Polaron theory

Such renormalization can be obtained in the framework of the small polaron theory [3]. Scoq is the energy gain of exciton localization. Let us note that the condition (20) and, therefore, Eq.(26) is correct for S 5/wo and arbitrary B/ujq for the lowest energy of the exciton polaron. So Eq.(26) can be used to evaluate the energy of a self-trapped exciton when the energy of the vibrational or lattice relaxation is much larger then the exciton bandwidth. 

In Fig. 1 the absorption spectra for a number of values of excitonic bandwidth B are depicted. The phonon energy Uq is chosen as energy unit there. The presented pictures correspond to three cases of relation between values of phonon and excitonic bandwidths - B < ujq, B = u)o, B > ujq- The first picture [B = 0.3) corresponds to the antiadiabatic limit B -C ljq), which can be handled with the small polaron theories [3]. The last picture(B = 10) represents the adiabatic limit (B wo), that fitted for the use of variation approaches [2]. The intermediate cases B=0.8 and B=1 can t be treated with these techniques. The overall behavior of spectra seems to be reasonable and... 

Many other time parameters actually enter - if the molecule is conducting through a polaron type mechanism (that is, if the gap has become small enough that polarization changes in geometry actually occur as the electron is transmitted), then one worries about the time associated with polaron formation and polaron transport. Other times that could enter would include frequencies of excitation, if photo processes are being thought of, and various times associated with polaron theory. This is a poorly developed part of the area of molecular transport, but one that is conceptually important. 

With mixed-valence compounds, charge transfer does not require creation of a polar state, and a criterion for localized versus itinerant electrons depends not on the intraatomic energy defined by U , but on the ability of the structure to trap a mobile charge carrier with a local lattice deformation. The two limiting descriptions for mobile charge carriers in mixed-valence compounds are therefore small-polaron theory and itinerant-electron theory. We shall find below that we must also distinguish mobile charge carriers of intennediate character. 

Because electron transfer is a rate phenomenon, considerations of timescale are unavoidable in the modeling and understanding of ET reactions. Indeed, even in simple polaron theory (vibronic theory for electron transfer between two sites), there are several timescales including... 

Generally, reactions are considered to be nonadiabatic when the effective splitting, IVyp, is smaller than the thermal energy. This is, however, an inexact prediction—polaron theory provides a more complete set of demarcations between the two limits [86]. 

V. G. Levich, in Physical Chemistry An Advanced Treatise, H. Eyring, D. Henderson, and W. Jost, eds., Vol. 9B, Ch. 12, Academic Press, New York (1970). A review stressing polaron theory in a rationalization in quantal terms of outer-sphere activation. 

It will be useful to express the self-energy of the polaron in the two models in units of a dimensionless coupling constant which is considerably important in polaron theory ... 

Gill (1972) was the first to suggest that charge transport in polymers occurred by polaron hopping. The application of polaron theory to transport in polymers was first described by Sahvun (1984). Schein et al. (1990), and Schein (1992). The models described by Sahvun and Schein and coworkers lead to a mobility that is a product of a Boltzmann probability of energy coincidence and the probability a carrier will hop to an adjacent site by thermal activation once... 

Schein et al. (1990) measured hole mobilities of TTA doped PC. The temperature dependence was described by an Arrhenius relationship. The results were described by a small-polaron argument, as proposed earlier by Schein and Mack (1988). The dependence of the activation energy on intersite distance is illustrated in Fig. 54. The authors argued that for p < 15 A the results are consistent with adiabatic small-polaron theory while for p > 15 A the results can be described by a nonadiabatic small-polaron argument. Schein et al. derived an expression for the zero-field polaron mobility as... 

In the small polaron theory the coupling constant St is much greater than 1, which allows the calculation of integrals in expression (219) [165]... 

A remarkable characteristic of carrier transport property in columnar and smectic phases is that carrier mobility is independent of temperature and electric field above room temperature [80,83,85]. A clear reason for this has not yet been confirmed although some models based on small polaron theory have been proposed [93]. 

