Temperature dependence harmonic vibration

The decrease in the heat capacity at low temperatures was not explained until 1907, when Einstein demonstrated that the temperature dependence of the heat capacity arose from quantum mechanical effects [1], Einstein also assumed that all atoms in a solid vibrate independently of each other and that they behave like harmonic oscillators. The motion of a single atom is again seen as the sum of three linear oscillators along three perpendicular axes, and one mole of atoms is treated by looking at 3L identical linear harmonic oscillators. Whereas the harmonic oscillator can take any energy in the classical limit, quantum theory allows the energy of the harmonic oscillator (en) to have only certain discrete values ( ) ... 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

K for myoglobin (Parak et al., 1981). Thus, measurements of (x ) at temperatures below this value should show a much less steep temperature dependence than measurements above, if nonharmonic or collective motions (whose mean-square displacement is denoted (x )c) are a significant component of the total (x ). Figure 21 illustrates the expected behavior of (x )v, x, and their sum for a simple model system in which a small number of substates are separated by relatively large barriers. In practice, the relative contributions of simple harmonic vibrations and coUective modes will vary from residue to residue within a given protein. 

From the plot of ln(I/I0) versus Q2 the values for the temperatures below Tc were derived. At higher temperatures deviations from linear behavior occur at large Q. Figures 29a and 29b show the values of the mean square vibration amplitude, , as a function of temperature for both copolymers. At low temperature the mean square displacement follows a nearly linear temperature dependence as expected for harmonic vibrations. A stronger and quasiexponential temperature dependence sets in around T = 250 K for the 60/40 copolymer and T = 230 K for the 80/20 copolymer. It should be noted that the temperatures where a deviation from the harmonic behavior occurs corresponds to the glass transition in the rase of both copolymers [6]. We can attribute this behaviour to the appearance of a new degree of freedom in this region. Similar... 

For centrosymmetric complexes the intensities of the parity-forbidden d< d bands arise through vibronic interactions and consequently show substantial temperature dependence. It can be shown that for the ideal case of a si ngle harmonic vibration of frequency y coupled to the electronic system the intensity of a band should be given by105-106... 

Even for barriers as high as 3.5kcal/mol, when 0 — 200 cm"1, (2) — 0.2 rad, so that zero-point linear displacements of H atoms are 0.20-0.22 A. Thus, the torsion vibrations, unlike stretching modes, are really motions with wide amplitudes in the full sense of these words. The temperature dependence of can be found in the harmonic approximation using (2.82) experimentally, it can be extracted from the temperature dependence of the width of INS peaks assigned to torsion vibrations. As a typical example of this dependence, the results of Trevino et al. [1980] for deuterated nitromethane crystal are represented in Figure 7.8. 

Assume that the activated complexes are non-linear. Determine the temperature dependence for hv C fc///, corresponding to classical partition functions for the harmonic vibrational degrees of freedom, as well as for hv 3> ksT. 

Despite its utility at room temperature, simple Marcus theory cannot explain the DeVault and Chance experiment. All Marcus reactions have a conspicuous temperature dependence except in the region close to where AG = —A. Marcus theory does not predict that a temperature-dependent reaction will shift to a temperature-independent reaction as the temperature is lowered. Hopfield proposed a quantum enhancement of Marcus theory that would permit the behavior seen in the experiment [11]. He introduced a characteristic frequency of vibration hco) that is coupled to electron transfer, in other words, a vibration that distorts the nuclei of the reactant to resemble the product state. This quantum expression includes a hyperbolic cotangent (Coth) term that resembles the Marcus expression at higher temperatures, but becomes essentially temperature independent at lower temperatures. Other quantized expressions, such as a full quantum mechanical simple harmonic oscillator behavior [12] and that of Jortner [13], give analogous temperature behavior. 

Participation of low-frequency intermolecular vibrations in the fluctuation preparation of the barrier changes principally the mechanism of temperature dependence K T) compared to the one-dimensional tunneling model. According to relation (5), to the Arrhenius relationship there corresponds the predominance of thermally activated over-the-barrier transitions over the tunneling ones, which is determined in the harmonic terms model (Figure 1) by the thermal occupation of the highest vibrational sublevels of the initial... 

It has been found that fluctuations in the orientation of the electric field gradient tensor due to torsional vibrations of the CIO3 group of KCIO3 account for the temperature dependence of the NQR frequency at low temperatures through changes in the values of the qa. Above 80 K, however, this does not adequately account for the variation since an expansion of the lattice occurs and the molecular vibrations cannot be approximated by harmonic oscillators. As the lattice expands, the distance between the ions increases, causing an additional decrease of qa and an increase in the sensitivity of the thermometer. 

Most of the earlier theoretical studies dealt with the simplest relaxation mechanism where the internal vibrational energy of the guest is dissipated directly into the delocalized and harmonic lattice phonons. The common results of these works " were, as we mentioned above, predictions of a strong temperature dependence for the relaxation and an exponential decrease in the rates with the size of the vibrational frequency. The former result has its origin in stimulated phonon emission the conversion of vibrational energy into lattice phonons is greatly facilitated if some excited phonon states are thermally populated. The energy-gap law is due to the fact that the order of the multiphonon relaxation increases with the size of... 

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