Fourier-transforms Patterson functions

Fourier-transforms (Patterson functions). This approach in a single-scattering version is widely used in X-ray crystallography, and has been adapted for use in the multiplescattering LEED situation by Adams and Landman and, in slightly different form, by Weinberg et The Adams-Landman approach, however, has been 

Although kinematic theory is unsuitable for a complete structure analysis by LEED, it remains important as the basis for a qualitative interpretation of three types of system which are beyond dynamical treatment at present  

---

Patterson function The Fourier transform of observed intensities after diffraction (e.g. of X-rays). From the Patterson map it is possible to determine the positions of scattering centres (atoms, electrons). 

An early strategy was to use measured intensities rather than structure factors as Fourier coefficients. Although the transform of I, known as a Patterson function, must, by definition contain the same information as the... 

The determination of the atomic structure of a reconstruction requires the quantitative measurement of as many allowed reflections as possible. Given the structure factors, standard Fourier methods of crystallography, such as Patterson function or electron-density difference function, are used. The experimental Patterson function is the Fourier transform of the experimental intensities, which is directly the electron density-density autocorrelation function within the unit cell. Practically, a peak in the Patterson map means that the vector joining the origin to this peak is an interatomic vector of the atomic structure. Different techniques may be combined to analyse the Patterson map. On the basis of a set of interatomic vectors obtained from the Patterson map, a trial structure can be derived and model stracture factor amplitudes calculated and compared with experiment. This is in general followed by a least-squares minimisation of the difference between the calculated and measured structure factors. Of help in the process of structure determination may be the difference Fourier map, which is... 

Here Mj is the Madelung constant based on I as unit distance, n is the number of molecules in the unit cell, zy is the charge number of atom j, V is the volume of the unit cell and h is the magnitude of the vector (hi, h2, ha) in reciprocal space or the reciprocal of the spacing of the planes (hihjha). The coordinates of atom j are a i/, X2, Xay. The sums over j are taken over all the atoms in the imit cell. F(h) is the Fourier transform of the Patterson function and (h) is the Fourier transform of the charge distribution /(r). F h) is given by... 

As suggested by Patterson in 1934, the complex coefficients in the forward Fourier transformation (Eqs. 2.129 and 2.132) may be substituted by the squares of structure amplitudes, which are real, and therefore, no information about phase angles is required to calculate the distribution of the following density function in the unit cell ... 

The interpretation of the Patterson function is based on a specific property of Fourier transformation (denoted as 3[...]) when it is applied to convolutions (<8>) of functions ... 

In the case of well-ordered crystals, It Is possible to deduce their atomic structures by appropriate manipulation of diffraction Intensities. In the case of x-ray scattering by liquids, direct use of measured intensities yields, at best, very limited structural Information (radial distribution functions). For ordered liquids, however, it is possible to posit structural models and to calculate what their scattering Intensities would be so that it is more productive to conduct the comparisons in diffraction space. To this end, it is possible to devise a point model to represent the spatial repetition of the constituent units in the ordered array and to compare its scattering maxima to the observed ones (6,9). More sophisticated analyses (10-12) make use of the complete electron densities (or projections onto the chain axis z), usually by calculating their Patterson functions P(z) since the scattering intensity function is its Fourier transform. 

Fig. 4a-d. Scheme of diffraction patterns and Fourier transforms arising from a lamellar stack with centrosymmetrical unit cell, (a) The discrete maxima at the reciprocal spacings s = n/d sample the continuous scattering function of one lamella, which would be observed without stacking (dotted line), (b) The structure factor or amplitude function, (c) The Patterson function obtained by Fourier transformation of I,(s). (d) The electron density profile obtained by Fourier transformation of Fj(s). Adapted from Ref. 3, with permission... 

The Patterson synthesis is a technique of data analysis in X-ray diffraction which helps to circumvent the phase problem. In it, a function P is formed by calculating the Fourier transform of the squares of the structure factors (which are proportional to the intensities) ... 

Another application in which useful information can be obtained in the absence of knowledge of the phase of the Fourier transform is the study of glasses and of crystals that contain short-range order but are disordered over long ranges. Here the objective is to determine a pair distribution function (PDF) [17], which is a generalization of the Patterson function that describes the probability of finding pairs of atoms separated by a vector r. - For various... 

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

In most cases, structural analysis of polymer crystals is carried out using uniaxially oriented samples (fibers or films). The basic procedures include (1) determination of the fiber period (2) indexing (hkl) diffractions and determining the unit cell parameters (5) determination of the space group symmetry (4) structural analysis and (5) Fourier transforms and syntheses and Patterson functions. The first three aspects of the procedure are discussed here, and the last two aspects are left for further references. 

P(x) is the radial distribution function (RDF) as introduced in Section I.I.2.I.4. It is the product of electronic densities attached to two points of the real space separated by vector x, with x being the distance between a pair of atoms in a set of atoms arranged disorderly. P(x) is thus a probability it is a generalization of the Patterson function of a crystal [10]. It corresponds to the Fourier transform of I(s). Reciprocally to Equation (1.8),... 

An important theorem relates the Fourier transform and convolution operations. The Convolution Theorem (8,9) states that the Fourier transform of a convolution is the product of the Fourier transfomis, or F (f g) = F(u)G(u). Applying this to the autocorrelation yields F Kx) f(-x)] = F(u)F(-u). If f(x) is real, F(u)F(-u)=F(u)F (u)= F(u)p. Thus, "the Fourier transform ofthe autocorrelation of a function frx) is the squared modulus of its transform" (Ref. 9, p. 81). Application to scattering replaces frx) with the electron density profile, p(x). We have then the important result that the Fourier transform of the autocorrelation of the electron density profile is exactly equal to the intensity in reciprocal space, F(u). The autocomelation function cf the electron density has a special name it is called the generalized Patterson fimction(8), P(x), given by ... 

The distribution of intensity of scattered radiation in a diffraction pattern is related by a Fourier transformation to the autocorrelation function of scattering density, p r)p(r )), where ( ) indicates an average over the sample. In crystallography the autocorrelation function is known as the Patterson function. It is very useful to factor out contributions to the total intensity from interfering waves scattered by single particles and from interparticle interferences ... 

The Fourier transform of equation (16) is known as the Patterson function ... 

Going back to the real space is possible by performing an inverse Fourier transform on the 7(s) [46, 60], The result of this operation is the so called Patterson function. 

---