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Equation Section 23University of Washington Department of Chemistry Chemistry 553 Fall Quarter 2003 Lecture 23: Correlation Functions and Spectral Line Shapes 12/04/02 Text Reading: Ch 21,22 A. Introduction Spectral line shapes and autocorrelation functions are related by a Fourier transform. At the heart of this relationship is the fact that energy absorbed from a weak field is dissipated by fluctuations that are characteristic of the system at equilibrium. In the following analysis we assume that A=B=u, where u is the electric or magnetic dipole moment. B. The Dielectric Constant We will be considering the interaction of radiation fields with electric dipoles. So let’s consider some properties of polarizeable materials. The simplest model is a parallel plate capacitor. Assume the plates are separated by a distance d and have area A. The normal vector to the plates is in the z direction. The electric field between the plates of the capacitor is E0 z (23.1) 0 where =q/A is the surface charge density of the plates.The constant 0 is the free A space permittivity. The capacitance is C0 0 where it is assumed that the space d between the plate is vacuum. If the space between the capacitor plates is filled by a A dielectric, the capacitance is C where is the permittivity of the dielectric. The d C E relative permittivity is defined as r , where E0 and E are the fields 0 C0 E0 that exist between the plates where the space is evacuated (0) or filled with dielectric. The electric polarization P of the material is related to the electric field E by (23.2) P 0 e E The subscript e indicates the electric susceptibility. The complex, frequencydependent electric susceptibility is related to the relative permittivity by (23.3) r 1 e Therefore the permittivity, also called the dielectric constant, is also a complex number i.e. r r i r . The relationship of the in-phase and out-ofphase components of the permittivity are related to the response function by 0 0 1 dt t cos t and dt t sin t (23.4) Now the expression for the rate of energy absorption from the field is… U r 0 U 1 where U 0 E02 and 0 is the vacuum permittivity. Then 2 U r 0 E02 U D U 2 see Lecture 21. BA 1 2 1 1 e i Q 1 e dteitBA t e m nm Amn Bnm m,n 1 i t e dte 2 i Q m,n 1 e i m 1 e e m Q m,n mn e i mn t Amn Bnm nm mn nm e m Q m,n 2 nm mn i (23.7) wher A=B=u which is the electric dipole moment. The r.hs. of (23.7) is pure imaginary so… i 1 e e m i i Q m,n nm mn 2 (23.8) e m 2 nm mn Q m,n Multiply (23.8) by N/V to make it per unit volume…and divide by the free space permittivity to make the l.h.s. a relative quantity… 1 e (23.6) We now calculate the complex susceptibility using linear response theory, see Lecture 22… BA (23.5) C. The Spectral Line Shape By definition the spectral line shape is: 3 0 I pi f u i i u f f ,i 1 e VN i , f Note the definition of the delta function 1 1 dt eit and so f ,i 2 2 dt e i f , i t (23.9) (23.10) Note the factor of three in (23.9) is introduced to indicate an average transition moment. In all calculations up to here the direction assumed is x. Then 2 mn x 13 mn nm . Combining (23.9) and (23.10) I 3 0 1 e N V 1 pi f u i i u f i, f 2 E f Ei exp it (23.11) where f ,i E f Ei (23.11) may be further reduced using standard methods… 1 iE t I dt pi f e f ueiEit i i u f e i t 2 i , f 1 2 1 2 dt p i f eiEHt ue iEHt i i u f e it i , f dt p i (23.12) f u t i e i t iu f i , f whereu t eiEHt ue iEHt We then use the closure property k k 1 to collapse the double k summation to a single summation 1 I dt pi i u f 2 i , f 1 2 1 2 dt p i i f u t i e i t i u u t i e i t dt (23.13) u u t e i t 1 2 dt C t e i t where C t u u t pi i u u t i i D. Summary (23.13) shows that the spectral lineshape and the correlation function C(t) are related by a Fourier transform. (23.13) can be inverted 1 i t I dt C t e C t (23.14) d I eit 2 (23.19) and (23.20) can be applied to a number of spectroscopies by making the appropriate substitution for the operator in the correlation function expression. For example o Microwave Spectroscopy: u=u0, the permanent dipole moment. u o Infrared: u Q , where Q is a normal coordinate Q 0 o Rayleigh Scattering: u uind eˆi eˆs where is the polarizability tensor, and eˆi , s are the unit vectors in the direction of the incident and scattered radiation. o Raman Scattering: u uind eˆi eˆs where Q 0 o Magnetic Resonance: correlation functions in magnetic resonance involve the magnetic dipole moment of the particle. Relaxation rates in magnetic resonance involve spectral densities that are Fourier transforms of correlation functions.