Fourier series calculation

Exercise 3.34 (For students of Fourier series) Check the Fourier series calculations about the function f in Section 3.4. 

The Rietveld method is employed both to finalize the model of the crystal structure, when necessary, e.g. to locate a few missing atoms in the unit cell by coupling it with Fourier series calculations, and to confirm the crystal structure determination by refining positional and other relevant parameters of individual atoms together with profile variables. The fully refined... 

The following five examples of completing the model of the crystal structure and Rietveld refinement are based on five materials discussed in sections 6.13 to 6.17. Therefore, in all cases the initial structural models will be employed exactly as they were derived in Chapter 6. When needed, the models will be completed by employing Fourier series calculation(s) using phase angles obtained after the initial models have been improved by using Rietveld refinement and the individual structure factors re-determined from the observed powder data after the refinement. ... 

We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

Cooley J W and ] W Tukey 1965. An Algorithm for the Machine Calculation of Complex Fourier Series Aiathemalics of Computation 19 297-301. 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

J. W. Cooley and J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comput., 19 (1965) 297-301. 

The excitation profile of multiple bands was also known for periodic RF pulses, such as the DANTE (delays alternating with nutation for tailored excitation) sequence.26 Similar to the PIP, all the phases and strengths of the effective RF fields can be obtained by expanding the periodic pulse into a Fourier series and properly rearranging the terms afterwards.27 Detailed calculation, comparison with the PIP, and the excitation profiles by a periodic pulse of fix sin(7tt/T) Ix and a DANTE sequence are presented in Section 3. 

A technical problem occurs when one attempts to apply this approach to study a surface. The calculations described for the bulk crystal assume perfect symmetry and a solid of infinite extent often described in terms of cyclic or periodic boundary conditions. However, for a surface, the translational symmetry is broken, and the usual expansions in Fourier series used for the bulk are not appropriate. For the bulk, a few atoms form a basis which is attached to a lattice cell, and this cell is... 

In order to estimate the presence of the atomic density of light Li atoms into the spinel structure it is necessary of first of all to estimate the effect of the cut off the Fourier- series on the view of the required (110) projections. It is necessary to do this because the projection of the potential on some plane can be influenced due to the limited amount of the reflections which forms the projection. In fig. 6 the theoretical projection has been calculated for enlarged set Fhki (up to sin G/k 1.6 A ) below. All atoms can be distinguished and are shown by the arrows. The heights of Li-atoms are small but are seen on the projection. 

In the three cases, the pseudopotential is calculated with the equation (9) and the values for each conformation have been fitted to a totally symmetric Fourier series. 

If the absolute intensities of the X-ray reflections are not available— but only relative intensities—the value of the constant term (the equivalent of 000 in the equations given previously) in relation to the other terms of the Fourier series (which are in this case in arbitrary units) is not known the figures for the electron density obtained by calculation, omitting the constant term, will all be wrong by this amount but for the purpose of locating atomic centres, this is of no consequence the image formed by the electron density contours is of precisely the same form. 

Since the responses that we are trying to calculate are linear in the force, it suffices to develop F(rm, t) in a spatial Fourier series and then to compute... 

Equation (5.18) tells us how to calculate p(jc,y,z) simply construct a Fourier series using the structure factors Fhkl. For each term in the series, h, k, and 1 are the indices of reflection hkl, and Fhkl is the structure factor that describes the reflection. Each structure factor Fhkl is a complete description of a diffracted ray recorded as reflection hkl. Being a wave equation, Fhkl must specify frequency, amplitude, and phase. Its frequency is that of the X-ray source. Its amplitude is proportional to (- j /)1/2, the square root of the measured intensity Ihkl of reflectionhkl. Its phase is unknown and is the only additional information the crystallographer needs in order to compute p(x,y,z) and thus... 

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