Powder, averaging

The reader will appreciate that the scattering law of a particular vibrational transition, given by Eq. (2.32), is written in terms of the scalar (or inner, or dot) product of the neutron momentum transfer vector, Q, and the atomic displacement vector, . This product is a function of the angle between the vectors and allows some interesting single crystal effects to be examined. 

Occasionally molecular crystals of suitable size are found that have all of their molecules aligned mutually parallel [8], It is worth investigating how the scattering law of one particular atom, vibrating in the normal mode, v, would change as a function of the relative 

The value of the pre-exponential factor is strongly dependent on the relative orientation of the two vectors. If they are parallel, cos = 1, the scattering strength reaches its maximum value but if they are perpendicular, cos = 0, the intensity in this transition disappears. (Except that, due to the finite thickness of samples, the observed intensity never falls to zero. This is similar to the effects of leakage in single crystal Raman spectroscopy.) In favourable cases the angular dependence of the intensity variation can be used to examine some vibrations in isolation from others, or alternatively to determine the relative orientations of different vibrational displacements. 

The angular variation in the exponential of the Debye-Waller factor is not as striking as that for the pre-exponential. This function depends on the sum over all y, and this will always provide some component of atomic displacement parallel to the momentum vector. 

Here we shall consider the simple case of a single hydrogen atom confined in a box. Such a system is akin to an octahedral interstitial 

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Geldart a group Powder Average particle size, (, )J.m Particle density, p, kg/m Angles Internal friction, deg of Repose, deg Sphericity, f... 

In contrast, soft magnetic solids and paramagnetic systems with weak anisotropy may be completely polarized by an applied field, that is, the effective field at the Mossbauer nucleus is along the direction of the applied field, whereas the EFG is powder-distributed as in the case of crystallites or molecules. In this case, first-order quadrupole shifts cannot be observed in the magnetic Mossbauer spectra because they are symmetrically smeared out around the unperturbed positions of hyperfine fines, as given by the powder average of EQ mj, d, in (4.51). The result is a symmetric broadening of all hyperfine fines (however, distinct asymmetries arise if the first-order condition is violated). 

Chocolate milk samples prepared from sweetened cocoa powders averaged 58 mg per serving of theobromine and 5 mg per serving of caffeine.28 Analysis of a "home-style" recipe resulted in higher methylxanthine values — 94 mg theobromine and 10 mg caffeine per serving. However, the authors noted that this recipe also had a stronger chocolate flavor. The lower values reported by Zoumas et al. and Blauch and Tarka compared to others was attributed to the inability of older methods to separate theobromine and caffeine, and the lack of precision and accuracy of the older methods. A compendium of theobromine and caffeine values reported for chocolate beverages from both published and unpublished studies has been compiled in Table 8. 

A more precise powder average is obtained by distributing the magnetic field over a number of grids (pi) lying in a sphere. Then... 

Fig. 9 Magnetization curves for various ZFS systems modeled as a powder average at T = 4.2 K top panel D/k = -5 (solid), - 10 (long dashed), - 15 (medium dashed),- 20 K (short dashed) bottom panel D/k = + 5 (solid), + 10 (long dashed), + 15 (medium dashed), + 20 K (short dashed) Brillouin function for a Curie paramagnet - dot-dashed...

The most comprehensive information obtained from a Mossbauer spectrum is contained in Bint that depends on the magnetic hyperfine tensor A and, through (S), on the ZFS, the electronic g tensor (and exchange couplings when we consider polynuclear systems). For samples containing randomly oriented molecules, such as poly crystalline powders or molecules in frozen solution, the Mossbauer spectrum depends on the orientation of the molecule relative to the direction of the applied field,4 6 which is fixed in the laboratory and is generally either parallel or perpendicular to the direction of Mossbauer radiation. As a consequence, the spectrum is a powder average from which we have to extract the various tensor quantities of... 

In NQR, composite pulses have been proposed for the compensation of resonance offset [107-111] and RF field inhomogeneity [107,108,112-114], Several reports on pulses of continuously variable shape for use in NQR have also appeared [115-117], Shaped pulses that compensate for RF field inhomogeneity also partially compensate for the powder average effect (Section 2.1.2.). Theoretically, shaped excitation pulses can enhance SNR by up to 15% in powder samples [117]. Combinations of shaped excitation and refocusing pulses may also provide better response from multi-echo pulse sequences [9],... 

Mononuclear Au(lll) species supported on ceria powder (averaged 46 nm) is catalytically active for CO oxidation, and would aggregate into gold clusters with even increased catalytic properties, proved by X-ray adsorption study (Aguilar-Guerrero and Gates, 2007). 

Powdered samples yield almost as much information as single crystal samples and they are much easier to prepare. There are several, more or less sophisticated, methods of incorporating powder-averaging effects into the response function, see [18]. 

Three important consequences follow from the powder average first, all vibrations have some component parallel to Q and are therefore observable second, the intensity of each mode is weakened, and third, combinations are observable. Simple combinations are two quantum events, which occur when Q has components along two atomic displacement directions of the same scattering atom. 

Fine powder average particle diameter 9 pm Solubility practically insoluble in ethanol, ether, and water soluble in dilute acids. 

Particle size distribution polycarbophils are produced from primary polymer particles of an average diameter of about 0.2 pm. These polymers are then flocculated, resulting in powders averaging 2-7 pm in diameter. Once formed, the flocculated agglomerates cannot be broken down into their primary particles. 

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