Atomic form factors

We can make a simple estimate of the atomic form factor by assuming a spherical distribution of electrons with radius Rj surrounding the /th atom and considering the phase shift from the scattering across the region. 

Angular part of the atomic form factor for various values of r = 4tt particle size/wavelength. X-rays with wavelengths much shorter than the particle size will be scattered in the forward direction. As the wavelength increases, the scattering becomes more omnidirectional. 

Actual values for the atomic form factors for various atoms as a function of scattering angle can be found in the International Tables for X-Ray Crystallography or from the online NIST database http //physics.nist.gov/PhysRefData/FFast/Text/cover.html. 

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B2.2.5.5 ATOMIC FORM FACTOR AND GENERALIZED OSCILLATOR STRENGTH... 

The parameters reported by Zintl Hauke were taken as the starting point of the parameter determination. Using these parameters, structure factors were calculated for all of the planes in the sphere of reflection. The atomic form factors of James Brindley (1935) were used. (Subsequent calculations made with two... 

The reliability factor B was 0276 after the first refinement and 0-211 after the fourth refinement. The parameters from the third and fourth refinements differed very little from one another. The final values are given in Table 1. As large systematic errors were introduced in the refinement process by the unavoidable use of very poor atomic form factors, the probable errors in the parameters as obtained in the refinement were considered to be of questionable significance. For this reason they are not given in the table. The average error was, however, estimated to be 0-001 for the positional parameters and 5% for the compositional parameters. The scattering power of the two atoms of type A was given by the least-squares refinement as only 0-8 times that of aluminum (the fraction... 

To obtain the atomic form factor according to the multipole formalism, we apply the Fourier transform... 

FIG. 5.9 Phase angles in an acentric X-N analysis phase angle as calculated with spherical-atom form factors and neutron positional and thermal parameters tpx is the unknown phase of the X-ray structure factors which must be estimated for the calculation of the vector AF. Use of FX — FN introduces a large phase error. Source Coppens (1974). 

Within an atomistic approximation, the structure factor can be expressed in terms of the atomic form factors, mean positions and mean-square displacements ... 

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...

Expressions of the form (9.34) enter, for example, in the calculation of atom form factors, where the operator L has the form... 

To the extent that our model holds true, one can use the sum of the expressions (9.39) in the case of the large strata of the atom on hand for the density matrix of the atom in momentum space. But the knowledge of the density matrix allows one — as f)irac especially has pointed out—to answer all questions about the atom, in particular the calculation of the atom form factors. 

As an example we cite here the atom form factor for the th biggest large stratum. In atomic units we have... 

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. 

Consider Fig. 9 showing the diffraction profile for the polymer Lucite (C5H802) taken from Tartari et al. [23], The dashed line represents the function F2 + S calculated from the mixture rule on the basis of the Independent Atom Model (IAM) using the tabulated atom form factors for H, C and O data presented in Fig. 8. The IAM curve is seen to approach the measured profile at high x values. 

J H Hubbell, W J Veigele, E A Briggs, R T Brown, D T Cromer and R J Howerton (1975) Atomic form factors, incoherent scattering functions and photon scattering cross-sections. J. Phys. Chem. Ref. Data 4, 471 Errata in 1977, 6, 615. 

Room temperature Spherical atom form factors (conventional refinement) 3104 270 4.69 3.85 2.07... 

Two or three most significant Ptm parameters of each atom type were then chosen from previous work on peptide molecules [76,22,29] and used as fixed parameters in least-squares refinements of PPP. Only the fractional coordinates and the thermal parameters of all atoms were adjusted the statistical indices of the refinement decreased dramatically as shown on Table 3, compared to those obtained from a conventional refinement using spherical atoms form factors. 

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 ). 

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