Neutron anomalous scattering

Vijayan, M.; Ramaseshan, S.

doi:10.1107/97809553602060000557

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.4, p. 275 | 1 | 2 |

Section 2.4.5.1. Neutron anomalous scattering

M. Vijayan^a ^* and S. Ramaseshan^b ^‡

^a Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India, and ^bRaman Research Institute, Bangalore 560 080, India
Correspondence e-mail: mv@mbu.iisc.ernet.in

2.4.5.1. Neutron anomalous scattering

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Apart from the limitations introduced by experimental factors, such as the need for large crystals and the comparatively low flux of neutron beams, there are two fundamental reasons why neutrons are less suitable than X-rays for the ab initio determination of crystal structures. First, the neutron scattering lengths of different nuclei have comparable magnitudes whereas the atomic form factors for X-rays vary by two orders of magnitude. Therefore, Patterson techniques and the related heavy-atom method are much less suitable for use with neutron diffraction data than with X-ray data. Secondly, neutron scattering lengths could be positive or negative and hence, in general, the positivity criterion (Karle & Hauptman, 1950) or the squarability criterion (Sayre, 1952) does not hold good for nuclear density. Therefore, the direct methods based on these criteria are not strictly applicable to structure analysis using neutron data, although it has been demonstrated that these methods could be successfully used in favourable situations in neutron crystallography (Sikka, 1969). The anomalous-scattering method is, however, in principle more powerful in the neutron case than in the X-ray case for ab initio structure determination.

Thermal neutrons are scattered anomalously at appropriate wavelengths by several nuclei. In a manner analogous to (2.4.3.1), the neutron scattering length of these nuclei can be written as $[b_{0} + b' + ib'' = b + ib''. \eqno(2.4.5.1)]$ The correction terms b′ and b″ are strongly wavelength-dependent. In favourable cases, $[b'/b_{0}]$ and $[b''/b_{0}]$ can be of the order of 10 whereas they are small fractions in X-ray anomalous scattering. In view of this pronounced anomalous effect in neutron scattering, Ramaseshan (1966) suggested that it could be used for structure solution. Subsequently, Singh & Ramaseshan (1968) proposed a two-wavelength method for unique structure analysis using neutron diffraction. The first part of the method is the determination of the positions of the anomalous scatterers from the estimated values of $[F_{Q}]$ . The method employed for estimating $[F_{Q}]$ is analogous to that using (2.4.4.9) except that data collected at two appropriate wavelengths are used instead of those from two isomorphous crystals. The second stage of the two-wavelength method involves phase evaluation. Referring to Fig. 2.4.3.2 and in a manner analogous to (2.4.3.5), we have $[\sin \psi_{1} = {F^{2}_{N1}(+) - F^{2}_{N1}(-)\over 4F_{N1}F''_{Q1}}, \eqno(2.4.5.2)]$ where $[\psi = \alpha_{N} - \alpha_{Q}]$ and subscript 1 refers to data collected at wavelength $[\lambda 1]$ . Singh and Ramaseshan showed that cos $[\psi_{1}]$ can also be determined when data are available at wavelength $[\lambda 1]$ and $[\lambda 2]$ . We may define $[F^{2}_{m} = [F^{2}_{N}(+) + F^{2}_{N}(-)]/2 \eqno(2.4.5.3)]$ and we have from (2.4.3.3), (2.4.3.4) and (2.4.5.3) $[F_{N} = (F^{2}_{m} - F''^{2}_{Q})^{1/2}. \eqno(2.4.5.4)]$ Then, $[\eqalignno{ &\cos \psi_{1} = {F_{m1}^{2} - F_{m2}^{2} - [(b_{1}^{2} + b''^{2}_{1}) - (b_{2}^{2} + b''^{2}_{2})] x^{2}\over 2(b_{1} - b_{2}) F_{N1} x} + {F_{Q1}\over F_{N1}}, \cr & &(2.4.5.5)}]$ where x is the magnitude of the temperature-corrected geometrical part of $[{\bf F}_{Q}]$ . $[\psi_{1}]$ and hence $[\alpha_{N1}]$ can be calculated using (2.4.5.2) and (2.4.5.5). $[\alpha_{N2}]$ can also be obtained in a similar manner.

During the decade that followed Ramaseshan's suggestion, neutron anomalous scattering was used to solve half a dozen crystal structures, employing the multiple-wavelength methods as well as the methods developed for structure determination using X-ray anomalous scattering (Koetzle & Hamilton, 1975; Sikka & Rajagopal, 1975; Flook et al., 1977). It has also been demonstrated that measurable Bijvoet differences could be obtained, in favourable situations, in neutron diffraction patterns from protein crystals (Schoenborn, 1975). However, despite the early promise held by neutron anomalous scattering, the method has not been as successful as might have been hoped. In addition to the need for large crystals, the main problem with using this method appears to be the time and expense involved in data collection (Koetzle & Hamilton, 1975).

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International Tables for Crystallography (2006). Vol. B. ch. 2.4, p. 275