Gaussian broadening

Figure J6 Real part of the dielectric function q for Si/CaF2 MQW s. Large dashed line seven Si DL s small dashed line four Si DL s dotted line two Si DL s dash-dotted line one Si DL. The values are compared with that of bulk Si solid line. Results are convolved with a Gaussian broadening of 0.1 eV.

For a Voigt function that is almost Lorentzian, the extent of Gaussian broadening can be visualized by plotting the dispersion of the lineshape, D f) against the absorption, A(f).76,77 For a pure Lorentzian lineshape, a circle is obtained. Hence, the extent of the departure from this circular shape indicates the extent of the Gaussian broadening. 

FIGURE 10.4 Discrete convolution of two functions (a) Gaussian broadening function (b) true signal (dotted line) and broadened result (solid line) of convolution with the Gaussian function. 

Fig. 1 Top Behavior of the electronic linear chiroptical response in the vicinity of an excitation frequency. Re = real part (e.g., molar rotation [< ]), Im = imaginary part (e.g., molar ellipticity [0]). Without absorption line broadening, the imaginary part is a line-spectrum (5-functions) with corresponding singularities in the real part at coex. A broadened imaginary part is accompanied by a nonsingular anomalous OR dispersion (real part). A Gaussian broadening was used for this figure [37]. Bottom Several excitations. Electronic absorptions shown as a circular dichroism spectrum with well separated bands. The molar rotation exhibits regions of anomalous dispersion in the vicinity of the excitations [34, 36, 37]. See text for further details...

Figure 2. Cluster density of states (in arbitrary units) generated by Gaussian broadening of the one-electron energies and cluster Fermi energy Cf a) icosahedral Niia (spin-polarized calculation, majority spin above the axis a = 0.2 eV). b) NiirNa (solid line a = 0.3 eV). Also shown are the sum of the contnbutions from the s and p populations of the nickel atoms (dashed line) and the population of the sodium atom (dotted line).

In the cited works [6,11-13], only the ability of several numerical inversion algorithms for recuperating w%V) was evaluated, and the MWD calculation was not considered. In the present article, the purely numerical Methods I-III are compared with Methods IV and V which require a linear molecular-weight calibration. For this reason, the calibration log M V) is included in Fig. lb. From that calibration and w%V), the true MWD w (log M) of Figs. Ic-lf was obtained. Note that the selection of a uniform and Gaussian broadening is not a limitation for an adequate evaluation of the (more general) Methods I-III. 

Fig. 22 a CP/MAS NMR spectra contact time 8 ms, spinning rate 5550 Hz the guest-free TNP, sample after o-xylene crystallization, b Protonated carbon suppression pulse sequence with a delay of 50 ps. a C-1 signals obtained by applying resolution enhancement line broadening =-40 Hz and Gaussian broadening=50%. (Adopted from [65] with permission)... 

Figure 3.34. Some commonly employed window functions. These are used to modify the acquired FID to enhance sensitivity and/or resolution (lb = line broadening parameter, gb = Gaussian broadening parameter i.e. the fraction of the acquisition time when the function has its maximum value see text)...

Bottom One-electron valenceband density of states for a one-dimenstional planar chain (1DTB, solid line), and for a three-dimensional OPW calculation based on the Lyon crystal structure (3D0PW, broken line). The ordinate scale refers only to the OPW result which was obtained using an 0.25 eV Gaussian broadening function. 

Fig. 28.10. Electronic density of states computed using 4k points and a 0.05 eV Gaussian broadening. The upper plot is for a bare (10,0) tube and the lower plot is for a (10,0) interacting with O2.

Fig. 17. Local density of surface states (solid line) for the Kahn et al. [166] model of the 110 surface of GaAs. Electronic states localized on the first two layers of Ga and As atoms are shown. The broken line is the bulk density of states, shown for comparison. Only strongly localized surface states are shown. The densities of states have been Gaussian broadened (after Chadi [194]).

Here the Gaussian broadening parameter a (a small number) secures that the DOS function is continuous the height of the DOS function is proportional to the spin multiplicity of the given state times its accidental degeneracy. 

Derbyshire et used a series of mathematical functions has been used to fit the proton free-induction decays of concentrated carbohydrate-water samples. For the solid protons, these functions included a sine function, as well as the Fourier transforms of single and multiple Pake functions multiplied by a Gaussian broadening. The NMR signal from the mobile protons is described by an exponential function. It is found that in most cases the sine function gives a satisfactory result and provides valuable information about the second moment... 

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