Scattering probability

In the diffuse mismatch model, the scattering destroys the correlation between the wave vector of the impinging phonon and that of the diffused one. In other words, the scattering probability is the same independent of which of the two materials the phonon comes from. This probability is proportional to the phonon state density in the material (Fermi golden rule). 

Figure 8.9 shows that the concentration of intermediate in reversible series reactions need not pass through a maximum, while Fig. 8.10 shows that a product may pass through a maximum concentration typical of an intermediate in the irreversible series reaction however, the reactions may be of a different kind. A comparison of these figures shows that many of the curves are similar in shape, making it difficult to select a mechanism of reaction by experiment, especially if the kinetic data are somewhat scattered. Probably the best clue to distinguishing between parallel and series reactions is to examine initial rate data—data obtained for very small conversion of reactant. For series reactions the time-concentration curve for S has a zero initial slope, whereas for parallel reactions this is not so. 

To reduce the divergence of the total cross section, which is obtained by integral of Eq. (17), the cutoff angle was set so as to limit the increase in the scattering probability at low scattering angle. Fig. 11 shows calculated total cross section for H (left) and He " (right). 

The Laue and the Bragg condition give us information about the angular distribution of the diffraction peaks. To calculate the peak intensities, we have to know more about the scattering properties of the atoms or molecules in the crystal. In the case of X-rays and electrons the scattering probability is proportional to the electron density ne(r) within the crystal. Since n,(r) has to have the same periodicity as the crystal lattice, we can write it as a three-dimensional Fourier series (using the notation eikx = cos kx + i sin kx) ... 

Up to this point we did not make any specific assumptions about the real space lattice. It could contain more than one atom per lattice point and more than more than one type of atoms. In such a case the lattice would be described using a Bravais lattice plus a basis (see Section 8.2.2. To obtain the intensity of the diffracted wave for crystals with a basis, we simply have to sum up the contributions from all scattering points within the unit cell. The scattering probability for a crystal of N unit cells with an electron density ne(r) is proportional to ... 

D.G. Truhlar, A. Kuppermann, Quantum mechanics of the H+TL reaction Exact scattering probabilities for collinear collisions, J. Chem. Phys. 52 (1970) 3841. 

Figure 4.38 Total cross sections for the scattering of electrons on rare gases. This cross section is given by

We return now to the connection between the S -matrix elements in Eq. (4.137) and the measurable cross-section. To that end, the scattering probability must be averaged over the relevant ) states, which all are assumed to be sharply centered around the momentum p = p0. 

The scattering probability into the solid angle element dQ about the direction of the final momentum p is... 

Since we have scattering from wave packets uniformly distributed over 6, the total scattering probability into the solid angle element dCl is / dbP(dil <— b). If this number is multiplied by the (relative) flux density of molecules in the beam, we get the flux of molecules that show up in dfl. Thus the cross-section is simply... 

Figure 11 A comparison of theoretical scattering probabilities (solid bnes) with those of experiment (symbols) [72], The top panel shows scattering into the same quantum state as the initial state for which there is good agreement. The bottom panel shows that theory greatly overestimates the probability of scattering into a rotationally excited state.

Figure 14 Experimental demonstration of the thermal effects in H2 dissociation and scattering on Cu surfaces. A shows the Arrhenius dependence of the dissociation probability at die translational energies indicated [44]. From these curves, a translational dependent activation can be extracted, as in B [44]. The dependence is clearly linear with a slope of — 1. Scattering probabilities for rotational excitation also show and Arrhenius dependence, as shown in C. The activation energy extracted from this has a very strong state dependence.

Most experiments do not depend on order parameters of higher rank, e.g. the influence of orientational order on an absorption band is completely described in terms of S and D (Luckhurst, 1993). On the other hand, Raman-spectra being based on a two-photon effect are influenced additionally by the order parameters of the next level, such as the Legendre polynomial P4 (Pershan, 1979). This is of considerable theoretical interest, however, up to now of less importance for practical applications. There are some further experimental techniques for gathering information on the orientational order, among these are fluorescence, neutron and electron scattering. Probably the most reliable method is NMR (Emsley, 1985), however this usually means deuteration of all hydrogen atoms but one. 

Although the theory of the SdH effect [256], which deals with the detailed problem of electron scattering in a magnetic field, is quite complicated a qualitative explanation is possible by virtue of a simple argument [257]. The probability for an electron to scatter is proportional to the number of states into which it can be scattered. As discussed above, for a metal in a magnetic field the density of states at the Fermi level N ep) will oscillate with the field and, therefore, the scattering probability and the electronic relaxation time r will oscillate, too. It can be shown that the oscillatory part of the density of states, Niep), has essentially the same analytic form as (3.6) with the... 

Berlin et al. (1970) and Ginzburg (1982) used differential scanning calorimetry to measure the heat of desorption of samples of varied initial water content, for several proteins. The data show considerable scatter, probably owing to broad endotherms that span about 100°C. 

---