Weighting functions line broadening

It is usual to specify the weighting function in terms of the extra line broadening it will cause. So, a linebroadening of 1 Hz is a function which will increase the linewidth in the spectrum by 1 Hz. For example if 5 Hz of linebroadening is required then (R q/tt) = 5 giving Rlb = 15.7 s-1. 

We noted in section 4.2 that the more rapidly the time domain signal decays the broader the lines become. A weighting function designed to improve the SNR inevitably leads to a broadening of the lines as such a function hastens the decay of the signal. In this section we will consider the opposite case, where the weighting function is designed to narrow the lines in the spectrum and so increase the resolution. 

The sine bell can be modified by shifting it the left, as is shown in Fig. 4.13. The further the shift to the left the smaller the resolution enhancement effect will be, and in the limit that the shift is by tt/2 or 90° the function is simply a decaying one and so will broaden the lines. The shift is usually expressed in terms of a phase (p (in radians) the resulting weighting function is ... 

Clearly these oscillations are undesirable as they may obscure nearby weaker peaks. Assuming that it is not an option to increase the acquisition time, the way forward is to apply a decaying weighting function to the FID so as to force the signal to go to zero at the end. Unfortunately, this will have the side effects of broadening the lines and reducing the SNR. 

The Thompson-Cox-Hastings function is often used to refine profiles with broad diffraction peaks because it is the more appropriate model for line-broadening analysis where the Lorentzian and Gaussian contributions for crystallite size and for microstrains are weighted. So in this case, the peak shape is simulated by the pseudo-Voigt function, which is a Unear combination of a Gaussian and a Lorentzian function (Table 8.5). 

The total peak profile is the sum of these two components, with a relative weight that depends on 0 and I. The Bragg component is mathematically a 5-function, but is in reality broadened by instrumental and sample imperfections. The shape of the diffuse component depends on the form of the correlation function. Many forms are possible, but two common ones are an exponential correlation function, leading to a Lorentzian line shape, and a Gaussian correlation function, leading to a Gaussian Hne shape. One can also use correlation functions that describe a preferred distance (e.g., island-island correlations) or with an in-plane anisotropy [42]. 

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