Obtaining Phases

The molecular image that the crystallographer seeks is a contour map of the electron density p(x,y,z) throughout the unit cell. The electron density, like all periodic functions, can be represented by a Fourier series. The representation that connects p(x,y,z) to the diffraction pattern is 

Equation (5.18) tells us how to calculate p(jc,y,z) simply construct a Fourier series using the structure factors Fhkl. For each term in the series, h, k, and 1 are the indices of reflection hkl, and Fhkl is the structure factor that describes the reflection. Each structure factor Fhkl is a complete description of a diffracted ray recorded as reflection hkl. Being a wave equation, Fhkl must specify frequency, amplitude, and phase. Its frequency is that of the X-ray source. Its amplitude is proportional to (- j /)1/2, the square root of the measured intensity Ihkl of reflectionhkl. Its phase is unknown and is the only additional information the crystallographer needs in order to compute p(x,y,z) and thus 

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. 

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A much better way would be to use phase contrast, rather than attenuation contrast, since the phase change, due to changes in index of refraction, can be up to 1000 times larger than the change in amplitude. However, phase contrast techniques require the disposal of monochromatic X-ray sources, such as synchrotrons, combined with special optics, such as double crystal monochromatics and interferometers [2]. Recently [3] it has been shown that one can also obtain phase contrast by using a polychromatic X-ray source provided the source size and detector resolution are small enough to maintain sufficient spatial coherence. 

This opens perspectives for obtaining phase contrast information in a microfocus tomographic system Recently we have developed a desktop X-ray microtomographic system [4] with a spot size of 8 micrometer (70 KeV) and equipped with a (1024) pixel CCD, lens coupled to a scintillator. The system is now commercially available [5], The setup is sketched in Figure 1 In this work we used the system to demonstrate the feasibility for phase contrast microtomography. 

Low-temperature solid-state synthesis is preferred in most cases, where appropriate, for obvious reasons such as energy and cost economy and process safety or for critical concerns regarding the accessibility of compounds that are stable only at low temperatures or non-equilibrium phases, i.e., compounds thermodynamically unstable with respect to the obtained phase (e.g., a ternary instead of binary phase). The use of low-temperature eutectics as solvents for the reactants, hydrothermal growth... 

The obtained phase changes correspond well to the introduced changes, demonstrating one of the multipurpose features of the multichannel device with which... 

Thus most of the time one obtains phase-separated systems in which the macromolecules of component A are not at all or only to a limited extent miscible with the macromolecules of component B, i.e., polymer A is incompatible or only partially compatible with polymer B. The synonymical terms polymer blend , polymer alloy , or polymer mixture denote miscible (homogeneous) as well as immiscible (heterogeneous) systems consisting of two or more different polymers. 

Alternative methods of solving the phase problem are also used now. When a transition metal such as Fe, Co, or Ni is present in the protein, anomolous scattering of X-rays at several wavelengths (from synchrotron radiation) can be used to obtain phases. Many protein structures have been obtained using this multiple wavelength anomalous diffraction (MAD phasing) method.404 407 408 Selenocysteine is often incorporated into a protein that may be produced in... 

In this Section we focus our attention on the development of the formalism for complex reactions with application to the formation of NH3. The results obtained (phase transition points and densities of particles on the surface) are in good agreement with the Monte Carlo and cellular automata simulations. The stochastic model can be easily extended to other reaction systems and is therefore an elegant alternative to the above-mentioned methods. 

Another vital type of ligand is a heavy-metal atom or ion. Crystals of protein/ heavy-metal complexes, often called heavy-atom derivatives, are usually needed in order to solve the phase problem mentioned in Chapter 2 (Section VI.F). I will show in Chapter 6 that, for the purpose of obtaining phases, it is crucial that heavy-atom derivatives possess the same unit-cell dimensions and symmetry, and the same protein conformation, as crystals of the pure protein, which in discussions of derivatives are called native crystals. So in most structure projects, the crystallographer must produce both native and derivative crystals under the same or very similar circumstances. 

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]. 

B. Obtaining phases from heavy-atom data... 

In order to resolve the phase ambiguity from the first heavy-atom derivative, the second heavy atom must bind at a different site from the first. If two heavy atoms bind at the same site, the phases of will be the same in both cases, and both phase determinations will provide the same information. This is true because the phase of an atomic structure factor depends only on the location of the atom in the unit cell, and not on its identity (Chapter 5, Section III.A). In practice, it sometimes takes three or more heavy-atom derivatives to produce enough phase estimates to make the needed initial dent in the phase problem. Obtaining phases with two or more derivatives is called the method of multiple isomorphous replacement (MIR). This is the method by which most protein structures have been determined. 

Before we can obtain phase estimates by the method described in the previous section, we must locate the heavy atoms in the unit cell of derivative crystals. 

A second means of obtaining phases from heavy-atom derivatives takes advantage of the heavy atom s capacity to absorb X rays of specified wavelength. As a result of this absorption, Friedel s law (Chapter 4, Section III.G) does not hold, and the reflections hkl and —h — k—l are not equal in intensity. This inequality of symmetry-related reflections is called anomalous scattering or anomalous dispersion. 

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