Electrons, anomalous dispersion

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. 

The a vaiues are a measure of the electron-density variation in the protein and solvent regions, and the ratio of these numbers is a measure of the contrast between the two regions. Since anomalous dispersion data were used to phase the maps, the map for the correct hand will show greater contrast. In this case, the original direct-methods sites give rise to greater contrast thereby indicating that these sites do correspond to the correct enantiomorph. 

The results obtained, which show that the same electron can emit photons with wavelength Xb in SRT, and electromagnetic waves with Xs in SLRT, suggest the existence of anomalous dispersion of light in cesium cell. These results, together with the results of further analysis, show that anomalous dispersion is consequence of superluminal phenomena in cesium cell, not vice versa, as the authors of the Refs. 6 and 7 claim. 

The results of this analysis show that anomalous dispersion of light in a cesium cell is a consequence of superluminal motion of electrons and superluminal propagation of electromagnetic waves. The Feynman diagram, presented in Fig. 8, is used in the analysis, to explain the phenomena that are taking place in cesium atomic cell and that cause superluminal effects [30]. 

Here, an energy transition of the electron will take place, and residual part of this process will be an electron with momentum pe2 = peo- Hence, the superluminal effects are taking place at vertices B and C. The possibility of an electron at this vertex emitting a photon with Xs A shows that anomalous dispersion in cesium cell is a result of superluminal processes, not vice versa. 

The most demanding element of macromolecular crystallography (except, perhaps, for dealing with macromolecules that resist crystallization) is the so-called phase problem, that of determining the phase angle ahkl for each reflection. In the remainder of this chapter, I will discuss some of the common methods for overcoming this obstacle. These include the heavy-atom method (also called isomorphous replacement), anomalous scattering (also called anomalous dispersion), and molecular replacement. Each of these techniques yield only estimates of phases, which must be improved before an interpretable electron-density map can be obtained. In addition, these techniques usually yield estimates for a limited number of the phases, so phase determination must be extended to include as many reflections as possible. In Chapter 7,1 will discuss methods of phase improvement and phase extension, which ultimately result in accurate phases and an interpretable electron-density map. 

In words, the desired electron-density function is a Fourier series in which term hkl has amplitude IFobsl, which equals (7/, /)1/2, the square root of the measured intensity Ihkl from the native data set. The phase ot hkl of the same term is calculated from heavy-atom, anomalous dispersion, or molecular replacement data, as described in Chapter 6. The term is weighted by the factor whU, which will be near 1.0 if ct hkl is among the most highly reliable phases, or smaller if the phase is questionable. This Fourier series is called an Fobs or Fo synthesis (and the map an Fo map) because the amplitude of each term hkl is iFobsl for reflection hkl. 

From Fig. 11.6 it can be seen that the polarisation (and so the refractive index) increases as it approaches a resonance frequency and temporarily falls to a "too" low value just beyond it. This remarkable and sudden change in behaviour was once considered anomalous and was called anomalous dispersion. The electro-magnetic wave theory showed that the "anomalous" dispersion is just as "normal" dispersion and can be explained as a direct consequence of the equation of motion of nuclei and electrons. 

Fig. 1 Top Behavior of the electronic linear chiroptical response in the vicinity of an excitation frequency. Re = real part (e.g., molar rotation [< ]), Im = imaginary part (e.g., molar ellipticity [0]). Without absorption line broadening, the imaginary part is a line-spectrum (5-functions) with corresponding singularities in the real part at coex. A broadened imaginary part is accompanied by a nonsingular anomalous OR dispersion (real part). A Gaussian broadening was used for this figure [37]. Bottom Several excitations. Electronic absorptions shown as a circular dichroism spectrum with well separated bands. The molar rotation exhibits regions of anomalous dispersion in the vicinity of the excitations [34, 36, 37]. See text for further details...

In the off-resonance region the radius of gyration is 42 A. This value lies well between those of iron-free apoferritin (51.5 A) and full ferritin (28 A) As saturated ferritin contains about 4300 iron atoms, an average iron content of about 3000 iron atom is estimated for this ferritin sample. From Eq. (65) and with reference to the radius of gyration of the FeOOH core, R = 28 A, the relative increase of R at the K-absorption edge indicates 14% decrease of the contrast q of ferritin, due to the anomalous dispersion of iron. The scattering density of the core decreases by as much as 17% and the atomic form factor of iron changes its value by one quarter (7 electrons in f ). 

Crystals of the compound of empirical formula FiiPtXe are orthorhombic with unit-cell dimensions a = 8-16, h = 16-81. c = 5-73 K, V = 785-4 A . The unit cell volume is consistent with Z = 4, since with 44 fluorine atoms in the unit cell the volume per fluorine atom has its usual value of 18 A. Successful refinement of the structure is proceeding in space group Pmnb (No. 62). Three-dimensional intensity data were collected with Mo-radiation on a G.E. spectrogoniometer equipped with a scintillation counter. For the subsequent structure analysis 565 observed reflexions were used. The platinum and xenon positions were determined from a three-dimensional Patterson map, and the fluorine atom positions from subsequent electron-density maps. Block diagonal least-squares refinement has led to an f -value of 0-15. Further refinements which take account of imaginary terms in the anomalous dispersion corrections are in progress. 

Our IBM 7040 least-squares programme was modified to cope with this situation. After several cycles of refinement of the gold atomic parameter, a c-axis difference electron-density projection was computed. On the resulting map it WM possible to locate the fluorine atoms there are only two independent fluorines, one in the general position x, y, z), 12(c), and the other in the special position (J, 0, 0), 6(o). Structure factors were calculated by use of the scattering factors of the International Tables for Au and F, that for gold being corrected for the real part of the anomalous dispersion effect B was taken as 2-0 A. ... 

The small maximum of the third order indicated by an arrow is ascribed by Davisson and Germer to anomalous dispersion of the electrons. Here, however, we shall not inquire further into this exceptional case. 

As the tables show, we can state a mean value for the internal potential of Ni which is independent of the electron velocity within the limits of experimental error. Here the electron velocity was varied between 40 and 350 volts. The author s experiments J on Cu, Ag, Au, Al, Pb, Fe, W , Mo, and Zr exhibit the same independence of the electron velocity. Here, however, we would expressly emphasize the fact that in the case of these latter metals also there may occur irregular variations in Eq, which may point to the existence of anomalous dispersion. It is extraordinarily difficult to prove that this anomalous dispersion really exists, and it will be found preferable to ascribe experimental diffraction maxima which give values of Eq not in agreement with the other values of Eq to unknown contamination of the surface rather than to anomalous dispersion, and to regard them as unidentifiable. 

The variation of S with wave-length has also been much investigated, and anomalous dispersion has definitely been found to exist a natural modification of the classical dispersion theory enables us to account for the latter. But since (according to Davisson and Germer) very few facts about these anomalous phenomena are meanwhile available in the case of electron reflection, we shall not trouble to describe these experiments. 

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