Structure factor amplitude

Diffracted amplitude structure factor andform factor... 

The Q and ft) dependence of neutron scattering structure factors contains infonnation on the geometry, amplitudes, and time scales of all the motions in which the scatterers participate that are resolved by the instrument. Motions that are slow relative to the time scale of the measurement give rise to a 8-function elastic peak at ft) = 0, whereas diffusive motions lead to quasielastic broadening of the central peak and vibrational motions attenuate the intensity of the spectrum. It is useful to express the structure factors in a form that permits the contributions from vibrational and diffusive motions to be isolated. Assuming that vibrational and diffusive motions are decoupled, we can write the measured structure factor as... 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

The main sources of error in charge density studies based on high-resolution X-ray diffraction data are of an experimental nature when special care is taken to minimise them, charge density studies can achieve an accuracy better than 1% in the values of the structure factor amplitudes of the simplest structures [1, 2]. The errors for small molecular crystals, although more difficult to assess, are reckoned to be of the same order of magnitude. 

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. 

When it is employed to specify an ensemble of random structures, in the sense mentioned above, the MaxEnt distribution of scatterers is the one which rules out the smallest number of structures, while at the same time reproducing the experimental observations for the structure factor amplitudes as expectation values over the ensemble. Thus, provided that the random scatterer model is adequate, deviations from the prior prejudice (see below) are enforced by the fit to the experimental data, while the MaxEnt principle ensures that no unwarranted detail is introduced. 

Errors in the low-density regions of the crystal were also found in a MaxEnt study on noise-free amplitudes for crystalline silicon by de Vries et al. [37]. Data were fitted exactly, by imposing an esd of 5 x 10 1 to the synthetic structure factor amplitudes. The authors demonstrated that artificial detail was created at the midpoint between the silicon atoms when all the electrons were redistributed with a uniform prior prejudice extension of the resolution from the experimental limit of 0.479 to 0.294 A could decrease the amount of spurious detail, but did not reproduce the value of the forbidden reflexion F(222), that had been left out of the data set fitted. 

Iversen et al, in their study of crystalline beryllium [32], were the first to make use of NUP distributions calculated by superposition of thermally-smeared spherical atoms. More recently, a superposition of thermally-smeared spherical atoms was used as NUP in model studies on noise-free structure factor amplitudes for crystalline silicon and beryllium by de Vries et al. [38]. The artefacts present in the densities computed with a uniform prior-prejudice distributions have been shown to disappear upon introduction of the NUP. No quantitative measure of the residual errors were given. 

When low-temperature studies are performed, the maximum resolution is imposed by data collection geometry and fall-off of the scattered intensities below the noise level, rather than by negligible high-resolution structure factor amplitudes. Use of Ag Ka radiation would for example allow measurement of diffracted intensities up to 0.35 A for amino-acid crystals below 30 K [40]. Similarly, model calculations show that noise-free structure factors computed from atomic core electrons would be still non-zero up to O.lA. 

Let us consider a collection H = (fr, h2,. . . , hA/) of symmetry-unique reflexions. We denote by Fj[ the target phased structure factor amplitude for reflexion h/, and with F rag the contribution from the known substructure to the structure factor for the same reflexion. We are interested in a distribution of electrons q( ) that reproduces these phased amplitudes, in the sense that, for each structure factor in the set of observations H,... 

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. 

The exact amount of error introduced cannot immediately be inferred from the strength of the amplitudes of the neglected Fourier coefficients, because errors will pile up in different points in the crystal depending on the structure factors phases as well to investigate the errors, a direct comparison can be made in real space between the MaxEnt map, and a map computed from exponentiation of a resolution-truncated perfect m -map, whose Fourier coefficients are known up to any order by analysing log(<7 (x) tm (x)). 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

The calculations discussed in the previous section fit the noise-free amplitudes exactly. When the structure factor amplitudes are noisy, it is necessary to deal with the random error in the observations we want the probability distribution of random scatterers that is the most probable a posteriori, in view of the available observations and of the associated experimental error variances. 

Under the simplifying assumption that the reflexions are independent of each other, K, can be written as a product over reflexions for which experimental structure factor amplitudes are available. For each of the reflexions, the likelihood gain takes different functional forms, depending on the centric or acentric character, and on the assumptions made for the phase probability distribution used in integrating over the phase circle for a discussion of the crystallographic likelihood functions we refer the reader to the description recently appeared in [51]. 

In this section we briefly discuss an approximate formalism that allows incorporation of the experimental error variances in the constrained maximisation of the Bayesian score. The problem addressed here is the derivation of a likelihood function that not only gives the distribution of a structure factor amplitude as computed from the current structural model, but also takes into account the variance due to the experimental error. 

The error-free likelihood gain, V,( /i Z2) gives the probability distribution for the structure factor amplitude as calculated from the random scatterer model (and from the model error estimates for any known substructure). To collect values of the likelihood gain from all values of R around Rohs, A, is weighted with P(R) ... 

This approximation has already proven very effective in the calculation of likelihood functions for maximum likelihood refinement of parameters of the heavy-atom model, when phasing macromolecular structure factor amplitudes with the computer program SHARP [53]. A similar approach was also used in computing the variances to be used in evaluation of a %2 criterion in [54]. 

