn + ions which arises from these deparmres is negligibly small. The non-stoichiometry then varies as in this region. When the departure from non-stoichio-... 

The Morse function which is given above was obtained from a study of bonding in gaseous systems, and dris part of Swalin s derivation should probably be replaced with a Lennard-Jones potential as a better approximation. The general idea of a variable diffusion step in liquids which is more nearly akin to diffusion in gases than the earlier treatment, which was based on the notion of vacant sites as in solids, remains as a valuable suggestion. 

FIG. 8 Plot of the fraction of vacant sites as a function of the coverage of inert species (X) during CO oxidation. The squares are determined using Monte Carlo simulations with a fixed X coverage using a grid of 256 x 256 sites. The arrows depict how the system evolves. The production of CO2 is proportional to the number of vacant sites. (From Ref. 68.)... 

NO, the monomer C is CO, and the products are A2 = N2 and CB = CO2. The adsorption probability of C species (Fc) is the parameter of the model. The slow rate-determining step in this sequence is the dissociation of NO which requires a neighboring site to proceed. Since product formation liberates more vacant sites than those necessary for the dissociation of NO, an autocatalytic production of vacant sites takes place. 

Figure 6.4 Crystal structure of ar-tetragonal boron. This was originally thought to be B50 (4Bi2 + 2B) but is now known to be either B50C2 or B50N2 in which the 2C (or 2N) occupy the 2(b) positions the remaining 2B are distributed statistically at other vacant sites in the lattice. Note that this reformulation solves three problems which attended the description of the or-tetragonal phase as a crystalline modification of pure B ...

The structures of boron-rich borides (e.g. MB4, MBfi, MBio, MB12, MBe6) are even more effectively dominated by inter-B bonding, and the structures comprise three-dimensional networks of B atoms and clusters in which the metal atoms occupy specific voids or otherwise vacant sites. The structures are often exceedingly complicated (for the reasons given in Section 6.2.2) for example, the cubic unit cell of YB e has ao 2344 pm and contains 1584 B and 24 Y atoms the basic structural unit is the 13-icosahedron unit of 156 B atoms found in -rhombohedral B (p. 142) there are 8 such units (1248 B) in the unit cell and the remaining 336 B atoms are statistically distributed in channels formed by the packing of the 13-icosahedron units. 

The structural relationships in Bi203 are more complex. At room temperature the stable fonn is monoclinic o -Bi203 which has a polymeric layer structure featuring distorted, 5-coordinate Bi in pseudo-octahedral iBiOs units. Above 717°C this transforms to the cubic -form which has a defect fluorite structure (Cap2, p. 118) with randomly distributed oxygen vacancies, i.e. [Bi203D]. The )3-form and several oxygen-rich forms (in which some of the vacant sites are filled... 

The result of this cts-insertion is that a vacant site is left behind, and this can be occupied by another ethylene molecule and steps (a) and (b) repeated indefinitely. 

Figure 1. Crossover scaling plot for tlie order parameter ( m > = ( ( ia - Bl / (<1>a + B)> of a symmetrical polymer mixture simulated by tlie bond fluctiiatioii model on tlie simple cubic lattice, with a concentration (jiv = 0.5 of vacant sites. Here N " ( m > is plotted vs. N t, and chain lengths from N = 32 to N = 512 are...

In polymerizing these compounds, a reaction between a-TiCls and triethylaluminum produces a five coordinate titanium (111) complex arranged octahedrally. The catalyst surface has four Cl anions, an ethyl group, and a vacant catalytic site ( ) with the Ti(lll) ion in the center of the octahedron. A polymerized ligand, such as ethylene, occupies the vacant site ... 

The next step is the cis insertion of the ethyl group, leaving a vacant site. In another step, ethylene occupies the vacant site. This process continues until the propagating chain terminates ... 

The propagating polymer then terminates, producing an isotactic polypropylene. Linear polyethylene occurs whether the reaction takes place by insertion through this sequence or, as explained earlier, by ligand occupation of any available vacant site. This course, however, results in a syndiotactic polypropylene when propylene is the ligand. 

At medium and high temperatures copper ultimately follows the parabolic law " . It has been shown " using radioactive tracers that the diffusion of copper ions in cuprous oxide is the rate-determining step at 8(X)-1 000°C, and there is considerable evidence in favour of the view that metal moves outwards through the film by means of vacant sites in the oxide lattice . 

Interstitial diffusion is rarely possible when two metals interdiffuse, since their atomic radii are usually of the same order. Several mechanisms have been proposed, but it is now generally accepted that interdiffusion is due to the motion of vacant sites within the lattice, solvent and solute atoms moving as the vacant sites migrate. The diffusion process is thus dependent upon the state of imperfection of the solvent metal and the alloy being formed. 

These are generally analogous to those of Wilkinson s compound, with the important difference that ligand dissociation cannot occur, so that the product of oxidative addition with H2 cannot have a vacant site to bind an alkene and will thus not act as a hydrogenation catalyst [53]. 

The ultimate purpose of mechanistic considerations is the understanding of the detailed reaction pathway. In this connection it is important to know the structure of the active catalyst and, closely connected with this, the function of the cocatalyst. Two possibilities for the action of the cocatalyst will be taken into consideration, namely, the change in the oxidation state of the transition metal and the creation of vacant sites. In the following, a few catalyst systems will be considered in more detail. 

The function of the tetraethyltin is to create vacant sites so that coordination of alkene molecules becomes possible, and to change the oxidation state of the tungsten atom from +6 to +4. Similar behavior of the aluminum compound in the system WCL-CgHsAlCb is not probable, because it has been demonstrated that WCle-AlClg is also an active catalyst (22, 44), which suggests that C2H5AICI2 functions as a Lewis acid. Vacant sites can be created by a Lewis acid as follows ... 

If the creation of vacant sites occurs in this way, it would be erroneous to conclude that the tungsten complex in its active form is not reduced, because reduction can also be accomplished by the reacting alkene molecules. 

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