The carrier transport characteristics were analyzed using Bassler s disorder formalism and Holstein s small polaron theory however, the temperature range in which the carrier mobility was measured precisely was not so wide that the transport mechanism could be determined clearly [108]. 

In fact, simple polaron theory is remarkably successful in describing the optica... 

On the basis of a definite analogy between e tx and F-centers we may expect the appearance under certain conditions of (e tr)2- particles of the type of F -centers. From the polaron model (57) it follows that the bipolaron (two electrons localized in a common polarization well) can not exist. In accordance with the work of Vinetskii and Giterman (63), in some cases the formation of the bipolarons becomes energetically possible in the result of interaction of the polarization wells of two separate polarons. However, the saving in energy for such bipolaron states is not large and hence they will not be stable in liquids under room temperature. Actually, up to the present time a series of attempts have been made to detect (e aq)2 in the irradiated liquid water but these attempts were not successful. The polaron theory (57) predicts that F -centers (two electrons in the anionic vacancy) may be stable. For this it is necessary that the ratio e/n2 (e and n2 are the static and optical dielectric constants, n—refraction index) should be more than 1.5. Evidently, in the glassy systems under consideration this requirement is fulfilled. 

Water slightly increased the conductivity of chromia in oxygen, air, and nitrogen, and caused no effect in hydrogen. Exposure to ultraviolet irradiation and X-rays produced no change in the electrical conductivity (188). Neutron irradiation reduced the conductivity, but did not affect the activation energy (189). Crawford and Vest (190) interpreted single-crystal data in terms of small polaron theory. 

A broad peak in the imaginary part of the dielectric constant, S2(co), above a threshold coinciding with the zero of the real component of the dielectric constant, ei((u) near 16,000 cm- (the plasma edge) has been assigned to indirect excitation of plasmons (577). An attempt has been made to apply small polaron theory to the optical data near the plasma edge with limited success (497). 

The small polaron theories for hydrogen tunnelling in metals assume that the asymmetry A is mostly caused by the H self-trapping energy, which has to be thermally overcome by a dynamic tilting of the potential before instantaneous tunnelling can occur in the coincidence configuration... 

We have used the dispersed-polaron theory to examine the reaction in which an electron moves from Hl" to the first quinone (Qa) and also the much slower back-reaction between Qa and P+ (26,27). The rate of electron transfer from Hl to Qa is predicted to be essentially independent of temperature, which agrees well with the experimental observations on Rps. viridis (31). 

Hl (18) shows a similiar insensitivity to AG°. Application of the dispersed-polaron theory to these initial steps is currently underway. 

The first two terms denote the reactant and the metal, the last term affects electron exchange between the metal and the reactant c denotes a creation and c an annihilation operator. Just like in Marcus (and polaron) theory, the solvent modes are divided into a fast part, which is supposed to follow the electron transfer instantly, and a slow part. The latter is modeled as a phonon bath after transformation to a single, normalized reaction coordinate q, with corresponding momentum p, the corresponding part of the Hamiltonian is... 

In 1954, Platzmann and Frank indicated the possibility of using the so-called radiationless theory of transitions developed by Lax for polyatomic molecules, and by Pekar for polar crystals, to the process involving charge transfer in liquids. The most general method in the theory of the radiationless transitions was suggested by Kubo and Toyozawa in 1955. Subsequently it was used in many other works. The first calculations for processes in polar liquids in the framework of the polaron theory were performed by Davydov and Deygen, " who investigated the properties of metal-ammonia solutions. 

Nevertheless, the first theory of ET, which also recognizes the importance of structural reorganization and mark the beginning of the polaron theory, was due to two Soviet physicists, L. D. Landau and S. 1. Pekar. In 1948, they calculated the effective mass of the polaron. 

On the other hand, the appearance of divalent iron ions clearly shows the increased number of oxygen vacancies. This implies that crystal defects indeed significantly affect the bulk ferromagnetic order in dilute ferromagnets, which fits exactly the polaronic theory of DMI materials [2]. When the number of doping iron ions is very small (in our case, x = 0.02), the maximum of the ferromagnetic moment was already reached in the as-prepared sample with a moderate... 

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