Phase of structure factor

Takayanagi K 1990 Surface structure analysis by transmission electron diffraction—effects of the phases of structure factors Acta. Crystalloger A 46 83-6... 

Cochran, W. (1955). Relations between the phases of structure factors. Acta Crystallogr. 8,473-478. 

Having located the heavy atom(s) in the unit cell, the crystallographer can compute the structure factors FH for the heavy atoms alone, using Eq. (5.15). This calculation yields both the amplitudes and the phases of structure factors Fh, giving the vector quantities needed to solve Eq. (6.9) for the phases ahkl of protein structure factors Fp. This completes the information needed to compute a first electron-density map, using Eq. (6.7). This map requires improvement because these first phase estimates contain substantial errors. I will discuss improvement of phases and maps in Chapter 7. 

A Fourier transform of the distortion-corrected image yields amplitudes and phases of structure factors to high resolution. The phase information is retained since the Fourier transform is generated by computer from an image of the crystal and not by recording a diffraction pattern, as in X-ray crystallography. 

Determining the phases of structure factors (especially for noncentrosymmetric structures) with the precision sufficient for a deformation density map is an extremely difficult task, as is combining X-ray and neutron data in a sensible way. The irrelevance of the valence electron density for high-order X-ray scattering is also only an approximation (it is correct for the spherically symmetric case only). 

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]. 

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. 

Bricogne, G. (1988) A Bayesian statistical theory ofthe phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors, Acta Cryst., A44, 517-545. 

Spackman, M.A. and Byrom, P.G. (1997) Retrieval of structure factor phases in acentric space groups. Model studies using multipole refinements, Acta Cryst., B53, 553-564. 

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

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. 

Equation 1 is a discrete Fourier transform, it is discrete rather than continuous because the crystalline lattice allows us to sum over a limited set of indices, rather than integrate over structure factor space. The discrete Fourier transform is of fundamental importance in crystallography - it is the mathematical relationship that allows us to convert structure factors (i.e. amplitudes and phases) into the electron density of the crystal, and (through its inverse) to convert periodic electron density into a discrete set of structure factors. 

In practice, recombination of structure factors involves first weighting of the phases of the modified structure factors in a resolution dependent fashion, according to their estimated accuracy or probability. Every phase also has an experimental probability (determined by experimental phasing techniques and/or molecular replacement). The two distributions are combined by multiplication, and the new phase is calculated from this combined probability distribution. The measured associated structure factor amplitude is then scaled by the probability of the phase, and we have our set of recombined structure factors. 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

As has become clear in previous sections, atomic thermal parameters refined from X-ray or neutron diffraction data contain information on the thermodynamics of a crystal, because they depend on the atom dynamics. However, as diffracted intensities (in kinematic approximation) provide magnitudes of structure factors, but not their phases, so atomic displacement parameters provide the mean amplitudes of atomic motion but not the phase of atomic displacement (i.e., the relative motion of atoms). This means that vibrational frequencies are not directly available from a model where Uij parameters are refined. However, Biirgi demonstrated [111] that such information is in fact available from sets of (7,yS refined on the same molecular crystals at different temperatures. 

Because Fhkl is a periodic function, it possesses amplitude, frequency, and phase. It is a diffracted X ray, so its frequency is that of the X-ray source. The amplitude of Fhkl is proportional to the square root of the reflection intensity lhkl, so structure amplitudes are directly obtainable from measured reflection intensities. But the phase of Fhkl is not directly obtainable from a single measurement of the reflection intensity. In order to compute p(x,y,z) from the structure factors, we must obtain, in addition to the intensity of each reflection, the phase of each diffracted ray. In Chapter 6,1 will present an expression for p(x,y,z) as a Fourier series in which the phases are explicit, and I will discuss means of obtaining phases. This is one of the most difficult problems in crystallography. For now, on the assumption that the phases can be obtained, and thus that complete structure factors are obtainable, I will consider further the implications of Eqs. (5.15) (structure factors F expressed in terms of atoms), (5.16) [structure factors in terms of p(x,y,z)], and (5.18) [p(x,y,z) in terms of structure factors]. 

In order to illuminate both the phase problem and its solution, I will represent structure factors as vectors on a two-dimensional plane of complex numbers of the form a + ib, where i is the imaginary number (—1)1/2. This allows me to show geometrically how to compute phases. I will begin by introducing complex numbers and their representation as points having coordinates (a,b) on the complex plane. Then I will show how to represent structure factors as vectors on the same plane. Because we will now start thinking of the structure factor as a vector, I will hereafter write it in boldface (FM,Z) instead of the italics used for simple variables and functions. Finally, I will use the vector representation of structure factors to explain a few common methods of obtaining phases. 

A representation of structure factors on this plane must include the two properties we need in order to construct p(x,y,z) amplitude and phase. Crystallog-raphers represent each structure factor as a complex vector, that is, a vector (not a point) on the plane of complex numbers. The length of this vector represents the amplitude of the structure factor. Thus the length of the vector representing structure factor Fhkl is proportional to The second prop-... 

In Chapter 4, Section HI.G, I mentioned Friedel s law, that lhkl = h k i-It will be helpful for later discussions to look at the vector representations of pairs of structure factors Fhkl and F h k l, which are called Friedel pairs. Even though hkl and l h k l are equal, Fhkl and F h k l are not. The structure factors of Friedel pairs have opposite phases, as shown in Fig. 6.3. 

The crystallographer can sometimes use the phases from structure factors of a known protein as initial estimates of phases for a new protein. If this method is feasible, then the crystallographer may be able to determine the structure of the new protein from a single native data set. The known protein in this case is referred to as a phasing model, and the method, which entails calculating initial phases by placing a model of the known protein in the unit cell of the new protein, is called molecular replacement. 

For the location search, the criterion is the correspondence between the expected structure-factor amplitudes from the model in a given trial location and the actual amplitudes derived from the native data on the desired protein. This criterion can be expressed as the R-factor, a parameter we will encounter later as a criterion of improvement of phases in final structure determination. The R-factor compares overall agreement between the amplitudes of two sets of structure factors, as follows... 

In the MEM/Rietveld analysis, each of the observed structure factors of intrinsically overlapped reflections (for instance, 333 and 511 in a cubic system) can be deduced by the structure model based on a free atom model in the Rietveld refinement. In such a case, the obtained MEM charge density will be partially affected by the free atom model used. In order to reduce such a bias, the observed structure factors should be refined based on the deduced structure factors from the obtained MEM charge density. The detail of the process is described in the review article [9,22-24]. In addition, the phased values of structure factors based on the structure model used in Rietveld analysis are used in the MEM analysis. Thus, the phase refinement is also done for the noncentrosymmetric case as P2, of Sc C82 crystal by the iteration of MEM analysis. The detail of the process is also described elsewhere [25]. All of the charge densities shown in this article are obtained through these procedures. 

Constraints of direct methods. The phase angles can be estimated by statistical methods that are based on the concept that the electron density is never negative, and that it consists of isolated, sharp peaks at atomic positions. The statistical methods for combining electron-density waves subject to these conditions are called direct methods. They make it possible to derive phases for a set of structure factors when only information on the magnitudes of F hkl) is available. At present this is the method of choice for small molecules. 

Both atoms have been confirmed on the Fourier map and it appears that the next missing atom is located in the 4(e) site with coordinates 0,0,0.368. All interatomic distances are normal and after including this atom into the computation of structure factors and phase angles, the corresponding Rp = 32.2%. This value is quite high, but it is still worthwhile to calculate a third Fourier map, which is shown in Table 6.15. 

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