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Introduction

The quantity usually measured in relation to each X-ray reflection is the intensity, which is proportional to |F(hkl)²| and hence it is |F(hkl)| that is determined experimentally. This quantity may be called the `geometrical structure factor' since it depends only on the positions of atoms and not on any differences in their scattering behaviour. If the nature of the scattering, including any phase change, is identical for all atoms, then |F(hkl)| = $\vert F(\=h\=k\=l)\vert$; this result is sometimes known as Friedel's law.¹

As long ago as 1930, Coster, Knol and Prins² performed a most elegant experiment with zinc blende, using X-ray wavelengths selected to lie close to an absorption edge for zinc, but not for sulphur. They were able to demonstrate a failure of Friedel's law and to show that circumstances arise in practice in which the phase change produced by each atom in a unit cell is not the same. The different resonance that leads to this effect has become known as anomalous dispersion.

As radiation counters have been steadily improved, measurements of X-ray intensities have attained such a degree of accuracy that it is no longer acceptable to neglect the resonant effects which are bigger than the experimental errors by several orders of magnitude. As a matter of fact the effect can be detected also with film techniques in non-centrosymmetrical crystals by measuring the integrated intensities of symmetry-equivalent reflexions when the incident wavelength is adequately selected. Friedel's law does not hold in these cases, a phenomenon which was first used by Bijvoet and collaborators^3-13 to find the absolute configuration of some crystals. The finding of methods to solve crystal structures directly by using this phenomenon by Pepinsky and collaborators,^14-19 Ramachandran and Raman,^20-23 Caticha-Ellis^24-27 prompted a keen interest in this field.

Let us state from the outset that the usual name of `anomalous dispersion' given to the effects studied in this article is entirely inadequate and misleading as it will soon be evident. However, I have kept it for convenience, since it has been used for years in the scientific literature, to avoid unnecessary confusions. It would certainly be more exact to rename the subject as `resonance effects in the scattering of radiation', or directly `resonance scattering of radiation' adding the qualifications nuclear or electronic according to the case. In the case of X-radiation there is always some resonant effect due to the continuous distribution of oscillator levels as we shall see below, so that the so-called normal scattering or non-resonant scattering is not normally found. The paradox then, is that `anomalous scattering' is absolutely normal while `normal scattering' occurs only as an ideal, oversimplified model, which can be used as a first approximation when studying scattering problems.

Calculations of the dispersion contribution to the atomic scattering factors made by several authors,^29-35 based on a method due to Parratt and Hempstead,³⁶ were used by many researchers to find the absolute structure of several crystals, as well as to completely solve many other crystal structures of increasing complexity including the solution of proteins.

Experimental determinations of a few dispersion corrections were performed by some authors in order to confirm the theoretical values and to investigate their dependence on the scattering angle and on the presence of more than one anomalous scatterer in the unit cell.^37-41

In an early paper, S. W. Peterson⁴² was able to measure differences in the intensities of symmetry equivalent reflexions in tyrosine hydrochloride and tyrosine hydrobromide. From measurements made on other crystals he established experimentally that this phenomenon was related to the presence of a heavy atom and of a polar axis. Peterson realized clearly the significance of these findings for direct structure determination by means of Fourier synthesis in non-centrosymmetrical crystals and in the same paper claims to have solved the crystal structure of both tyrosine hydrochloride and hydrobromide using only the information contained in the measurements themselves. It is interesting to note that the paper by Okaya, Saito and Pepinsky¹⁵ is dated only 23 days before that of Peterson.⁴²

In this article only the field of X-ray resonant scattering is reviewed. We shall mention here only in passing two other related resonant effects:

• (a) Neutron resonant scattering
• (b) Resonant scattering of gamma rays (Mossbauer effect).
• (a) Resonant scattering of thermal neutrons produces in some elements

appreciable in-quadrature components of the scattering factors for neutron wavelengths of about 1 Å. Peterson and Smith⁴³ have shown the possibility of the crystallographic use of the phenomenon for substances containing such light elements as Li⁶ and B¹⁰ that can hardly produce any sizeable resonant effect with X-rays. Also some heavy elements such as Cd¹¹³, Sm¹⁴⁹, Eu¹⁵¹ and Gd¹⁵⁷ have appreciable in-quadrature components in the 1 Å region. Peterson and Smith have also observed that the number of elements with high in-quadrature components would be greatly enlarged by using neutrons of about 0.1 Å. The practical difficulty being of course the fact that the neutron flux in this region is very low. A hot source built in the reactor might be used to produce the required shift to higher thermal energies in the Maxwellian distribution.

