Tables for
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:

2.3.4. Anomalous dispersion

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The physical basis for anomalous dispersion has been well reviewed by Ramaseshan & Abrahams (1975)[link], James (1965)[link], Cromer (1974)[link] and Bijvoet (1954)[link]. 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 [f_{0} + \Delta f' + i\Delta f'',] where [f_{0}] is the scattering factor of the atom without the anomalous absorption and re-scattering effect, [\Delta f'] is the real correction term (usually negative), and [\Delta f''] is the imaginary component. The real term [f_{0} + \Delta f'] is often written as f′, so that the total scattering factor will be [f' + if'']. Values of [\Delta f'] and [\Delta f''] are tabulated in IT IV (Cromer, 1974[link]), although their precise values are dependent on the environment of the anomalous scatterer. Unlike [f_{0}], [\Delta f'] and [\Delta f''] 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 [{\bf F}_{{\bf h}} = {\textstyle\sum\limits_{j = 1}^{N}}\; f'_{j} \exp (2\pi i{\bf h}\cdot {\bf x}_{j}) + i {\textstyle\sum\limits_{j = 1}^{N}}\; f''_{j} \exp (2\pi i{\bf h}\cdot {\bf x}_{j}). \eqno(] (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 [A + iB] and the second as [a + ib]. Then, from (,[link] [{\bf F} = (A + iB) + i(a + ib) = (A - b) + i(B + a). \eqno(] Therefore, [|{\bf F}_{{\bf h}}|^{2} = (A - b)^{2} + (B + a)^{2}] and similarly [|{\bf F}_{{{\bar {\bf h}}}}|^{2} = (A + b)^{2} + (- B + a)^{2},] demonstrating that Friedel's law breaks down in the presence of anomalous dispersion. However, it is only for noncentrosymmetric reflections that [|{\bf F}_{{\bf h}}| \neq |{\bf F}_{{\bar{\bf h}}}|].

Now, [\rho ({\bf x}) = {1 \over V} {\sum\limits_{{\bf h}}^{\rm sphere}} {\bf F}_{{\bf h}} \exp (2\pi i{\bf h}\cdot {\bf x}).] Hence, by using ([link] and simplifying, [\eqalignno{\rho ({\bf x}) = &{2 \over V} {\sum\limits_{{\bf h}}^{\rm hemisphere}} [(A \cos 2\pi {\bf h}\cdot {\bf x} - B \sin 2\pi {\bf h}\cdot {\bf x})\cr &+ i(a \cos 2\pi {\bf h}\cdot {\bf x} - b \sin 2\pi {\bf h}\cdot {\bf x})]. &(}] The first term in ([link] 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. The [P_s({\bf u})] function

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Expression ([link] gives the complex electron density expression in the presence of anomalous scatterers. A variety of Patterson-type functions can be derived from ([link] for the determination of a structure. For instance, the [P_{s} ({\bf u})] function gives vectors between the anomalous atoms and the `normal' atoms.

From ([link] it is easy to show that [\eqalign{{\bf F}_{{\bf h}} {\bf F}_{{\bf h}}^{*} = &\ |{\bf F}_{{\bf h}}|^{2}\cr = &\ {\textstyle\sum\limits_{i, \,  j}} (f'_{i} f'_{j} + f''_{i} f''_{j}) \cos 2\pi {\bf h}\cdot ({\bf x}_{i} - {\bf x}_{j})\cr &+ {\textstyle\sum\limits_{i, \,  j}} (f'_{i} f''_{j} - f''_{i} f'_{j}) \sin 2\pi {\bf h}\cdot ({\bf x}_{i} - {\bf x}_{j}).}] Therefore, [|{\bf F}_{{\bf h}}|^{2} + |{\bf F}_{{\bar{\bf h}}}|^{2} = 2 {\textstyle\sum\limits_{i, \,  j}} (f'_{i} f'_{j} + f''_{i} f''_{j}) \cos 2\pi {\bf h}\cdot ({\bf x}_{i} - {\bf x}_{j})] and [|{\bf F}_{{\bf h}}|^{2} - |{\bf F}_{{\bar{\bf h}}}|^{2} = 2 {\textstyle\sum\limits_{i, \,  j}} (f'_{i} f''_{j} - f''_{i} f'_{j}) \sin 2\pi {\bf h}\cdot ({\bf x}_{i} - {\bf x}_{j}).]

