Density of states models

The multiple trapping model of transport in an exponential band tail is described by Eq. (3.20) in Section 3.2.1 and a fit to this expression is given in Fig. 7.8. TTie free carrier mobilities are 13 and 1 cm V s" for electrons and holes respectively, with the band tail slopes of 300 °C and 450 C (Tiedje el al. 1981). Implicit in the analysis is the assumption that the exponential band tail extends up to the mobility edge, but the density of states model developed in Fig. 3.16 shows that this is a poor approximation. The band taU changes slope below E. and this may change the estimated values of the mobility. 

Exchange interaction,, in cV, for the transition metals, determined from experimental optical spectra (Moore, 1949, 1952), as indicated in the text. Also listed is the critical value of l/ = WJ5, above which the uniform-density-of-states model predicts ferromagnetism H /5 was obtained from the Solid State Table. 

Rectangular Density of States Model for Electronic Entropy In this problem we imagine two competing structures, both characterized by electronic densities of states that are of the rectangular-band-type. In... 

Fig. 6. Optical absorption coefficient a for the a-Si H/a-SiN H superlattice for values of wbll width Lg = 400,20, and 8 A plotted as (otE) versus photon energy E. The experimental data aregivenby the full circles. The solid curves were calculated for the density-of-states model shown in Fig. Sb. The extrapolation of the gap E, is shown for the case of 1. — 20 A by the dashed line. (From Tiedje et al. (1984).]...

The intensity of the Ni L2 peak decreases as the concentration of La increases (i.e. going from LaNis to LaNi). The band structure calculations of LaNij indicate the transfer of 1.5 electrons per La atom to Ni (Malik et al. 1982). The 1.5 electrons will be sufficient to fill the 0.6 d hole of Ni in LaNi and 1.2 d holes of Ni in LaNi2. As a result, Ni L2 SXAPS in these intermetallics is not expected. However, in no case is a complete disappearance of the Ni L2 peak observed. The appearance of a Ni L2 peak consistently in all the La-Ni intermetallics indicates the presence of 3d unoccupied states in the conduction band of Ni. The mere charge transfer is, therefore, not considered adequate to explain the present SXAPS results. The BIS of pure La shows a peak at 5eV above p, corresponding to 4f states (Lang et al. 1981). At this energy the variation in DOS of Ni is minimal (Speier et al. 1984). According to the joint density of states model the APS spectra of Ni (as well as of La) would show, apart from the threshold peak, a structure for the peak in DOS at 5 eV above p. Since no such structure has been observed in SXAPS spectra of La and Ni in the... 

The one-electron theory, discussed previously, explains satisfactorily the features observed in the spectra of simple and 3d transition metals. The theory is valid for systems having a continuous DOS above Ep. Discrepancies between theory and experiment were observed for lanthanides. The breakdown of the one-electron model occurs because the excited core electron and/or projectile electron may occupy 4f orbitals, which are quite localized about the excited ion. Wendin (1974) has made an attempt to explain the spectral features on the basis of a two densities of states model one for the scattered projectile electron and the other for the excited ion with an electron-hole pair. This model is able to explain some of the spectral features. More theoretical work, taking into account the core-level widths, core-hole lifetime broadening, many-body and other effects contributing to the spectrum, is needed to provide a more plausible explanation for the APS spectra. 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

The complete thennodynainics of a system can now be obtained as follows. Let die isolated system withAi particles, which occupies a volume V and has an energy E within a small uncertainty E, be modelled by a microscopic Flamiltonian Ti. First, find the density of states p( ) from the Flamiltonian. Next, obtain the entropy as S(E, V, N) = log V E) or, alternatively, by either of the other two equivalent expressions... 

If K is adiabatic, a molecule containing total vibrational-rotational energy E and, in a particular J, K level, has a vibrational density of states p[E - EjiJ,K). Similarly, the transition state s sum of states for the same E,J, and Kis [ -Eq-Ef(J,K)]. The RRKM rate constant for the Kadiabatic model is... 

Mixing the 2J + 1 Ai levels, for the K active model, results in the following sums and densities of states ... 

In the above section a hannonic model is described for calculating RRKM rate constants with hamionic sums and densities of states. This rate constant, denoted by Ic iE, J), is related to the actual anhamionic RRKM rate constant by... 

Figure A3.13.15. Master equation model for IVR in highly excited The left-hand side shows the quantum levels of the reactive CC oscillator. The right-hand side shows the levels with a high density of states from the remaining 17 vibrational (and torsional) degrees of freedom (from [38]).

Figure A3.13.16. Illustration of the level populations (eorresponding to the C-C oseillator states) from various treatments in the model of figure A3.13.15 for C2Hg at a total energy E = (he) 41 000 em and a tlneshold energy = (he) 31 000 em The pomts are mieroeanonieal equilibrium distributions. The erosses result from the solution of the master equation for IVR at steady state and the lines are thennal populations at the temperatures indieated (from [38] quant, is ealeulated with quantum densities of states, elass. with elassieal meehanieal densities.).

Figure 6 shows the field dependence of hole mobiUty for TAPC-doped bisphenol A polycarbonate at various temperatures (37). The mobilities decrease with increasing field at low fields. At high fields, a log oc relationship is observed. The experimental results can be reproduced by Monte Carlo simulation, shown by soHd lines in Figure 6. The model predicts that the high field mobiUty follows the following equation (37) where d = a/kT (p is the width of the Gaussian distribution density of states), Z is a parameter that characterizes the degree of positional disorder, E is the electric field, is a prefactor mobihty, and Cis an empirical constant given as 2.9 X lO " (cm/V). ...

The role of disorder in the photophysics of conjugated polymers has been extensively described by the work carried out in Marburg by H. Bassler and coworkers. Based on ultrafast photoluminescence (PL) (15], field-induced luminescence quenching [16J and site-selective PL excitation [17], a model for excited state thermalizalion was proposed, which considers interchain exciton migration within the inhomogenously broadened density of states. We will base part of the interpretation of our results in m-LPPP on this model, which will be discussed in some detail in Sections 8.4 and 8.6. 

The density of states (DOS) of lattice phonons has been calculated by lattice dynamical methods [111]. The vibrational DOS of orthorhombic Ss up to about 500 cm has been determined by neutron scattering [121] and calculated by MD simulations of a flexible molecule model [118,122]. 

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