Fourier series syntheses

We wish to obtain an image of the scattering elements in three dimensions (the electron density). To do this, we perform a 3-D Fourier synthesis (summation). Fourier series are used because they can be applied to a regular periodic function crystals are regular periodic distributions of atoms. The Fourier synthesis is given in O Eq. 22.2 ... 

About 1915 W.H. Bragg suggested to use Fourier series to describe the arrangement of the atoms in a crystal [1]. The proposed technique was somewhat later extended by W. Duane [2] and W.H. Zachariasen was the first who used a two-dimensional Fourier map in 1929 for structure determination [3], Since then Fourier synthesis became a standard method in almost in every structure determination from diffraction data. 

In this demonstration of a Fourier series we will use only cosine waves to reproduce the shadow image of the black squares. The procedure itself is rather straightforward, we just need to know the proper values for the amplitude A and the index h for each wave. The index h determines the frequency, i.e. the number of full waves trains per unit cell along the a-axis, and the amplitude determines the intensity of the areas with high (black) potential. As outlined in Figure 4, the Fourier synthesis for the present case is the sum of the following terms ... 

If the Fourier synthesis is carried out by adding in the strong reflections first, we will see how fast the Fourier series converges to the projected potential. The positive potential contribution from the reflection is shown in white, whereas the negative potential contribution is shown in black. Most of the atoms are located in the white regions of each cosine wave, but the exact atomic positions will not become evident until a sufficiently large number of structure factors have been added up. 

As I described earlier, this entails extracting the relatively simple diffraction signature of the heavy atom from the far more complicated diffraction pattern of the heavy-atom derivative, and then solving a simpler "structure," that of one heavy atom (or a few) in the unit cell of the protein. The most powerful tool in determining the heavy-atom coordinates is a Fourier series called the Pattersonfunction P(u,v,w), a variation on the Fourier series used to compute p(x,y,z) from structure factors. The coordinates (u,v,w) locate a point in a Patterson map, in the same way that coordinates (x,y,z) locate a point in an electron-density map. The Patterson function or Patterson synthesis is a Fourier series without phases. The amplitude of each term is the square of one structure factor, which is proportional to the measured reflection intensity. Thus we can construct this series from intensity measurements, even though we have no phase information. Here is the Patterson function in general form... 

In words, the desired electron-density function is a Fourier series in which term hkl has amplitude IFobsl, which equals (7/, /)1/2, the square root of the measured intensity Ihkl from the native data set. The phase ot hkl of the same term is calculated from heavy-atom, anomalous dispersion, or molecular replacement data, as described in Chapter 6. The term is weighted by the factor whU, which will be near 1.0 if ct hkl is among the most highly reliable phases, or smaller if the phase is questionable. This Fourier series is called an Fobs or Fo synthesis (and the map an Fo map) because the amplitude of each term hkl is iFobsl for reflection hkl. 

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). 

When a diffraction grating, such as a crystal, interacts with X rays, the electron density that causes this diffraction can be described by a Fourier series, as discussed in Chapter 6. The diffraction experiment effects a Fourier analysis, breaking down the Fourier series (of the electron density) into its components, that is, the diffracted beams with amplitudes, F[hkl). The relative phases a(hkl) are, however, lost in the process in all usual diffraction experiments. This loss of the phase information needed for the computation of an electron-density map is referred to as the phase problem. The aim of X-ray diffraction studies is to reverse this process, that is, to find the true relative phase and hence the true three-dimensional electron density. This is done by a Fourier synthesis of the components, but it is now necessary to know both the actual amplitude F[hkl) and the relative phase, a[hkl), in order to calculate a correct electron-density map (see Figure 8.1). We must be able to reconstruct the electron-density distribution in a systematic way by approximating, as far as possible, a correct [but so far unknown) set of phases In this way the crystallographer, aided by a computer, acts as a lens for X rays. 

Possible phase angles are constrained by these two conditions (Figure 8.3), so that relative phase determination hinges on the mathematical expressions for Fourier series. In the total Fourier synthesis involving F[hkl) with the correct value of a[hkl) these two conditions should apply. This can be appreciated by an examination of Figure 6.9 (Chapter 6), where the negative features of all the electron-density waves have disappeared in the final summation. Exceptions occur in neutron scattering where... 

Lipson. H., and Beevers, C. A. An improved numerical method of two-dimensional Fourier synthesis for crystals. Proc. Phys. Soc. 48, 772-780 (1936). Patterson, A. L., and Tunell, G. A method for the summation of the Fourier series used in the X-ray analysis of crystal structures. Amer. Mineralogist 27. 655-679 (1942). 

