Inverse Fourier transform calculation

This inverse Fourier transform calculation of the correlations of density of scattering centres of the sample gives particularly precise results when this sample is a crystal. In this case p(f) is periodic. The scattered intensities are then 8 functions , or Dirac s functions, that are zero almost everywhere, except for well-defined values of 2 where they take on great amplitudes. They are known as Bragg peaks for which all scattered waves have the same phase. Interferences of all these waves are consequently constructive in the directions where Bragg s peaks appear. This is the consequence of the mathematical result that the... 

The fast Fourier transform (FFT) is used to calculate the Fourier transform as well as the inverse Fourier transform. A discrete Fourier transform of length N can be written as the sum of two discrete Fourier transforms, each of length N/2. 

We calculate the quantity V(k) by inverse Fourier transform, by summing V up to the six shell of neighbors. This method favorably contrasts with the evaluation of V(k) directly in k-space and is justified by the fast convergence of V with the shell number... 

The calculation of e in momentum space is analogous to that in position space. Starting with the r-representation, and expressing the quantity F(r)(pi(r) as the inverse Fourier transform of [F(r) (pi(r)]T(p), one easily finds that ... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

A 3D potential map was calculated from the 144 independent reflections by inverse Fourier transformation (Figure 4)[9]. All 24 unique Si positions but no oxygens could be determined directly from the peaks in this 3D potential map. 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

Seven trial Aj j(u) sets in all were obtained from the corresponding trial defocus values from -405 A to -375 A with a step of 5 A, so seven corrected trial li(u) sets were then calculated. Inverse Fourier transform of those... 

If an inverse Fourier transform is calculated using the amplitudes and phases extracted from the FT for all the reflections, a lattice averaged map with pi symmetry is obtained (Fig. 5a). This map is not yet proportional to the projected potential. The various distortions introduced by the electron-optical lenses, crystal tilt etc. must first be corrected for. 

The projected potential of the crystal can be calculated by inverse Fourier transformation ... 

Different maps were calculated by inverse Fourier transformation of the amplitudes and phases of all reflections at different stages of image processing. Lattice averaged maps were obtained from the amplitudes and phases directly as extracted from the images, before the CTF correction and imposing the S5mimetry. 

The inverse Fourier transform of G again yields a Gaussian function g(t) = linewidth Tg of which can be numerically calculated from... 

Fig. 7a- . Plot of Log[Xj(77 K)/ X,(300K)] as a function of k. Points are the experimental values of the logarithm calculated at the extrema of the inverse Fourier transform of the first neighbour peak, a One monolayer of cobalt on copper (111), normal incidence b one monolayer of cobalt on copper (111), grazing incidence c one monolayer of cobalt on copper (110). Continuous lines are the linear regressions corresponding to each spectrum. Related values of = fj2(300 K) - cP(77 K)] are indicated...

It is clear that S(kx, ky) is the two-dimensional Fourier transform of the nuclei density function P (x,y) (i.e. the volume density function p(x,y,z) averaged normal to the slice). Reconstruction of p(x,y) from S(kx, ky) simply requires that we calculate the inverse Fourier transform... 

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