Acoustic long-wavelength

For short wave vectors (or long wavelengths) corresponding to waves in the acoustic or ultrasonic range, eq. (8.17) reduces to... 

Novel sample preparation techniques include ultrasonic extractions that use high frequency acoustic waves to heat and break up samples (9), as well as microwave-assisted extractions (MAE) that use long wavelength radiation for faster and less energy intensive extractions of thermally sensitive analytes (JO-13). Other innovations treat samples with high pressure and high temperature solvents in the liquid or in the supercritical state. These adaptations reduce the overall solvent use and speed the extractions. These methods include accelerated solvent extraction (ASE) (14) and supercritical fluid extraction (SEE) (8). 

Circular dichroism is always a small fraction of the direct absorption. Using visible, UV or infrared ratios as a guide we would expect that the ratio will be — aiX THz, where a is of the order of the atom or ion spacing. This is a very small number at terahertz frequencies. But the molecular unit, terahertz chromophore that is responsible for the terahertz absorption involves many atoms and ions. For a helix 7 periods in length, the long wavelength acoustic mode in our simple model is — 140 and we expect to recover circular dichroism/absorption ratios that are reasonable relative to those of electronic excitations or near infrared vibrations. For this particular, rather... 

Earlier Toyozawa et al. 153> formulated a dynamic model for the behaviour of the vibrational excitation. This clarified and improved upon the classical description of a non-diffusing wave-packet. Such description appears justifiable for F-centres whose optical bands are broad and diffuse not exhibiting any fine structure that might be indicative of discrete, quantised vibrational levels. (The relatively large spatial extent of the F-centre electronic wave functions also favours the involvement of long wavelength acoustic vibrations. For these a classical description can be appropriate.)... 

The acoustic phonons of (SN) have been studied by inelastic neutron scattering. These investigations are extremely difficult because of the small size and poor quality of the crystals and therefore had to be restricted to the long wavelength region. 

On the other hand, the two modes of CeBeij show a softening of about 2% with respect to the reference line. All other symmetry modes of CeBe j do not show any anomaly. This applies also to the bulk modulus of CeBcij, partly indicative of the behavior of the long-wavelength acoustic phonons as shown in fig. 36. As a function of Q/V the bulk modulus of the RBe j compounds follows a straight line with CeBe j right on it (Mock et al. 1985). However, an even stronger softening of the two F modes is found upon Ce dilution in... 

The elastic constants determine the acoustic phonon dispersion for long wavelengths according to the equations of motion in the continuum limit ... 

A plot of the orthorhombic unit cell for polyethylene is illustrated in Fig. 4, with the 50% probability density surfaces shown for each atom of the unit cell. The anisotropic dynamics of the polyethylene crystal are immediately apparent. Significantly, the largest contributions to B,- come from the long wavelength acoustic modes, due to the inverse dependence on a/. In experimental studies on polymers, the temperature factor is often... 

Here, the simple dispersion (oiq) v g was used for the acoustic phonons (if whole molecules vibrate with respect to each other, like the change of the stacking distance in a stack one speaks of acoustic phonons, because they have a long wavelength comparable to those of acoustic waves) and the sound velocity v, is determined by the relation o, = Ci/p), where c, is the longitudinal elastic constant and p is the mass density. One should remark that this very simple linear dispersion relation (o q) = v,g is not necessarily correct. With the help of the FG method described in Section 9.1 one can obtain more accurate dispersion curves. Equation (9.48) can now be used to calculate the charge carrier mobilities and free paths, defined in this case hy p= e xlm ) and A = (t ), respectively, where... 

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