Gaussian convolution, derivative

We found it quite simple instead to write and deploy a Gaussian convolution program that was applied to every data point in turn and took only a fraction of a second to run with our limited number of 512 data points. Although in its simplest form this is mathematically equivalent to the Fourier transformation described earlier, we modified it so that the convolution was taken with the first-or second derivative of a Gaussian profile of arbitrarily chosen width (Sections 4.2 and 4.3). With this we were able to discriminate against pedestal, linear and... 

As an example of a convolution procedure which gives a result in closed form, the convolution of two Gaussian functions will be treated first by means of a direct calculation of the convolution integral. Then the more powerful approach of Fourier transformations will be used to derive the same result for the Gaussian functions, but extending the application to the convolution of two Lorentzian functions. 

In this appendix we describe a stencil algorithm which avoids many of the drawbacks of quadrature rules used in classical lattice models, while the extra computational cost is modest. The derivation consists of finding a unique and optimal set of stencil coefficients for a convolution with a Gaussian kernel, adapted to the special case of off-lattice density functional calculations. Stencil coefficients are the multipliers of the function values at corresponding grid points. 

Depth resolution is most commonly defined as the depth over which a signal from some abruptly appearing layer climbs from 16% of its maximum intensity to 84% when plotted on a hnear scale as this represents two standard deviations ( lGaussian function with a step function. This is illustrated in Figure 5.5. This definition is also applied to decaying signals from abruptly terminated layers. Caution must, however, be exercised when matrix effects and/or radiation-enhanced segregation are active, as these can modify the value derived relative to the absolute sputter-induced depth resolution. 

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