The method has been analyzed by Ramaseshan,⁴⁴ applied by McDonald and Sikka⁴⁵ to solve the crystal structure of cadmium nitrate tetradeuterate and further extended by Sikka^46-48 and others.

(b) Certain nuclei, notably Fe⁵⁷ and Sn¹¹⁹ and also some rare earths have nuclear resonance levels that can absorb and emit gamma radiations with wavelengths in the useful range for crystallographic uses. The use of the emitted lines would have some definite advantages due to the fact that their widths are much smaller than those of the X-ray emission lines by a factor of the order of 10⁹. The main disadvantage is the very low intensity of the Mossbauer radiation in comparison with X-rays, a complicating factor from the point of view of the detectability and the statistics of the measurements. However, in some experiments where the sharpness of the line is essential, great advantage could be obtained from the use of such radiation.

In the list of references some papers on this subject have been included.^49-60

It is curious that in spite of the widespread use of anomalous scattering, no book had been written on this subject until 1974. The proceedings of an Inter-Congress Conference organized by the Commission on Crystallographic Apparatus of the International Union of Crystallography held in Madrid in that year thus became an all important reference which covers most of the crystallographic aspects of anomalous scattering.⁷⁵

In this chapter we review briefly the necessary theory to understand the origin of the dispersion effects and the basis for their calculations. The treatment is based on the four chapter of James' book: The Optical Principles of the Diffraction of X-Rays ,⁶¹ which contains a detailed account of the subject.

In the classical approach to calculate the (normal) atomic scattering factors the hypothesis is made that the frequency of the incident wave $\omega_i$ is large in comparison with the resonance frequencies of the atom ($\omega_K, \omega_L$, etc., where the subindex refers to the electron shell), that is, those associated to the absorption edges. In practice, for the wavelengths normally used in crystallographic studies, this hypothesis can be approximately fulfilled only in the case of light atoms, but it is not generally true in most real cases. This is readily seen by comparing the values of the atomic absorption edges with those of the characteristic wavelengths of anticathodes normally used.

 

$$\alpha \beta$$ Element K absorption edge (Å) wavelengths (Å)

3-Li 226.6 240.0 -

4-C 43.6 44.0 -

11-Na 11.48 11.91 11.62

19-K 3.436 3.744 3.454

23-V 2.269 2.505 2.284

24-Cr 2.070 2.291 2.085

26-Fe 1.743 1.937 1.756

27-Co 1.608 1.790 1.607

28-Ni 1.488 1.659 1.500

29-Cu 1.380 1.5418 1.392

42-Mo 0.6197 0.7106 0.6322

55-Cs 0.345 0.402 0.354

57-La 0.318 0.372 0.328

In Table 1 we quote some values of the K absorption edges and of the emitted K wavelengths for some elements. It is obvious from Table 1 that, for instance, if one has iron or cobalt atoms in a sample, Cu$K\alpha$ radiation will produce strong resonance effects since this wavelength is slightly smaller than the K-absorption edges of these elements and consequently heavily absorbed by them. This is also the case for V atoms with Cr$K\beta$ radiation or Cr atoms with Fe$K\alpha$, etc. Obviously in these cases the calculation of the atomic scattering factor f₀ without resonance effects is no longer valid since $\omega_i$ is comparable to $\omega_K$.

It is then necessary to study in what way the values of f₀ will be altered to f by resonance. Three main different approaches have been used to calculate the dispersion corrections:

• (a) Hönl^62-64 used hydrogen-like eigenfunctions to obtain the oscillator strengths and from them the photoelectric absorption cross-sections. Hönl's method, restricted to the K-electrons contribution, was extended by other authors.
• (b) Parratt and Hempstead's approach³⁶ used semi-empirical relations for the photoelectric absorption cross-section from which f' and f'' are obtained.
• (c) Cromer and Liberman³⁵ used relativistic Slater-Dirac wave functions.

Recently these methods were reviewed by Wagenfeld.¹⁰⁶

There is a close parallelism between the classical and the quantum treatment; we give next a scheme of James' classical treatment.

Copyright © 1981, 1998 International Union of Crystallography