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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*)| =
; this result is sometimes known as Friedel's law.^{1}

As long ago as 1930, Coster, Knol and Prins^{2} 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. Peterson^{42} 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 Pepinsky^{15} 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

The method has been analyzed by Ramaseshan,^{44} applied by McDonald and
Sikka^{45} to solve the crystal structure of cadmium nitrate tetradeuterate
and further extended by Sikka^{46-48} and others.

(b) Certain nuclei, notably Fe^{57} and Sn^{119} 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 10^{9}. 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 is large in comparison with the resonance frequencies of the atom (, 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.

Element | K absorption edge (Å) |
K |
K |

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 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 radiation
or Cr atoms with Fe, etc. Obviously in these cases the calculation of
the atomic scattering factor *f _{0}* without resonance effects is no longer valid
since is comparable to .

It is then necessary to study in what way the values of *f _{0}* will be altered to

- (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 approach
^{36}used semi-empirical relations for the photoelectric absorption cross-section from which*f*' and*f*'' are obtained. - (c) Cromer and Liberman
^{35}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**