Phase determination, diffracted beams

Direct methods, direct phase determination A method of deriving relative phases of diffracted beams by consideration of relationships among the Miller indices and among the structure factor amplitudes of the stronger Bragg reflections. These relationships come from the conditions that the structure is composed of atoms and that the electron-density map must be positive or zero everywhere. Only certain values for the phases are consistent with these conditions. 

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. 

The intensity differences obtained in the diffraction pattern by illuminating such a crystal by x-rays of different wavelengths can be used in a way similar to the method of multiple isomorphous replacement to obtain the phases of the diffracted beams. This method of phase determination which is called Multiwavelength Anomalous Diffraction, MAD, and which was pioneered by Wayne Hendrickson at Columbia University, US, is now increasingly used by protein cystallographers. 

Crystallization of proteins can be difficult to achieve and usually requires many different experiments varying a number of parameters, such as pH, temperature, protein concentration, and the nature of solvent and precipitant. Protein crystals contain large channels and holes filled with solvents, which can be used for diffusion of heavy metals into the crystals. The addition of heavy metals is necessary for the phase determination of the diffracted beams. 

The present chapter deals first with all the preliminary steps which must be taken to obtain suitable data for structure determination (whether by direct or indirect methods)—the measurement of the intensities of diffracted beams, and the application of the corrections necessary to isolate the factors due solely to the crystal structure from those associated with camera conditions. It then goes on to deal with the effect of atomic arrangement on the intensities of diffracted beams, the procedure in deducing the general arrangement, and finally the methods of determining actual atomic coordinates by trial. It follows from what has been said that, as soon as atomic positions have been found to a sufficient degree of approximation to settle the phases of the diffracted beams, then the direct method can be used this, in fact, is the normal procedure in the determination of costal structures. 

The reason why it is not usually possible to employ this direct method for the solution of crystal structures has already been indicated at the beginning of Chapter VII it is that we do not usually know, and cannot determine experimentally, the phases of the various diffracted beams with respect to a chosen point in the unit of pattern. However, for certain crystals we can from the start be reasonably certain of the phase relations of the diffracted beams, or can deduce them from crystallographic evidence, and in these circumstances we can proceed at once to combine the information, either mathematically or by experimental methods in which light waves are used in place of X-rays. Otherwise, it is necessary to find approximate positions by trial, the approximation being taken as far as is necessary to be certain of the phases of a considerable number of reflections as soon as the phases are known, the direct method can be used. 

Phase problem The problem of determining the phase angle (relative to a chosen origin) that is to be associated with each diffracted wave that is combined to give an electron-density map. The measured intensities of diffracted beams give only the squares of the amplitudes, but the relative phases cannot normally be determined experimentally (see Chapter 8). The determination of the relative phases of the Bragg reflections is crucial to the calculation of the correct electron density map. 

The simplest type of crystal diffraction, and that considered here, is called kinematical diffraction. In such diffraction the X-ray beam, once diffracted, is not further modified by additional diffraction in its passage through the crystal. The phase differences between radiation scattered at different points in the crystal depend only on differences in the path lengths of the incident and diffracted waves. The summation of these waves with the appropriate amplitudes and relative phases determines the intensities. 

The amplitudes of diffracted beams are summed with a periodicity inversely proportional to their order (and with a relative phase, which needs to be determined) to give an electron density map. 

This Chapter is concerned with methods for obtaining the relative phase angles for each Bragg reflection so that the correct electron-density map can be calculated and, from it, the correct molecular structure determined. When scattered light is recombined by a lens, as described in Chapters 3 and 6, the relationships between the phases of the various diffracted beams are preserved. In X-ray diffraction experiments, however, only the intensities of the Bragg reflections are measured, and information on the relative phases is lost. An attempt is maxle to remedy this situation by deriving relative phases by one of the methods to be described in this Chapter. Then Equation 6.3 (Chapter 6) is used to obtain the electron-density map. Peaks in this map represent atomic positions. 

The structure of gas-phase clusters has not been determined. The adoption of gas-phase electron diffraction techniques to cluster beams has provided some data, mostly averaged structural parameters. " There is experimental evidence in several systems that structures do exist with unusual stability, especially for alkali metals.Theoretical calculations suggest that these stabilities are due to electronic shell closings in the evolution of the electronic structure of a globular, close-packed cluster. Calculations suggest that aluminum also exhibits this behavior. 

The latter instrument is of particular value in work of this kind because it allows continuous observation of a diffraction line. For example, the temperature below which a high-temperature phase is unstable, such as a eutectoid temperature, can be determined by setting the diffractometer counter to receive a prominent diffracted beam of the high-temperature phase, and then measuring the intensity of this beam as a function of temperature as the specimen is slowly cooled. The temperature at which the intensity falls to that of the general background is the temperature required, and any hysteresis in the transformation can be detected by a similar measurement on heating. 

The use of a sample holder requires the sample to be reduced to a fine powder. This condition is not a problem when determining, for example, the structure of a new phase that has just been synthesized and this type of equipment is used for that purpose by solid state chemists. However, it is sometimes necessary to directly study bulk samples for which grinding could cause phase transitions. In those cases, the sample is a plane object that can be studied with the same diffractometer. Different configurations are then possible, with different paths for diffracted beams and different angular resolutions depending on the relative positions of the various elements included in the apparatus. 

All methods of deduction of the relative phases for Bragg reflections from a protein crystal depend, at least to some extent, on a Patterson map, commonly designated P(uvw) (46, 47). This map can be used to determine the location of heavy atoms and to compare orientations of structural domains in proteins if there are more than one per asymmetric unit. The Patterson map indicates all the possible relationships (vectors) between atoms in a crystal structure. It is a Fourier synthesis that uses the indices, l, and the square of the structure factor amplitude f(hkl) of each diffracted beam. This map exists in vector space and is described with respect to axes u, v, and w, rather than x,y,z as for electron-density maps. 

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