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Unformatted text preview: Todays Lecture 10) Mon, Oct. 23: Relaxation Measurements a) Correlation functions b) Causes of relaxation c) NMR measurements of correlation functions d) Example of relaxation measurements Consider random field fluctuations along the X axis: Lets monitor two of these spins for awhile: Assuming these are completely random: ( 29 = t B x How can we describe the magnitude of these fluctuations? ( 29 = t B x ( 29 2 t B x How does the value of B x at one point correlate with its value at another point in time? Short spacing: Long spacing: Lets formalize this with an autocorrelation function: ( 29 ) ( ) ( + = t B t B G x x ( 29 t B G x 2 ) ( = ( 29 C e t B G x / 2 ) ( = The correlation time, c , is the time constant for the exponential decay of the function How does this translate into frequencies? The spectral density function ( 29 { } i G J = exp ) ( 2 For a randomly fluctuating field along the xaxis: ( 29 2 2 2 1 2 1 2 ) , ; ( 2 c c x c x B A B J + = = This is a Lorentzian lineshape the natural linewidth seen in an NMR spectrum Lets graph this:  = c t t G exp 5 1 ) ( What about less random events? A correlation function can describe the motion of a molecule or part of a molecule. The correlation function above is for the isotropic diffusion of a rigid rotor. Again, the correlation time, c , is the time constant for the exponential decay of the function. c is approximately the amount of time the molecule takes to rotate 1 radian. Notice that short correlation times cause the correlation function to decay rapidly and long times cause the function to decay more slowly. The correlation time depends primarily on molecular size and shape as well as solvent viscosity, temperature, A. S. Edison University of Florida ns c 1 = ns c 10 = ns c 100 = + = 2 2 1 5 2 ) ( c c J Spectral Density function The spectral density function, J( ) , is the Fourier transform of the correlation function. Just as rapidly relaxing time domain signals give rise to broad lines, short correlation times have a broad spectral density function. This makes sense: molecules that tumble very rapidly can sample a wide range of frequencies. Molecules that tumble slowly and have very long correlation times only sample lower frequencies....
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 Spring '08
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