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Phonons Debye-Waller factor

Thermal properties of overlayer atoms. Measurement of the intensity of any diffracted beam with temperature and its angular profile can be interpreted in terms of a surface-atom Debye-Waller factor and phonon scattering. Mean-square vibrational amplitudes of surfece atoms can be extracted. The measurement must be made away from the parameter space at which phase transitions occur. [Pg.261]

According to the model, a perturbation at one site is transmitted to all the other sites, but the key point is that the propagation occurs via all the other molecules as a collective process as if all the molecules were connected by a network of springs. It can be seen that the model stresses the concept, already discussed above, that chemical processes at high pressure cannot be simply considered mono- or bimolecular processes. The response function X representing the collective excitations of molecules in the lattice may be viewed as an effective mechanical susceptibility of a reaction cavity subjected to the mechanical perturbation produced by a chemical reaction. It can be related to measurable properties such as elastic constants, phonon frequencies, and Debye-Waller factors and therefore can in principle be obtained from the knowledge of the crystal structure of the system of interest. A perturbation of chemical nature introduced at one site in the crystal (product molecules of a reactive process, ionized or excited host molecules, etc.) acts on all the surrounding molecules with a distribution of forces in the reaction cavity that can be described as a chemical pressure. [Pg.168]

A AH < kT has important consequences. As the temperature is lowered to where AHg, kT, strong electron-phonon interactions must manifest themselves. Direct evidence for mode softening and strong electron-phonon coupling in the internal Ty < T < 250 K has been provided by measurements of the Mdssbauer recoiless fraction and the X-ray Debye-Waller factor as well as of muon-spin rotation Therefore, it would be... [Pg.18]

Taking the phonon source out of equilibrium at a certain frequency range may lead to enhancement in Ipc. On a speculative level, one may visualize shining the electrons with a high intensity beam of non-equilibrium phonons with a narrow frequency range around, say, w0. Icc, resulting from resonant transitions, will be significantly affected only when u>o is close to the differences e — Cj or c( — C(. The effect on the Debye-Waller factor will be small for a narrow-band beam. In this way, Icc will initially increase with the intensity of this radiation, until decoherence effects will take over and Ipc will disappear. [Pg.79]

For both heavy and Tight polaron a dependence of y on the nonadiabaticity parameter fl/T appears. It implies the dependence of the Debye-Waller factor and consequently of the polaron mass on the phonon frequency O. This can be thought of as an analogy of the isotope effect at zero temperature. [Pg.640]

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... [Pg.138]

The rate constant for ET can mathematically be regarded as the optical spectrum of a localized electron in the limit where the photon energy to be absorbed or emitted approaches zero. Erom the theory of radiative transitions [10, 12] and r / -b 1) = / for a positive integer /, we see that the factor multiplied to on the right-hand side of Eq. 27 represents the thermally renormalized value of the Franck-Condon factor [i.e., the squared overlap integral between the lowest phonon state in Vy(Q) and the ( AG /te)-th one in piQ)] for ET. The renormalization manifests itself in the Debye-Waller factor exp[—,vcoth( / (y/2)], smaller than e which appears also in neutron or X-ray scattering 12a]. Therefore, yen in Eq- 27 represents the effective matrix element for electron tunneling from the lowest phonon state in the reactant well with simultaneous emission of i AG /liw) phonons. [Pg.150]

We have shown, in later sections, how precise INS measurements of the DOS provide the most stringent means of testing the model potential functions that lie at the heart of any LD or MD simulation. In the last a few years, we have systematically studied the vibrational dynamics of a large verity of phases of ice using above instruments at ISIS. These spectra were obtained at very low temperatures (< 15 K) on the recoverable high-pressure phases of ice and a few forms of amorphous forms of ice, in order to reduce the Debye-Waller factor and avoid multiphonon excitations. Hence the one-phonon spectra, g(co), can be extracted from the experimental data for the theoretical simulations. [Pg.501]

The last two terms in equation 1 are, in effect, linearly related to the frequency distribution, g(Ho). All of the remaining factors are known or are established by the experimental conditions, except the Debye-Waller factor, exp (—2 IF). It has been customary to set this factor equal to unity, because 2 IF is small under the conditions where the one-phonon approximation is valid (14). [Pg.4]

