Series-termination errors

The use of E magnitudes and the limits we shall impose on the reflections entering the summation mean that the electron density is only approximate (At the very least there are serious series termination errors.) but hopefully is sufficient to reveal stmctural features so that model building can begin. 

Fig. 19. The relative unpaired spin density in antiferromagnetic NiO. The solid and dashed contours denote positive and negative density respectively. The circle-like contours in the center arise from series termination errors [after Ref. (27)]...

FIGURE 9.12. Series-termination errors, (a) A normal atomic scattering factor curve and (b) the atomic peak obtained by Fourier transformation, (c) A truncated atomic scattering factor curve, such as that used for data that are measured to a lower sin 6/ value than advisable. The missing portion of the scattering curve is indicated, (d) The atomic peak obtained by Fourier transformation. Note the ripples caused by loss of the missing portion of the atomic scattering curve. 

Series-termination errors Errors that result from a limitation in the number of terms in a Fourier series. Ideally an infinite number of data is required to calculate a Fourier series. In practice, the number of data depends on the resolution (reciprocal radius or sin0/A) to which the data have been measured. Because of truncation of the Fourier series at the highest value of sin 0/X of the data, peaks in the resulting Fourier syntheses are surrounded by a series of ripples. These are especially noticeable around a heavy atom because its scattering factor is still appreciable at the highest values of sin 9/X measured. Difference maps (q-v.) can be used to obviate most of the effects of series-termination errors. 

Termination-of-series errors See Series-termination errors. 

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. 

A resolution individual atoms (C, N, O, S) are almost resolved. Hydrogen atoms do not become visible until about 1.2 A resolution (Fig. 4). Series termination errors are significantly reduced, which allows water molecules to be placed with confidence. 

The exponential convergence factor minimizes series termination errors. To the extent that 1//(s) )/(j) /(/b - /app) in the case of Eq. (7a) and FiFjfFkF s in the case of Eq. (7b) are nearly constant, and 117,(5) -17,(5), smin and k are equal to zero, the individual peaks of the radial distribution curves will have nearly Gaussian shapes. In these cases D(r) will be given to good approximation by... 

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. 

A valence density map of uracil is shown in Figure 3. Notice that contrast between O and N is clearly evident, but the density about the protons and C nuclei does not appear as a distinct maximum. For data that extend to v 1.0 A-1 in sin0/X and for root-mean-square amplitudes of vibration larger than 0.14 A, these valence density maps should be free from series termination error.15... 

Dielectric constants of metals, semiconductors and insulators can be determined from ellipsometry measurements [38, 39] Since the dielectric constant can vary depending on the way in which a film is grown, the measurement of accurate film thicknesses relies on having accurate values of the dielectric constant. One common procedure for determining dielectric constants is by using a Kramers-Kronig analysis of spectroscopic reflectance data [39]. This method suffers from the series-termination error as well as the difficulty of making corrections for the presence of overlayer contaminants. The ellipsometry method is for the most part free of both these sources of error and thus yields the most accurate values to date [39],... 

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