Number of vacancies

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... 

Polymers are a little more complicated. The drop in modulus (like the increase in creep rate) is caused by the increased ease with which molecules can slip past each other. In metals, which have a crystal structure, this reflects the increasing number of vacancies and the increased rate at which atoms jump into them. In polymers, which are amorphous, it reflects the increase in free volume which gives an increase in the rate of reptation. Then the shift factor is given, not by eqn. (23.11) but by... 

At the moment it is not clear how far our first results concerning the role of defects are influenced by the vacancy concentration of 2%. We have used this rather high concentration in order to have a significant number of vacancies in our systems. It will be necessary to make further simulations using more moderate vacancy concentrations and probably other kinds of defects to get a real understanding of the role of defects in martensitic transformations. 

The formation of the combination of defects may be described as a chemical reaction and thermodynamic equilibrium conditions may be applied. The chemical notations of Kroger-Vink, Schottky, and defect structure elements (DSEs) are used [3, 11]. The chemical reactions have to balance the chemical species, lattice sites, and charges. An unoccupied lattice site is considered to be a chemical species (V) it is quite common that specific crystal structures are only found in the presence of a certain number of vacancies [12]. The Kroger-Vink notation makes use of the chemical element followed by the lattice site of this element as subscript and the charge relative to the ideal undisturbed lattice as superscript. An example is the formation of interstitial metal M ions and metal M ion vacancies, e.g., in silver halides ... 

In order to use this method, one starts by defining the number of states that maybe present. Let be the total number of lattice sites and Ny be the number of vacancies. The number of wa that Ny vacancy-sites can be arranged on Nl sites will be a simple combination, defined by a mixing entropy ... 

A plot of this equation (Fig. 2.1 b) closely resembles Figure 2.1a. The minimum in the curve gives the equilibrium number of vacancies present and confirms that vacancies exist in all crystals at temperatures above 0 K. For this reason these defects cannot be removed by thermal treatment but are always present in a crystal. Such defects are thus intrinsic defects. At equilibrium, AGy will be equal to zero and the minimum in the AGy versus tiy curve is given by... 

Figure 2.3 Basis of a lithium iodide cell (schematic) (a) electrons are liberated at the lithium metal anode and re-enter the cell via the I2/P2PV cathode (b) lithium ion transport across the electrolyte via Li vacancies, to form Lil at the anode. The number of vacancies (Schottky defects) has been grossly exaggerated.

The formation energies of point defects in a pure metal are 1.0 eV (vacancies) and 1.1 eV (interstitials). The number of vacancies is ... 

Defect clusters are similarly prominent in hydrated phases. For example, anatase nanocrystals prepared by sol-gel methods contain high numbers of vacancies on titanium sites, counterbalanced by four protons surrounding the vacancy, making a (Vxi 4H ) cluster. In effect the protons are associated with oxygen ions to form OH- ions, and a vacancy-hydroxyl cluster is an equally valid description. Similar clusters are known in other hydrated systems, the best characterized being Mn4+ vacancies plus 4H in y-Mn02, known as Reutschi defects. 

At ever higher partial pressures the number of nonmetal vacancies will diminish more and more and the number of holes will exceed the number of vacancies in this regime. Thus the approximate electroneutrahty Eq. (7.13) will no longer be representative. A more appropriate form of the electroneutrality equation for the high-pressure region is obtained by ignoring the minority species, nonmetal vacancies and electrons, to give... 

The donor defects can be compensated by cation vacancies or electrons (Section 8.2.1). That is, the addition of donors to the right-hand side of Eq. (8.7) requires an increase in the number of vacancies ... 

As illustrated in Fig. 7.15, the electromagnetic radiation measured in an XRF experiment is the result of one or more valence electrons filling the vacancy created by an initial photoionization where a core electron was ejected upon absorption of x-ray photons. The quantity of radiation from a certain level will be dependent on the relative efficiency of the radiationless and radiative deactivation processes, with this relative efficiency being denoted at the fluorescent yield. The fluorescent yield is defined as the number of x-ray photons emitted within a given series divided by the number of vacancies formed in the associated level within the same time period. 

The notion of point defects in an otherwise perfect crystal dates from the classical papers by Frenkel88 and by Schottky and Wagner.75 86 The perfect lattice is thermodynamically unstable with respect to a lattice in which a certain number of atoms are removed from normal lattice sites to the surface (vacancy disorder) or in which a certain number of atoms are transferred from the surface to interstitial positions inside the crystal (interstitial disorder). These forms of disorder can occur in many elemental solids and compounds. The formation of equal numbers of vacant lattice sites in both M and X sublattices of a compound M0Xft is called Schottky disorder. In compounds in which M and X occupy different sublattices in the perfect crystal there is also the possibility of antistructure disorder in which small numbers of M and X atoms are interchanged. These three sorts of disorder can be combined to give three hybrid types of disorder in crystalline compounds. The most important of these is Frenkel disorder, in which equal numbers of vacancies and interstitials of the same kind of atom are formed in a compound. The possibility of Schottky-antistructure disorder (in which a vacancy is formed by... 

The number of vacancies per cubic centimeter is a very strong function of temperature and is given by... 

The effect of atomic motion in the solid state on nuclear resonance line width is illustrated by the behavior of Na resonance from NaCl as a function of temperature 97). In Fig. 9 is shown the variation of the Na line width with temperature for pure NaCl and NaCl doped with an atomic fraction concentration of 6 X 10 of CdCU. As discussed in Section II,A,2 the low-temperature, rigid-lattice line width will narrow when the frequency of motion of the nuclei under observation equals the line width expressed in sec.-. The number of vacancies present should be equal to the concentration of divalent impurities and the jump frequency of Na+ is the product of the atomic vacancy concentration and the vacancy jump frequency... 

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