As the value of Q continues to increase the centre of the Gaussian envelope moves out to higher frequencies and its width expands. The envelope s central intensity maximum decreases dramatically and the total intensity falls, since the Debye-Waller factor is smaller. Eventually the envelope will broaden and weaken to such an extent that it disappears into the experimental background. This simple picture nicely summarises the effects of phonon wings but it will be considerably modified by the introduction of more realistic treatments of the external vibrations of molecular crystals, see Chapter 5. However, the model remains sufficiently robust to provide an introduction to the effects of molecular recoil. [Pg.59]

Subsequently both curves decrease in intensity at high Q, due to the Debye-Waller factor and from Eq. (5.25) a = 0.0247 A (= 2/81). Because of this unusual sample the determination of a is imclouded by concerns over the impact of phonon wings ( 5.3.3). [Pg.212]

This can be seen in Fig. 9.9 where the phonon-wings and the higher overtones are shown and the fundamental transition spectra of H2O and D2O ice-VIII are shown separately. The effect of an improved Debye-Waller factor can be seen in the spectrum of D2O. It has a heavier Sachs-Teller mass and the bands appear at lower frequency and so lower Q. [Pg.406]

Fig. 10.5 Comparison of the INS spectrum of polyethylene (a) experimental, (b) calculated at the SVWN/6-31G level and convoluted with a Gaussian function whose full width at half-maximum is 30 cm" and (c) calculated INS spectrum including the Debye-Waller factors and phonon wings. Reproduced from [9] with permission from the American Institute of Physics. Fig. 10.5 Comparison of the INS spectrum of polyethylene (a) experimental, (b) calculated at the SVWN/6-31G level and convoluted with a Gaussian function whose full width at half-maximum is 30 cm" and (c) calculated INS spectrum including the Debye-Waller factors and phonon wings. Reproduced from [9] with permission from the American Institute of Physics.
Fig. 10.17 (a) INS spectrum of /raw-polyacetylene, (b) calculated density-of-states convoluted with a Gaussian lineshape and the instrument resolution function and (c) as (b) including the effects of the Debye-Waller factor and phonon wings. Reproduced from [29] with permission of Elsevier. [Pg.457]

In this section we will consider polydimethylsiloxane (PDMS) as an example of the type of work that is possible with amorphous polymers. The structure and INS spectrum of PDMS are shown in Fig. 10.21a [40]. The repeat unit shown in Fig. 10.21b was used to model the spectrum using the Wilson GF matrix method [41]. The major features are reproduced skeletal bending modes below 100 cm", the methyl torsion and its overtone at 180 and 360 cm respectively, the coupled methyl rocking modes and Si-0 and Si-C stretches at 700-1000 cm and the unresolved methyl deformation modes 1250-1500 cm. The last are not clearly seen because the intensity of the methyl torsion results in a large Debye-Waller factor, so above 1000 em or so, most of the intensity occurs in the phonon wings. [Pg.462]

The absorption spectrum of atoms and molecules in low temperature solids is composed of a sharp zero-phonon line and a phonon side band (Table 2.12 ). The phonon side band corresponds to light absorption accompanied by phonon absorption or emission. The absorption shown in bold in Table 2.12 is a zero-phonon line. The sum of the absorption, drawn in a finer line, yields the phonon side band. The phonon side band appears on the higher energy side of the zero-phonon line at low temperatures. A measure of the interaction between guest molecule and host matrix is given by the Debye-Waller factor, DW(T), defined as a function of temperature, T, in Eq. (2.3), using the areas of the zero-phonon line, S0(T), and the phonon side band, SP(T). [Pg.97]

Recoil-free Fraction and Debye-Waller Factor We have already seen qualitatively that the recoil-free fraction or probability of zero-phonon events will depend on three things ... [Pg.9]

However, low-temperature spectroscopy, in particular the fluorescence-narrowed emission, can in principle tell one further thing about the influence of proteins on chromophores. There can be a quasi-continuous tail at energies below the ZPL peak, and it represents coupling of the dipole transition to the low-frequency modes (the phonons) of the protein or the solvent. The relative strength of the ZPL to the phonon tail is the Debye-Waller factor and is a measure of the strength of the phonon-chro-mophore coupling. The phonon band is expected to be homogeneously broadened. Unfortunately, at this point no one has been able to use low-temperature spectroscopy to resolve the ZPL and the Debye spectrum in a protein, or observe the Debye factor as a function of temperature. We look forward to these important measurements. [Pg.163]

