Isotropic values

Fig.3 -NMR spectra at different sample orientations. The oriented sample of Fig. 2B was manually tilted in the static magnetic field Bq to acquire spectra at different angles, where 0 = 0° corresponds to the usual alignment of the sample normal parallel to the direction of Bq. The arrows indicate the shift of the anisotropic signals with respect to the isotropic value (dotted line), according to the factor 3(cos - l)/2 as expected for a molecule undergoing fast long-axial rotation about the membrane normal...

Here cto is the isotropic shift (1/3 [cti, 0-21 -1- ct3,]) and tree is the projection of the chemical shift tensor along the spinning axis and defined analogously to Eq. (3). In general this latter term produces a powder pattern when summed over nuclei at all orientations. However, when cos 0 = l/J3, the powder pattern collapses and only the isotropic value remains, i.e., CTjj/ = 0-5,. If this magic angle is misset by e radians, then (9)... 

If the isotropic value of g for the hydrated cation in eq. 8 above is used, we obtain the relation between the interion distance d (in nm) and the observed broadening. 

Free diffusion of molecules in solution is characteristically a haphazard process with net directionality determined only by solute gradients and diffusion coefficients. Within cellular and extracellular spaces, however, diffusion can be strongly influenced by noncovalent interactions of solvent and solute molecules with membranes as well as the cellular and extracellular matrix. Channels and orifices can also alter the movement of solute and solvent molecules. These interactions can greatly alter the magnitude of the diffusion coefficient for a molecule from its isotropic value D in water to apparent diffusion coefficient D (which often can be directionally resolved into D, Dy, and D ). The parameter A, known as the tortuosity, equals DID y. In principle, A has X, y, and z components that need not be equal if there is any anisotropy in the local electrical fields or porosity of the matrix. 

An enlarged view of the variation of 5iso in the vicinity of Tn is shown in Fig. 9. hi this plot, the low temperature side corresponds to the average of the data of the four peaks in Fig. 8. Notice that very close to Tn, the spectra are a superposition of those from both phases, but aroimd T = 373 K there is a sudden jump in the isotropic value, showing the onset of the near first-order phase transition. Here the whole plot is thus S-shaped, the jump in the isotropic value due to the phase transition is indicated as AS in the figure. The shape is consistent with the theoretical model developed in Chap. 1 in this volume. Some pertinent details are included here for completeness. 

These authors not only consider the isotropic value of such a coupling but also study its tensor character, 2hJ(N,N), where all four anisotropic terms were considered, i.e., FC/SD, PSO, SD and DSO. They studied systematically the dependence of the 2hJ(N, N) coupling with the hydrogen bond length, d(N-N), and bond angle, 9. For the latter fixed at 0=180°, they find that the isotropic part of 2hJ(N, N), 2hJiso(N,N), decays exponentially when increasing... 

Extensive magnetic measurements showed that both have an 5 = 12 ground state. The best fit to the HFEPR powder spectrum gave an isotropic (/-value of 2.15 and D = 0.015cm 1 The positive sign of D showed that these materials would not form SMM. 

Many spectroscopic phenomena depend to first order on symmetric second-rank tensors, that is, on six independent numbers [9]. The information contained in these six numbers, and its experimental accessibility in different media are illustrated in Figure 3. In fluids, five of these numbers are usually inaccessible to experiment, and the surviving isotropic value of... 

FIGURE 31. Summary chart of the principal components for the 29Si shielding tensors of the compounds studied. Axial symmetry is indicated by the double intensity of lines at The arrows indicate the isotropic values. The shielding scale is relative to the isotropic value for tetramethylsilane. Reproduced by permission of the Royal Society of Chemistry from Reference 140... 

The final refinement used anisotropic thermal parameters for all positions except 0(5), for which only isotropic values were used. The largest peak (x = 0.21, = z H, height 1.6 eX 3) on the final difference Fourier function (esd = 0.13 eX 3 at a general position) was unstable in least-squares refinement. The structure of partially dehydrated Bag-A is shown in Figure 1. 

Table I presents the comparison between the calculated and experimental, 5N chemical shifts in purine. The assignment of the NMR resonances is straight-forward using the liquid state, in 7-methyl substituted purine (22), and calculated values. As expected, in the predominat N7-H form, N7 experiences a large up field shift while N9 shows a similar down field shift as a consequence of the migration of the proton from N9 in the liquid to N7 in the solid. The isotropic values for the N9-H form in the solid are quite similar to the values observed in the liquid, the largest difference between corresponding chemical shift values is 12 ppm.

Most of the ab initio studies of NMR chemical shifts in peptides and proteins have focused on the average or isotropic value of the NMR chemical shift. The chemical shift is a tensor quantity and is, therefore, capable of providing six independent pieces of information, namely, the magnitude and direction of each of the three principal components. In general, the shielding tensor can be antisymmetric, leading to nine independent components. However, only the symmetric part of the shielding tensor is relevant to the experiments... 

In order to improve on the RHF values, electron correlation is included in the ab initio method by using MP2. Comparing the results for the 6-31 lG(d,p) basis set for all the diatomics, electron correlation results in a sizable correction to the RHF results differences of 8.7 ppm for A1H, 8.0 ppm for A1F, 3.7 ppm for A1C1 and 7.4 ppm for A1NC are noted. In the case of A1H, the isotropic value of the aluminum shielding calculated at the MP2 level is greater than the result from RHF theory however, for the fluoride, chloride and isocyanide, the opposite trend holds. Gauss et al. (42) also observed this trend for the hydride, fluoride and chloride. 

Not all 9 components of a are typically reported when describing the chemical shielding tensor. Instead, the 3 principal components (or eigen values) in a principal axis system (PAS) are reported. The CSA tensor can also be described by three additional parameters 1) the isotropic value (or trace), aiso, of the shielding tensor and is defined as... 

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