The atomic scattering factor

An atom consists of a nucleus plus an electron density distribution, given in the form of a radial function p(jc,y,z)  

The atomic wavefimction ir x,y, z) can be calculated to a very high degree of accuracy by quantum mechanics (Section 3.5). If each infinitesimal unit of a continuous electron density is taken as a scattering unit, the summation of equation 5.15 becomes an integral. In addition, the scattered wave is averaged over the spherical distribution around the nucleus, and application of 5.16 gives 

A sum of partial scattered waves amounting to the total scattering power of an object is called the structure factor of that object. Equation 5.18 is the structure factor of an 

Calling/k the atomic scattering faetor of atom A , an obvious extension of the previous reasoning leads to the following expression for the structure factor of an array of atoms, or a molecule containing m atoms  

This expression neglects deformations due to chemical bonding (see Section 5.7) The intensity of a diffracted beam is proportional to the square of the module of the structure factor. Since the structure factor is a complex number, the square of the module is obtained by multiplying the structure factor by its complex conjugate  

Equation (3.25), which was derived for the intensity of the radiation scattered by a unit cell, contains the atomic scattering factor / . This factor depends principally on the nature of the radiation (and therefore the scattering mechanism), the nature of the scattering point, and the scattering angle. 

It will be helpful to discuss the scattering of x-rays by a single atom before considering the atomic scattering factor for electrons. 

To derive an expression for /(x) using classical arguments, we make the following assumptions  

1 The scattering atom contains electrons that are distributed throughout a volume comparable to atomic dimensions and to the wavelength of the radiation. 

2 Each electron is loosely bound in the atom so that it scatters as a free electron. This means that if the frequency v of the incident radiation is very large compared with the natural frequency of oscillation (vq), then the scattered wave will be exactly x radians out of phase with the incident wave for all the electrons. 

X-rays are diffracted by the electrons on each atom. The scattering of the X-ray beam increases as the number of electrons, or equally, the atomic number (proton number), of the atom, Z, increases. Thus heavy metals such as lead, Pb, Z = 82, scatter X-rays far more strongly than light atoms such as carbon, C, Z = 6. Neighbouring atoms such as cobalt, Co, Z = 27, and nickel, Ni, Z = 28, scatter X-rays almost identically. The scattering power of an atom for a beam of X-rays is called the atomic scattering factor, /a. 

Atomic scattering factors were originally determined experimentally, but now can be calculated using quantum mechanics. The atomic scattering factors, derived from quantum mechanical calculations of the electron density around an atom, are (approximately) given by the equation  

The nine constants, bt and ch called the Cromer-Mann coefficients, vary for each atom or ion. The Cromer-Mann coefficients for sodium, Na, with an atomic number, Z= 11, for use in equation (6.2) are given in Table 6.1. The units of /a are electrons, and the wavelength of the radiation is in A. To use the SI unit of nm, (10 A = 1 nm), for the wavelength, use the same values of at and c, but divide the values of bt by 100. 

These and other values used in this Chapter are taken from http //www-structure.llnl.gov. 

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Figure Bl.8.1. The atomic scattering factor from a spherically synnnetric atom. The volume element is a ring subtending angle a with width da at radius r and thickness dr.

Because the neutron has a magnetic moment, it has a similar interaction with the clouds of impaired d or f electrons in magnetic ions and this interaction is important in studies of magnetic materials. The magnetic analogue of the atomic scattering factor is also tabulated in the International Tables [3]. Neutrons also have direct interactions with atomic nuclei, whose mass is concentrated in a volume whose radius is of the order of... 

The atomic scattering factor for electrons is somewhat more complicated. It is again a Fourier transfonn of a density of scattering matter, but, because the electron is a charged particle, it interacts with the nucleus as well as with the electron cloud. Thus p(r) in equation (B1.8.2h) is replaced by (p(r), the electrostatic potential of an electron situated at radius r from the nucleus. Under a range of conditions the electron scattering factor, y (0, can be represented in temis... 

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

Potassium chloride actually has the same stnicture as sodium chloride, but, because the atomic scattering factors of potassium and chlorine are almost equal, the reflections with the indices all odd are extremely weak, and could easily have been missed in the early experiments. The zincblende fonn of zinc sulphide, by contrast, has the same pattern of all odd and all even indices, but the pattern of intensities is different. This pattern is consistent with a model that again has zinc atoms at the comers and tlie face centres, but the sulphur positions are displaced by a quarter of tlie body diagonal from the zinc positions. 

A teclmique that employs principles similar to those of isomorphous replacement is multiple-wavelength anomalous diffraction (MAD) [27]. The expression for the atomic scattering factor in equation (B1.8.2h) is strictly accurate only if the x-ray wavelength is well away from any characteristic absorption edge of the element, in which case the atomic scattering factor is real and Filiki) = Fthkl V- Since the diffracted... 

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

Herein denotes fj the atomic scattering factor of atom j in the unit cell, the Xy jy Zj are the corresponding fractional coordinates of the atom j and the hkl are the Miller) indices of the Fourier component (see below). If the structure is... 

In direct methods calculations we use normalised structure factors E(hkl), which are the structure factors compensated for the fall-off of the atomic scattering factors f hkl). In fact this procedure tries to simulate point-like scattering centres. 

The range of values of the atomic scattering factors for electrons is less than that of X-rays. The Patterson method works optimally when... 

The /, are the atomic scattering factors of the atoms of type /. These depend npon both and. The phases may canse the waves to add, as in for example Si 004, or cancel, as in for example Si 002. The larger the stracture factor, the broader is the rocking cnrve. 

The atomic scattering factors /, are ustrally calculated in terms of the scattering of an individtral free electron. This is calculated as if the electron were a classical oscillator—since the assrrmption is that the electron is a free charged... 

Figure 4.2 The variation of the atomic scattering factor/, with scattering angle 2. Values for silicon (Z=14) and germanium (Z=32) are shown...

Figure 4.4 The real (scattering) and imaginary (absorption) parts of the atomic scattering factor for gallium and arsenic near the K absorption edges. After Cockerton etal. ...

Including resonance effects, the atomic scattering factor for a many-electron atom is written as... 

Here, ac, the real component of <5, is related to the atomic scattering factors / by... 

When the atomic scattering factor is real (as it is when bonding effects on the charge density are neglected), and resonance scattering has been corrected for, the harmonic structure factor expression is equal to... 

As first shown by Dawson (1967), Eq. (11.3) can be generalized by inclusion of anharmonicity of the thermal motion, which becomes pronounced at higher temperatures. We express the anharmonic temperature factor of the diamond-type structure [Chapter 2, Eq. (2.45)] as 71(H) = TC(H) -f iX(H), in analogy with the description of the atomic scattering factors. Incorporation of the temperature... 

Equation (B. 11) implies that /(H ) = /(H), that is, the rotational symmetry of the space group, is repeated in the diffraction pattern. In addition, if the atomic scattering factors / are real, which is the case when resonance effects are negligible, a center of symmetry is added to the diffraction pattern, that is, /(H) = F(H) F (H) = /( —H) even in the absence of an inversion center, which is Friedel s law. 

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