Let us now consider the significance of a Patterson in the presence of anomalous dispersion. The normal Patterson definition is given by [\eqalign{ P({\bf u}) &= {\textstyle\int\limits_{V}} \rho^{*} ({\bf x}) \rho ({\bf x} + {\bf u})\;\hbox{d}{\bf x}\cr &= {1 \over V^{2}} {\sum\limits_{{\bf h}}^{\rm sphere}} |{\bf F}_{{\bf h}}|^{2} \exp (-2 \pi i {\bf h} \cdot {\bf u})\cr &\equiv P_{c} ({\bf u}) - iP_{s} ({\bf u}),}] where [P_{c} ({\bf u}) = {2 \over V} {\sum\limits_{{\bf h}}^{\rm hemisphere}} (|{\bf F}_{{\bf h}}|^{2} + |{\bf F}_{{\bar{\bf h}}}|^{2}) \cos 2 \pi {\bf h} \cdot {\bf u}] and [P_{s} ({\bf u}) = {2 \over V} {\sum\limits_{{\bf h}}^{\rm hemisphere}} (|{\bf F}_{{\bf h}}|^{2} - |{\bf F}_{{\bar{\bf h}}}|^{2}) \sin 2 \pi {\bf h} \cdot {\bf u}.]

The [P_{c}({\bf u})] 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 [P_{c}({\bf u})] are proportional in height to [(f'_{i} f'_{j} + f''_{i} f''_{j})].

The [P_{s} ({\bf u})] component is more interesting. It represents vectors between the normal atoms in the unit cell and the anomalous scatterers, proportional in height to [(f'_{i} f''_{j} - f''_{i} f'_{j})] (Okaya et al., 1955[link]). 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.[link]. In a unit cell containing N atoms, n of which are anomalous scatterers, the [P_{s}({\bf u})] function contains only [n(N - n)] positive peaks and an equal number of negative peaks related to the former by anticentrosymmetry. The analysis of a [P_{s}({\bf u})] synthesis presents problems somewhat similar to those posed by a normal Patterson. The procedure has been used only rarely [cf. Moncrief & Lipscomb (1966)[link] and Pepinsky et al. (1957)[link]], probably because alternative procedures are available for small compounds and the solution of [P_{s}({\bf u})] is too complex for large biological molecules.


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(a) A model structure with an anomalous scatterer at A. (b) The corresponding [P_{s}({\bf u})] function showing positive peaks (full lines) and negative peaks (dashed lines). [Reprinted with permission from Woolfson (1970, p. 293)[link].] 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[link]). 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 [{\bf F}_{{\bf h}} (N)]. In addition, it contains a few heavy atoms which have a significant anomalous-scattering effect. The non-anomalous component will be [{\bf F}_{{\bf h}} (H)] and the anomalous component is [{\bf F}''_{{\bf h}} (H) = i(\Delta f''/ f') {\bf F}_{{\bf h}} (H)] (Fig.[link]). The total structure factor will be [{\bf F}_{{\bf h}}]. The Friedel opposite is constructed appropriately (Fig.[link]). Now reflect the Friedel opposite construction across the real axis of the Argand diagram (Fig.[link]). Let the difference in phase between [{\bf F}_{{\bf h}}] and [{\bf F}_{{\bar{\bf h}}}] be ϕ. Thus [4 |{\bf F}''_{\bf h} (H)|^{2} = |{\bf F}_{\bf h}|^{2} + |{\bf F}_{{\bar{\bf h}}}|^{2} - 2|{\bf F}_{\bf h}| |{\bf F}_{{\bar{\bf h}}}| \cos \varphi,] but since ϕ is very small [|{\bf F}''_{\bf h} (H)|^{2} \simeq {\textstyle{1 \over 4}} (|{\bf F}_{\bf h}| - |{\bf F}_{{\bar{\bf h}}}|)^{2}.] Hence, a Patterson with coefficients [(|{\bf F}_{\bf h}| - |{\bf F}_{{\bar{\bf h}}}|)^{2}] will be equivalent to a Patterson with coefficients [|{\bf F}''_{\bf h} (H)|^{2}] which is proportional to [|{\bf F}_{\bf h} (H)|^{2}]. Such a Patterson (Rossmann, 1961a[link]) will have vectors between all anomalous scatterers with heights proportional to the number of anomalous electrons [\Delta f'']. This `anomalous dispersion' Patterson function has been used to find anomalous scatterers such as iron (Smith et al., 1983[link]; Strahs & Kraut, 1968[link]) and sulfur atoms (Hendrickson & Teeter, 1981[link]).


