Exploiting the anisotropy of anomalous scattering boosts the phasing power of SAD and MAD experiments

Schiltz, M.; Bricogne, G.

doi:10.1107/S0907444908010202

Notes

¹These relations are equivalent to equation (3) in Templeton & Templeton (1982). In Fanchon & Hendrickson (1990) they also appear in equation (3) and in Kirfel et al. (1991) they appear in equation (8). Both Templeton & Templeton (1982) and Kirfel et al. (1991) explicitly chose for the vectors u, v, u' and v' directions that are perpendicular and parallel, respectively, to the plane of diffraction. These directions are denoted s, s', p and p' by Templeton & Templeton (1982) or [sigma] , [sigma] ', [pi] and [pi] ' in Kirfel et al. (1991). However, the choice of these directions is completely free, as was noticed by Fanchon & Hendrickson (1990). Below, we show that a particular choice of vectors u, v, u' and v' can considerably simplify the expressions in the case of linearly polarized X-rays.

²This also explains why in macromolecular crystallography the effects of AAS are much more pronounced in nonforbidden reflections than in forbidden ones. In forbidden reflections, the normal part of the structure factor, F(h), is always zero, so that the AAS-induced intensity modulations only arise from the terms G_p'p(h) and $[G_{p^{\prime}_{\perp}p}({\bf h})]$ , which are always small. However, for nonforbidden reflections (22) can be explicitly written as I(h) [approximately equal to] $[k \psi^2 \{|F({\bf h})| ^2 + |G_{p^{\prime}p}({\bf h})|^2 + 2|F({\bf h})|\cdot |G_{p^{\prime}p}({\bf h})|\cdot\cos[\varphi_{F({\bf h})} - \varphi_{G_{p^{\prime}p}({\bf h})}]\}]$ . As a consequence of the quadrature relationship between intensities and structure factors, the modulations arising from G_p'p(h) in the third term of the right-hand side of this equation are `boosted' by the multiplication by the usually much larger |F(h)|.

³We will henceforth use the term symmetry-related reflections rather than symmetry-equivalent reflections, since in the presence of AAS these reflections are no longer equivalent.

⁴This operation is similar to the reflection of F(-h) through the real axis on the complex plane when building the Harker construction with Friedel pairs in the case of anomalous scattering (North, 1965). In that case, the Harker construction is actually set up with F(h) and $[\tilde F({\bf -h})]$ , where $[\tilde F({\bf -h})]$ is the complex conjugate of F(-h).

⁵It also holds exactly, at any scattering angle, for reflections h [perpendicular] p, i.e. for diffraction in the vertical plane. This was, for example, the case for all measurements that were carried out on four-circle diffractometers with a point detector by Templeton and Templeton.

⁶To put it more succinctly: anomalous scattering is similar to X-ray absorption, but with phase shifts (Bricogne & Schiltz, 2000).