Projection-cross-section theorem

Equation (5.4.13) states that the NMR spectrum Ti(o)) acquired in the presence of a magnetic-field gradient G produces a projection P —u>/ yGx)) of the spin density Mo x,y,z)- In comparison to that, the FID (5.4.12) provides a trace through k space. This relationship is known as the projection-cross-section theorem. 

Here the space variables r and s in the Cartesian coordinate frame of the projection have been replaced by the Cartesian coordinates x and y in the laboratory frame (cf. Fig. 5.4.1), and

rotation angle between both frames. This equation is another formulation of the projection cross-section theorem (cf. eqn (5.4.12)), which states that the Fourier transform p(k, cp) of a projection P(r, (p) is defined on a line p kcos(p, k sin p) at an angle

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The projection-cross-section theorem states that an w-dimensional cross section. Cm t), through A-dimensional time domain data (m < N) is related by an /M-dimensional Fourier transformation to an /n-dimensional orthogonal projection... 

Fig. 1 Illustration of the projection-cross-section theorem [17-19] for a 2D frequency space with two indirect dimensions k and j. ID data cf (t) on a straight line in the 2D time domain (ty, tj) (left) is related to a ID orthogonal projection (of) of the spectrum in the 2D frequency domain (

It is thus not possible to obtain a phase-sensitive 45 projection of a normal homonuclear 2D J spectrum, and instead it is usual to project the absolute value or the power spectrum. This gives rise to distorted intensities and peak positions, but nevertheless is an extremely useful aid to assignment. The subject of phase-sensitive proton spectra without multiplet splittings is not however entirely closed, despite the clear message of the projection-cross-section theorem, and will be returned to later. In the interim, some of the many data acquisition and handling methods discussed above will be illustrated with some experimental spectra. 

The quantity f q) is the diffraction pattern, and a R) is the Patterson function (cf. Fig. 5.4.3). This interpretation relates to the description of PFG NMR in terms of probability densities or average propagators. In fact, the condition (5.4.20) defines a cross-section in the 2D k space spanned by fei and k2, so that by the projection-crosssection theorem (5.4.12) the Patterson function can be interpreted as a projection of the corresponding signal 5(r] )S (r2) onto the subdiagonal in the space defined by r and rz. 

Because the projectile provides a perturbation which is treated as a sum of single particle perturbations, each orbital electronic wave function, develops independently from the others the correct description of the system is given by forming an appropriate antisymmetrized product of these time dependent orbitals. To calculate the required cross section we have to project onto all multielectron states that have a K-shell vacancy, and sum the resulting probabilities. The simple single electron theorem results. There is no role played by the passive electrons in the IPM except in defining the initial Fermi sea. For example, to attempt to include statistical correlation in a classical calculation of an inclusive cross section would be unproductive. 

Numerical execution of cross-correlations on a conventional computer is demanding in time. The processing time could be significantly reduced by use of dedicated processors for parallel computing, but computation in the time domain can be avoided altogether if time-invariant field gradients are employed. Projections of the spin density are obtained by Fourier transformation of the ID (4.3.2) and 3D (eqn (4.3.3) with n—y) cross-correlation functions. With the correlation theorem (cf. Section 4.3.3),... 

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