Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 2.4, pp. 265-269   | 1 | 2 |

Section 2.4.3. Anomalous-scattering method

M. Vijayana* and S. Ramaseshanb

aMolecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India, and bRaman Research Institute, Bangalore 560 080, India
Correspondence e-mail:

2.4.3. Anomalous-scattering method

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Atomic scattering factors are normally calculated on the assumption that the binding energy of the electrons in an atom is negligible compared to the energy of the incident X-rays and the distribution of electrons is spherically symmetric. The transition frequencies within the atom are then negligibly small compared to the frequency of the radiation used and the scattering power of each electron in the atom is close to that of a free electron. When this assumption is valid, the atomic scattering factor is a real positive number and its value decreases as the scattering angle increases because of the finite size of the atom. When the binding energy of the electrons is appreciable, the atomic scattering factor at any given angle is given by [f_{0} + f' + if'', \eqno(] where [f_{0}] is a real positive number and corresponds to the atomic scattering factor for a spherically symmetric collection of free electrons in the atom. The second and third terms are, respectively, referred to as the real and the imaginary components of the `dispersion correction' (IT IV, 1974[link]). f′ is usually negative whereas f″ is positive. For any given atom, f″ is obviously [90^{\circ}] ahead of the real part of the scattering factor given by [f = f_{0} + f'. \eqno(]

The variation of f′ and f″ as a function of atomic number for two typical radiations is given in Fig.[link] (Srinivasan, 1972[link]; Cromer, 1965[link]). The dispersion effects are pronounced when an absorption edge of the atom concerned is in the neighbourhood of the wavelength of the incident radiation. Atoms with high atomic numbers have several absorption edges and the dispersion-correction terms in their scattering factors always have appreciable values. The values of f′ and f″ do not vary appreciably with the angle of scattering as they are caused by core electrons confined to a very small volume around the nucleus. An atom is usually referred to as an anomalous scatterer if the dispersion-correction terms in its scattering factor have appreciable values. The effects on the structure factors or intensities of Bragg reflections resulting from dispersion corrections are referred to as anomalous-dispersion effects or anomalous-scattering effects.


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Variation of (a) [f'] and (b) [f''] as a function of atomic number for Cu [K\alpha] and Mo [K\alpha] radiations. Adapted from Fig. 3 of Srinivasan (1972[link]). Violation of Friedel's law

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Consider a structure containing N atoms of which P are normal atoms and the remaining Q anomalous scatterers. Let [{\bf F}_{P}] denote the contribution of the P atoms to the structure, and [{\bf F}_{Q}] and [{\bf F}''_{Q}] the real and imaginary components of the contribution of the Q atoms. The relation between the different contributions to a reflection h and its Friedel equivalent [-{\bf h}] is illustrated in Fig.[link]. For simplicity we assume here that all Q atoms are of the same type. The phase angle of [{\bf F}''_{Q}] is then exactly [90^{\circ}] ahead of that of [{\bf F}_{Q}]. The structure factors of h and [-{\bf h}] are denoted in the figure by [{\bf F}_{N}(+)] and [{\bf F}_{N}(-)], respectively. In the absence of anomalous scattering, or when the imaginary component of the dispersion correction is zero, the magnitudes of the two structure factors are equal and Friedel's law is obeyed; the phase angles have equal magnitudes, but opposite signs. As can be seen from Fig.[link], this is no longer true when [{\bf F}''_{Q}] has a nonzero value. Friedel's law is then violated. A composite view of the vector relationship for h and [-{\bf h}] can be obtained, as in Fig.[link], by reflecting the vectors corresponding to [-{\bf h}] about the real axis of the vector diagram. [{\bf F}_{P}] and [{\bf F}_{Q}] corresponding to the two reflections superpose exactly, but [{\bf F}''_{Q}] do not. [{\bf F}_{N}(+)] and [{\bf F}_{N}(-)] then have different magnitudes and phases.


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Vector diagram illustrating the violation of Friedel's law when [{\bf F}''_{Q} \neq 0].


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A composite view of the vector relationship between [{\bf F}_{N}(+)] and [{\bf F}_{N}(-)].

