X-ray scattering amplitude

According to Eq. (1.16), the elastic coherent X-ray scattering amplitude is the Fourier transform of the electron density in the crystal. The crystal is a three-dimensional periodic function described by the convolution of the unit cell density and the periodic translation lattice. For an infinitely extended lattice,... 

The mean-square displacements of each of the atoms in the crystal, which affect the the X-ray scattering amplitudes, are obtained by summation over the displacements due to all normal modes, each of which is a function of ea(j Icq), as further discussed in section 2.3. The eigenvalues of D are the frequencies of the normal modes. 

A. The Relationship between the Electron-Density Distribution in a Crystal and the X-Ray Scattering Amplitudes from a Single... 

At present the X-ray diffraction method is used mostly to determine atomic positions in a crystal. However, X-ray scattering amplitudes depend directly on the electron-density distribution in a crystal, from which atomic positions can be derived on the assumption of coincidence of the nuclear positions and the center of gravities of total electron densities around atomic nuclei. 

In relativistic quantum theory, it is the second term in Eq. (51) that is entirely responsible for scattering as it is second order in the vector potential. It is convenient to divide this term into four components, and the X-ray scattering amplitude in the case of elastic scattering may be written as (Arola et al., 1997 Arola and Strange, 2001)... 

The formal resemblance to Bragg diffraction can now begin to be seen. The function F(Xab) defined in Eq. (4) is the analog of the x-ray scattering amplitude (5)... 

To understand the way in which this is achieved we first recall from eq. 2 that the X-ray scattering amplitude may be written as... 

How is that knowledge used to find the phase of the contribution from the protein in the absence of the heavy-metal atoms We know the phase and amplitude of the heavy metals and the amplitude of the protein alone. In addition, we know the amplitude of protein plus heavy metals (i.e., protein heavy-metal complex) thus we know one phase and three amplitudes. From this we can calculate whether the interference of the x-rays scattered by the heavy metals and protein is constructive or destructive (Figure 18.10). The extent of positive or negative interference plus knowledge of the phase of the heavy metal together give an estimate of the phase of the protein. 

The Maxwell theory of X-ray scattering by stable systems, both solids and liquids, is described in many textbooks. A simple and compact presentation is given in Chapter 15 of Electrodynamics of Continuous Media [20]. The incident electric and magnetic X-ray helds are plane waves Ex(r, f) = Exo exp[i(q r — fixO] H(r, t) = H o exp[/(q r — fixO] with a spatially and temporally constant amplitude. The electric field Ex(r, t) induces a forced oscillation of the electrons in the body. They then act as elementary antennas emitting the scattered X-ray radiation. For many purposes, the electrons may be considered to be free. One then finds that the intensity /x(q) of the X-ray radiation scattered along the wavevector q is... 

Once a suitable crystal is obtained and the X-ray diffraction data are collected, the calculation of the electron density map from the data has to overcome a hurdle inherent to X-ray analysis. The X-rays scattered by the electrons in the protein crystal are defined by their amplitudes and phases, but only the amplitude can be calculated from the intensity of the diffraction spot. Different methods have been developed in order to obtain the phase information. Two approaches, commonly applied in protein crystallography, should be mentioned here. In case the structure of a homologous protein or of a major component in a protein complex is already known, the phases can be obtained by molecular replacement. The other possibility requires further experimentation, since crystals and diffraction data of heavy atom derivatives of the native crystals are also needed. Heavy atoms may be introduced by covalent attachment to cystein residues of the protein prior to crystallization, by soaking of heavy metal salts into the crystal, or by incorporation of heavy atoms in amino acids (e.g., Se-methionine) prior to bacterial synthesis of the recombinant protein. Determination of the phases corresponding to the strongly scattering heavy atoms allows successive determination of all phases. This method is called isomorphous replacement. 

In Chapter 3 we went as far as we could in the interpretation of rocking curves of epitaxial layers directly from the features in the curves themselves. At the end of the chapter we noted the limitations of this straightforward, and largely geometrical, analysis. When interlayer interference effects dominate, as in very thin layers, closely matched layers or superlattices, the simple theory is quite inadequate. We must use a method theory based on the dynamical X-ray scattering theory, which was outlined in the previous chapter. In principle that formrrlation contains all that we need, since we now have the concepts and formtrlae for Bloch wave amplitude and propagatiorr, the matching at interfaces and the interference effects. 

The influence of molecular vibrations on the interference pattern was first studied by James in 1932 He discussed the intensity distribution (of X-ray scattering) by vibrating molecules, and determined the dependence of the intensity distribution on the vibrational amplitudes. 

Moreover, the above discussion assumes that the experimental technique measures exactly what the computational technique does, namely, the separation between the nuclear centroids defining a bond. X-ray crystallography, however, measures maxima in scattering amplitudes, and X-rays scatter not off nuclei but off electrons. Thus, if electron density maxima do not correspond to nuclear positions, there is no reason to expect agreement between theory and experiment (for heavy atoms this is not much of an issue, but for very light ones it can be). Furthermore, the conditions of the calculation typically correspond to an isolated molecule acting as an ideal gas (i.e., experiencing no intermolecular interactions), while a technique... 

Fig. 4.11 Small-angle X-ray scattering patterns from a face-centred cubic phase formed by a PEO127PPO48PEO127 (F108) Pluronic solution (35 wt% in water) at 30 °C during oscillatory shear at 10rad s 1 with a strain amplitude of 40% (Diat et al. 1996). The patterns correspond to (a) the (q qt) plane and (b) the (<7v,4V) plane. In (c) the pattern was recorded in the (qv,qt) plane but with the beam incident close to the outer rotor. It corresponds to one of the FCC twins giving the diffraction pattern in Fig. 4.12(b).

Interference between X-rays scattered at different atomic centres occurs in exactly the same way as for an atom. The scattered amplitude becomes a function of an atomic distribution function. In an amorphous fluid, a gas or non-crystalline solid the function is spherically symmetrical and the scattering independent of sample orientation. It only depends on the radial distribution of scattering centres (atoms). 

Because the amplitude of the lattice distortion in the Peierls insulator is very small, the X-ray scattering associated with it is weak. Furthermore the one-dimensional nature of this distortion gives rise to diffuse Bragg planes instead of the usual well-defined Bragg reflections. These two facts have led to the development of a special diffuse X-ray photographic technique often referred to as the monochromatic Laue technique or XDS (for X-ray diffuse scattering) (64). 

Caspar et al. (1988) described an elegant analysis of the diffuse X-ray scattering from insulin crystals. They found two types of coupled motion one with a characteristic length of about 6 A and amplitude of about 0.4 A, the other with a characteristic length of about 20 A and smaller amplitude. The latter motion represents the jiggle of neighboring molecules of the lattice. The former represents the coupled fluidlike fluctuations within a protein molecule. The short-range motions appear to be similar to those detected by Mossbauer spectroscopy. 

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