Sample spherical

Yh fyT = X)( z) > as cubic and completely disordered lattices (incidentally, the type of arrangements for which in the classical Lorentz cavity-field calculation the contribution of the dipoles inside the small sphere vanishes) these lattice sums vanish for large spherical samples. The sums and... 

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

For the hnear susceptibility, the zero-held specihc heat as well as the dipolar helds, the anisotropy dependence cancels out in the case of randomly distributed anisotropy (at least for sufficiently symmetric lathees). In other cases the anisotropy is a very important parameter as shown for the linear susceptibility in Figure 3.3 for an inhnite (macroscopic) spherical sample. The susceptibility is divided by Xiso = order to single out effects of anisotropy and dipolar... 

Figure 3.7. Imaginary component of the dynamical susceptibiUty versus temperature (the real component is shown in the inset) for a spherical sample and spins placed in a simple cubic lattice. The anisotropy axes are all parallel, and the response is probed along their common direction. The dipolar interaction strength hi = 5j/2a is hi = 0 (thick lines), 0.004, 0.008, 0.012, and 0.016 from (a) right to left and (b) top to bottom. The frequency is coxo/tr = 2k x 0.003.

Equivalent Incident Power Densities at Different Frequencies. The average electric field intensity squared that is induced within a spherical sample of radius (a) much less than the wavelength ( ) of the radiofrequency field in air is... 

X-ray intensities were measured photometrically. To reduce the effect of absorption as much as possible, the sulfide was diluted with magnesium oxide. The value of the coefficient julR was experimentally determined. Moreover, to eliminate the effect of preferred orientation, which arises when the sample assumes the form of a cylindrical rod, we adopted a spherical sample shape of 0.3-mm. diameter. 

That no indication of the significant influence of particle shape on FFF elution behavior has been published until recently may be attributed to the fact that the majority of the approximately 500 papers so far published have reported on spherical or nearly spherical samples, and that the studies on the non-spherical samples focused only on sample fractionation rather than on a quantitative assessment of physicochemical quantities. This problem can be solved if fractions from FFF are further characterized, for example, by dynamic fight scattering or if an independent detector for diffusion coefficients is available. 

Despite numerous applications, conventional CRAMPS still remains one of the most demanding solid state NMR experiments as it requires the use of specially prepared spherical samples to minimise radiofrequency inhomogeneity effects and the careful calibration and setting of pulse widths and phases. Further modifications of the experiment that do not require the complicated and extended set-up procedures have been suggested recently. These are known as rotor-synchronised CRAMPS, which combines a new multiple pulse sequence [21], and its modification which uses a standard WHH-4 sequence at ultrafast MAS frequencies (e.g. 35 kHz) [22]. 

In his semi-macroscopic theory Kirkwood considers a dielectric sphere of macroscopic size, of volume V containing N molecules. The spherical sample of the isotropic, homogeneous dielectric of electric permittivity e is immersed in a uniform external field 0 applied in vacuum (of permittivity The mean macroscopic electric field E existing within the sphere is given by ... 

Fig. 19. The lower trace (b) is the response of a 3-mm-diameter spherical sample of water to a string of 90° pulses in the new coil. Each negative spike corresponds to a full rotation of the nuclear magnetization. The upper trace (a) was obtained with the old self-supporting eight-turn coil whose pitch was also optimized.

The chronology of the experimental data is from first to last. The parameters have a mean of 0.7470872, maximum deviation from the mean of 0.0370847, and standard deviation of 0.0216772. There is no systematic trend in the deviation which is as it should be if we have treated the problem properly. The standard deviation is only 2.9% of the mean. We consider this to be a very satisfactory resolution of a difficult experimental problem. The small size and near-spherical nature of the crystal permitted a spherical sample absorption correction to be applied, for which fir was taken to be <0.5. 

A dielectric sphere of dielectric coefficient e embedded in an infinite dielectric of permittivity 82 is an important case from many points of view. The idea of a cavity formed in a dielectric is routinely used in the classical theories of the dielectric constant [67-69], Such cavities are used in the studies of solvation of molecules in the framework of PCM [1-7] although the shape of the cavities mimic that of the molecule and are usually not spherical. Dielectric spheres are important in models of colloid particles, electrorheological fluids, and macromolecules just to mention a few. Of course, the ICC method is not restricted to a spherical sample, but, for this study, the main advantage of this geometry lies just in its spherical symmetry. This is one of the simplest examples where the dielectric boundary is curved and an analytic solution is available for this geometry in the form of Legendre polynomials [60], In the previous subsection, we showed an example where the SC approximation is important while the boundaries are not curved. As mentioned before, using the SC approximation is especially important if we consider curved dielectric boundaries. The dielectric sphere is an excellent example to demonstrate the importance of curvature corrections . 

Data from ref. 61 l4N spectra at 4-3345785 MHz ( 0-5 Hz), sample temperature 30 2°C concentric spherical sample containers are used in order to eliminate bulk susceptibility effects on shifts and signal shape external standards are used, CH3NO, (neat liquid) and, for signals within 15 ppm of that of neat CH,N02, tetranitromethane (neat liquid) reported shifts represent values obtained from iterative fitting of theoretical and experimental lineshapes using a differential saturation method, ref. 63, reported errors are standard deviations for the fitting of at least 200 data points, and represent 68% confidence limits for shifts which are recalculated from values referred to C(N02)4, the error of the shift of C(N02)4 relative to 0H,NO2 is included. 

In order to provide a means for the precise recalculation of nitrogen chemical shifts reported since 1972, it is necessary to have accurate values of the differences in the screening constants between neat CH3N02 and the large number of reference compounds which have so far been used. Table VII shows the results of precise, 4N measurements (61) which have been carried out in concentric spherical sample and reference containers in order to eliminate bulk susceptibility effects on the shifts. Since the technique adopted (61, 63) involves the accumulation of a large number of individually calibrated spectra with the subsequent use of a full-lineshape analysis by the differential saturation method, (63) the resulting random errors comprise those from minor temperature variations, phase drifts, frequency instability, sweep nonlinearity, etc. so that systematic errors should be insignificant as compared with random errors. 

Ref. 61, l4N spectra, neat CH3N02 external reference in concentric spherical sample containers in order to eliminate bulk susceptibility effects differential saturation technique and full lineshape analysis as in ref. 63, errors quoted are standard deviations for about 200 data points. 

Ref. 61, see also Table VII for comments 14N spectra, concentric spherical sample containers, full lineshape analysis. 

This behavior for I and A can be predicted on classical grounds. The work function for bulk metal would be modified for small spherical samples by the Born charging energy. The ionization potential would be increased and the electron affinity would be decreased by the same amount... 

Both carbonyl iron and iron oxide particles were dispersed in the Elas-tosil 604 A. After mixing it with the Elastosil 604 B component, the solution was transferred into a cube-shaped mould. The cross-linking reaction was carried out at ambient temperature for 4.5 h to obtain the magnetic composites. After cross-linking polymerization, the cubed, cylindrical, and spherical samples were removed from the moulds [77,78]. 

The choice of sample-holder shape will affect the interaction of the gas atmosphere with the sample. If a glass capillary-tube-type holder is used, the changing of the furnace atmosphere will have little effect on the DTA curve due to the long gas diffusion path between the sample and the furnace atmosphere. A flat-dish-iype holder is perhaps ideal for control of the gas-solid reaction but may cause loss of AT sensitivity due to radiant heat loss. For reactions in which the gas atmosphere plays no part, a spherical sample holder might be ideal, but would cause difficulty in introducing the sample. 

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