10-Divacanices

10-Divacanices - 114 Divacancies and other vacancy clusters...

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Unformatted text preview: 114 Divacancies and other vacancy clusters Just as single vacancies, or monovacancies, exist in crystals at equilibrium, so do divancancies, trivacancies, quadravacancies and other point defect clusters. Considering monovacancies and divacancies, we may expect the equilibrium state of the crystal to be determined by ¡¢ G ¡ n v = and ¡¢ G ¡ n 2 v = , where n v and n 2 v are the number of monovacancies and divacancies, respectively, in a crystal with N lattice sites. The results would be x v = exp ¡ ¢ g v kT £ ¤ ¥ ¦ § ¨ and x 2 v = exp ¡ ¢ g 2 v kT £ ¤ ¥ ¦ § ¨ , where x 2 v is the fraction of divacancy sites occupied by divacancies and ¡ g 2 v is the Gibbs free energy of formation of the divacancy. We may symbolize these lattice defects as follows The fraction of divacancies can be expressed as x 2 v = n 2 v n 2 v sites = exp ¡ ¢ g 2 v kT £ ¤ ¥ ¦ § ¨ , where n 2 v sites is the number of divacancy sites in the crystal having N lattice sites. Because there are N lattice sites and z nearest neighbors for each lattice site (coordination number) then n 2 v sites = Nz 2 , where the factor of 2 is needed to avoid double counting indistinguishable divacancies. Thus the number of divacancies in a crystal is then 115 n 2 v = Nz 2 exp ¡ ¢ g 2 v kT £ ¤ ¥ ¦ § ¨ = Nz 2 exp ¢ s 2 v k £ ¤ ¥ ¦ § ¨ exp ¡ ¢ e 2 v kT £ ¤ ¥ ¦ § ¨ exp ¡ p ext ¢ v 2 v kT £ ¤ ¥ ¦ § ¨ . For comparison with monovacancies it is convenient to express the formation energy of the divacancy as ¡ e 2 v = 2 ¡ e v + ¡ e binding , where the following process of creating the divacacy is envisioned. The divacancy is created by first creating two separate monovacancies that are then bound together to make the divacancy. For this comparison we can approximate the less important terms in the divacancy formation free energy as ¡ s 2 v ¢ 2 ¡ s v and ¡ v 2 v ¢ 2 ¡ v v ....
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10-Divacanices - 114 Divacancies and other vacancy clusters...

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