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Anomalous-dispersion effect for a molecule whose light atoms contribute [{\bf F}_{\bf h}(N)] and heavy atom [{\bf F}_{\bf h}(H)] with a small anomalous component of [{\bf F_h''}(H)], [90^{\circ}] ahead of the non-anomalous [{\bf F}_{\bf h}(H)] component. In (a) is seen the construction for [{\bf F}_{\bf h}] and [{\bf F}_{\bar{\bf h}}]. In (b) [{\bf F}_{\bar {\bf h}}] has been reflected across the real axis.

It is then apparent that a Patterson with coefficients [\Delta F_{\rm ANO}^{2} = (|{\bf F}_{\bf h}| - |{\bf F}_{{\bar{\bf h}}}|)^{2}] (Rossmann, 1961a[link]), as well as a Patterson with coefficients [\Delta F_{\rm ISO}^{2} = (|{\bf F}_{NH}| - |{\bf F}_{H}|)^{2}] (Rossmann, 1960[link]; Blow, 1958[link]), represent Pattersons of the heavy atoms. The [\Delta F_{\rm ANO}^{2}] Patterson suffers from errors which may be larger than the size of the Bijvoet differences, while the [\Delta F_{\rm ISO}^{2}] Patterson may suffer from partial lack of isomorphism. Hence, Kartha & Parthasarathy (1965)[link] have suggested the use of the sum of these two Pattersons, which would then have coefficients [(\Delta F_{\rm ANO}^{2} + \Delta F_{\rm ISO}^{2})].

However, given both SIR and anomalous-dispersion data, it is possible to make an accurate estimate of the [|{\bf F}_{H}|^{2}] magnitudes for use in a Patterson calculation [Blundell & Johnson (1976, p. 340)[link], Matthews (1966)[link], Singh & Ramaseshan (1966)[link]]. 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[link]) 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[link]; North, 1965[link]) that [F^{2}_{N} = F^{2}_{NH} + F^{2}_{N} \mp {2 \over k} (16k^{2} F^{2}_{P} F^{2}_{H} - \Delta I^{2})^{1/2},] where [\Delta I = {F^{+}_{NH}}^{2} - {F^{-}_{NH}}^{2}] and [k = \Delta f''/f']. The sign in the third-term expression is − when [|(\alpha_{NH} - \alpha_{H})|  \lt  \pi/2] or + otherwise. Since, in general, [|{\bf F}_{H}|] is small compared to [|{\bf F}_{N}|], it is reasonable to assume that the sign above is usually negative. Hence, the heavy-atom lower estimate (HLE) is usually written as [F^{2}_{\rm HLE} = F^{2}_{NH} + F^{2}_{H} - {2 \over k} (16 k^{2} F^{2}_{P} F^{2}_{H} - \Delta I^{2})^{1/2},] 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.


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