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Unformatted text preview: 21Jan2011 Chemistry 21b Spectroscopy Lecture # 8 Spectroscopic Line Shapes Now that we have established the strength and time-dependence of the interaction of light with matter, the last general set of trends we will need to think about before beginning a detailed examination of specific spectroscopic systems are that of the shapes of the spectroscopic lines that arise from the transitions between various eigenstates. We start with the transition rate between two levels in terms of the electric field strength, that is, Eq. (7.9), which may be rewritten R fi ( ) = 2 h 2 | < f | E | i > | 2 i ( fi ) , where we have now included the population density of initial states i in the transition rate (since in real life we will be dealing with an ensemble of atoms or molecules in a sample although we will look at the rapidly developing field of single molecule spectroscopy later in the course). R fi ( ) is proportional to the field squared, and thus to the intensity I of the incident radiation field ( I is the energy flux per unit area per unit time). Energy should therefore be removed from a light beam at a constant rate, and phenomenologically this can be written in the form- dI dt = k abs ( ) I , where k abs is the macroscopic, frequency dependent absorption coefficient (in the language of the Kramers-Kroning relationship, k abs is the real part of the I and the imaginary part of the refractive index). In terms of the rate noted above,- ( dI/dt ) = f h fi R fi . From the Heisenberg form of the Schr odinger equation, the time evolution of an operator may be written as dF dt = h [ H, F ] . (8 . 1) Equation (8.1) leads to a general solution like (provided H is independent of t ) F ( t ) = e Ht/ h F (0) e- Ht/ h . Using this formalism and the integral form of the delta function it is possible to show that the frequency dependence of the intensity profile for a spectroscopic transition is given by I ( ) = 3 2 integraldisplay - summationdisplay i summationdisplay f i < i |- - | f >< f |- - ( t ) | i > dt , where- is the dipole moment of the system at t = 0 and- is a dimensionless unit vector which indicates the direction, but not the magnitude, of the electric field. Using the 65 completeness relation ( f | f >< f | = 1) and by treating the sum over the populations of the initial states i as an ensemble, we have I ( ) = 1 2 integraldisplay - e t <- (0) - ( t ) > avg dt , (8 . 2) where the factor of one-third comes from averaging the electric field over an isotropic absorbing medium. The quantity <- (0) - ( t ) > avg is very important in spectroscopy, and is called the dipole correlation function G(t) . Eq. (6.2) tells us the general result that the line shape or frequency response of a system interacting with light is just the Fourier transform of its dipole correlation function, or...
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