It is easily seen that Friedel's law is obeyed in centric data even when anomalous scatterers are present. [{\bf F}_{P}] and [{\bf F}_{Q}] are then parallel to the real axis and [{\bf F}''_{Q}] perpendicular to it. The vector sum of the three components is the same for h and [-{\bf h}]. It may, however, be noted that the phase angle of the structure factor is then no longer 0 or [180^{\circ}]. Even when the structure is noncentrosymmetric, the effect of anomalous scattering in terms of intensity differences between Friedel equivalents varies from reflection to reflection. The difference between [{\bf F}_{N}(+)] and [{\bf F}_{N}(-)] is zero when [\alpha_{P} = \alpha_{Q}] or [\alpha_{Q} + 180^{\circ}]. The difference tends to the maximum possible value [(2F''_{Q})] when [\alpha_{P} = \alpha_{Q} \pm 90^{\circ}].

Intensity differences between Friedel equivalents depend also on the ratio (in terms of number and scattering power) between anomalous and normal scatterers. Differences obviously do not occur when all the atoms are normal scatterers. On the other hand, a structure containing only anomalous scatterers of the same type also does not give rise to intensity differences. Expressions for intensity differences between Friedel equivalents have been derived by Zachariasen (1965[link]) for the most general case of a structure containing normal as well as different types of anomalous scatterers. Statistical distributions of such differences under various conditions have also been derived (Parthasarathy & Srinivasan, 1964[link]; Parthasarathy, 1967[link]). It turns out that, with a single type of anomalous scatterer in the structure, the ratio [{|F^{2}_{N} (+) - F^{2}_{N} (-)|\over F^{2}_{N} (+) + F^{2}_{N} (-)}] has a maximum mean value when the scattering powers of the anomalous scatterers and the normal scatterers are nearly the same (Srinivasan, 1972[link]). Also, for a given ratio between the scattering powers, the smaller the number of anomalous scatterers, the higher is the mean ratio. Friedel and Bijvoet pairs

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The discussion so far has been concerned essentially with crystals belonging to space groups P1 and [P\bar{1}]. In the centrosymmetric space group, the crystal and the diffraction pattern have the same symmetry, namely, an inversion centre. In P1, however, the crystals are noncentrosymmetric while the diffraction pattern has an inversion centre, in the absence of anomalous scattering. When anomalous scatterers are present in the structure [({\bf F}''_{Q} \neq 0)], Friedel's law breaks down and the diffraction pattern no longer has an inversion centre. Thus the diffraction pattern displays the same symmetry as that of the crystal in the presence of anomalous scattering. The same is true with higher-symmetry space groups also. For example, consider a crystal with space group P222, containing anomalous scatterers. The magnitudes of [{\bf F}_{P}] are the same for all equivalent reflections; so are those of [{\bf F}_{Q}] and [{\bf F}''_{Q}]. Their phase angles, however, differ from one equivalent to another, as can be seen from Table[link] When [{\bf F}''_{Q} = 0], the magnitudes of the vector sum of [{\bf F}_{P}] and [{\bf F}_{Q}] are the same for all the equivalent reflections. The intensity pattern thus has point-group symmetry [2/m\ 2/m\ 2/m]. When [{\bf F}''_{Q} \neq 0], the equivalent reflections can be grouped into two sets in terms of their intensities: [hkl], [h\bar{k}\bar{l}], [\bar{h}k\bar{l}] and [\bar{h}\bar{k}l]; and [\bar{h}\bar{k}\bar{l}], [\bar{h}kl], [h\bar{k}l] and [hk\bar{l}]. The equivalents belonging to the first group have the same intensity; so have the equivalents in the second group. But the two intensities are different. Thus the symmetry of the pattern is 222, the same as that of the crystal.

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Phase angles of different components of the structure factor in space group [P222]

ReflectionPhase angle [(^{\circ})] of
[{\bf F}_{P}][{\bf F}_{Q}][{\bf F}''_{Q}]
[hkl,\ h\bar{k}\bar{l},\ \bar{h}k\bar{l},\ \bar{h}\bar{k}l] [\alpha_{P}] [\alpha_{Q}] [90 + \alpha_{Q}]
[\bar{h}\bar{k}\bar{l},\ \bar{h}kl,\ h\bar{k}l,\ hk\bar{l}] [- \alpha_{P}] [- \alpha_{Q}] [90 - \alpha_{Q}]

In general, under conditions of anomalous scattering, equivalent reflections generated by the symmetry elements in the crystal have intensities different from those of equivalent reflections generated by the introduction of an additional inversion centre in normal scattering. There have been suggestions that a reflection from the first group and another from the second group should be referred to as a `Bijvoet pair' instead of a `Friedel pair', when the two reflections are not inversely related. Most often, however, the terms are used synonymously. The same practice will be followed in this article. Determination of absolute configuration

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The determination of the absolute configuration of chiral molecules has been among the most important applications of anomalous scattering. Indeed, anomalous scattering is the only effective method for this purpose and the method, first used in the early 1950s (Peerdeman et al., 1951[link]), has been extensively employed in structural crystallography (Ramaseshan, 1963[link]; Vos, 1975[link]).

