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Unformatted text preview: 2.57 Fall 2004 – Lecture 8 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 8 In the last lecture, we have talked about the primitive unit cell . There is only one lattice point (equivalently speaking) per primitive unit cell. The smallest space formed by all the bisecting planes is a Wigner-Seitz cell, as indicated in the figure. a 1 a 2 a 1 a 2 a 1 ’ a 2 ’ 1 A Wigner-Seitz Primitive Unit Cell A Primitive Unit Cell A Conventional Unit Cell For the bonding potential, two often-used empirical expressions for the repulsive potential between the atoms separated by a distance r are 12 R r B ) r ( U = (Lennard-Jones) and ζ − = / r o R e U ) r ( U (Born-Mayer) where B, ζ , and U o are empirical constants determined from experimental data, such as the interatomic spacing and the binding energy. x Repulsion Attraction Interatomic potential Φ 2.57 Fall 2004 – Lecture 8 2 Combining this attractive potential (van der Waals potential) with the Lennard-Jones potential for the repulsive force, we obtain the Lennard-Jones interaction potential between a pair of atoms i and j in a crystal as 6 12 ij ij ij r A r B U − = . What makes a crystal structure a favorable structure is that the total potential energy of the system reaches a minimum, as required by the second law of thermodynamics for a stable system. In ionic crystals , such as NaCl, the single valence electron in the sodium atom moves to the chlorine atom such that both Na + and Cl- have closed electron-shells but meanwhile, become charged. The Coulomb potential among the ions becomes the major attractive force. The potential energy of any ion i in the presence of other ions j is then, o o 2 j i ij o 2 A , i r 4 q r 4 q U πε α − = ∑ πε ± = ≠ where q is the charge per ion, ε o the dielectric constant, and r o the nearest-neighbor separation. The parameter α is called the Madelung constant and is related to the crystal structure. This attractive potential, combined with an appropriate repulsive potential, gives a description of the potentials for ionic crystals. Covalent bonds are formed when electrons from neighboring atoms share common orbits, rather than being attached to individual ions as in ionic crystals. Diamond, silicon, and germanium are all covalent crystals . Each atom has four electrons in the outer shell and forms a tetrahedral system of covalent bonds with four neighboring atoms, as...
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This note was uploaded on 01/12/2011 for the course ME 305 taught by Professor Wright,j during the Spring '10 term at Birla Institute of Technology & Science, Pilani - Hyderabad.
- Spring '10