Convolution procedures

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. 

---

However for several of the molecules shown in Figures 1 and 2, DNA has only a small effect on the observed fluorescence lifetime. These molecules include trans-7,8-dihydroxy-7,8-dihydro-BP (15,18,19), trans-4,5-dihydroxy-4,5-dihydro-BP (15,18), BPT (7,18), 1,2,3,4-tetrahydro-BA (12), 8,9,10,11-tetrahydro-BA (14), 5,6-dihydro-BA (12), anthracene (12) and DMA (14). Typical decay profiles obtained in fluorescence lifetime measurements of trans-7,8-dihydroxy-7,8-dihydro-BP and of 8,9,10,11-tetrahydro-BA are shown in Figure 6. The lifetimes extracted from the decay profiles shown here have been obtained by using a least-squares de-convolution procedure which corrects for the finite duration of the excitation lamp pulse (77). 

The bandpass of the incoming radiation has already been considered in connection with the monochromatization of synchrotron radiation, Section 1.4, and the finite resolution of the electron spectrometer, introduced in Section 1.5 (equ. (1.49)), will be taken up again in Section 4.2.2. Therefore, only the level width rnquantitative analysis of the linewidth obtained for the Is photoline in neon. 

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

As asserted in the previous section, the height of the photolines shown in Fig. 2.4 does not provide the correct measure of the intensity of a photoline. It will now be demonstrated that the appropriate measure for intensities is the area A under the line, recorded within a certain time interval, at a given intensity of the incident light, and corrected for the energy dispersion of the electron spectrometer. This quantity, called the dispersion corrected area AD, then depends in a transparent way on the photoionization cross section er and on other experimental parameters. In order to derive this relation, the photoionization process which occurs in a finite source volume has to be considered, and the convolution procedures described above have to be included. In order to facilitate the formulation, it has to be assumed that certain requirements are met. These concern ... 

As demonstrated in Section 10.4.2, the convolution procedure leads to a simple and impressive result if an integration is performed over all spectrometer voltages (for the corresponding treatment of the experimental data see below). One derives (see also [WTW77])... 

As an example of a convolution procedure which gives a result in closed form, the convolution of two Gaussian functions will be treated first by means of a direct calculation of the convolution integral. Then the more powerful approach of Fourier transformations will be used to derive the same result for the Gaussian functions, but extending the application to the convolution of two Lorentzian functions. 

Extension of this procedure to the ion relaxation N1 (t) is not as straightforward. The rate of charge pumping (3.432) is maximal immediately after 8-pulse excitation, but changes and turns to zero very soon. Since the stationary pumping consists of the infinite sequence of such events, the source term is actually the sum of them, which can be obtained by application of the same convolution procedure as in Eq. (3.437) ... 

InSe and GaSe crystals are characterized with a weak interaction of 3D Wannier excitons with homopolar optical A -phonons [18, 19]. Therefore, when calculating the exciton absorption spectra, we took into consideration effects of broadening the exciton states using the standard convolution procedure (see in [18]) for theoretical values of a(7jco) the absorption coefficient in the Elliott s model [20] with y /io>) — 77 [n(E 2+/ 2)] the Lorentzian function in the Toyozawa s model [21], where r is the half-width at half-maximum which is usually associated with the lifetime tl/2r. 

Nn- = exp(—N"jK) (exponential). We conclude that the most probable N" value of the experimental distribution must lie close to ACad of the Q1 + R12 branch. The convolution procedure also provides a check on the method for determining band intensities. It is found that the band head areas of the computer-synthesized spectra (Trot 800 K) agree with the input vibrational populations to 2 %. 

Chronopotentiometry has found only little use in mechanistic organic electrochemistry. This is primarily due to experimental difficulties in the accurate evaluation of the transition time. A solution to this problem includes the application of a convolution procedure [230]. Another extension includes the application of exponential current-time functions and theoretical data for this method are now available for a number of mechanisms [231]. 

The method described above is called a two-stage procedure (3). An alternative approach is based on a convolution procedure that attempts to model the relationship between in vitro dissolution and plasma concentrations in a single step. The model predicted plasma concentrations are directly compared to the actual plasma concentrations obtained in the studies (4). 

This phenomenon comes from the fact that the spectrum of the hole is actually a convolution of the single molecule line shape (Fig. lb) with itself, reflecting the two procedures associated with the registration of a hole burning spectrum, namely, burning and reading. That the overall hole shape is asymmetric with respect to the zero-phonon hole is a saturation phenomenon. It should also be stressed that, because of this convolution procedure, the width of the zero-phonon hole Fh is actually twice the homogeneous line width ... 

The convolution procedure starts from e and proceeds down to s =0 (as... 

The structure of a molecular or a liquid crystal is a result of convolution of two density functions, the density of a group of atoms in a molecule and periodic density function of a lattice. Let us look at the convolution procedure. By definition, the... 

Going back to solid or liquid crystals we can say that the convolution procedure distributes molecular density over the sites of the crystal lattice. On the left side of Fig. 5.14, the two functions, the electron density of a molecule pmoi(r) and discrete points of the lattice density piattice(r) =28(ri Fj) are shown separately (before convolution). On the right side we see the result of their convolution. Note that the convolution operation/i(x) /2(x) is dramatically different from the multiplication operation/i(x)/2(x). An example is illustrated by Fig. 5.15, in which function/2(x) is the same piattice(r) function as in the previous picture and/i(x) is the so called box-function. The latter is equal to 1 within its contour and 0 outside. The multiplication selects only few 8-functions from the whole lattice. On the contrary, the convolution translates pmoi into new functional space, namely the space of piattice-... 

Pilo, Maria I., Sanna, G. and Seeber, R. (1992) Analysis of cyclic voltammetric responses by Fourier transform-based deconvolution and convolution procedures , J.Electroanal.Chem. 323, 103-115. See references therein for semiintegral and semidifferential electroanalysis, as well as for convolutive potential sweep voltammetry. 

---