Gaussian resolution function

Assume a two-component mixture of some type, in which one of the components is concentrated at either the surface, or at a substrate interface, or both. Then, instead of a sharp break in the concentration at the surface or interface expected in reality, Gaussian resolution functions with a characteristic full width at half maximum (FWHM) are recorded see Figure 12.12 (39). While this leads to a powerful analysis mode, the composition profile sometimes has unrealistic edges. 

Fig. 6. The difference Compton profile for Ti and TiH experimental and theoretical for several models. The line A shows the experimental line profile difference normalised to two electrons before applying deconvolution for instrumental broadening and smoothing on row data of Ti and TiH. The solid line B shows the difference profile of Bandstructure calculations according to APW self-consistent approximation after a convolution with a Gaussian resolution function of a=0.30 a.u. The remaining lines C-F are the theoretical difference profiles of indicated model after convolution with the previous resolution function. 

Thus far, the discussion has been restricted to triangular window functions. However, it has been discovered that windows of many other functional forms are capable of bringing about improvement in the spectral lines. In this research the author has found that the window of Gaussian shape has produced the best overall results. With the same interferogram and extension by the same amount as in the previous example, premultiplication by the Gaussian window function shown in Fig. 14(a) produced the restored interferogram shown in Fig. 18(a). The restored spectral line shown in Fig. 18(b) has a resolution much improved over that of Fig. 17(b), where the triangular window function was used, yet the artifacts are no worse. The researcher should explore the various functional forms of the window function to find the one best suited for his or her particular data. 

If we multiply the interferogram by the triangular window function of Fig. 14(b) before extension, we obtain the interferogram and spectral lines shown in Fig. 29. The artifacts have been considerably reduced, but a slight loss of resolution has occurred. The best overall results are obtained by premultiplying the interferogram by the Gaussian window function of Fig. 14(a) before extension. These results are shown in Fig. 30. The spectral lines of... 

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) ... 

A detailed analysis of the UV-VIS spectrum of (spinach) plasto-cyanin in the Cu(II) state has been reported (56). A Gaussian resolution of bands at 427, 468, 535, 599, 717, 781, and 926 nm is indicated in Fig. 7. Detailed assignments have been made from low-temperature optical absorption and magnetic circular dichroic (MCD) and CD spectra in conjunction with self-consistent field Xa-scattered wave calculations. The intense blue band at 600 nm is due to the S(Cys) pvr transition, which is intense because of the very good overlap between ground- and excited-state wave functions. Other transitions which are observed implicate, for example, the Met (427 nm) and His (468 nm) residues. These bonds are much less intense. The low energy of the d 2 orbital indicates a reasonable interaction between the Cu and S(Met), even at 2.9 A. It is concluded that the S(Cys)—Cu(II) bond makes a dominant contribution to the electronic structure of the active site, which is strongly influenced by the orientation of this residue by the... 

The approximation of the resolution function components as Gaussian or Lorentzian functions is an approximation. Furthermore incorporation of the resolution function as a convolution in t is an approximation. 

The resolution function for these components is represented as a Gaussian in t, with standard deviation A/., //. 

Gaussian shape with standard deviation AE 63 meV. For the gold filter, the total resolution function RmU) is therefore represented as a convolution of a Lorentzian of HWHM AImel and a Gaussian of standard deviation At mi, whereas with the U filter analyser Rm(1) is represented as a Gaussian function... 

All direct depth profiling techniques used to study the surface segregation from binary polymer mixtures have a depth resolution [29] p limited to some 5-40 nm HWHM (half width at half maximum of the related Gaussian function). They cannot observe the real composition profile < )(z) (for the sake of comparison mimicked by mean field prediction (dashed line) in Fig. 16a) but rather its convolution (solid line in Fig. 16a) with an instrumental resolution function characterized by p. The total surface excess z however provides a good parameter, independent of resolution, as it has been concluded based on experimental data obtained using different direct techniques [170]. 

Fig. 10.17 (a) INS spectrum of /raw-polyacetylene, (b) calculated density-of-states convoluted with a Gaussian lineshape and the instrument resolution function and (c) as (b) including the effects of the Debye-Waller factor and phonon wings. Reproduced from [29] with permission of Elsevier. 

Fig. n. A stacked plot of 87Rb NMR spectra of human erythrocytes as a function of time, showing the uptake of Rb+. The initial extracellular Rb+ concentration was lOmM, and the Dy(TTHA)3- concentration was 25mM. Each spectrum took 4min to acquire, and the time between spectra was 20 min. Spectra were processed with Gaussian resolution enhancement. From Ref. 58, with permission. 

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