Atomic form factor, electron

T(S) is the Debye-WaUer factor introduced in (2). The atomic form factors are typically calculated from the spherically averaged electrcai density of an atom in isolation [24], and therefore they do not contain any information on the polarization induced by the chemical bonding or by the interaction with electric field generated by other atoms or molecules in the crystal. This approximation is usually employed for routine crystal stmcture solutions and refinements, where the only variables of a least square refinement are the positions of the atoms and the parameters describing the atomic displacements. For more accurate studies, intended to determine with precisicai the electron density distribution, this procedure is not sufficient and the atomic form factors must be modeled more accurately, including angular and radial flexibihty (Sect. 4.2). 

Fig. 4 The atomic form factor of a C atom (in ls 2s 2p electronic configuration). Core electron scattering is in blue. Valence electron scattering is in red and total scattering in black...

The atomic form factor accounts for the internal structure of the different atoms or molecules. It will also be different for X-rays and neutrons, since the former probe the electron distribution of the target, while the latter interact with the nuclei of the atoms. Therefore, the analysis of the positions of the reflexes indicates mainly the lattice constants and angles. The intensity of the reflexes contains mainly information about the atomic configuration within an unit cell (structure factor) and the scattering behavior of the single atoms (form factor). 

Two advanced techniques have been proposed and applied to some crystal structures (Section IV,C), in which aspherical distributions of valence electrons around an atom are directly taken into account in the least-squares calculations. Aspherical atomic form factors are introduced in the least-squares refinement in the first method (29, 38, 80) and multipole parameters describing the aspherical valence distributions are used in the second method (31, 34, 46). 

The first factor in square brackets represents the Thomson cross-section for scattering from a free electron. The second square bracket describes the atomic arrangement of electrons through the atomic form factor, F, and incoherent scatter function, S. Finally, the last square bracket contains the factor s(x), the molecular interference function that describes the modification to the atomic scattering cross-section induced by the spatial arrangement of atoms in their molecules. 

The form factors fo at Q = 0 are an approximation of the number of electrons Z of an atom. When modeling neutron PDFs, the appropriate nuclear scattering lengths or magnetic form factors replace these atomic form factors. In equation (6), rik is the distance between atoms i and k summed over all the atoms in the sample. 

Fig. 1. Dispersion of the imaginary part f of the atomic form factor of uranium, f" (in units of electrons) reaches about one third of the non-resonant atomic form factor f = 92. This schematic representation has been taken from The International Tables of Crystallography, IV ...

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 ). 

From this one can define the atomic form factor/(0 as the Fourier transform of the electron density, and the structure factor F Q) as... 

A.O.Williams Jr. noted in his Hartree-calculations on the closed shell atom Cu as early as 1940 (Phys. Rev. 58, 723) The charge density of each single electron turns out to resemble that for the nonrelativistic case, but with the maxima "pulled in " and raised.. .. The size of the relativistic corrections appear to be just too small to produce important corrections in atomic form factors or other secondary characteristics of the whole atom.. .. However, it must be noticed that copper is a relatively light ion, and the corrections for such an ion as mercury would be enormously greater. S.Cohen in 1955 and... 

Fig. 3. Distribution of the uncompensated electron density in the [001] direction for MnBi (1), MnSb (2), and MnAs (3). pj, is the density of all the electrons in the neutral mai anese atom, calculated from the theoretical atomic form factor [4].

The inner diffraction effect is produced when the individual particles of the atom capable of vibration, i.e. the electrons contained in the atoms, are dispersed and give rise to secondary radiations which interfere with one another. In a liquid built up of single atoms— A, Kr, Xe, Hg, Ga— the result is that, on account of intra-atomic interference, the dependence of intensity distribution on the angle of diffraction is already affected. This influence is generally expressed by a factor, which, because of its origin, is called the atomic form factor its action is that more intensity is scattered in the directions near the primary beam than if the interaction of the individual electrons is not taken into account. 

As we have seen, on the whole the agreement with theory for the localized form factor associated with the 4f electrons in lanthanide metals and compounds is satisfactory provided one is careful to use relativistic calculations. The situation for the conduction electron polarization distribution is less clear. Conduction electron form factors were obtained for Gd by Moon et al. (1972) and for Er by Stassis et al. (1976). In both cases, these were obtained by separating from the measured form factor the localized 4f contribution, and in both cases appear to be different from either a 5d or 6s atomic form factor. A spin-polarized augmented-plane-wave (APW) calculation of the conduction electron polarization in ferromagnetic Gd was performed by Harmon and Freeman (1974). Their results are, however, only in qualitative agreement with the results of Moon et al. The theoretical form factor of Harmon and Freeman is in somewhat better agreement with the experimental results of Stassis et al. on Er. 

If the frequency of the incident X rays is close to the frequency of the K absorption edge of the irradiated atom, scattering can no longer be regarded as elastic some of the incident energy is used to excite the electrons in the atom. The atomic form factors have to be corrected for anomalous dispersion the tables in [29] include this correction. The significance of these corrections for the determination of the absolute configuration is discussed in Section 15.2.4. 

Neutrons are scattered isotropically from individual nuclei, whereas, for LS and SAXS, the scattering originates in the electron cloud, so the atomic form factors are in principle (2-dependent. However, the variation is small in practice (<1% for Q < 0.1 A ) and is usually neglected for SAXS and LS [36]. The Thompson-scattering amplitude of a classical electron is rj = 0.282 x 10 cm [65], so the X-ray scattering length of an atom, /, is proportional to the atomic number (/ = rjZ) and increases with the number of electrons per atom. For neutrons, values of b vary from isotope to isotope (see below). If the nucleus has nonzero spin, it can interact with the neutron spin, and the total cross section (atot) splits into coherent and incoherent components as explained below. 

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