Three-phonon spectrum

The shape of the two-phonon spectrum is given by the convolution of the one-phonon spectrum with itself, g(a )ext g((i))ext- The shape of the three-phonon spectrum, n = 3, is obtained by convolving the one- and two-phonon spectra, and so on. The weight of each contribution is given by... 

In the molecular approximation used in (14) only the L = 3W — 6 (W is the number of atoms) discrete intramolecular vibrations of the molecular complex in vacuo are considered. In general these vibrations correspond to the L highest optical branches of the phonon spectrum. The intermolecular vibrations, which correspond to the three acoustical branches and to the three lowest optical branches are disregarded, i.e., the center of mass and - in case of small amplitudes - the inertial tensor of the complex are assumed to be fixed in space... 

There is only one known acceptor in diamond, responsible for the p-type conductivity of the lib diamonds. For some time, it was assumed that this acceptor was aluminium [49], but it has been suggested [43] and finally shown conclusively [38] that boron was indeed responsible for the p-type conductivity and the spectroscopic properties of type lib blue diamonds. Natural lib diamonds had been identified ca. 1954 (see Sect. 2.11), and synthetic lib diamonds were obtained at the beginning of the 1960s [80]. Boron is commonly introduced as a dopant in synthetic diamonds and its ionization energy ) is 370 meV [177]. The discrete acceptor spectrum of B extends approximately 70 meV below ) and is superimposed on the two- and three-phonon spectra of Cdiam- Boron acceptor absorption lines are observed at 305, 347 and 363 meV ( 2780, 2800, and 2930 cm 1) at RT, giving phonon-assisted transitions near 464 and 504meV (see [140], and references therein). 

The first measurement of the temperature dependence of an optical line width in an actinide system, Np + in LaC, was recently completed (47). The fluorescence transitions at 671.4 and 677.2 nm were studied from 10 to 200 K. The low temperature limit for the line width of the 677.2 nm transition is 16.5 GHz and is a measure of the width of the first excited crystal-field level of the ground manifold. The 671.4 nm transition has a line width of 0.55 GHz at 10 K. Its temperature dependence is described in terms of an effective three-level scheme for the excited manifold. The parameters are comparable to those found for Pr + in LaF. Further comparison depends upon the details of the phonon spectrum and the electronic states. At low temperatures, the residual width of the 671.4 nm transition was limited by the laser line width. This is consistent with the very narrow line widths observed in Pr +. Additional detailed studies of this type and proper contrast and comparison between lanthanides and actinides may provide the additional information needed to describe the electron-phonon and electron-ligand interactions of the actinides. 

In the above diagram, the emission intensities of the three bands remain constant. It is the peak Intensity which changes as the band broadens. To this point, we have aceepted the fact that vibronic coupling leads to broad band excitation and emission in a phosphor. Take note that the above diagram is the result of experimental measurements which prove that as the temperature increases, the phonon spectrum becomes broadened, thereby leading to broadening of the bands. Thus, at - 200 °C., the number of phonon vibrations is restricted and a rather sharp emission band is seen. As the temperature rises, the number of separate phonon branches increases (the empty phonon levels become occupied) and the emission (excitation) band is further broadened. Note that at some temperature above 300 °C. (in this example), the phonon vibrations increase to the... 

We performed a calculation of the relaxation rates using the phonon Green s functions of the perfect (CsCdBr3) and locally perturbed (impurity dimer centers in CsCdBr3 Pr ) crystal lattices obtained in Ref. [8]. The formation of a dimer leads to a strong perturbation of the crystal lattice (mass defects in the three adjacent Cd sites and large changes of force constants). As it has been shown in Ref. [8], the local spectral density of phonon states essentially redistributes and several localized modes appear near the boundary of the continuous phonon spectrum of the... 

The above-mentioned relationships may be due to the form of the initial part of the phonon spectrum, i.e., its acoustical branch for three-dimensional Debye solids the spectrum is g8(v) oc for two-dimensional structures one-dimensional structures... 

