Density of translational states

The quantity 2/u/( + — y ) 1/2 /h is the density of translational states per unit length of the reaction coordinate [eqn. (1)], 2( +— i y)lfi v2 is the translational velocity, the reciprocal lifetime of the complex per unit length of the reaction coordinate, the leading factor of selects only those trajectories which lead to product formation, and i is the reduced mass associated with the reaction coordinate. 

For this choice, an exact solution of the population evolution was possible and the result was expressed in terms of v, b, and the density of translational states Po which was assumed to be constant. In the limit b- co, t)j2(e) = t) (constant coupling) and the population of state 1> decreases exponentially, but for smaller values of b nonexponential decay is predicted. 

SO that the one-ditnensional density of translational states [p -1/2] jg multiplied by... 

The density of translational states (per unit volume) is defined so as to enable us to go from a momentum to an energy scale. The definitions... 

Here, ej f are the vibration-rotation energies of the initial (anion) and final (neutral) states, and E denotes the kinetic energy carried away by the ejected electron (e.g., the initial state corresponds to an anion and the final state to a neutral molecule plus an ejected electron). The density of translational energy states of the ejected electron is p(E) = 4 nneL (2meE) /h. We have used the short-hand notation involving P P/p to symbolize the multidimensional derivative operators that arise in the non BO couplings as discussed above ... 

The preceding suggests that the structure of the density of vibrational states in the hindered translation region is primarily sensitive to local topology, and not to other details of either structure or interaction. This is indeed the case. Weare and Alben 35) have shown that the density of vibrational states of an exactly tetrahedral solid with zero bond-bending force constant is particularly simple. The theorem states that the density of vibrational states expressed as a function of M (o2 (in our case M is the mass of a water molecule) consists of three parts, each of which contains one state per molecule. These arb a delta function at zero, a delta function at 8 a, where a is the bond stretching force constant, and a continuous band which has the same density of states as the "one band Hamiltonian... 

Thus, we have only a rough identification of the factor exp(—E /kBT) with the probability of finding relative translational energies that exceed E. If one ignores the different density of energy states for free translational motion in two and three dimensions, one can claim that exp -E / kBT) is the probability of finding relative translational energies that exceed E. ... 

The internal supercurrent corresponding to operator (12) emerges, when the SC solution is modifi ed by adding an extra phase dhi, to pr) in the Bogoli-ubov transformation for b-states. If the phase 4>ab is the same for all stripe elements, then the translational supercurrent equals zero. However, when 4>ab has a weak position dependence, the zero-temperature density of translational supercurrent can be expressed as ... 

With the use of eq. 7, phonon dispersion curyes Vj(k) can be calculated [4] phonons with k = 0 are the infrared and/or Raman actiye modes to be matched with the experiments. From the data obtained from eq. 7, the one- and multi-phonon density of yibrational states g (v) can be calculated. When translational periodicity of long chains is lost, a suitable modification of eq. 5 can be introduced, in order to calculate g (v) for disordered chains [5]. 

For bimolecular reactions, it is necessary to include the density of reactant states i.e. the number per unit volume per energy interval) associated with the relative translational motion ... 

The essential technical tool is the result, from Chapter 5, that the density of translational quantum states in the three-dimensional space of the experiment is dp/h. ... 

The most common grouping is one that integrates over aU directions of the final velocity leading to populations of different (groups of) states. As already discussed this approach brings in the density (per unit volume) of translational states... 

It should be pointed out that for organic semiconductors mobilities of up to 15 cmVVs at room temperature have been reported. However, these high values relate to single crystals with translation symmetry in contrast to PEDOTrPSS as an amorphous solid. Additionally, free-charge carrier interaction can be neglected in such crystalline systems, whereas in PEDOTPSS such an interaction is anticipated due to the high density of oxidized states. This undetermined contribution makes it difficult to translate conductivity models applied for organic semiconductors to PEDOT PSS. 

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