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- Imperfections In Nearly Perfect Crystals Symposium Held At Pocono Manor October 12 14 1950?
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Diamond Flaws to Avoid – Diamond Inclusions & Imperfections
Try again? Defects are then said to be annihilated. The disordering of crystals requires the expenditure of energy see for example references ; one might well ask why defects should exist a t all in solids which would otherwise be perfect. An answer can be provided from two different points of view. Neglecting higher vibrational energy levels, one may think of every lattice atom as being in its ground state, and of every defect as representing an excited state. According to the Boltzmann principle, there is a finite probability that a t any temperature above OmK.
I n applying this reasoning to Schottky crystals, let n be the number of defects existing in the crystal of N atoms and let r. If n A complementary point of view can be established on the basis of thermodynamic considerations.
Imperfections in nearly perfect crystals symposium held at Pocono manor October 12 14 1950
One of the fundament. Now it is true that the perfect crystal represents a state in which the internal energy E has the lowest possible value. Now a growth in the randomness corresponds to an augmentation of the entropy, S; because of the negative sign in equation 5 , this leads to a lowering of the free energy.
Up to a certainpoint, therefore, the increase in E brought, about by the generation of defects is more than offset by the increase in S, and F is thereby reduced. The creation of defects proceeds until the further gain in entropy no longer matches the energy expenditure; a t this point F is a minimum. Using the above concepts as a basis for. In order to arrive a t these simple expressions it is, however, necessary to assume a that the disorder is small, so that no interaction among the defects is encountered, and b that the properties of the regions close to the defects do not differ appreciably from those of a perfect lattice.
I n considering these systems it becomes necessary to distinguish between cation and anion defects; furthermore, the presence of charges on the ionic constituents must he taken into account. Whether a given ionic crystal will exhibit Frenkel or Schottky type disorder will depend on the energetic requirements for the creation of the two types of defects. A systematic study of these energy requirements was undertaken by Jost, Schottky, Mott, and by others Three types of interactions have been considered: a the coulomb attraction between a given ion and the remainder of the crystal; this energy must be overcome to create a vacancy.
This polarization gives rise to attractive forces, which largely compensate the energy expenditures in a. For the alkali halides it has been shown that the Coulomb and polarization energies for the production either of Frenkel or of Schottky defects are roughly the same.
Diamond Flaws to Avoid – Diamond Inclusions & Imperfections
Accordingly, in such lattices, the presence of Schottky defects should be energetically favored. I t has been experimentally confirmed that alkali halides are subject to Schottky-type disorder. In silver halides it is found that van der Waals forces contribute appreciably to the binding energy of the crystal. Calculations which take account of this factor 11 indicate that the formation of Frenkel defects then requires the lesser amount of energy. Several investigators have claimed that the disorder in silver halides is also due to Schottky defects. Several other points should be noted.
First, because each ion carries a charge, the law of electroneutrality must hold; this implies that cation and anion vacancies must be formed in equivalent amounts in Schottky crystals. Second, it is found from the above calcula- tions that lattice ions in the immediate vicinity of a vacancy will move away from the empty site.
This is because the ions no longer experience a Coulomb force in the direction of such a site after the defect has been created. Third, a t sufficiently high temperatures, ions can move from one interstitial position to an adjacent one, or from a lattice position into an adjoining vacancy.
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I n the presence of an externally applied field these ions d l drift either with or against the field; thus, current. We turn now to a study of defects in solids which are contaminated by impurities. Since a very systematic investigation of these matters has been given by Hauffe 4a we shall take up only one example which, however, illustrates all the general principles that are involved.
Consider, then, a AgBr crystal to which a small amount of CdBrz has been added.
Because of unfavorable energetic requirements, these anions cannot be placed in interstitial positions; if, onthe other hand, they enter the anion sublathice substitutionally then they merely displace two other Br- ions which require placement.