KaSap Electronic Materials_Chapter_1_problem_solutions

KaSap Electronic Materials_Chapter_1_problem_solutions -...

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1.1 1.5 Madelung constant If we were to examine the NaCl crystal in three dimensions, we would find that each Na + ion has 6 Cl ions as nearest neighbors at a distance r 12 Na + ions as second nearest neighbors at a distance 2 r 8 Cl ions as third nearest neighbors at a distance 3 r and so on. Show that the electrostatic potential energy of the Na + atom can be written as r M e r e r E o o πε 4 3 8 2 12 6 4 ) ( 2 2 = + = L Madelung constant M for NaCl where M , called Madelung constant , is given by the summation in the square brackets for this particular ionic crystal structure (NaCl). Calculate M for the first three terms and compare it with M = 1.7476, its value had we included the higher terms. What is your conclusion? Solution From Coulomb’s law of electrostatic attraction we know that the PE between two charges Q 1 and Q 2 separated by a distance r is given by r Q Q PE 0 2 1 4 = First we consider the interaction between Na + ion and 6Cl ions at distance r . Applying Coulomb’s law we have r e r e e r Q Q PE 0 2 0 0 2 1 1 4 6 4 ) )( 6 ( 4 = + = = Similarly, we now consider 12 Na + ions as second nearest neighbors at a distance 2 r 2 4 12 2 4 ) )( 12 ( 4 0 2 0 0 2 1 2 r e r e e r Q Q PE = + + = = and 8 Cl ions as third nearest neighbors at a distance 3 r 3 4 8 3 4 ) )( 8 ( 4 0 2 0 0 2 1 3 r e r e e r Q Q PE = + = = and similarly we can consider the next nearest set of neighbors and so on. Therefore, the overall PE of the Na + ion is . . . 3 4 8 2 4 12 4 6 ) ( 0 2 0 2 0 2 + + = r e r e r e r E or r Me r e r E 0 2 0 2 4 . . . 3 8 2 12 6 4 ) ( = + = where clearly + + = 3 8 2 12 6 M Considering just the first three terms we have M = 2.133. This is considerably different from the value M = 1.7464, the value obtained when higher order terms are considered. This implies that the next nearest neighbors have substantial effect on the potential energy.
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1.2 1.7 Van der Waals bonding Below 24.5 K, Ne is a crystalline solid with an FCC structure. The interatomic interaction energy per atom can be written as = 12 6 13 . 12 45 . 14 2 ) ( r r r E σ ε (eV/atom) where and are constants that depend on the polarizability, the mean dipole moment, and the extent of overlap of core electrons. For crystalline Ne, = 3.121 × 10 -3 eV and = 0.274 nm. a . Show that the equilibrium separation between the atoms in an inert gas crystal is given by r o = (1.090) . What is the equilibrium interatomic separation in the Ne crystal? b . Find the bonding energy per atom in solid Ne. c . Calculate the density of solid Ne (atomic mass = 20.18). Solution a. Let E = potential energy and x = distance variable between the atoms. The energy E is given by = 12 6 13 . 12 45 . 14 2 ) ( x x x E The force F on each atom is given by = = 2 5 2 11 7 . 86 56 . 145 2 ) ( ) ( x x x x dx x dE x F = 7 6 13 12 7 . 86 56 . 145 2 ) ( x x x F When the atoms are in equilibrium, this net force must be zero. Using
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This note was uploaded on 04/24/2008 for the course MSE MSE 350 taught by Professor Morelli during the Spring '08 term at Michigan State University.

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KaSap Electronic Materials_Chapter_1_problem_solutions -...

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