Many molecules, particularly biologically important ones, are chiral in that the molecular structure is not superimposable on its mirror image. Chirality (handedness) arises primarily on account of the presence of asymmetric carbon atoms in the molecule. A tetravalent carbon is asymmetric when the four atoms (or groups) bonded to it are all different from one another. The substituents can then have two distinct arrangements which are mirror images of (or related by inversion to) each other. These optical isomers or enantiomers have the same chemical and physical properties except that they rotate the plane of polarization in opposite directions when polarized light passes through them. It is not, however, possible to calculate the sign of optical rotation, given the exact spatial arrangement or the `absolute configuration' of the molecule. Therefore, one cannot distinguish between the possible enantiomorphic configurations of a given asymmetric molecule from measurements of optical rotation. This is also true of molecules with chiralities generated by overall asymmetric geometry instead of the presence of asymmetric carbon atoms in them.

Normal X-ray scattering does not distinguish between enantiomers. Two structures [A\ (x_{j}, y_{j}, z_{j})] and [B\ (-x_{j}, -y_{j}, -z_{j})\ (\;j = 1, \ldots ,N)] obviously produce the same diffraction pattern on account of Friedel's law. The situation is, however, different when anomalous scatterers are present in the structure. The intensity difference between reflections h and [-{\bf h}], or that between members of any Bijvoet pair, has the same magnitude, but opposite sign for structures A and B. If the atomic coordinates are known, the intensities of Bijvoet pairs can be readily calculated. The absolute configuration can then be determined, i.e. one can distinguish between A and B by comparing the calculated intensities with the observed ones for a few Bijvoet pairs with pronounced anomalous effects. Determination of phase angles

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An important application of anomalous scattering is in the determination of phase angles using Bijvoet differences (Ramachandran & Raman, 1956[link]; Peerdeman & Bijvoet, 1956[link]). From Figs. and, we have [F^{2}_{N} (+) = F^{2}_{N} + F''^{2}_{Q} + 2F_{N} F''_{Q} \cos \theta \eqno(] and [F^{2}_{N} (-) = F^{2}_{N} + F''^{2}_{Q} - 2F_{N} F''_{Q} \cos \theta. \eqno(] Then [\cos \theta = {F^{2}_{N} (+) - F^{2}_{N} (-)\over 4F_{N} F''_{Q}}. \eqno(] In the above equations [F_{N}] may be approximated to [[F_{N} (+) + F_{N} (-)]/2]. Then [\theta] can be evaluated from ([link] except for the ambiguity in its sign. Therefore, from Fig.[link], we have [\alpha_{N} = \alpha_{Q} + 90^{\circ} \pm \theta. \eqno(] The phase angle thus has two possible values symmetrically distributed about [{\bf F}''_{Q}]. Anomalous scatterers are usually heavy atoms and their positions can most often be determined by Patterson methods. [\alpha_{Q}] can then be calculated and the two possible values of [\alpha_{N}] for each reflection evaluated using ([link].

In practice, the twofold ambiguity in phase angles can often be resolved in a relatively straightforward manner. As indicated earlier, anomalous scatterers usually have relatively high atomic numbers. The `heavy-atom' phases calculated from their positions therefore contain useful information. For any given reflection, that phase angle which is closer to the heavy-atom phase, from the two phases calculated using ([link], may be taken as the correct phase angle. This method has been successfully used in several structure determinations including that of a derivative of vitamin B12 (Dale et al., 1963[link]). The same method was also employed in a probabilistic fashion in the structure solution of a small protein (Hendrickson & Teeter, 1981[link]). A method for obtaining a unique, but approximate, solution for phase angles from ([link] has also been suggested (Srinivasan & Chacko, 1970[link]). An accurate unique solution for phase angles can be obtained if one collects two sets of intensity data using two different wavelengths which have different dispersion-correction terms for the anomalous scatterers in the structure. Two equations of the type ([link] are then available for each reflection and the solution common to both is obviously the correct phase angle. Different types of Patterson and Fourier syntheses can also be employed for structure solution using intensity differences between Bijvoet equivalents (Srinivasan, 1972[link]; Okaya & Pepinsky, 1956[link]; Pepinsky et al., 1957[link]).