A test of the computational strategy outlined in the previous paragraph has been performed on a set of synthetic noisy structure factor amplitudes. The diffraction data were computed from the same model density for L-alanine at 23 K as the one used for the noise-free calculations described in Section 3.1. 

Gaussian noise has been added onto the structure factor amplitudes squared as computed from the L-alanine model density for each datum, the amount of noise added was proportional to the experimental esd for the corresponding intensity measurement ... 

From these noisy structure factor amplitudes squared, a sample of 2532 noisy structure factor amplitudes F Noisy up to 0.463 A, and the associated standard deviations cr(/F Noisy), have been computed using the computer program BAYES [55]. A number... 

BUSTER has been run against the L-alanine noisy data the structure factor phases and amplitudes for this acentric structure were no longer fitted exactly but only within the limits imposed by the noise. As in the calculations against noise-free data, a fragment of atomic core monopoles was used the non-uniform prior prejudice was obtained from a superposition of spherical valence monopoles. For each reflexion, the likelihood function was non-zero for a set of structure factor values around this procrystal structure factor the latter acted therefore as a soft target for the MaxEnt structure factor amplitude and phase. 

BUSTER chooses the minimal grid necessary to avoid aliasing effects, based on the prior prejudice used and on the fall-off of the structure factor amplitudes with resolution for the 23 K L-alanine valence density reconstruction the grid was (64 144 64). The cell parameters for the crystal are a = 5.928(1)A b = 12.260(2)A c = 5.794(1) A [45], so that the grid step was shorter than 0.095 A along each axis. 

Figure 5. L-Alanine. Fit to noisy data. Calculation A. Distribution of residual structure factor amplitudes at the end of the MaxEnt calculation on 2532 noisy data up to 0.463A. Residuals plotted ... 

We briefly discuss in this section the results of the valence MaxEnt calculation on the noisy data set for L-alanine at 23 K we will denote this calculation with the letter A. The distribution of residuals at the end of the calculation is shown in Figure 5. It is apparent that no gross outliers are present, the calculated structure factor amplitudes being within 5 esd s from the observed values at all resolution ranges. 

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

Spence, J.C.H. (1993) On the accurate measurement of structure factor amplitude and phases by electron diffraction, Acta. Cryst. A, 49, 231-260. 

Since the phase angles cannot be measured in X-ray experiments, structure solution usually involves an iterative process, in which starting from a rough estimate of the phases, the structure suggested by the electron density map obtained from Eq. (13-3) and the phase computed by Eq. (13-1) are gradually refined, until the computed structure factor amplitudes from Eq. (13-1) converge to the ones observed experimentally. 

For Comparison Notions of Normal Scattering. As the electron density is assumed to be a real quantity, it directly follows the central symmetry of scattering patterns known by the name Friedel s law [244], Friedel pairs are Bragg reflections hkl and hkl that are related by central symmetry. Concerning their scattering amplitudes, Friedel pairs have equal amplitude Aha = A-g and opposite phase (phki = -scattering intensity the phase information on the structure factor is lost. 

The electron crystallography method (21) has been used to characterize three-dimensional structures of siliceous mesoporous catalyst materials, and the three-dimensional structural solutions of MCM-48 (mentioned above) and of SBA-1, -6, and -16. The method gives a unique structural solution through the Fourier sum of the three-dimensional structure factors, both amplitude and phases, obtained from Fourier analysis of a set of HRTEM images. The topological nature of the siliceous walls that define the pore structure of MCM-48 is shown in Fig. 28. 

The structure factor S(q as defined in Eq. (54) in terms of the Ising pseudospins Si, in the framework of the first Bom approximation describes elastic scattering of X-rays, neutrons, or electrons, from the adsorbed layer. SCq) is particularly interesting, since in the thermodynamic limit it allows to estimate both the order parameter amplitude tj/, the order parameter susceptibility X4, and correlati length since for q near the superstructure Bragg reflection q we have (k = q— q%)... 

The method of strueture analysis developed by the Soviet group was based on the kinematieal approximation that ED intensity is directly related (proportional) to the square of structure factor amplitudes. The same method had also been applied by Cowley in Melbourne for solving a few structures. In 1957 Cowley and Moodie introdueed the -beam dynamical diffraction theory to the seattering of eleetrons by atoms and crystals. This theory provided the basis of multi-sliee ealeulations whieh enabled the simulation of dynamieal intensities of eleetron diffraetion patterns, and later electron microscope images. The theory showed that if dynamical scattering is signifieant, intensities of eleetron diffraetion are usually not related to strueture faetors in a simple way. Sinee that day, the fear of dynamical effects has hampered efforts to analyze struetures by eleetron diffraction. 

In HREM images of inorganic crystals, phase information of structure factors is preserved. However, because of the effects of the contrast transfer function (CTF), the quality of the amplitudes is not very high and the resolution is relatively low. Electron diffraction is not affected by the CTF and extends to much higher resolution (often better than lA), but on the other hand no phase information is available. Thus, the best way of determining structures by electron crystallography is to combine HREM images with electron diffraction data. This was applied by Unwin and Henderson (1975) to determine and then compensate for the CTF in the study of the purple membrane. 

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