Density of internal states

Other classical-path treatments have been formulated for atom-molecule [43,95-101] and molecule-molecule [102-107] collisions however, they were mostly based on internal-state expansions that become computationally impractical as the collision energy increases. Semiclassical calculations have also been implemented by means of time-dependent wavepackets [39], whose propagation can become expensive for motions with widely differing time scales. The following TCF-semiclassical approach encompasses very high densities of internal states as well as fast and slow motions. 

Troe has described how one can estimate the value of the partition function basic expression for the density of internal states at the dissociation limit, which treats the vibrations in RadiRad ) as harmonic. Multiplicative factors are then estimated to allow, in turn for (i) the anhar-monicity of the vibrations (ii) the energy dependence of the density of vibrational states (iii) an overall rotation factor, which allows for the existence of centrifugal barriers and (iv) an internal rotation factor allowing for the barriers associated with internal rotors. 

We begin with the density of internal states. First, we need the number, Ni(Ei), of internal states, whose energy is less than or equal to i. If the reactants are an atom in its groimd electronic state and a diatomic molecule, then the only internal energy is the rovibrational energy of the diatom. We need to coimt how many states have a rovibrational energy less than or equal to Ei. We can formally write this cormting as... 

The density of internal states is defined such that pi Ei)AE is the number of internal states in the narrow energy range E to Ei + 6Ei. For the systems of interest to us, the number of states per energy interval is sufficiently high so that the number of states is essentially a smooth function. Therefore the density of states is also a continuous function and is to be understood as... 

Figure 8.16 (a) IR and (b) Raman spectra for the mineral calcite, CaC03. The estimated density of vibrational states is given in (c) while the deconvolution of the total heat capacity into contributions from the acoustic and internal optic modes as well as from the optic continuum is given in (d). 

Second, we calculate the unimolecular rate constant at the internal energy E via the RRKM theory. We use Eq. (7.54), where the rotational energy is neglected and where the sum and density of vibrational states are evaluated classically. Thus at E = 184 kJ/mol we get... 

It is quite clear from the above that noticeable hopping conductivity can exist when there is a large density of localized states (traps for electrons) only, which can be provided by initial structure defects or by radiation damages. As far as hopping parameters (and possible free electron concentration) do depend on temperature to a large extent, the temperature increase will lead to a redistribution of internal electric fields and currents. The results of some pertinent experiments are presented below. 

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