When this evolutionary synthesis was formulated, people did not know what are genes. In this sense, the development of synthetic evolution theory can be compared to the initial steps of thermodynamics Fourier, for instance, formulated correct heat conduction equation (and developed powerful methods to solve it — Fourier series) without any knowledge of what was heat. But once molecular nature of heat was understood, the science of thermodynamics received its natural foundation in statistical mechanics. Similarly, once the nature of genes, as the sequences of DNA coding for particular proteins, was understood — it opened up the doors for molecular understanding of evolution. And since the molecules involved are, of course, biopolymers of DNA, RNA, and proteins — we should touch upon this topic in this book. That is why we invite you to the discussion of physics underpinnings of evolution. 

Equation 10.6 is a specific form of the Fourier series, one the most important concepts in signal processing. The process as just described is known as Fourier synthesis, that is the process of synthesising signals by using appropriate values and applying the Fourier series. It should be noted lhat while the sinusoids must have frequencies that are harmonically related, the amplitudes and phases of each sinusoid can take on any value. In other words, the form of the generated waveform is determined solely by the amplitudes and phases. 

According to the Fourier series Eq. 8.25, any periodic waveform is the sum of a fundamental sinusoid and a series of its harmonics. Notice that, in general, each harmonic component consists of a sine and cosine component. Of course, either of them may be zero for a given waveform in the time domain. Such a waveform synthesis (summation) is done in the time domain, but each wave is a component in the frequency domain. The frequency spectrum of a periodic function of time f(t) is therefore a line spectrum. The amplitudes of each discrete harmonic frequency component is ... 

Because of the limitation intrinsic to the adoption of an explicit parametrised density model, many crystallographers have been dreaming of disposing of such models altogether. The thermally-smeared charge density in the crystal can of course be obtained without an explicit density model, by Fourier summation of the (phased) structure factor amplitudes, but the resulting map is affected by the experimental noise, and by all series-termination artefacts that are intrinsic to Fourier synthesis of an incomplete, finite-resolution set of coefficients. 

Cullis, R., et al. (1961). The structure of haemoglobin, VIII. A three-dimensional Fourier synthesis at 5.5 A resolution determination of the phase angles. Proceed. Royal Soc. London Series A 265,15-38. 

While it is very easy, when one knows the structure of the crystal and the wavelength of the rays, to predict the diffraction pattern, it is quite another matter to deduce the crystal structure in all Its details from the observed pattern and the known wavelength. The first step is lo determine the spacing of the atomic planes from the Bragg equation, and hence the dimensions of the unit cell. Any special symmetry of the space group of the structure will be apparent from space group extinction. A Irial analysis may (hen solve the structure, or it may be necessary to measure the structure factors and try to find the phases or a Fourier synthesis. Various techniques can be used, such as the F2 series, the heavy atom, the isomorphous series, anomalous atomic scattering, expansion of the crystal and other methods. 

Time scaling. Because the phase-vocoder (the short-time Fourier transform) gives access to the implicit sinusoidal model parameters, the ideal time-scale operation described by Eq. (7.3) can be implemented in the same framework. Synthesis time-instants / are usually set at a regular interval / +1 - / = R. From the series of synthesis time-instants / analysis time-instants / are calculated according to the desired time warping function tua = T l(t ). The short-time Fourier transform of the time-scaled signal is then ... 

Lastly it should be noted that the time or scan rate issue equally plagues time as well as frequency domain methods for obtaining Rf, since in the time domain measurement, the triangle waveform is simply the Fourier synthesis of a series of sinusoidal signal functions. However, voltage sweep, potential step, and impedance methods should all yield the same value of Rf when all the scan... 

A more selective approach consists in trying to influence the kinetics of formation of at least one network in this case, the two networks are formed more or less simultaneously, and the resulting morphology and properties can be expected to vary to some extent without changing the overall composition. The same system as previously studied, PUR/PAc, has been utilized in order to prepare a series of in situ simultaneous IPNs (SIM IPNs), by acting essentially on two synthesis parameters the temperature of the reaction medium and the amount of the polyurethane catalyst. Note that the term simultaneous refers to the onset of the reactions and not necessarily to the process. The kinetics of the two reactions are followed by Fourier transform infra-red (FTIR) spectroscopy as described earlier (7,8). In this contribution, the dynamic mechanical properties, especially the loss tangent behavior, have been examined with the aim to correlate the preceding synthesis parameters to the shape and temperature of the transitions of the IPNs. 

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