Anharmonic scattering

The reason why the spacings are equal, and not the 1-0, 2-1, 3-2,... anharmonic intervals, is explained in Figure 9.21. The laser radiation of wavenumber Vg takes benzene molecules into the virtual state Fj from which they may drop down to the v = level. The resulting Stokes scattering is, as mentioned above, extremely intense in the forward direction with about 50 per cent of the incident radiation scattered at a wavenumber of Vg — Vj. This radiation is sufficiently intense to take other molecules into the virtual state V2, resulting in intense scattering at Vg — 2vj, and so on. 

A dynamic transition in the internal motions of proteins is seen with increasing temperamre [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24]. 

The observation of the departure from cubic symmetry above Tm co-incident with the appearance of the central peak scattering serves to resolve the conflict between dynamic and lattice strain models. The departure from cubic symmetry may be attributed to a shift in the atomic equilibrium position associated with the soft-mode anharmonicity. In such a picture, the central peak then becomes the precusor to a Bragg reflection for the new structure. 

The second set of problems in dynamics are those of scattering theory where the Hamiltonian is of the form H = H0 + V and the interaction V vanishes when the colliding particles are far apart. It is usually assumed that the H0 part is already solved and that the interesting or the hard part is to account for the role of V. For realistic systems, which are anharmonic, even the role of H() can be quite significant. An example that has received much recent attention is the reaction of vibrationally excited HOD with H atoms (Sinha et al., 1991 Figure 8.2). The large difference in the OH and OD vibrational frequencies means that the stretch overtones of HOD are primarily local in character (cf. Section 4.21). It follows that one can excite HOD to overtones localized preferentially on either one of the two bonds and that an approaching H atom will abstract prefer-... 

To conclude, it seems that the nature of the anharmonic coupling between a high frequency intramolecular mode and a thermally excited low frequency mode is understood. It turns out that the strength of the influence on the infrared spectrum critically depends on the values of (Oq, Sm and t. However, we have to wait for more experimental data on these low frequency modes, probably obtained with the helium atom scattering technique, bdbre we can make more definite conclusions. 

Here, AT is a constant, f is the incoming intensity, R is the distance of the scattered wave from the molecule (in practical terms, it is the distance between the scattering center and the point of observation), i and j are the labels of atoms in the jV-atomic molecule, g contains the electron scattering amplitudes and phases of atoms, 5 is a simple function of the scattering angle and the electron wavelength, I is the mean vibrational amplitude of a pair of nuclei, r is the intemuclear distance r is the equilibrium intemuclear distance and is an effective intemuclear distance), and k is an asymmetry parameter related to anharmonicity of the vibration of a pair of nuclei. 

With the availability of lasers, Brillouin scattering can now be used more confidently to study electron-phonon interactions and to probe the energy, damping and relative weight of the various hydro-dynamic collective modes in anharmonic insulating crystals.The connection between the intensity and spectral distribution of scattered light and the nuclear displacement-displacement correlation function has been extensively discussed by Griffin 236). 

As first shown by Dawson (1967), Eq. (11.3) can be generalized by inclusion of anharmonicity of the thermal motion, which becomes pronounced at higher temperatures. We express the anharmonic temperature factor of the diamond-type structure [Chapter 2, Eq. (2.45)] as 71(H) = TC(H) -f iX(H), in analogy with the description of the atomic scattering factors. Incorporation of the temperature... 

In the hydrate lattice structure, the water molecules are largely restricted from translation or rotation, but they do vibrate anharmonically about a fixed position. This anharmonicity provides a mechanism for the scattering of phonons (which normally transmit energy) providing a lower thermal conductivity. Tse et al. (1983, 1984) and Tse and Klein (1987) used molecular dynamics to show that frequencies of the guest molecule translational and rotational energies are similar to those of the low-frequency lattice (acoustic) modes. Tse and White (1988) indicate that a resonant coupling explains the low thermal conductivity. 

The LHA is exact for potentials that contain, at most, quadratic terms but obviously an approximation for anharmonic potentials. Thus, a single Gaussian wave packet within a local harmonic approximation can, e.g., not tunnel or bifurcate, i.e., there will be no simultaneously reflected and transmitted part in scattering off barriers. 

With regards to the second feature of real crystals mentioned earlier, there are different types of anharmonicity-induced phonon-phonon scattering events that may occur. However, only those events that result in a total momentum change can produce resistance to the flow of heat. A special type, in which there is a net phonon momentum change (reversal), is the three-phonon scattering event called the Umklapp process. In this process, two phonons combine to give a third phonon propagating in the reverse direction. 

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