Fourier synthesis waves

Since we deal with a periodic pattern, it is possible to apply a technique that was originally invented by the French physicist and mathematician Jean Baptiste Joseph Fourier (1768-1830). Fourier was the first who showed that every periodic process (or an object like in our case) can be described as the sum (a superposition) of an infinite number of individual periodic events (e.g. waves). This process is known as Fourier synthesis. The inverse process, the decomposition of the periodic event or object yields the individual components and is called Fourier analysis. How Fourier synthesis works in practice is shown in Figure 4. To keep the example most simple, we will first consider only the projection (a shadow image) of the black squares onto the horizontal a-axis in the beginning (Figure 3). 

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. 

The image intensity /(x, y) at the image plane of the objective lens results from two-dimensional Fourier synthesis of the diffracted beams (the square of the FT of the waves at the exit face of the crystal), modified by a phase-contrast transfer function factor (CTF, sin /), given by Scherzer (1949), as... 

The structure factor, which is nothing but the wave function of the density, cannot be measured directly and the intensity of the diffracted wave I = F2(hkl), does not contain the phase information required for Fourier synthesis of the density. 

Fourier synthesis A method of summing waves (such as scattered X rays) to obtain a periodic function (such as a representation of the electron density in a crystal). (See Chapter 6 glossary for a more detailed definition.)... 

A Fourier synthesis, illustrated in Figure 6.7, is the reverse of a Fourier analysis. A Fourier synthesis involves the summation of waves of known frequency, amplitude and phase in order to obtain a more complicated, but still periodic function. The relationship between a Fourier synthesis and a Fourier analysis is evident from the use of the same set of waves in both Figures 6.6 and 6.7. In a Fourier synthesis, if everything except the relative phases of the component waves are known, there will still be an almost infinite number of ways in which the waves can be combined. 

FIGURE 6.7. Fourier synthesis. Take a series of waves of different amplitudes, frequencies, and relative phases. Sum them. Because the waves are periodic, their sum is also periodic. Note the relative phase of each term. 

The electron density in a crystal precisely fits the definition of a periodic function in which an exact repeat occurs at regularly fixed intervals in any direction (the crystal lattice translations). Therefore the electron density in a crystal with a periodicity d can be described by a Fourier synthesis in which each component cosine wave (which we will call an electron-density wave) has a periodicity (i.e., wavelength) d/n, and the amplitude of the rath-order Bragg reflection. 

FIGURE 6.9. Contributions to terms in a Fourier synthesis, (a) Individual terms from 000 to 10 00 h =10, k = 0, I = 0) are represented with positive areas shaded. These ten electron-density waves combine to give the electron density shown at the bottom of the diagram. This electron density is dependent on the phases (-f == 0°, - = 180 ) which are 000 + 100 200 300 -h 400 500 600 + 700 800... 

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). 

Electron-density map A contour representation of electron density in a crystal structure. Peaks appear at atomic positions. The map is computed by a Fourier synthesis, that is, the summation of waves of known amplitude, periodicity, and relative phase. The electron density is expressed in electrons per cubic A. 

Fourier synthesis The summation of sine and cosine waves to give a periodic function an example is the computation of an electron density map from waves of known phase, frequency, and amplitude F (. See also the definition in Chapter 1. 

FIGURE 8.1. Four waves with the same amplitude and periodicity are combined in three different ways [(a), (b), and (c)] as a result of different relative phase angles In each case the result of the Fourier synthesis (addition of waves) is different. Shown at the top of this Figure is a crystallographer with information on amplitudes and periodicities of the electron-density waves to be summed (on cardboard strips), but no information on relative phases (how to align the cardboard strips). 

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... 

The situation, in truth, is somewhat more involved than this explanation would suggest. The individual reflections of the diffraction pattern are the interference sum of the waves scattered by all of the atoms in the crystal in a particular direction and, therefore, are themselves waves. Being waves they have not only an amplitude, but also a unique phase angle associated with each of them. This too depends on the distribution of the atoms, their xj, yj, Zj. The phase angle is independent of the amplitude of the reflection, but most important, it is an essential part of the individual terms that contribute to the Fourier synthesis, the electron density equation. Unfortunately, the phase angle of areflection cannot be recorded, as we record the intensity. In fact we have no practical way (and rather few impractical ways either) to directly measure it at all. But, without the phase information, no Fourier summation can be computed. In the 1950s, however, it became possible, with persistence, skill, and patience (and luck), to recover this elusive phase information for... 

Fourier demonstrated that any periodic function, or wave, in any dimension, could always be reconstructed from an infinite series of simple sine waves consisting of integral multiples of the wave s own frequency, its spectrum. The trick is to know, or be able to find, the amplitude and phase of each of the sine wave components. Conversely, he showed that any periodic function could be decomposed into a spectrum of sine waves, each having a specific amplitude and phase. The former process has come to be known as a Fourier synthesis, and the latter as a Fourier analysis. The methods he proposed for doing this proved so powerful that he was rewarded by his mathematical colleagues with accusations of witchcraft. This reflects attitudes which once prevailed in academia, and often still do. 

As remarked previously, a crystal acts to decompose the continuous Fourier transform of the electron density in the unit cells into a discrete spectrum, the diffraction pattern, which we also call the weighted reciprocal lattice. Thus a crystal performs a Fourier analysis in producing its diffraction pattern. It remains to the X-ray crystallographer to provide the Fourier synthesis from this spectrum of waves and to recreate the electron density. 

The physical interpretation is that each Fourier term i/ /is a wave in a plane defined by h, k, and /. By summing these waves in different directions, we can approximate any three-dimensional function, just like in the one-dimensional case above. This is the inverse FT, also known as Fourier synthesis, as the function is being built from component waves. The normal FT is called Fourier analysis, since the function is being broken down to its component waves. 

A Fourier synthesis is a mathematical calculation whereby, in the case ofX-ray diffraction, the scattered waves (with correct amplitudes and relative phases) are recombined to give the electron density in the crystal. It is essentially the opposite of a Fourier analysis and is the equivalent of image formation by a lens. It is the stage of the experiment in which the crystallographer and the computer act as the lens of a microscope. Provided the relative phases can be found, an electron density map can be calculated (Fig. 14). 

Figure 10.5 Fourier synthesis of a square wave for different numbers of harmonics...

To demonstrate some of the properties of Fourier synthesis consider the task of synthesising a square wave, as shown in Figure 10.4, and defined by ... 

Figure 10.6 Fourier synthesis of a square wave for different numbers of harmonics. We can see that the curve in figure d has symmetry about the x-axis and so there is an equal amount of area above and below the x-axis. The total area is therefore 0, hence the contribution of the second harmonic to a square wave is 0. Figures b and f have more area beneath the x-axis than above, hence the total area will sum to a non-zero value.

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