Gaussian line broadening function

The composite filter 7(g)) may either be the true inverse filter, truncated for oo large if necessary, or any of the variations described in Section IV. In their original work, Rendina and Larson chose 7(g)) = (co)/t(co), where //(co) is a Gaussian line-broadening function that limits the ultimate resolution obtainable but yields a manageable 7(g)). For their studies Rendina and Larson used Ns = 4. 

The type of the line-broadening function (LBF) may be either Lorentzian or Gaussian. The appropriate type depends on the nature of the line broadening mechanism. If in doubt it is recommended to start with a Lorentzian shape. After you have selected a file and wavenumber range as usual in the dialog box (Fig. 10.50), specify on the Adjust Parameter page a deconvolution factor and a suppression factor for the noise, or alternatively, a factor for the bandwidth and the resolution enhancement. Start the function using the Deconvolute button. 

Note that observed Une shapes may not be purely Lorentzian or Gaussian when more than one broadening mechanism contributes in the interaction. The combinations of Lorentzian and Gaussian line-shape functions can normally be approximated by a so-called Voigt profile. 

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

However, the Lorentzian form of the dipolar broadening function, which has the advantage of mathematical simplicity, is not suitable for an interpretation in terms of second moments it is replaced with a Gaussian dipolar function S(oa, AG), where the parameters AG correspond to the appropriate fractions of the square root of the intra-group rigid lattice second moments. With appropriate values for AG, calculated and experimental line shapes I(oo) are found to be in a good agreement for cross-linked polyethylene oxide) swollen in chloroform 1U). 

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...

Distortions along low-frequency modes and small frequency changes between the neutral and cation states, if present, contribute to the width of the vibrational lines in the photoelectron spectra after taking into account instrumental line broadening. Such band profiles can be treated semiclassi-cally using equation (8), where IE is the ionization energy and D is related to the transition moment. The Gaussian functions used to fit the experimental spectra can be described... 

Due to the finite propagation time T of the wavepackets, the Fourier transformation causes artifacts known as the Gibbs phenomenon [122]. In order to reduce this effect, the autocorrelation function is first multiplied by a damping function cos jtt/IT) [81,123]. Furthermore, to simulate the experimental line broadening, the autocorrelation functions will be damped by an additional multiplication with a Gaussian function exp — t/xd)% where zj is the damping parameter. This multiplication is equivalent to a convolution of the spectrum with a Gaussian with a full width at half maximum (FWHM) of /xd- The convolution thus simulates... 

Stoichiometry fluctuations are also responsible for a line broadening effect that, with the exception of cubic phases, is also hkl) dependent. The effect on the line profile is directly related to the nature of the fluctuation if the compositional variation is described by a suitable function, e.g. a Gaussian curve, then the resulting peak profile component is also Gaussian and the FT for this effect can be written as ... 

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. 

The use of certain apodization functions improves the frequency resolution we obtain in our Fourier-transformed spectrum, but caution should be exercised when employing this technique. The use of negative line broadening and shifted Gaussian or squared sine bells (with the maximum to the right of the start of the FID) can be used to resolve a small peak that formerly appeared as the shoulder of a larger peak, but supervisors and reviewers frown upon the excessive application of these methods the starting NMR spectroscopist would do well to exercise restraint in this area. 

---