Fourier transform isotropic molecules

The isotropic nature of a liquid implies that any structure factor, S(k), obtained from a scattering experiment (typically X-ray or neutron) on that liquid will contain no angular dependence (of the molecules). Thus, the Fourier transform of any S(k) will yield a radial distribution function. Recently developed techniques of isotopic substitution [5-7] have been utilized in neutron diffraction experiments in order to extract site-site partial structure factors, and hence site-site radial distribution functions, gap(r). Unfortunately, because g p(r) represents integrals (convolutions) over the full pair distribution function, even a complete set of site-site radial distribution functions can not be used to reconstruct unambiguously the full molecular pair distribution function [2]. However, it should be mentioned at... 

Fig. 9. Two-dimensional isotropic-anisotropic chemical shift conelation spectrum of L-tyrosine, showing the isotropic projection and the assignment of the peaks to each site in the molecule. A set of 6S sign were digitized at rates of 12 kHz over a range of angles. Experimental data were then interptdated over a regular grid ctnnpos of 128 X 256 points and Fourier transformed. (Adapted with permission from Frydman et al. )...

Figure 2.28 (A) Observed (left) and calculated (right) NMR spectra of Leu5-enkephalin crystallized from water (A), methanol/H20 (B) and DMF/H2O (C). The isotropic signal at 0 Hz is due to natural abundant solvent molecules. Lorentzian line broadening (lb=1000 Hz) was applied prior to Fourier transformation. The 180° flip frequencies for the calculations are 5.0 x 103 (A), 3.0 x 104 (B), 2.4 x 106 (Q. The asymmetry parameters, tj, for the calculations are 0.02 (A), 0.05 (B), 0.05 (C). t=30 ps was used for the calculations. Reprinted from Ref. [166]. Copyright 1999 American Chemical Society.

In studies of isotropic and anisotropic liquids, key to relating the diffraction pattern to the scatterers in real space are two properties of Fourier transforms additivity and rotation [1]. Thus the transform of a sum is equal to the sum of the transforms also rotation in real space causes an equal rotation in reciprocal space. The reciprocal space, i.e. F(Q), will therefore accurately reflect the averaging over the irradiated volume of the specimen. For a spherically averaged rigid molecule the scatto-ed intensity is... 

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