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lecture22_2011

lecture22_2011 - 25Feb2011 Chemistry 21b Spectroscopy...

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25Feb2011 Chemistry 21b – Spectroscopy Lecture # 22 – Electronic Spectroscopy of Periodic Solids Previously we have investigated the vibrational modes of periodic solids using the harmonic potential approximation. Here we are interested in the electronic properties of such materials. From the LCAO-MO perspective, we could think about combining hydrogen orbitals in a periodic potential that mimics atomic structure. This can be done, but for the purposes of illustration we will look at a one-dimensional model that replaces the long range potential associated with Coloumb attraction with a simpler (positive) square-well potential whose height and width are variable. This is called the Kronig- Penney model for the electronic structure of periodic solids. The Atkins & Friedman text on reserve has a nice summary of this model, from which the following notes are derived. The Kronig-Penney Model of the Electronic Structure of Solids When we thought about the vibrational spectroscopy of crystalline solids, the periodicity of the lattice assumed paramount importance. Not surprisingly, the same is true for the electronic behavior of such materials. Here we briefly outline the simplest one-dimensional model of the so-called band structure of solids, called the Kronig-Penney model. A pictorial representation of this approach is shown below. Figure 22.1 – A schematic of the 1-D potential that defines the Kronig-Penney model of the electronic structure of crystalline solids. Briefly, we consider an infinite array of square wells, with a potential barrier height of V 0 , a barrier width of b , and a well spacing of a . Formally, the solutions we will present below are valid for periodic potentials where V 0 b =constant. In this model, it is clear that V ( x + a ) = V ( x ). Under such conditions, the time independent Schr¨ odinger equation for 175
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the electrons in the solid has a solution of the form ψ q ( x ) = u q ( x ) e iqx u q ( x + a ) = u q ( x ) . (22 . 1) The periodic functions u q ( x ) are called Bloch functions . As in our analysis of the vibrations
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