At low temperatures the width of the pure electronic transition is much narrower than that of the associated phonon transitions because electronic relaxation is much slower than vibronic relaxation. This is the reason why the zero-phonon line is so prominent in the spectrum. With increasing temperature the line shape will lose its characteristic features because (1) the Debye-Waller factor drops rapidly with increasing temperature, so that for many systems the intensity in the zero-phonon line is close to zero above 50 K and (2) the width y of the transition increases strongly with temperature. As a consequence of this thermal broadening, the peak... [Pg.229]

The selected high-resolution spectra and the simulated spectra (as shown in Fig. 23) did not directly reveal any reason why the appearance of PMI and PDI bulk spectra was so different. The calculations showed, however, that PMI has a significant static dipole moment in the So state (around 6 Debye units), which increases by 1 Debye unit upon excitation into the Si state. By symmetry, PDI has no dipole moment in either state. This pointed to strong linear electron-phonon coupling in the case of PMI, which was in line with the measured Debye-Waller factors for PMI ( d = 0.15) and PDI ( d = 0.4). [Pg.105]

The fundamentals of SSS are based on the theory of impurity centers in a crystal. The optical spectrum of an organic molecule embedded in a matrix is defined by electron-vibrational interaction with intramolecular vibrations (vibronic coupling) and interaction with vibrations of the solvent (electron-phonon coupling). Each vibronic band consists of a narrow zero-phonon line (ZPL) and a relatively broad phonon wing (PW). ZPL corresponds to a molecular transition with no change in the number of phonons in the matrix (an optical analogy of the resonance -line in the Mossbauer effect). PW is determined by a transition which is accompanied by creation or annihilation of matrix phonons. The relative distribution of the integrated intensity of a band between ZPL and PW is characterized by the Debye-Waller factor ... [Pg.749]

Here we have introduce he phonon frequencies UJj (q) and the polarization vectors ej (0,j). n( w) is the Bose ractor, phonon creation (energy loss) correspond to 03. phonon annihilation (energy gain) corresponds to (q)>0. V(q,z) is the 2-dimensional Fourier transform of the potential and Q now is considered in the extended zone scheme. The Debye Waller factor 2W is given by ... [Pg.430]

If the Mossbauer atom is bound in a solid, the recoil energy may be taken up by the matrix via excitation of lattice vibrations. The recoil energy is then reduced by a factor given by the atom and the solid mass ratio. If the phonon energy is low enough, there will be a finite probability, f, that the emission (absorption) will take place with no creation or annihilation of phonon in the lattice, that is, with no recoil energy loss, and this is the Mossbauer effect. The / factor (recoil-free fraction, Debye-Waller factor, Lamb-Mossbauer factor) is given by... [Pg.477]

Accordingly, if multi-phonon processes were not involved, the phonon density corresponds to the amplitude-weighted frequency distribution as multiplied with the Debye-Waller factor. Actually, however, multiphonon scattering is not negligible and peaks due to multi-phonon processes appear in the phonon density curves. [Pg.402]

As usual, we will turn the sum over q into an integral over the first BZ. We will also employ the Debye approximation in which all phonon frequencies are given by cuq = vq with the same average v there is amaximum frequency cod, and related to it is the Debye temperature ks D = < d- TWs approximation leads to the following expression for the Debye-Waller factor ... [Pg.233]


See other pages where Phonons Debye-Waller factor is mentioned: [Pg.14]    [Pg.154]    [Pg.30]    [Pg.162]    [Pg.79]    [Pg.646]    [Pg.158]    [Pg.588]    [Pg.384]    [Pg.121]    [Pg.202]    [Pg.214]    [Pg.196]    [Pg.121]    [Pg.154]    [Pg.154]    [Pg.199]    [Pg.148]    [Pg.150]    [Pg.152]    [Pg.229]    [Pg.9]    [Pg.65]    [Pg.75]    [Pg.95]   
See also in sourсe #XX -- [ Pg.232 ]




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