Series termination

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. 

Not only is the choice of a uniform prior-prejudice distribution not sensible it also exposes the calculation to two main sources of computational errors, both connected with the functional form of the MaxEnt distribution of scatterers, and with its numerical evaluation namely series termination ripples and aliasing errors in the numerical sampling of the exponential modulation of mix). The next two paragraphs will illustrate these issues in some detail. 

As already pointed out by Jauch [30], the series appearing in the exponential factor that modulates m (x) in (6) has a finite number of terms, and can therefore give rise to series termination artefacts. In particular, although the exponentiation will ensure positivity of the resulting density, series termination ripples will be present in the reconstructed map whenever the spectrum of the modulation required by the observations extends significantly past the resolution of the series appearing in the exponential. This in turn will depend both on the true density whose Fourier coefficients are being fitted, and on the choice for the prior prejudice. 

Figure 1 shows the average strength of the Fourier coefficients of log( (x)/m(x)), with q(x) a multipolar synthetic density for L-alanine at 23 K, and two different prior-prejudice distributions mix). It is apparent that the exponential needed to modulate the uniform prior still has Fourier coefficients larger than 0.01 past the experimental resolution limit of 0.463 A. Any attempt at fitting the corresponding experimental structure factor amplitudes by modulation of the uniform prior-prejudice distribution will therefore create series termination ripples in the resulting MaxEnt distribution. 

For a sequential test, the number of experiments is not predefined. Rather, experiments are performed sequentially (surprise ), and the series terminated as soon as enough data is available that a decision can be made as to whether the difference is large enough . True, it is theoretically possible for such a sequence of experiments to be indefinitely long in practice, however, it is far more common for the situation to become decidable after fewer experiments than are required for the case of a fixed number of experiments. 

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. 

Marlon B24 Series Terminally blocked linear alcohol ethoxylates... 

The series must terminate, or else the power series will diverge. The value k for which the series terminates is... 

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. 

Figure 1. High-order photofragmentation pattern of CjqK (top) and CjoCs (bottom) detected by FT-ICR mass spectrometry. Note the CnK" fragment series breaks off at 44, while the C Cs series terminates at 48. The bare C clusters seen in the top panel are fragments from injected into the ICR trap along with CmK providing an internal calibration, while in the bottom panel these C, fragments result from C 2 parent ions. Conditions (top panel) 200 shots at 10 Hz of ArF excimer laser radiation (193 nm) at 6 mJ cm" shot" (bottom panel) 1600 shots ArF at 3 mJ cm" shot". During laser excitation the pressure in the ICR trap was less than 1 x 10" Torr. A few noise spikes have been removed to simplify the figure.

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. 

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Detailed instructions for this modification to the operating system were already available to us through purchase of the PLOT-10 Terminal Control System software from Tektronix, Inc., for support of Tektronix 4010 series terminals on IBM computers. 

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 maximum entropy method also offers the possibility of super-resolution , i.e., better resolution than might be anticipated than the simple analogy with optical systems (section 2(a)). Series termination effects in conventional Fourier syntheses lead to negative regions around the peaks. The maximum entropy principle ensures that the density is everywhere positive and gives much sharper peaks in which series termination effects have been suppressed [261,263]. 

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... 

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