Self-convolution

Let us make the inverse Fourier transform of the scattering intensity (5.23) and use the properties of the Fourier integral  

As a result, we obtain the convolution of the density functirai p(r) with the same function inverted with respect of the origin of the reference frame p( r). Note that the minus sign appears due to different signs in the exponents for two complex conjugates in (5.28). The P(r) function is known as density autocorrelation function or the Paterson function when used in structural analysis. Thus, we may write the inverse and direct Fourier transforms as follows  

It means that the scattering (or diffraction) intensity and the autocorrelation function are reciprocal Fourier transforms similar to the reciprocal transforms of scattering amplitude F(q) and density p(r). It should be noted that in statistical physics one widely uses the density correlation function G(r) mentioned earlier (5.26) that is related to the structure factor S(q) exactly as the Paterson function is related to intensity of scattering 7(q). Below we prefer to use G(r). 

Resuming this section, remember that there are two approaches to calculate the scattering intensity 7(q)  

To make a Fourier transform of density p(r), in order to find the scattering field 

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We have seen that the intensities of diffraction are proportional to the Fourier transfomi of the Patterson fimction, a self-convolution of the scattering matter and that, for a crystal, the Patterson fimction is periodic in tln-ee dimensions. Because the intensity is a positive, real number, the Patterson fimction is not dependent on phase and it can be computed directly from the data. The squared stmcture amplitude is... 

In SXAPS the X-ray photons emitted by the sample are detected, normally by letting them strike a photosensitive surface from which photoelectrons are collected, but also - with the advent of X-ray detectors of increased sensitivity - by direct detection. Above the X-ray emission threshold from a particular core level the excitation probability is a function of the densities of unoccupied electronic states. Because two electrons are involved, incident and the excited, the shape of the spectral structure is proportional to the self convolution of the unoccupied state densities. 

To illustrate the self-convolution operation, draw two identical boxcars and evaluate the area in common as a function of their relative separation along the abscissa. 

A particular case of convolution is that of a function with itself. From Eq. (13) this self-convolution can be expressed by... 

Note that we obtain a very nice ramp function through multiplying H(x) by a straight line of unit slope, and that the same result can be obtained by the self-convolution of H(x) ... 

Fig. 10. The fraction r0/yei for random crosslinking of high molecular weight primary chain distribution. 1 monodisperse or Poisson, 2 most probable, 3 self-convoluted random [Dobson and Gordon (4/)]...

Multiple self-convoluted random, Schulz-Zimm... 

The phase problem can be solved, that is, phases of the scattered waves determined, either by Patterson function or by direct methods. The Patterson function P is a self-convolution of the electron density p, and its magnitude at a point u, v, w can be obtained by multiplying p (x, y, z) hy p (x + u, y + V, z + w) and summing these products for every point of the unit cell. In practice, it is calculated as... 

AQt (i() designates the self convolution or auto correlation of AQe(il) which is by definition... 

According to Equation (24) AQj (il) is related to I(s) by a three dimensional Fourier transformation. Since the phase of the scattered waves is lost in passing from A(5) over to I( ) only the self convolution of Q,(jl) and not Qe(X) itself as in Equation (20) is obtained from the scattering experiment. Therefore, additional information is needed for a unique structural description. 

As follows from Eq. 2.137, the multiplication of functions in the reciprocal space (e.g. structure amplitudes) results in a convolution of functions (e.g. electron or nuclear density) in the direct space, and vice versa. Since Eq. 2.136 contains the structure amplitude multiplied by itself, the resultant Patterson function, P yy, represents a self-convolution of the electron (nuclear) density. Hence, it may be described as follows ... 

If we take the scalar product of an atom s velocity with its velocity a short time later, t, and take an average over all the atoms of the same t)q)e then we may represent it as a convolution v(0) v(i), also commonly written (v(O).v(t)). The Fourier transform of a convolution is equivalent to product of the Fourier transforms of the two functions taken separately. Self convolution, autocorrelation, yields the square of the functions imder Fourier transformation and only the real part of the transformation is available, the power spectrum, p (a). 

In systems where inter-hydrogen forces are important, usually because of the close proximity of hydrogen neighbours, dispersion is clearly seen in the one quantum spectrum, as in Fig. 6.22. The higher order transitions from such systems would not then be expected to appear as sharp transitions, since the two phonon spectrum should resemble the self convolution of the one phonon spectrum ( 2.6.3). Occasionally, however, sharp transitions are observed, at E(2 o just below the energy of 2E(i o) If -d is the full width at half height of the (1<—0) transition. Then, provided that... 

Dy(r) results from the self-convolution of the shape function and is thus always positive. Dyj, r) and Dp(r) result from the convolution of the shape and fluctuation functions and the self-convolution of the fluctuation function, respectively. Both these can be positive or negative depending on the distribution of densities within the particle. 

The BE in APS is obtained directly from the recorder plots by applying the correction for the work function of the thermionic electron soiuce. To avoid the imcertainty introduced due to this correction in BE measiuements, Fukuda et al. [34] have used a field-emission soiurce. In earlier measurements in APS, the BE was determined in a simple way by the intersection of the extrapolated projection of the bac ound and positive going low energy slope of the peak. Since the APS yield is proportional to the self-convolution of the density of the final electron states broadened by the finite lifetime of the core hole and other effects stated earlier, precise knowledge about BE can be obtained by using deconvolution techniques. Successful deconvolution techniques have been developed by Fukuda et al. [34], Dose et al. [35,36], and Schulz et al. [37]. 

Based on the three-dimensional fimction proposed by Patterson in 1934, a new Fourier series that could be calculated directly from the measured intensities. This function is defined as the self-convolution of the electron density, p r), and has the same periodicity as the electron density ... 

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