Some interesting and important conclusions were drawn by separating the phonon spectrum in accordance with the polarization of the oscillations [15]. The whole spectrum was divided into six branches, each of which has an almost Gaussian form of the distribution curve g( ). For cubic crystals, these six branches consist of three acoustical branches (one branch of longitudinal and two branches of transverse waves) and three optical branches (one longitudinal and two transverse waves). The acoustical vibrations can be compared with the vibrations of atoms in a unit cell, and the optical vibrations with mutual oscillations of the sublattices in relation to one another. The curves of the density distribution of oscillations in each [Pg.180]

For crystals with a cubic structure, the elastic properties are characterized by three elastic moduli cj, Cj2, and C44. When considering the first two coordination spheres in the central-force approximation, these three moduli are sufficient to reproduce the whole phonon spectrum. When the next coordination spheres are considered, additional force constants must be known. The number of force constants which must be known depends on the number n of the spheres beii considered. 

Fig. 10.7. Shift of three characteristic peaks of the phonon spectrum of La under pressure. The negative Grilneisen constant ya= -1.3 reflects the phonon softening (WUhl et al., 1973).

Another advantage of this model is that expression (3.1.6) involves only three quantities related to the smface characteristics and does not depend on any specific model of the phonon spectrum. The contributions of specific surface phonons and adsorbate vibrational modes are additive ... 

The reflection spectrum measured at room temperature from 90 to 600 cm" has a maximum near 130 cm (R 85%) and a minimum near 185 cm (R 1%). These two extrema were attributed to the lattice vibrations cojo and (Olq, respectively. Axe [1], see p. 187. Absorption measurements on an EuSe film on LiF at 2 K gave o)to = 134.0 cm" Ikezawa, Suzuki [2]. Bulk samples have their characteristic reststrahlen band between (Ojo and colo, see Fig. 116 which shows reflection and transmission spectra of stoichiometric EuSe obtained by sublimation for the paramagnetic state at 300 K and for the antiferromagnetic NNSS state at 3 K. The numerous intrinsic structures in T(v) (and weaker in R(v)) observed beside the reststrahlen band were explained by one-, two-, and three-phonon processes consistent with measurements of second-order Raman scattering (cf. p. 248). Absorption spectra calculated from R and T at 300, 80, and 3 K (K ax S x10" cm" at v 130 cm ) are given in the paper, Mutzenich etal. [3]. The reststrahlen wavelength is calculated by — 76 im... 

This mechanism concerns the relaxation of vibrationally excited molecules near surfaces. It depends strongly on the ratio between the vibrational frequency of a molecule, coq, and the Debye frequency of the substrate, cup, which determines the upper limit of the phonon spectrum. If a>o < cod, relaxation through creation of a single phonon is possible. Usually the corresponding decay rates are of the order of lO -lO s . For (n — 1)cud < o o < ncup, the relaxation is accompanied by the generation of n phonons. The probability of an n-phonon process to occur rapidly decreases with order n. Typical values for the two-, three- and four-phonon decay rates are 10 -10 10 -10 and 10 -10 s, respectively (Zhdanov and Zamaraev 1982). 

Figure 7 Absorption spectrum at 77 K in the near-infrared spectral region associated with transitions from the horon acceptor to the valence band (full curve). The dotted curve shows the two- and three-phonon absorption that is present in aU diamonds. (From Ref. 29.)...

As in the case of metals and semi-conductors, there exist specific surface excitations in insulating oxides. Three types of surface phonon modes may be distinguished the Rayleigh mode, the Fuchs and Kliewer modes and the microscopic surface modes. The first two modes have a long penetration length into the crystal. They are located below the bulk acoustic branches and in the optical modes, respectively. The latter are generally found in the gap of the bulk phonon spectrum. 

Further dehydration of boehmite at 600 0 produces y-alumina, whose spectrum is shown in Figure 3b. There is a loss in surface area in going from boehmite to y-alumina. The sample shown here has a surface area of 234 m /g (this sample was obtained from Harshaw A23945 the calcined Kaiser substrate gave an identical infrared spectrum). The y-alumina sample shows two major differences from o-alumina. First, there is a more intense broad absorption band at 3400 cm" due to adsorbed water on the y-alumina. Second, the y-alumina does not show splitting of the phonon bands between 400 and 500 cm" as was observed for o-alumina. The y-alumina is a more amorphous structure and has much smaller crystallites so the phonon band is broader. The y-alumina also shows three features at 1648, 1516 and 1392 cm" due to adsorbed water and carbonate. 

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