Tunnelling transitions

The above discussion concerning eq. (2.1) implied that tunneling transitions were incoherent and characterized by a rate constant. This is predetermined by assumption (ii) mentioned at the... 

The transition is fully classical and it proceeds over the barrier which is lower than the static one, Vo = ntoColQl- Below but above the second cross-over temperature T 2 = hcoi/2k, the tunneling transition along Q is modulated by the classical low-frequency q vibration. The apparent activation energy is smaller than V. The rate constant levels off to its low-temperature limit k only at 7 < Tc2, when tunneling starts out from the ground state of the initial parabolic term. The effective barrier in this case is neither V nor Vo,... 

The requirement I > 2 can be understood from the symmetry considerations. The case of no restoring force, 1=1, corresponds to a domain translation. Within our picture, this mode corresponds to the tunneling transition itself. The translation of the defects center of mass violates momentum conservation and thus must be accompanied by absorbing a phonon. Such resonant processes couple linearly to the lattice strain and contribute the most to the phonon absorption at the low temperatures, dominated by one-phonon processes. On the other hand, I = 0 corresponds to a uniform dilation of the shell. This mode is formally related to the domain growth at T>Tg and is described by the theory in Xia and Wolynes [ 1 ]. It is thus possible, in principle, to interpret our formalism as a multipole expansion of the interaction of the domain with the rest of the sample. Harmonics with I > 2 correspond to pure shape modulations of the membrane. 

We remind the reader that the tunneling transition energy could also be thought of as an eigenenergy of the wall s motion, but of a lower, 1=1, order, associated with the translational motion of the shell s center of mass. 

Finally, we show that the second-order coupling between direct tunneling transitions is subdominant to the already computed quantities. Consider an interaction of the form yy l,-2 )(2,l + H.C. If one repeats simple-mindedly the steps leading to Eq. (75), one obtains the following simple expression for the free energy correction due to interaction between the underlying structural transitions ... 

Tetramethylplumbane mixtures with tetramethylstannane were studied at 2 K by high resolution INS (inelastic neutron scattering), revealing rotational tunnelling transitions for the methyl groups146. 

We now show that the two-mode model can account for the observed intervalence band asymmetry of the C T ion, the weak tunneling transitions and a Ar value of 0.04 8. We fix the... 

Energy of the particle undergoing a tunneling transition through... 

The excited n-electron may tunnel through a potential barrier in the free state of the neighbouring molecule preserving the energy. The probability for tunnel transition is as a rule, more than the probability of the returning to the initial state. Apparently the energy of the potential barrier may be considered equal to the molecule ionization potential. The barrier form depends on the coulomb potential between the electron and positive ion and affinity of the neutral molecule. 

Secondly, the model of thermal diffusion does not allow one to explain the independence of the reaction rate on temperature observed for many low-temperature electron transfer processes. Indeed, the thermal diffusion of molecules in liquids and solids is known to be an activated process and its rate must be dependent on temperature. True, at low temperatures when activated processes are very slow, diffusion itself can be assumed to become a non-activated process going on via a mechanism of nuclear tunneling, i.e. by tunneling transitions of atoms over very short (less than 1 A) distances. A sequence of such transitions can, in principle, result in a diffusional approach of reagents in the matrix. Direct tunneling of the electron, whose mass is less than that of an atom by a factor of 10 or 104, can, however, be expected to proceed much faster. 

In this formula, V is the electron matrix element for electron tunneling transition, l is the distance between the centres of the D and A particles, a is the width of the charge transfer band, and EmSLX is the position of the maximum of this band. Emax = Eu — EA + A, where (ED — EA) is the difference of the redox potentials of the donor and the acceptor and A is the energy spent on the excitation of the vibrational degrees of freedom. 

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