Peak shape function Gauss

The two simplest peak shape functions (Eqs. 2.49 and 2.50) represent Gaussian and Lorentzian distributions, respectively, of the intensity in the Bragg peak. They are compared in Figure 2.42, from which it is easy to see that the Lorentz function is sharp near its maximum but has long tails on each side near its base. On the other hand, the Gauss function has no tails at the base but has a rounded maximum. Both functions are centrosymmetric, i.e. G(x) = G -x) and L x) = L -x). 

Figure 2.42. The illustration of Gauss (dash-dotted line) and Lorentz (solid line) peak shape functions. Both functions have been normalized to result in identical definite integrals...

It is worth noting, that when software on hand does not employ a Gauss peak shape function, it can be easily modeled by the pseudo-Voigt function using the fixed mixing parameter, t) = 1. 

There are generally three types of peaks pure 2D absorption peaks, pure negative 2D dispersion peaks, and phase-twisted absorption-dispersion peaks. Since the prime purpose of apodization is to enhance resolution and optimize sensitivity, it is necessary to know the peak shape on which apodization is planned. For example, absorption-mode lines, which display protruding ridges from top to bottom, can be dealt with by applying Lorentz-Gauss window functions, while phase-twisted absorption-dispersion peaks will need some special apodization operations, such as muliplication by sine-bell or phase-shifted sine-bell functions. 

The weighting functions used to improve line shapes for such absolute-value-mode spectra are sine-bell, sine bell squared, phase-shifted sine-bell, phase-shifted sine-bell squared, and a Lorentz-Gauss transformation function. The effects of various window functions on COSY data (absolute-value mode) are presented in Fig. 3.10. One advantage of multiplying the time domain S(f ) or S(tf) by such functions is to enhance the intensities of the cross-peaks relative to the noncorrelation peaks lying on the diagonal. 

Asymmetric peaks can be described by an exponentially modified Gauss function (EMG). The deviation from Gauss shape needs the expansion of the function by a time constant term r. 

The default setting is Lorentz, i.e. a pure Lorentzian function. A single click on the upper arrow key switches immediately to a pure Gaussian function. The next click on the same arrow sets the peak to Baseline. If, beginning again with the Lorentzian type, the down arrow is clicked on instead, the band shape changes to 100% Lorentz + Gauss. In principle this band... 

Fig. 9.1 (a) Gauss, Lorentz and Voigt (ABl/ABq = 1.0) lines as 1st derivatives adjusted to the same peak-peak amplitudes, (b) Integrated area of derivative lines with the same amplitude as function of the ABl/ABq ratio of a Voigt Une. For ABl/ABq 1 the line approaches a Lorentz shape with an area which is 3.51 times larger than that of a Gaussian with the same amplitude [13]... 

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