Atomic number diffracting power

The following discussion relates chiefly to complex crystals in w hich all the atoms have much the same diffracting powers—crystals such as those of many organic compounds for it is in these circumstances chat the indirect method of trial and error must often be used. For crystals containing a minority of heavy atoms together with a larger number of lighter atoms in the unit cell, the direct or semi-direct methods described in Chapter X are more appropriate. 

In the Fourier series method the weighted radial distribution func -tion, which represents the number of atoms at a distance r from any atom, weighted by the products of the diffracting powers, is given by (he expression... 

Recall that X rays are diffracted by the electrons that surround atoms, and that images obtained from X-ray diffraction show the surface of the electron clouds that surround molecules. Recall also that the X-ray diffracting power of elements in a sample increases with increasing atomic number. Neutrons are diffracted by nuclei, not by electrons. Thus a density map computed from neutron diffraction data is not an electon-density map, but instead a map of nuclear mass distribution, a "nucleon-density map" of the molecule (nucleons are the protons and neutrons in atomic nuclei). 

Ignoring for the present ionization and contraction of the lattice, the structure of cesium chloride may be considered similar to that of cesium metal, but with the cesium atoms removed from the body centers and chloride ions inserted. In the diffraction pattern for cesium chloride, the 100 reflections, 111 reflections, and other reflections absent from the pattern for cesium metal are present but are weak. Wave interference similar to that occuring for cesium metal must occur, but here interference is not complete. The planes of chloride ions are not as strong reflectors or scatterers as the planes of cesium ions (the scattering power of an atom rises sharply with atomic number). Thus, interference in cesium... 

For many years it was believed that hydrogen atoms could not be seen" in the electron density maps produced by X-ray diffraction. The reason for this is that the atomic X-ray scattering power is proportional to the square of the atomic number. This statement was generally, but not invariably, true until the demise of film-recording methods in the mid-1960 s and the advent of the computer-controlled X-ray diffractometers which could provide very accurate X-ray diffraction intensities. 

It is wise at this point to introduce one additional refinement into the expression for the diffraction pattern that further expands its application. Above, we made the tacit assumption that all points scattered with the same intensity for instance, they all represented atoms that had the same number of electrons. This, of course, is unrealistic, for most molecules are composed of different kinds of atoms. We can easily correct for this assumption, however, by making the amplitude of the wave scattered by each point equal or proportional to its scattering power, its atomic number Zj. In doing this, we rewrite the Fourier transform as... 

To take into account the disparity in electrons, and hence scatting power, for the various atoms in the unit cell, the atomic number Zj has been introduced as a means of defining amplitudes for the component waves. The total diffracted wave for the entire crystal will be the product of the equation above with the total number of unit cells in the crystal.1 For a single unit cell, like that in Figures 5.12a and 5.12b, an atom s contribution to the total structure factor of one unit cell can also be illustrated in vector terms as in Figure 5.13. To put the expression in the correct units, it is necessary to multiply the summation by a constant, the volume of the unit cell. Here, that is simply V = a x b x c. Thus... 

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. 

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