Scattering by an atom

X-ray diffraction. The mechanism by which atoms diffract or scatter electromagnetic radiation via the coupling of the electron cloud of the atom to the incident oscillating electric field was discussed in the section on SERS. The X-rays scattered by an atom are the resultant of the waves... 

For typical value of s 0.2 1/A, the ratio f l(e lmc f Thus electrons are scattered by an atom much more strongly than X-ray. 

If k is in atomic units ap1. the differential cross section is in units per steradian. Differential and total cross sections for multichannel scattering by an atomic target can be derived from general formulas [27, 184], The partial cross section for scattering from channel q to channel p is... 

The next step is the scattering by an atom. This effect is basically the addition of the scattering of the electron cloud around the nucleus, since each electron in the atom scatters part of the incident radiation in a coherent form in agreement with the Thomson equation. Owing to the fact that the electrons in an atom are located at different points within the atom, and the fact that the x-ray wavelength is of the same order as the atomic dimensions, there will be path differences between waves scattered by different electrons if these path differences are less than one wavelength, then the interference will be partially destructive [20,22,26], To describe this effect, the parameter/is defined, also called the atomic scattering factor, which is the ratio of the amplitude scattered by an atom, Aa, to the amplitude scattered by an electron, Ae, that is [21]... 

To obtain the amplitude scattered by an atom contmnmz a single electron (whose charge is spread diffusely throughout the volume of the atom), we must integrate Eq. (3.29) over the volume of the atom. Hence,... 

Electrons scatter x-rays. The amplitude of the wave scattered by an atom is proportional to its number of electrons. Thus, a carbon atom scatters six times as strongly as a hydrogen atom does. 

In 1951, Johannes Martin Bijvoet used such differences in intensity, resulting from anomalous scattering by an atom in a noncentrosymmetric crystal, to determine the chirality (absolute configuration) of the tartrate ion. Details of this method, which has been used extensively for finding the absolute configurations of natural products and for determining macromolecular structures, are given in Chapter 14. 

The amplitude of the scattered beam is therefore, a gradually decaying function of the scattered angle and it varies with cp and with 0. The intrinsic angular dependence of the x-ray amplitude scattered by an atom is called the atomic scattering function (factor),/ and its behavior is shown schematically in Figure 2.25, left as a function of the phase angle. 

Figure 2.25. The schematic showing the dependence of the intensity scattered by an atom, i.e. the atomic scattering factor,/

The working vector v having unit length and which is normal to the planes can be drawn. It passes not only through this unit cell but identically through all unit cells in the crystal. We use this unitary vector simply to define a direction. The phase of a wave scattered by an atom with respect to the set of planes will be 0 = 2ji(D/dhia), where D is the atom s distance from the nearest plane. But D is the projection of 3ri onto i>, that is, l) = x Ft, so that 0 = 2jt(xi v/dhu)-... 

To this point we have assumed that an atom, be it heavy or otherwise, scatters as a point source of scattering power fj having phase 0j. Although the detailed physical explanation is outside the scope of this book and involves quantum mechanical properties, it must be pointed out that this is not entirely true. An atom scatters X rays in a somewhat more complex fashion, in that its scattered radiation is composed of two components. The major component, which arises from normal Thompson scattering, and is by far the largest component, has phase 0 dependent on the atom s position as we have assumed. But there is also a minor component of the scattering that has phase 0 + jt/2. This is because the electrons of the atom also absorb a small amount of radiation due to electron resonance phenomena and re-emit it with a phase change. This second component is called the anomalous dispersion, and to be entirely correct, we should properly describe the radiation scattered by an atom as a complex number,... 

The following question then arises is the wave scattered by an atom simply the sum of the waves scattered by its component electrons More precisely, does an atom of atomic number Z, i.e., an atom containing Z electrons, scatter a wave... 

This problem is most simply approached by finding the phase difference between waves scattered by an atom at the origin and another atom whose position is variable in the jc direction only. For convenience, consider an orthogonal unit cell, a section of which is shown in Fig. 4-8. Take atom A as the origin and let diffraction occur from the (AGO) planes shown as heavy lines in the drawing. This means that the Bragg law is satisfied for this reflection and that the path difference between ray 2 and ray 1, is given by... 

The amplitude (i.e. the strength ) of the diffracted radiation scattered by an atom in a plane (hkl) will be given by the value of fa appropriate to the correct value of sin9/X (=1/2dhlci) for the (hkl) plane. However, because the scattering atoms are at various locations in the unit cell, the waves scattered by each atom are out of step with each other as they leave the unit cell. The difference by which the waves are out of step is called the phase difference between the waves. 

Figure 6.11 The representation of scattered waves as vectors (a) a scattered wave vector in the x, y plane (b) the wave scattered by an atom A at (x, y, z) (c) the wave scattered by atom A at the origin, (000) (d) the addition of waves scattered by five atoms, A, B, C, D, and E (e) represention of fA as a complex number on an Argand diagram...

This terminology allows the scattering of X-rays by atoms in a unit cell to be added algebraically, by writing the scattering by an atom in a unit cell as a complex amplitude f ... 

To understand how a diffraction pattern may be generated, consider the scattering of X rays by atoms in two parallel planes (Figure 11.24). Initially, the two incident rays are in phase with each other (their maxima and minima occur at the same positions). The upper wave is scattered, or reflected, by an atom in the first layer, while the lower wave is scattered by an atom in the second layer. In order for these two scattered waves to be in phase again, the extra distance traveled by the lower wave must be an integral multiple of the wavelength (A) of the X ray that is,... 

---