Intensity dimensions

D-NMR methods are highly useful for structure elucidation. Jeener described the principles of the first 2D-NMR experiment in 1971 [31]. In standard NMR nomenclature, a data set is referred to by one, i.e., less than the total number of actual dimensions, since the intensity dimension is implied. The 2D-data matrix therefore can be described as a plot containing two frequency dimensions. The inherent third dimension is the intensity of the correlations within the data matrix. This is the case in ID NMR data as well. The implied second dimension actually reflects the intensity of the peaks of a certain resonance... 

There is a depth or intensity dimension within some b-SoCs (discussed in chapter 14). So we could speak of the b-SoC having reached its deepest (or shallowest) extreme, (back)... 

An LC—MS chromatogram is essentially a three-dimensional data set with separate chromatographic, mass-to-charge (m/z) and ion count (intensity) dimensions. The combination of chromatography and mass spectral approaches can take on many incarnations and are dependent on the nature of the metabonomics investigation. Because of the complexity of most metabonomics samples, the final solution is often a compromise between good science and expediency since the ideal analytical conditions for 1000+ metabolome components is an um-ealistic goal. In this section, we will discuss each of these elements separately with an emphasis on biomedical metabonomic applications. 

Figure 5 Effect of free-induction decay (FID) truncation in the (a) time or (b) intensity dimension. Truncation effects have been exaggerated to make the distortions more obvious. The sample is squalane in CDCI3.

Multidimensional analyses of ratings of similarity typically yield a dominant hedonic dimension pleasant odors cluster on one side of an odor space and unpleasant odors cluster on the other ([75,97,99,102 104,106] cf, [96]). Analyses often yield an intensity dimension as well, and occasionally one or two others that are unimportant in statistieal terms and defy clear interpretation. 

We have seen that the intensities of diffraction are proportional to the Fourier transfomi of the Patterson fimction, a self-convolution of the scattering matter and that, for a crystal, the Patterson fimction is periodic in tln-ee dimensions. Because the intensity is a positive, real number, the Patterson fimction is not dependent on phase and it can be computed directly from the data. The squared stmcture amplitude is... 

For molecules having dimensions comparable with the wavelength, phase differences will occur between waves scattered from different regions of the molecule. These phase differences result in an angular dependence of the scattered intensity. The reduction may be expressed in temis of a particle interference factor P(2Q) such that... 

The remarkable stability and eontrollability of NMR speetrometers penults not only the preeise aeeiimulation of FIDs over several hours, but also the aequisition of long series of speetra differing only in some stepped variable sueh as an interpulse delay. A peak at any one ehemieal shift will typieally vary in intensity as this series is traversed. All the sinusoidal eomponents of this variation with time ean then be extraeted, by Fourier transfomiation of the variations. For example, suppose that the nomial ID NMR aequisition sequenee (relaxation delay, 90° pulse, eolleet FID) is replaeed by the 2D sequenee (relaxation delay, 90° pulse, delay i -90° pulse, eolleet FID) and that x is inereased linearly from a low value to ereate the seeond dimension. The polarization transfer proeess outlined in die previous seetion will then eause the peaks of one multiplet to be modulated in intensity, at the frequeneies of any other multiplet with whieh it shares a eoupling. 

Figure Bl.15.16. Two-pulse ESE signal intensity of the chemically reduced ubiqumone-10 cofactor in photosynthetic bacterial reaction centres at 115 K. MW frequency is 95.1 GHz. One dimension is the magnetic field value Bq, the other dimension is the pulse separation x. The echo decay fiinction is anisotropic with respect to the spectral position.

The intensity of the vertically polarized scattered light is proportional to sin 0 which, in polar coordinates, is described by a figure 8-shaped curve centered at the origin and having maximum values of 1 at 0 = 90°, Because 0 is symmetrical with respect to the z axis, this component of scattered light is described in three dimensions by a doughnut-shaped surface in which the hole has shrunk to a point - centered symmetrically in the xy plane. 

In Example 10.5 we extracted both the molecular weight and the radius of gyration from Ught-scattering data. There may be circumstances, however, when nothing more than the dimensions of the molecule are sought. In this case a simple alternative to the analysis discussed above can be followed. This technique is called the dissymmetry method and involves measuring the ratio of intensities scattered at 45° and 135°. The ratio of these intensities is called the dissymmetry ratio z ... 

For a two-dimensional array of equally spaced holes the difftaction pattern is a two-dimensional array of spots. The intensity between the spots is very small. The crystal is a three-dimensional lattice of unit cells. The third dimension of the x-ray diffraction pattern is obtained by rotating the crystal about some direction different from the incident beam. For each small angle of rotation, a two-dimensional difftaction pattern is obtained. 

---