The phase problem

Thus all sin terms, the imaginary terms, cancel in pairs, while all cos terms are added. 

We should note in passing that the original expression for p(x, y, z) was a product of two complex numbers. This implied that p(x, y, z) was also complex. We know, of course, that electron density has no imaginary component and that / (x, y, z) must always be a strictly real function. We see here that because the Fourier transform is defined as a series summed from -oo to +oo, and because of Friedel s law, p(x, y, z) does indeed conform to reality. It s somehow reassuring to know that mathematics and reality are not in conflict, isn t it  

At this point the most problematic feature of the process emerges. Inspection of the electron density equation as it was initially stated shows that the coefficient of each term in the summation for p(x, y, z) at any value of x, y, z is T ki The structure factor Ff,u is, as we have seen, a wave. It is a complex number it has an amplitude and a phase. In the final form of the equation we see that this feature persists in the form of the phase angle for each structure factor that must be included in the kernal. To calculate p(x, y, z), then, 

Here we have encountered the crucial, essential difficulty in X-ray diffraction analysis. It is not experimentally possible to directly measure the phase angles / hki of the structure factors. The best that our sophisticated detectors can provide are the amplitudes of the structure factors Fhki but not their phases. Thus we cannot proceed directly from the measured diffraction pattern, the measured intensities, through the Fourier equation to the crystal structure. We must first find the phases of the structure factors. This central obstacle in structure analysis has the now infamous name, The Phase Problem. Virtually all of X-ray diffraction analysis, not only macromolecular but for all crystals, is focused on overcoming this problem and by some means recovering the missing phase information required to calculate the electron density. 

Given these unknowns, it might appear that X-ray data collection would be a very difficult process indeed. It is not, in fact. X-ray crystallographers only rarely think about planes in the crystal, or their orientation. They use instead the diffraction pattern to guide them when they orient and manipulate a crystal in the X-ray beam. Recall that the net, or lattice, on which the X-ray diffraction reflections fall is the reciprocal lattice, and that every reciprocal lattice point, or diffraction intensity, arises from a specific family of planes having unique Miller indexes hki. 

Additional discussion and examples of the usefulness of the information that can be obtained from X-ray structures will be furnished in Chapter 10. 

The situation is even worse when it comes to locating hydrogens by the X-ray method. A hydrogen atom has only one electron. When it forms a bond, that electron is largely pulled away from the simple spherical Is orbital and into the bond. Hence when one measures a C-H bond length using X rays, the C-H bond is about 0.12 A shorter than that defined by the nuclear positions. In the case of an 0-H group, the bond is shorter by approximately 0.18 A Other differences in structure that result when X-ray structures are compared with nuclear position structures are further discussed under neutron diffraction, in the next section. 

Unfortunately, many chemists do not realize the fact that bond lengths determined by X-ray crystallography are significantly different from bond lengths that 

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The question of determination of the phase of a field (classical or quantal, as of a wave function) from the modulus (absolute value) of the field along a real parameter (for which alone experimental determination is possible) is known as the phase problem [28]. (True also in crystallography.) The reciprocal relations derived in Section III represent a formal scheme for the determination of phase given the modulus, and vice versa. The physical basis of these singular integral relations was described in [147] and in several companion articles in that volume a more recent account can be found in [148]. Thus, the reciprocal relations in the time domain provide, under certain conditions of analyticity, solutions to the phase problem. For electromagnetic fields, these were derived in [120,149,150] and reviewed in [28,148]. Matter or Schrodinger waves were... 

These stmctures do not diffract as weH as smaH molecules, and as a result, there are many weak reflections and data coHection takes much longer than for smaH molecules. Also, the solution of the phase problem is more difficult and usuaHy requires the coHection of data sets with monochromatic radiation at several different wavelengths. Because of the much longer data coHection times, area detectors are almost always used. Also because of the long... 

In small-molecule crystallography the phase problem was solved by so-called direct methods (recognized by the award of a Nobel Prize in chemistry to Jerome Karle, US Naval Research Laboratory, Washington, DC, and Herbert Hauptman, the Medical Foundation, Buffalo). For larger molecules, protein aystallographers have stayed at the laboratory bench using a method pioneered by Max Perutz and John Kendrew and their co-workers to circumvent the phase problem. This method, called multiple isomorphous replacement... 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

In this section we first (Section IV A) derive a formal expression for the channel phase, applicable to a general, isolated molecule experiment. Of particular interest are bound-free experiments where the continuum can be accessed via both a direct and a resonance-mediated process, since these scenarios give rise to rich structure of 8 ( ), and since they have been the topic of most experiments on the phase problem. In Section IVB we focus specifically on the case considered in Section III, where the two excitation pathways are one- and three-photon fields of equal total photon energy. We note the form of 8 (E) = 813(E) in this case and reformulate it in terms of physical parameters. Section IVC considers several limiting cases of 813 that allow useful insight into the physical processes that determine its energy dependence. In the concluding subsection of Section V we note briefly the modifications of the theory that are introduced in the presence of a dissipative environment. 

Hauptman, H. (in press) The phase problem of X-ray crystallography. In Direct Methods for Solving Macromolecular Structures, Fortier, S. (Ed.), Kluwer, Dordrecht. 

Application. Anomalous X-ray diffraction (AXRD), anomalous wide-angle X-ray scattering (AWAXS), and anomalous small-angle X-ray scattering (ASAXS) are scattering methods which are selective to chemical elements. The contrast of the selected element with respect to the other atoms in the material is enhanced. The phase problem of normal X-ray scattering can be resolved, and electron density maps can be computed. 

Once the phase problem is solved, a model of the protein can be built and refined, so that the model best fits the experimental data. A refined model will accurately represent the positions of the atoms in the unit cell. 

Solving the phase problem in protein crystallography is a requirement for any structural study. The three... 

Hauptman H. and J. Karle, Solution of the phase problem I. The centrosymmetric crystal. ACA monograph No. 3. Ann Arbor, (Edwards Brothers, Inc., 1953). 

The use of blank-disc CBED patterns for solving crystal stmctures by electron diffraction (in conjunction with direct methods for the phase problem) would seem to have many advantages ... 

Similar to X-Ray and neutron diffraction analysis, electron dilFraction structure analysis consists of such main stages as the obtaining of appropriate diffraction patterns and their geometrical analysis, the precision evaluation of diffraction-reflection intensities, the use of the appropriate formulas for recalculation of the reflection intensities into the structure factors, finally the solution of the phase problem, Fourier-constructions. 

A survey of conventional methods for solving the phase problem... 

Abstract This introduction to the phase-problem in crystallography is addressed to those... 

If the Patterson method cannot be applied because the structure has no or too many heavy atoms, it is possible to use another approach for phase determination, the so-called direct methods. By the term direct methods is meant that class of methods which exploits relationships among the structure factors in order to go directly from the observed magnitudes E to the needed phases < ) (Herbert A. Hauptman, Nobel lecture, 9. Dec., 1985). The direct method approach for solving the phase problem uses probability... 

The electron density is everywhere positive (a probability can not be negative ). Nevertheless, that the positivity of the density function is not a necessary prerequisite for solving the phase problem via direct methods was recently shown [20]... 

Zou, X. D. (1999) On the Phase Problem in Electron Microscopy The Relationship Between Structure Factors, Exit Waves, and HREM lmages . Microscopy Research and Technique 46,202-219. 

XD Zou. On the phase problem in electron microscopy the relationship between structure factors, exit waves, and HREM images. Microsc. Res. Tech. 46 202-219,... 

Solving the phase problem using isomorphous replacement... 

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