Fourier series error synthesis

Figure 4. Principle of Fourier synthesis in one dimension. In this simple example of a Fourier series with cosine waves we need to know the amplitude A and the index h for each wave. The index h gives the frequency, i.e. the number of full wave trains per unit cell along the a-axis. The left row of images shows how the intensity within the unit eell ehanges for each Fourier component. The last image at the bottom gives the result after superposition of the waves with index /z = 2 to 10 (areas with high potential are shown in black, brighter areas in the map indicate low potential). The corresponding intensity profiles along the a-axis for one unit cell are shown in the middle row. The ripples in the profile of the Fourier sum arise from the limited number of eomponents that have been used in the synthesis (termination errors). If the...

Some points are noteworthy. According to Fourier, formally, the series must be summed over all integral frequencies from —oo to +00 to be mathematically exact. In practice of course, this is never possible. As the number of terms increases, however, as higher frequency terms are included, the approximation to the exact resultant wave function becomes more nearly correct. As shown in Figure 4.9, it often doesn t require all that many terms before a quite acceptable result is obtained. The difference between the exact waveform and the one we obtain from summing a limited series of Fourier terms is known as series termination error. As illustrated by the two-dimensional case in Figure 4.11, the phases of the component waves in the synthesis play a crucial role in determining the form of the resultant wave. 

FIGURE 1 An F0 Fourier synthesis for melamine that shows series termination error. Maximum sind/X is 1.0 A 1. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

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