Spectral Density upon Frequency

G(t) is often, but not necessarily, a simple function decaying exponentially to zero on either side of t = 0 with a time constant x, the correlation time for the motion under consideration. For example, random jumps of otherwise fixed interactions will give G(t) the same first-order decay as occurs with random radioactive fission of nuclei. Random angular diffusion will achieve the same. In the exponential case, for any frequency co, 

Many molecular motions are more complex than simple angular jumps or isotropic diffusion. Although this means that G x) will often not be a simple exponen- 

Another tractable case is that of relatively rapid internal rotations or librations superimposed on a slower overall molecular tumbling. The presence of such motions is revealed by unexpected NOE and Tj s, and by a nonstandard variation of with spectrometer frequency. In this case also the angular correlation function does not decay exponentially. Instead, it decays at a rate determined by the internal motion to some fraction of its original value. This fraction quantitatively represents the mean angular constraint on the internal motion, in that it is the angular correlation that remains when the internal motion alone has completed its randomizing work. G x) then continues to decay more slowly to zero at a rate (here assumed to be much slower) determined by the overall molecular tumbling. 

Analyses of this type reveal plausible and consistent internal motions for such diverse cases as proline ring-puckering, floppy peptides, proteins including the mobile binding of chloride ion, nucleic acids, and even solid rubber.  

They can also measure very low-energy barriers to the rotation of methyl groups, and they may find application in the study of highly fluxional inorganic compounds where the internal exchanges make a measurable contribution to relaxation. 

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Because the fast and slow relaxation components do not depend upon the same spectral densities J(0) and J(u) versus J(u) and J(2(d) respectively, they change In unlike manner with the observing Larmor frequency. For sodlum-23, when the correlation time x in the bound state exceeds 5 ns the observed resonance acquires enough non-Lorentzlan character that It Is feasible to deconvolute reliably the llneshape Into Its two Lorentzlan components. Then the most accurate method Is to perform another deconvolution at a widely dlfferen Larmor frequency w. For Instance, the slow relaxation rate yir varies by about 700 between observing frequencies of 23.81 Ind 62.86 MHz for long x values (> 5 nsX61. 

Chang and Woessner, using the spin-echo technique, made and TU measurements on eletal muscle (115). While the curve Is cfbse to a single exponential decay, the T2 curve can be fitted with a double exponential function. The fast and slow (f,s) components have T values of 16 and 1.6 ms, respectively, for fresh muscle. The value for the same sample Is 18 ms. Hence, T. - Tg- < Tgg and Tgg = T. The reader will recall (Section 2 that T g depends upon the spectral density at twice the Larmor frequency, T depends on J((o whereas Tgf depends upon spectral densities at zero emd at Larmor frequency and Tgg depends upon the sum J(o) + J(2a>). Thus, the experimental results obtained by Chang emd Woessner Imply that J(0)>>J(o) ) = J(2oj ). This last equality. In turn. Indicates... 

The connection between the rate constants W and molecular parameters is a complex subject that can only be outlined here. Basically, the dipole-dipole interaction is a through-space effect in which one nuclear magnetic dipole interacts with the local field created by a second nucleus. The value of W depends upon the component of this field that fluctuates at the frequency of the transition, that is at 0, (o and 2Larmor frequency. Working out the algebra shows that the cross-relaxation rates are proportional to spectral densities, which are Fourier transforms of time correlation functions that describe molecular motions ... 

This is Planck s famous radiation law, which predicts a spectral energy density, p , of the thermal radiation that is fully consistent with the experiments. Figure 2.1 shows the spectral distribution of the energy density p for two different temperatures. As deduced from Equation (2.2), the thermal radiation (also called blackbody radiation) from different bodies at a given temperature shows the same spectral shape. In expression (2.2), represents the energy per unit time per unit area per frequency interval emitted from a blackbody at temperature T. Upon integration over all frequencies, the total energy flux (in units of W m ) - that is, Atot = /o°° Pv Av - yields... 

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