Convolution of energy distribution functions

The Lorentzian distribution arises from the decaying hole-state, but for an analysis of the observed shape of a photoline, the energy distribution of the incoming light and the spectrometer function must be known and taken into account. In general, the latter functions cannot be presented in a closed form however, quite often they are approximated by a Gaussian distribution  

If the Gaussian function is normalized at its maximum to unity, Gmax = 1, one gets 

The combined effect of the individual energy distribution functions which are of relevance for the photoionization process and for the detection of photoelectrons, can now be discussed. As a first step, the photoionization process alone, i.e., without any detection device, is discussed with the help of Fig. 2.9. The y-axis represents an energy scale with respect to the ground state, and different states of neon are plotted along the x-direction. For photons with sufficient energy hv and with a distribution function GB( ph, Eph), one reaches the continuum above the Is 

Such convolution procedures are usually performed with a computer. For practical work, however, it is helpful to note three convolution results  

From the general discussion of the width of photolines, the observed width (fwhm)exp in neon can now be related to individual contributions. The main interest 

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Figure 2.8 Several energy-distribution functions, all normalized to the same height. The Lorentzian and Gaussian distributions are shown for equal fwhm values. The Voigt profile results from the convolution of the shown Lorentzian and Gaussian functions.

Note that such convolutions are commutative, associative, and distributive.) If these photoelectrons, created with the distribution function Fs( kin, kin) in the source region, are detected with an energy analyser, a further convolution is necessary to account for the instrumental resolution as described by the spectrometer functions Gsp( kin, pass), equ. (1.48). The convolution yields the distribution function fexp( pass, kin), or equivalently Fexp(Usp, l/°p), of photoelectrons observed at a preselected pass-energy pass or, equivalently, at a given spectrometer voltage Gsp. One has (see equ. (10.54b))... 

For a correct analysis of photoionization processes studied by electron spectrometry, convolution procedures are essential because of the combined influence of several distinct energy distribution functions which enter the response signal of the electron spectrometer. In the following such a convolution procedure will be formulated for the general case of photon-induced two-electron emission needed for electron-electron coincidence measurements. As a special application, the convolution results for the non-coincident observation of photoelectrons or Auger electrons, and for photoelectrons in coincidence with subsequent Auger electrons are worked out. Finally, the convolutions of two Gaussian and of two Lorentzian functions are treated. 

Equation (3.7) shows that k is the convolution of an energy distribution function E exp( —E/ZcgT) with the excitation function o-(E). The contributions of the factors appearing in Eq. (3.7) are presented graphically in Fig. 2 for two different temperatures. We call the product of the energy distribution function and the excitation function the reaction function. 

The association rate data determined in this study can be used to make quite a precise binding energy estimate for the aluminum ion-benzene complex. The relation between the association rate constant and the binding energy was made with use of phase space theory (PST) to calculate as a function of E, with a convolution over the Boltzmann distribution of energies and angular momenta of the reactants (see Section VI). PST should be quite a reasonable approximation for... 

The smoothed energy level distribution g(E) is defined through a convolution of g(E) with an averaging function /(x) ... 

Fig. 4. Total cross section for the CID of Cr(CO)6+ with Xe, extrapolated to zero pressure, as a function of collision energy in the center-of-mass frame (lower x-axis) and laboratory frame (upper x-axis). The solid line shows a representative fit to the data using the model of Eq. (4) convoluted over the energy distributions of the two reactants. The dashed line shows the same model in the absence of energy convolution, for reactants with an internal temperature of 0 K. A 50x magnification of the threshold region of the cross section is presented in the upper left side of the figure. Adapted from [9]...

Fig. 8. Cross sections for reaction of Pt+(2D5/2) with H2 as a function of kinetic energy in the center-of-mass frame (lower axis) and laboratory frame (upper axis). The best fit of Eq. (3) with parameters given in the text to the data is shown as a dashed line. The solid line shows this model convoluted over the kinetic and internal energy distributions of the reactant neutral and ion. Adapted from [97]...

All experiments result in a spectriun which represents the energy distribution of photo electrons convoluted with instrumental functions from the analyzer and the detector system. These functions are fundamentally different for the two common analyzer types these are the cylindrical mirror analyzer (CMA) and the hemispherical analyzer (HSA). A description of the function of both devices can be found in the literature [2]. In commercial analytical instnunents, the HSA is more common than the CM A and all further discussion is limited to this device. 

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