Real space structures, Fourier transform

Figure 10. Real space structures and their Fourier transforms. From [391].

For protons, n p) is related to the Fourier transform of the proton wavefunction and an NCS measurement of n(p) can be used to determine the wavefunction in a way analogous to the determination of a real space structure from a diffraction pattern. If n p) is known then, in principle, both the proton wavefunction and the exact form of the potential energy well can be reconstructed. 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

In this work I choose a different constraint function. Instead of working with the charge density in real space, I prefer to work directly with the experimentally measured structure factors, Ft. These structure factors are directly related to the charge density by a Fourier transform, as will be shown in the next section. To constrain the calculated cell charge density to be the same as experiment, a Lagrange multiplier technique is used to minimise the x2 statistic,... 

PE, the united atom model. We considered a sufficiently long PE chain made up of 5000 united atoms under periodic conditions in each direction. The initial amorphous sample prepared at 600 K was quenched to 100 K and drawn up to 400%. The sample was then quickly heated to various crystallization temperatures, and the molecular processes of fiber formation were monitored in situ via the real-space image and its Fourier transform, the structure function S3d([Pg.79]

The static structure of three-dimensional colloidal suspensions is usually determined experimentally, not by measuring directly g(r) in real space, but by measuring the static structure factor S(k) in the reciprocal space, which is the Fourier transform of the local particle-concentration correlation function. The radial distribution function is directly related to the Fourier transform of S(k), as it is explained below. Let us consider a system of N particles in a volume V. The local particle concentration p(r) at the position r is given by... 

The determination of crystal structure is then immediate, in principle, since any standard diffraction pattern will be related to, e.g., the product of an appropriate combination of three such delta functions (periodic in x,y,z directions), with atomic form factors. Inversion to get the real space atomic positions from the diffraction pattern is then possible via the convolution theorem for Fourier transforms, provided the purely technical problem of the undetermined phase can be solved. 

The Fourier transform equations show that the electron density is the Fourier transform of the structure factor and the structure factor is the Fourier transform of the electron density. Examples are worked out in Figures 6.14 and 6.15. If the electron density can be expressed as the sum of cosine waves, then its Fourier transform corresponds to the sum of the Fourier transforms of the individual cosine waves (Figure 6.16). The inversion theorem states that the Fourier transform of the Fourier transform of an object is the original object, hence the opposite signs in Equations 6.12.1 and 6.12.2. This theorem provides the possibility of using a mathematical expression to go back and forth between reciprocal space (structure factors) and real space (electron density), so that the phrase and vice versa is applicable here. 

Fourier series are used in crystal structure analysis in several ways. An electron-density map is a Fourier synthesis with measured values of F hkl) and derived values of phase angles 0 1. A Fourier analysis is the breakdown to component waves, as in the diffraction experiment. Fourier transform theory allows us to travel computationally between real space, p xyz), and reciprocal space, F hkl). 

Real-space averaging A computational method for improvement of phases, when there are two or more identical chemical units in the crystallographic asymmetric unit. The electron densities of these identical units are averaged. Then a new set of phases is computed by Fourier transformation of the averaged structure, and with these a new map is synthesized with the observed F values. By iteration of this procedure, the electron density is improved. 

We have seen that the diffracted waves Fhki, from a particular family of planes hkl, when Bragg s law is satisfied, depends only on the perpendicular distances of all of the atoms from those hkl planes, which are h xj for all atoms j. Therefore each Fhki carries information regarding atomic positions with respect to a particular family hkl, and the collection of Fhki for all families of planes hkl constitutes the diffraction pattern, or Fourier transform of the crystal. If we calculate the Fourier transform of the diffraction pattern (each of whose components Fhki contain information about the spatial distribution of the atoms), we should see an image of the atomic structure (spatial distribution of electron density in the crystal). What, then, is the mathematical expression that we must use to sum and transform the diffraction pattern (reciprocal space) back into the electron density in the crystal (real space) ... 

Finally, we see from the Fourier transform equations, for the structure factor Fhu and the electron density p x, y, z), that any change in real space (e.g., the repositioning of an atom) affects the amplitude and phase of every reflection in diffraction space. Conversely, any change in the intensities or phases in reciprocal space (e.g., the inclusion of new reflections) affects all of the atomic positions and properties in real space. There is no point-to-point correspondence between real and reciprocal space. With the Fourier transform and diffraction phenomena, it is One for all, and all for one (Dumas, The Three Musketeers, 1844). 

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