Lecture 9 Point Defects - Point Defects Monovacancy...

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Point Defects Monovacancy The (free) energy of a vacancy has four terms (in an oxide it can have more): 1) The local change in the bonding associated with removing the atom – “broken bonds” 2) A local energy term associated with increased local bonding. For instance, if we consider the bonding in terms of atoms connected by pairwise bonds, if we remove one bond the others can get stronger. 3) A long-range elastic strain energy. Energy term 2) above may want to bring the atoms closer together; this is balanced by elastic strains. 4) An entropy term. Part of this is an entropy of mixing, i.e. associated with the number of locations one can put the vacancy into the material. A second part is the change in the phonon free energy. In most cases we will combine 1), 2) and 3) as an enthalpy, and add an extra entropy for 4). For completeness, in a ceramic where the presence of a vacancy can lead to an additional electron/hole there is an electronic entropy term as well which might be fairly large; in metals this is normally considered to be fairly small. Configurational entropy (see also, for instance Kelly et al Chpt 9) Suppose we have n vacancies randomly distributed among N sites. The number of possible configurations is )! ( ! ! ! ]) 1 [ ) .... ( 1 )( 1 ( n N n N n n N N N N - = - - - - = The entropy associated with this is S=k*ln Ω. We use Stirling’s approximation of ln(x!) = xlnx – x (works if x is large enough) ) ln( ) ( ln ln / n N n N n n N N k S - - - - = Dividing by N, and doing a little algebra: )) / 1 ( ln( ) / 1 ( ) / ( ln ) / ( ln / N n N N n N n N N n N kN S - - - - = The lnN term does not matter, it is constant for a given N atoms, so for the defect M n C C C C C kN S / ); 1 ln( ) 1 ( ln / = - - - - = Note that S/N is the entropy per vacancy
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This is a Fermi-Dirac entropy term for a relative concentration of C with S/N the entropy per site. (You can only have one vacancy per site, similar to one electron per site and unlike the case for bosons (e.g. phonons) where you can have many per site.) Except when C is large, which is rarely the case, the term (1-C)ln(1-C) can be ignored.
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