# Stoichiometric Defects

Defects in Stoichiometric Solids

It is generally known that all the compounds follow law of definite proportion. But there are certain solid compounds which, refuse to obey this universal law. Such solid compounds which actually do not possess the exact compositions according to the electronic considerations have been given the name Berthollide or non-stoichiometric compounds. For example, in certain oxides such as $TiO_{1.7- 1.8}, WO_{2.88- 2.92}, Fe_{0.95}O$ Metallic hydrides as , $CeH_{2.69}, VH_{0.56}$ ,sulphides as $CuFeS_{1.94} , Cu_{1.7} S , Cu_{1.65} , Te, Cu_{1.6} Se$; tungsten bronzes as $Na_x WO_8$ , etc., there is between the atoms and hence they are termed non-stoichiometric compounds.

It is clear that such an unbalance of composition is only possible if the structure is in some way irregular that it possesses defects. Non-stoichiometry means there is an excess either of metal or of non-metal atoms.

In a compound $XY_n$,the general concept that each atom is on an appropriate lattice point, and every lattice point tenanted by the right kind of atom, is an idealization of the real crystal which at absolute zero of temperature represents the equilibrium state. Now, at this absolute temperature in a real crystal, the thermal vibrations of the atoms facilitate the occurrence of lattice defects. These defects in crystal lattice amount to variations from the regularity which characterizes the material as a whole. The defects are of two types.

1.       Frenkel Defect

This defect generally arises when an ion occupies an interstitial position between lattice points.

Here positive ions occupy interstitial positions being smaller than negative ions. In the figure given above, it is clear that one of the positive ions occupies a position in interstitial space rather than at its own appropriate site in the lattice due to which a ‘hole’ is created in the lattice as shown in the figure.

The defect mostly appears in those compounds where positive and negative ions differ largely in their radii and coordination number is low.

2.       Schottky Defect

This is a defect which mainly arises if some of the lattice points are unoccupied. Such points which are unoccupied have been given the name lattice vacancies or ‘holes’. The figure exhibits Schottky defect of crystals when existence of two holes, one due to a missing positive ion and the other due to missing negative ion in crystal lattice is there.

It is also found that this defect is generally observed in strong ionic compounds having a high coordination number and the radius ratio r / R is not far below unity. Examples are cesium chloride and sodium chloride. Although both types of defects probably characterize crystals of non-stoichiometric compounds, the Schottky defects are more important.

Rees has introduced some symbolism for the constitution of imperfect crystals. He gave an idea that a lattice site appropriate presented. The nature of atom and type of site are specified for an occupied lattice; in this way an atom of type x on its proper lattice site is represented by $x / o_x$. The symbol $x_{1- A} /o_x$ represents the fraction 1 – A(A < 1) the x lattice sites is occupied by correct species of atom. Now the fraction A remains vacant unless some other species of atom is specified as also located on x lattice sites.

Generally, interstitial positions are represented by $\Delta$ Hence interstitial positions occupied by a particular species of atom may then be symbolized by $x / \Delta$ With the help of this symbolism it is possible to know the concentration and nature of lattice defects in any system. The reactions by which lattice defects are formed can be represented by quasi chemical equations.

Lattice vacancies (holes) occur in almost all types of ionic solids. However Schottky defect appears more often than Frenkel defect. The reason is that the energy needed to form a Schottky defect is much less than that needed to form a Frankel defect.

3.       Substitutional Defects

Interchange of atoms between lattice sites produces this type of defect.

 Type Example Nature of disorder Frenkel defect AgBr Interstitial atoms and vacancies of same kind Schottky defect KCL Vacancies in anion and Cation lattices Substitutional disorder CuAu Interchange of atoms between lattice sites

Consider a polar compound of the formula AB. Now, this can incorporate excess of metal B due to the following reasons:

(i)                 By having more vacant A sites than vacant B sites;

(ii)               By having greater concentration of interstitial A atoms less than the concentration of vacant A sites;

(iii)               By having greater concentration of interstitial B atoms than of vacant B sites.

The equation $\Delta G = \Delta H- T \Delta S$

where $\Delta G$ = free energy change

$\Delta H$ =enthalpy change

T = temperature in Kelvin

$\Delta S$ =entropy change

suggests that the production of lattice defects of Frenkel or Schottky is an endothermic process. The calculations lead to give positive value of $\Delta S$. Therefore it is clear that the defects create certain randomness into the crystal and which thereby increases the entropy.

A peculiar phenomenon has been noticed in case of ionic compounds. It is found that substances of high melting points such as CaO, KCl, the value of equilibrium constant is very small at ordinary temperature. In such case also the concentration of defects becomes noticeable as melting point. At$700^0, C, SrCl_2$ has been found to have 0.1%. Frenkel defect.

Non-stoichiometry in solids is a very general phenomenon, indeed stoichiometry is exceptional and it has been shown thermodynamically that a condensed phase even at equilibrium is not of unique composition except at its singular points and at a temperature near absolute zero. In crystalline substances which have been prepared at high temperatures, it is not unusual for an abnormally high concentration of lattice defects to be retained on cooling.

The nature of lattice defects which gives non-stoichiometric character can be ascertained by comparing the calculated density of solid compounds from X-ray measurements with observed density. Take the case of FeS which has range from $FeS_{1.00} \text{to} FeS_{1.14}$.  It is considered that this range is due to interstitial sulphur atoms. While according to Hagg and Sucksdorf, there is deficiency of iron therefore the average formula is $Fe_{1.00} m\text{to} Fe_{0.88}S$ and there should be diminution in density as sulphur percentage increases.

In general, the tolerance of a crystal lattice for defects increases markedly at elevated temperatures and the stable range of compositions of non-stoichiometric phases usually become progressively broader at high temperatures.

The compound titanium monoxide exhibits the structure like that of sodium chloride. The difference between measured and calculated density indicates a high concentration of Schottky defects. The concentration of vacant cation sites and vacant anion sites gives an idea of wide variation in Ti : O ratio:

Composition    Tio0.69   Tio1.00      Tio1.12       Tio1.25      Tio1.33

O sites vacant   34            15                9                  4               2

(%)

Ti sites vacant   4               15               19              23              26

(%)

Generally, transition metals and heavy metal oxides provide an evidence for the formation of non-stoichiometric compounds. Non-stoichiometric compounds also show fluorescence, semi-conductivity and centers of colour. They own their existence to high activation energy of invariable phases when they are annealed. It has also been noticed that non-stoichiometric compounds are stable above critical temperature below which they break up into phases approximating to rational formulae. For example, tantalum hydride at lower temperature breaks up into $Ta_2H$ and nearly hydrogen-free tantalum.

Researches have shown that at high temperatures, solid ionic compounds become ionic conductors through migration of ions within crystalline solid. For example, $Cr_2O_3$  has unpaired electrons in partly filled of levels of transition metal ions. These electrons are not mobile.

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