Density-of-states functions

The density of states is the central function in statistical thermodynamics, and provides the key link between the microscopic states of a system and its macroscopic, observable properties. In systems with continuous degrees of freedom, the correct treatment of this function is not as straightforward as in lattice systems - we, therefore, present a brief discussion of its subtleties later. The section closes with a short description of the microcanonical MC simulation method, which demonstrates the properties of continuum density of states functions. 

Since Eo, the minimum energy required for E in order to react, is usually quite large compared to the vibrational spacing in the reactant, the spacing between the active levels in A is small and it makes sense to employ the density of states function p(Ev ) for A (i.e. instead of attempting to sum discrete individual states). For... 

The two peaks at E and E2 are related to transitions (indicated by arrows in Figure 4.7) from the vicinity of specific points of high symmetry in the Brillouin zone, denoted by L and X. However, in order to explain the exact locations of these two absorption peaks, the density of states function must be also taken into account. Thus, it appears obvious that the full interpretation of interband spectra is far from simple. 

In (14), N(E) is the density of state function, i.e. the number of eigenstates of the electron (in the given energy scheme) between E tmd E + dE. Equation (14) is then just a conservation condition for the number of electrons N. 

Pauli magnetism is induced in a free electron gas by the application of an external magnetic field H. This magnetic field, acting as a perturbation, creates different energy states for electrons with spin aligned to the field, the concentration of which is n+, and electrons with spin opposite to the field, the concentration of which is n. Also the density of state function will be different for the two populations (see Fig. 13) (N+(E) and N (E)). 

The density of states function given here follows from the assumption of a parabolic energy band and includes spin degeneracy (Bube, 1974, p. 172). (Note that the Fermi function effectively cuts off the integrand at a few kT above eF, so that for kT the upper limit may be extended to infinity.) Also we have set c n = 0. 

By referencing eF and sD to the conduction band edge, we can use Eq. (B14) to write n = Nc exp(eF//cT). Also we define what we will call a modified density of states function ... 

The numerator A represents the density of states function, which in practice is set equal to a constant [326]. In this way, profiles may be obtained for the collision-induced spectra. However, one should not expect the IT shape to approximate real line shapes well [83, 274]. The basic assumption of IT mentioned above ignores the wealth of information... 

The observed second harmonic signal is the double convolution of the oscillator density of states function Ds(e) with a thermal broadening function F(eVQ+E-h 0 ( Fig.5a ) and a modulation broadening function G(E) ( Fig.Sb ). 

One particular case of Eq. (A 12) has attracted considerable attention. If one sets M = E and considers the infinite temperature limit, the probabilities of the macrostates ) and Ej can be replaced by the associated values of the density-of-states function G(Ej) and G(Ej). The resulting equation has been christened the broad-histogram relation [128] it forms the core of extensive studies of transition probability methods referred to variously as flat histogram [129] and transition matrix [130]. Applications of these formulations seem to have been restricted to the situation where the energy is the macrovariable, and the energy spectmm is discrete. 

The energy states in a band are not evenly distributed, and the band structure is better described by a plot of energy E against a density of states function N(E) which represents the number of energy states lying between E and SE, as shown in Fig. 4.3.2. 

So far, we have fairly extensively discussed the general aspects of static and dynamic relaxation of core holes. We have also discussed in detail methods for calculating the selfenergy (E). Knowing the self-energy, we know the spectral density of states function A (E) (Eq. (10)) which describes the X-ray photoelectron spectrum (XPS) in the sudden limit of very high photoelectron kinetic energy (Eq. (6)). We will now present numerical results for i(E) and Aj(E) and compare these with experimental XPS spectra and we will find many situations where atomic core holes behave in very unconventional ways. 

