Phase Problems

Despite the apparent simplicity with which a crystal structure can be restored by applying Fourier transformation to diffraction data (Eqs. 2.132 to 2.135), the fact that the structure amplitude is a complex quantity creates the so-called phase problem. In the simplest case (Eq. 2.133), both the absolute values of the structure amplitudes and their phases (Eq. 2.105) are needed to locate atoms in the unit cell. The former are relatively easily determined from powder (Eq. 2.65) or single crystal diffraction data but the latter are lost during the experiment. 

Determination of the crystal structure of an unknown material is generally far from a straightforward procedure, especially when only powder diffraction data are available. It is truly a problem solving process and not a simple refinement, which often may be fully automated. The latter is a technique, which improves structural parameters of the approximately or partially known model, usually by using a least squares minimization against available diffraction data. It is worth noting that the least squares method is 

A large variety of methods, developed with a specific goal to solve the crystal structure from diffraction data, can be divided into two major groups. The first group entails techniques that are applicable in direct space by constructing a model of the crystal structure from considerations other than the available array of structure amplitudes. These include  

When one or more models are constructed, they are tested against the experimental diffraction data. Often some of these approaches are combined together but they always stem from the requirement that the generated model must make physical, chemical and crystallographic sense. Thus, their successful utilization requires a certain level of experience and knowledge of how different classes of crystals are built, e.g. what to expect in terms of coordination and bond lengths for a particular material based solely on its chemical composition. Direct space modeling approaches will be discussed, to some extent, in Chapter 6. 

However, there is a major difficulty in carrying out this inversion process. The electric field vector E h,k/) of the scattered x-rays is directly proportional to Sc(hM at the present there is no method for directly measuring this E-field. For x-rays, we are limited to the intensity I as the observable, which is given by I E / The E is a complex function involving both amplitude and phase, but the phase irrformation is lost unless one uses a coherent source to record the phase information as is done in holography. Since we do not yet have x-ray lasers suitable for this purpose, this is known as the phase problem in crystallography (see McPherson, 1999). 

The quantities (1 /V)F hkl) are the Fourier coefficients. Here V is the volume of the unit cell. The summation is extended over all diffraction maxima. 

If the values of the structure factor F hkt) were all known, we would calculate the values of p and thereby determine the locations of the atoms in a crystal. However, in an actual recorded diffraction pattern, only the intensities and, consequently, the amplitudes of the diffracted rays may be measured. We do not have specific information on the crystal s phase. Since all atoms are not at the same place, the scattering wave from one atom may be in phase or out of phase with that of its neighbor. A value of ) has to be assigned to each value of F hkl). This poses a thorny problem, which has been called the phase problem. At one time it was generally believed that crystal structures could not be determined from diffraction intensities alone. In this section we describe two ingenious and successful approaches to this problem. 

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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... 

Each diffracted beam, which is recorded as a spot on the film, is defined by three properties the amplitude, which we can measure from the intensity of the spot the wavelength, which is set by the x-ray source and the phase, which is lost in x-ray experiments (Figure 18.8). We need to know all three properties for all of the diffracted beams to determine the position of the atoms giving rise to the diffracted beams. How do we find the phases of the diffracted beams This is the so-called phase problem in x-ray crystallography. 

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. 

For the case of a three-phase problem, where the solute is accessible to the a, (3, and y phases, Whitaker [427] finds the overall average phase concentration for the case of local mass equilibrium given by... 

When a diffracted X-ray beam hits a data collection device, only the intensity of the reflection is recorded. The other vital piece of information is the phase of the reflected X-ray beam. It is the combination of the intensity and the phase of the reflections that is needed to unravel the contributions made to the diffraction by the electrons in different parts of the molecule in the crystal. This so-called phase problem has been a challenge for theoretical crystallographers for many decades. For practical crystallography, there are four main methods for phasing the data generated from a particular crystal. 

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. 

We began our analysis in Section II and ended it in Section VC2 by making the connection of the time- and energy-domain approaches to both coherence spectroscopy and coherent control. It is appropriate to remark in closing that new experimental approaches that combine time- and energy-domain techniques are currently being developed to provide new insights into the channel phase problem. We expect that these will open further avenues for future research. 

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. 

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

In an earlier study calorimetry achieved this objective for the compositional boundaries between two and three phases (2). Such boundaries are encountered both in "middle-phase microemulsion systems" of low tension flooding, and as the "gas, oil, and water" of multi-contact miscible EOR systems (LZ). The three-phase problem presents by far the most severe experimental and interpretational difficulties. Hence, the earlier results have encouraged us to continue the development of calorimetry for the measurement of phase compositions and excess enthalpies of conjugate phases in amphiphilic EOR systems. 

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. 

These instruments employ a continuous flow of persulfate solution to promote oxidation prior to ultraviolet irradiation, and have a low system blank and low detection limit. Since all reactions take place in the liquid phase, problems suffered by combustion techniques, such as catalyst poisoning, reactor corrosion, and high-temperature element burnouts, are obviated. However, the ultraviolet-promoted chemical oxidation technique is not designed to handle particulate-containing samples, and tends to give incomplete oxidation for certain types of compounds such as cyanuric acid. 

This illustrates the statement made earlier that the most convenient choice of standard state may depend on the problem. For gas-phase problems involving A, it is convenient to choose the standard state for A as an ideal gas at 1 atm pressure. But, where the vapor of A is in equilibrium with a solution, it is sometimes convenient to choose the standard state as the pure liquid. Since /a is the same for the pure liquid and the vapor in equilibrium... 

A number of EIA theorists believe in incorporating formal RA methods into EIA as a way to cope with uncertainties, especially in impact prediction where a formal framework for ecological risk assessment (EcoRA) is already developed. It includes three generic phases problem formulation, analysis, and risk characterization followed by risk management. The analysis phase includes an exposure assessment and an ecological effects assessment (see, e.g., US EPA (1998)). 

When retention times of mixture components decrease, there may be problems with either the mobile or stationary phase. It may be that the mobile phase composition was not restored after a gradient elution, or it may be that the stationary phase was altered due to irreversed adsorption of mixture components, or simply chemical decomposition. Use of guard columns may avoid stationary phase problems. 

In imaging, faster accumulation by means of the steady-state free procession (SSFP) has been used successfully for 15N, but the method has so far not been reported for spectroscopy, although it should prove useful there also, provided proper attention is paid to phasing problems. 

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