An interesting situation occurs when the replaceable atoms in a pair of isomorphous structures are anomalous scatterers. The phase angles can then be uniquely determined by combining isomorphous replacement and anomalous-scattering methods. Such situations normally occur in protein crystallography and are discussed in Section[link]. Anomalous scattering without phase change

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The phase determination, and hence the structure solution, outlined above relies on the imaginary component of the dispersion correction. Variation in the real component can also be used in structure analysis. In early applications of anomalous scattering, the real component of the dispersion correction was made use of to distinguish between atoms of nearly the same atomic numbers (Mark & Szillard, 1925[link]; Bradley & Rodgers, 1934[link]). For example, copper and manganese, with atomic numbers 29 and 25, respectively, are not easily distinguishable under normal X-ray scattering. However, the real components of the dispersion correction for the two elements are −1.129 and −3.367, respectively, when Fe [K\alpha] radiation is used (IT IV, 1974[link]). Therefore, the difference between the scattering factors of the two elements is accentuated when this radiation is used. The difference is more pronounced at high angles as the normal scattering factor falls off comparatively rapidly with increasing scattering angle whereas the dispersion-correction term does not.

The structure determination of KMnO4 provides a typical example for the use of anomalous scattering without phase change in the determination of a centrosymmetric structure (Ramaseshan et al., 1957[link]; Ramaseshan & Venkatesan, 1957[link]). f′ and f″ for manganese for Cu [K\alpha] radiation are [-0.568] and 2.808, respectively. The corresponding values for Fe [K\alpha] radiation are [-3.367] and 0.481, respectively (IT IV, 1974[link]). The data sets collected using the two radiations can now be treated as those arising from two perfectly isomorphous crystals. The intensity differences between a reflection in one set and the corresponding reflection in the other are obviously caused by the differences in the dispersion-correction terms. They can, however, be considered formally as intensity differences involving data from two perfectly isomorphous crystals. They can be used, as indeed they were, to determine the position of the manganese ion through an appropriate Patterson synthesis (see Section[link]) and then to evaluate the signs of structure factors using ([link] when the structure is centrosymmetric. When the structure is noncentrosymmetric, a twofold ambiguity exists in the phase angles in a manner analogous to that in the isomorphous replacement method. This ambiguity can be removed if the structure contains two different subsets of atoms Q1 and Q2 which, respectively, scatter radiations [\lambda_{Q1}] and [\lambda_{Q2}] anomalously. Data sets can then be collected with [\lambda], which is scattered normally by all atoms, [\lambda_{Q1}] and [\lambda_{Q2}]. The three sets can be formally treated as those from three perfectly isomorphous structures and the phase determination effected using ([link] (Ramaseshan, 1963[link]). Treatment of anomalous scattering in structure refinement

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The effect of anomalous scattering needs to be taken into account in the refinement of structures containing anomalous scatterers, if accurate atomic parameters are required. The effect of the real part of the dispersion correction is largely confined to the thermal parameters of anomalous scatterers. This effect can be eliminated by simply adding f′ to the normal scattering factor of the anomalous scatterers.

The effects of the imaginary component of the dispersion correction are, however, more complex. These effects could lead to serious errors in positional parameters when the space group is polar, if data in the entire diffraction sphere are not used (Ueki et al., 1966[link]; Cruickshank & McDonald, 1967[link]). For example, accessible data in a hemisphere are normally used for X-ray analysis when the space group is P1. If the hemisphere has say h positive, the x coordinates of all the atoms would be in error when the structure contains anomalous scatterers. The situation in other polar space groups has been discussed by Cruickshank & McDonald (1967[link]). In general, in the presence of anomalous scattering, it is desirable to collect data for the complete sphere, if accurate structural parameters are required (Srinivasan, 1972[link]).

Methods have been derived to correct for dispersion effects in observed data from centrosymmetric and noncentrosymmetric crystals (Patterson, 1963[link]). The methods are empirical and depend upon the refined parameters at the stage at which corrections are applied. This is obviously an unsatisfactory situation and it has been suggested that the measured structure factors of Bijvoet equivalents should instead be treated as independent observations in structure refinement (Ibers & Hamilton, 1964[link]). The effect of dispersion corrections needs to be taken into account to arrive at the correct scale and temperature factors also (Wilson, 1975[link]; Gilli & Cruickshank, 1973[link]).


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