International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 2.3, pp. 246-248   | 1 | 2 |

## Section 2.3.4. Anomalous dispersion

M. G. Rossmanna* and E. Arnoldb

aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and  bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail:  mgr@indiana.bio.purdue.edu

### 2.3.4. Anomalous dispersion

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#### 2.3.4.1. Introduction

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The physical basis for anomalous dispersion has been well reviewed by Ramaseshan & Abrahams (1975), James (1965), Cromer (1974) and Bijvoet (1954). As the wavelength of radiation approaches the absorption edge of a particular element, then an atom will disperse X-rays in a manner that can be defined by the complex scattering factor where is the scattering factor of the atom without the anomalous absorption and re-scattering effect, is the real correction term (usually negative), and is the imaginary component. The real term is often written as f′, so that the total scattering factor will be . Values of and are tabulated in IT IV (Cromer, 1974), although their precise values are dependent on the environment of the anomalous scatterer. Unlike , and are almost independent of scattering angle as they are caused by absorption of energy in the innermost electron shells. Thus, the anomalous effect resembles scattering from a point atom.

The structure factor of index h can now be written as (Note that the only significant contributions to the second term are from those atoms that have a measurable anomalous effect at the chosen wavelength.)

Let us now write the first term as and the second as . Then, from (2.3.4.1), Therefore, and similarly demonstrating that Friedel's law breaks down in the presence of anomalous dispersion. However, it is only for noncentrosymmetric reflections that .

Now, Hence, by using (2.3.4.2) and simplifying, The first term in (2.3.4.3) is the usual real Fourier expression for electron density, while the second term is an imaginary component due to the anomalous scattering of a few atoms in the cell.

#### 2.3.4.2. The function

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Expression (2.3.4.3) gives the complex electron density expression in the presence of anomalous scatterers. A variety of Patterson-type functions can be derived from (2.3.4.3) for the determination of a structure. For instance, the function gives vectors between the anomalous atoms and the normal' atoms.

From (2.3.4.1) it is easy to show that Therefore, and

Let us now consider the significance of a Patterson in the presence of anomalous dispersion. The normal Patterson definition is given by where and

The component is essentially the normal Patterson, in which the peak heights have been very slightly modified by the anomalous-scattering effect. That is, the peaks of are proportional in height to .

The component is more interesting. It represents vectors between the normal atoms in the unit cell and the anomalous scatterers, proportional in height to (Okaya et al., 1955). This function is antisymmetric with respect to the change of the direction of the diffraction vector. An illustration of the function is given in Fig. 2.3.4.1. In a unit cell containing N atoms, n of which are anomalous scatterers, the function contains only positive peaks and an equal number of negative peaks related to the former by anticentrosymmetry. The analysis of a synthesis presents problems somewhat similar to those posed by a normal Patterson. The procedure has been used only rarely [cf. Moncrief & Lipscomb (1966) and Pepinsky et al. (1957)], probably because alternative procedures are available for small compounds and the solution of is too complex for large biological molecules.

 Figure 2.3.4.1 | top | pdf |(a) A model structure with an anomalous scatterer at A. (b) The corresponding function showing positive peaks (full lines) and negative peaks (dashed lines). [Reprinted with permission from Woolfson (1970, p. 293).]

#### 2.3.4.3. The position of anomalous scatterers

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Anomalous scatterers can be used as an aid to phasing, when their positions are known. But the anomalous-dispersion differences (Bijvoet differences) can also be used to determine or confirm the heavy atoms which scatter anomalously (Rossmann, 1961a). Furthermore, the use of anomalous-dispersion information obviates the lack of isomorphism but, on the other hand, the differences are normally far smaller than those produced by a heavy-atom isomorphous replacement.

Consider a structure of many light atoms giving rise to the structure factor . In addition, it contains a few heavy atoms which have a significant anomalous-scattering effect. The non-anomalous component will be and the anomalous component is (Fig. 2.3.4.2a). The total structure factor will be . The Friedel opposite is constructed appropriately (Fig. 2.3.4.2a). Now reflect the Friedel opposite construction across the real axis of the Argand diagram (Fig. 2.3.4.2b). Let the difference in phase between and be ϕ. Thus but since ϕ is very small Hence, a Patterson with coefficients will be equivalent to a Patterson with coefficients which is proportional to . Such a Patterson (Rossmann, 1961a) will have vectors between all anomalous scatterers with heights proportional to the number of anomalous electrons . This anomalous dispersion' Patterson function has been used to find anomalous scatterers such as iron (Smith et al., 1983; Strahs & Kraut, 1968) and sulfur atoms (Hendrickson & Teeter, 1981).

 Figure 2.3.4.2 | top | pdf |Anomalous-dispersion effect for a molecule whose light atoms contribute and heavy atom with a small anomalous component of , ahead of the non-anomalous component. In (a) is seen the construction for and . In (b) has been reflected across the real axis.

It is then apparent that a Patterson with coefficients (Rossmann, 1961a), as well as a Patterson with coefficients (Rossmann, 1960; Blow, 1958), represent Pattersons of the heavy atoms. The Patterson suffers from errors which may be larger than the size of the Bijvoet differences, while the Patterson may suffer from partial lack of isomorphism. Hence, Kartha & Parthasarathy (1965) have suggested the use of the sum of these two Pattersons, which would then have coefficients .

However, given both SIR and anomalous-dispersion data, it is possible to make an accurate estimate of the magnitudes for use in a Patterson calculation [Blundell & Johnson (1976, p. 340), Matthews (1966), Singh & Ramaseshan (1966)]. In essence, the Harker phase diagram can be constructed out of three circles: the native amplitude and each of the two isomorphous Bijvoet differences, giving three circles in all (Blow & Rossmann, 1961) which should intersect at a single point thus resolving the ambiguity in the SIR data and the anomalous-dispersion data. Furthermore, the phase ambiguities are orthogonal; thus the two data sets are cooperative. It can be shown (Matthews, 1966; North, 1965) that where and . The sign in the third-term expression is − when or + otherwise. Since, in general, is small compared to , it is reasonable to assume that the sign above is usually negative. Hence, the heavy-atom lower estimate (HLE) is usually written as which is an expression frequently used to derive Patterson coefficients useful in the determination of heavy-atom positions when both SIR and anomalous-dispersion data are available.

### References

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Blow, D. M. & Rossmann, M. G. (1961). The single isomorphous replacement method. Acta Cryst. 14, 1195–1202.
Blundell, T. L. & Johnson, L. N. (1976). Protein crystallography. New York: Academic Press.
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