141A_PS10_sp07 - b) Show that these four functions are the...

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PHYSICS 141A S. G. LOUIE SPRING 2007 Problem Set #10 Due: Friday, 04/20/07 Reading: Chapter 8 of ISSP. (25) 1. ISSP, Ch. 8, Problem 1: Impurity orbits. (25) 2. ISSP, Ch. 8, Problem 2: Ionization of donors. (25) 3. ISSP, Ch. 8, Problem 3: Hall effect with two carrier types. (25) 4. Tetrahedral hybridization. Let s be a normalized s-wavefunction and p x , p y , p z be normalized p-wavefunctions. Consider the four hybrid functions: φ 1 = 1 2 s + 1 2 p x + 1 2 p y + 1 2 p z 2 = 1 2 s 1 2 p x 1 2 p y + 1 2 p z 3 = 1 2 s + 1 2 p x 1 2 p y 1 2 p z 4 = 1 2 s 1 2 p x + 1 2 p y 1 2 p z a) Show that under rotations by 120 ° and 240 ° about the [111] direction, 2 , 3 , and 4 , transform into each other.
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Unformatted text preview: b) Show that these four functions are the appropriate tetrahedral hybrid functions, that is, their maximum electron density is directed in the four tetrahedral directions. c) Show that the functions are normalized to unity; show that all of the s and p electron densities are used up by these four functions. d) Show that the angle between the adjacent directions of maximum electron density is 109 30 . (Note: These hybrids are responsible for the covalent bonds that form the important semiconductor crystals Si, Ge, GaAs, etc.)...
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