One of the two integrals (4.67) may be evaluated analytically the second one (elliptic) may be calculated numerically by the trapezoidal method with the introduction of a small imaginary part in (4.67) to avoid spurious oscillations due to the finite number of integration points. The resulting density-of-states function is shown in Fig. 4.9. The band is asymmetric because of the equivalent term Vx j it exhibits three van Hove singularity points two discontinuities at the boundaries, and one logarithmic divergence corres-... 

Therefore, in this particular case, Aj(u)) provides a direct measure of the density of occupied states at the adsorbate, lmGaa(co — coo), if there is no pronounced structure in the density of final states and if the variation of the matrix element (J t a) with the final state energy can be neglected over the width of the adsorbate density of states function. The angular distribution of Aj(m) is determined by the matrix element f r a and reflects the symmetry of the adsorbate orbital. 

Langhoff and Robinson have expressed the absorption coefficient A (E) in terms of the energy-dependent density-of-states function Dq(E) and the energy-dependent interaction V(E) ... 

Lastly, it is generally assumed that 0.5eV is the best possible resolution for solid state XPS measurements and the experimental resolution function is reasonably well reproduced by a Gaussian of full width f at half maximum of 0.7eV. A final "theoretical XPS spectrum" is obtained after correction of the basic density of states function by cross-section effects and convolution by the experimental resolution function (16) ... 

Another quantity that has to be taken into consideration is the density of states at the Fermi level, g E, and any alterations caused to it by the charge transfer process. The importance of g E on the adsorptive and catalytic properties of a metal surface has been stated by some investigators [131-133]. More specifically, the density of states defines the ability of the surface to respond to the presence of an adsorbate [132]. Theoretical calculations for the density of states function, g E), have been reported in the literature for certain metals [134]. The g E) function for the d metals Ru, Rh, and Pd is characterized by the participation of the d electrons. All three metals have a high density of states at the Fermi level (1.13 for Ru, 1.35 for Rh,... 

The entropy can be calculated by taking the logarithm of the amplitude of the density of states (the height of the Gaussian distribution). The energy can then be evaluated by determining the value at which the density of states function reaches its maximum value. 

Figure 1 A.2 shows this equivalence, together with typical density-of-states functions for the energy levels in a semiconductor electrode and a solntion containing a redox conple. As this figure ihustrates, the Fermi level does not necessarily correspond to an actnal electronic energy level in the phase.

Figure 4.21 shows the reason for this in terms of the densities-of-states functions of the metal and the solution. On the metal side at normal temperatures, the probability Pe(E) that a state is occupied swings relatively sharply from 1 to 0 across the Fermi level... 

The gaussian form of the solution density-of-states functions is explained in Appendix 4A. 

Figure 4.21 Density-of-states functions fortiie reaction Ox + e" Rd at a metal electrode at (a) equilibrium (b) a modestly negative overpotential (c) an extreme negative overpotential. The arrows indicate die direction and magnitude of ET, as determined by the overlap of the function p E which is the probability that a state in die elecd ode of energy u widi respect to die Fermi level is occupied, with the functions (E) andDo ( ),...

Figure 4.25 shows the relevant densities-of-states functions, again for the example of an n-type electrode. On the solution side, the density-of-states functions DofE) and of the Rd, Ox couple have the usual gaussian form (and we again assume these to be independent of electrode potential). On the semiconductor side, the density DfE) of states in the conduction band increases parabohcally from the band-edge energy E. The occupancy functionpfE) has the same Fermi-Dirac form as for a metal electrode but only its tail lies in the conduction band, where the Boltzmann approximation usually suffices. 

Figure 4.25 Densities-of-states functions for the solution and an n-type semiconductor electrode at (a) equilibrium and (b) a negative overpotential.

In the case of electrochemical ET, the relevant overlap was between the gaussian density-of-states function of the reacting species in solution and the Fermi-Dirac distribution function of the charge carriers in the electrode (Fig. 4.21). Figure 4B. 1 shows the analogous density-of-states functions for a homogeneous ET reaction. The rate... 

Figure 4B. 1 Density-of-states functions for the reaction D + A + A in homogeneous solution.

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