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Unformatted text preview: 71 Equilibrium Concentration of Vacancies We now apply the Boltzmann analysis to vacancies in a crystal lattice. The problem is the same as the one we have solved for solutes. In this case, vacancies (or vacant lattice sites) can be created by removing atoms from the crystal interior and placing them on the surface of the crystal, much like we moved A atoms from the lattice and replaced them with B atoms taken from the A/B interface. Using the Boltzmann analysis for solutes as an example, we can immediately write the equilibrium vacancy fraction as x eq = exp ¡ ¢ g v kT £ ¤ ¥ ¦ § ¨ x eq = exp ¡ ¢ h v ¡ T ¢ s v kT £ ¤ ¥ ¦ § ¨ x eq = exp ¢ s v k £ ¤ ¥ ¦ § ¨ exp ¡ ¢ h v kT £ ¤ ¥ ¦ § ¨ x eq = exp ¢ s v k £ ¤ ¥ ¦ § ¨ exp ¡ ¢ e v + p ext ¢ v v kT £ ¤ ¥ ¦ § ¨ where ¡ e v = formation energy of the vacancy, (It is instructive to make a little estimate of the formation energy of the vacancy. Consider a simple cubic lattice with lattice parameter a , where one atom is missing. We can think of this vacancy as a cubic hole in the lattice. The cubic hole has six surfaces, each of area a 2 . If these “surfaces” have the surface energy of the solid, ¡ s , (think of them as free surfaces), then we might estimate the formation energy of the vacancy as ¡ e v = 6 a 2 ¢ s . Taking a = 2 Å = 0.2 nm and ¡ s = 1 J / m 2 (a typical surface energy) leads to a 72 vacancy formation energy of ¡ e v = 2.4 x 10 ¢ 19 J or ¡ e v = 1.5 eV . Thus we see that the formation energy for the vacancy should be of the order of one eV ) ¡ v v = formation volume of the vacancy ( ¡ v v ¢ £ = atomic volume for a perfectly rigid lattice (the lattice grows by one atomic site when each vacancy is formed). Actually lattices are not perfectly rigid because the lattice relaxes inward a little when the vacancy is formed, so that typically ¡ v v ¢ 0.5 £ ¤ 0.7 £ ), and ¡ s v = formation entropy of the vacancy ( ¡ s v = change in entropy associated with changes in the vibrational frequencies of the atoms surrounding the vacancy. Typically ¡ s v ¢ 2 k £ 8 k . This can be understood in a semiquantitative way by considering the changes in vibrational frequencies of the atoms next to a vacancy, as shown below. All of these fundamental quantities can be measured in various ways. For pure metals and many other crystals these quantities are well known. Rough estimate of ¡ s v The formation entropy of a vacancy can be estimated by considering the vibrational frequencies of the atoms next to the vacancy. We can use the Einstein model of a solid and treat each atom as three harmonic oscillators. Each atom is considered to be held in place by three orthogonal springs, each with spring constant k s ....
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 Spring '08
 nix
 Thermodynamics, Entropy, vacancy

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