Momentum transfer, intensity

Inertial forces are developed when the velocity of a fluid changes direction or magnitude. In turbulent flow, inertia forces are larger than viscous forces. Fluid in motion tends to continue in motion until it meets a sohd surface or other fluid moving in a different direction. Forces are developed during the momentum transfer that takes place. The forces ac ting on the impeller blades fluctuate in a random manner related to the scale and intensity of turbulence at the impeller. 

Figure 9-7. Elastic electron-diffraction pattern of a highly textured hcxaphenyl film. The Miller indices arc assigned using the intcrplauar spacings calculated in Kef. 11371. Inset Intensity of the f020) peak as a function of the angle between momentum transfer and the Teflon rubbing direction (see text) - taken from Ref. 138. ...

Fig. 11. Measured decrease of inelastic scattered He intensity (Rayleigh wave) from Pt(l 11) with increasing momentum transfer.

While all vibrational transitions arg allowed by Eq. (1), the intensity of a mode is governed by the (Q c.j term which expresses the component of the neutron momentum transfer along the direction of the atomic displacements. To an extent, this feature can be exploited with substrates such as Grafoil which have some preferred orientation. By aligning Q parallel or perpendicular to the predominant basal plane surfaces, the intensity of the "inplane" and "out-of-plane" modes, respectively, can be enhanced. In practice, while this procedure can be useful in identifying modes (9), the comparison with calculated intensities can be complicated by uncertainties in the particle-orientation distribution function. In this respect, randomly oriented substrates are to be preferred (10). 

Some of the alternative TOF instrument designs involve replacing the beryllium filter with either a crystal or a mechanical chopper to monochromate the incident beam. With this change, the spectrometer can be used with a higher incident neutron energy (typically E 50 meV) so that a smaller momentum transfer Q is possible for 5 the same energy transfer (21,22). With a monochromatic incident beam, a beryllium filter is sometimes substituted for the chopper after the sample in order to increase the scattered intensity but with a sacrifice in the,minimum Q attainable. Energy transfers up to 100 meV (800 cm" ) can be achieved with TOF spectrometers at steady state reactors before the incident neutron flux is limited by the thermal spectrum of the reactor. (With hot moderators such as at the Institut Laue-... 

The intensive momentum transfer and the strong interaction between the opposing streams lead to the global results below ... 

Neutron reflectivity is based on the interaction between a neutron beam and a surface [8]. In such an experiment, a collimated neutron beam, of wavelength k and incident angle 0, is directed onto an interface and the reflected intensity, R, is measured as a function of the momentum transfer Q, as given by the following equation ... 

Fig. 19 Main plot SAXS intensity (I) vs momentum transfer for a solution of 51 in acetonitrile (5.1 g L 1). The symbols and the solid line correspond to the experimental data points and the numerical fit using GNOM/DAMMIN simulated annealing, constraining the symmetry to the point group P432 (% = 1.397). Inset reconstructed low resolution particle shape for 51 obtained by the GNOM/DAMMIN fit (semitransparent spheres) superimposed onto the PM3 stationary point (space-filling model, iso-butyl groups substituted by methyl groups)...

Fig.7 SANS intensity (I) versus momentum transfer (q), for a 22.9 mM solution of supramolecular polymer 3 in rfg-toluene at several temperatiues (22 °C ( ) 36 °C (0) 52 °C ( ) 68 °C (A)). Reprinted with permission from [40]...

For hydrocarbons in zeolites, only incoherent scattering has to be considered because of the large incoherent cross section of hydrogen. The neutron intensity scattered follows the incoherent scattering law 5i c(Q, (o), which is related to the self-motion of protons, where AQ and Aw denote the neutron momentum transfer and the neutron energy transfer, respectively. 

The intensity variation along CTR s is particularly sensitive to the difference between the bulk and surface structures. Let us take for instance a surface whose last interplanar distance is b instead of a. This produces a large asymmetry of the CTR intensity around Bragg peaks. The larger the maximum value of the perpendicular momentum transfer, the larger the interference term and thus the asymmetry, and the better the accuracy on this relaxation. Hence, the measurement of CTR s allows the determination of the atomic structure of a surface. 

Small-angle x-ray scattering requires a special detector arrangement to measure at small and very small angles (0.0002° to 6°). In this technique, the intensity of the scattered photons, I q), is monitored as a function of the absolute value of the momentum transfer vector, q, for an elastic scattering event ... 

We see that the observed INS intensity of a vibrational transition is directly related to the MSD of the scattering atom in the mode of interest and the momentum transferred to the system, Q, during the scattering event. The MSD is a function only of the sample, i.e. of the forces acting on the atom and is a function only of the spectrometer, i.e. how the experiment was performed. 

In a SANS experiment a monochromatic neutron beam, intensity lo, is directed on the sample and scattered intensity I(Q) is measured as a function of angle 20, to the incident direction. Here Q is the momentum transfer (Q = 47Tsin0/ ). An important and recent development of the SANS technique concerns the investigation of materials which contain an oriented porous texture, such as fibres and layer-like materials [11-13]. 

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