Structure factors

Chandrasekaran, R.; Stubbs, G.

doi:10.1107/97809553602060000702

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 19.5, p. 445 | 1 | 2 |

Section 19.5.3.3. Structure factors

R. Chandrasekaran^a ^* and G. Stubbs^b

^aWhistler Center for Carbohydrate Research, Purdue University, West Lafayette, IN 47907, USA, and ^bDepartment of Molecular Biology, Vanderbilt University, Nashville, TN 37235, USA
Correspondence e-mail: [email protected]

19.5.3.3. Structure factors

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Cochran et al. (1952) showed that the structure factor on layer line l of a helix made up of repeating subunits is $[{\hbox{{\bf F}}(R, \psi, Z) = \!\textstyle\sum\limits_{j}\displaystyle \textstyle\sum\limits_{n}\displaystyle f_{j} J_{n} (2\pi Rr_{j}) \exp \{i[n(\psi + \pi/2) - n\varphi_{j} + 2\pi lz_{j}/c]\}.} \eqno(19.5.3.1)]$ Diffraction occurs only for $[Z = l/c.\ r_{j}, \varphi_{j}]$ and $[z_{j}]$ are the real-space coordinates of atom j in the repeating unit of the helix; $[f_{j}]$ is the atomic scattering factor of that atom. $[J_{n}]$ is the Bessel function of the first kind of order n. The summation over n includes only those values of n that satisfy the selection rule $[l = tn + um, \eqno(19.5.3.2)]$ where m is any integer. In practice, the summation may be limited to values of [|n|] less than $[2\pi r_{\max} R + 2]$ , where $[r_{\max}]$ is the radius of the outermost atom in the polymer, because the value of a Bessel function $[J_{n}(x)]$ is negligible for n greater than about x + 2. For low-order Bessel functions or applications requiring greater accuracy, slight variations of this limitation are used.

The structure factor F is a complex number with an amplitude and phase, and is fully equivalent to that derived using the trigonometric functions in crystallography. The expression for intensity $[I = \hbox{{\bf FF}}^{*} = |\hbox{{\bf F}}|^{2}]$ holds good.

Equation (19.5.3.1) can be rewritten (Klug et al., 1958) as $[\hbox{{\bf F}}(R, \psi, l/c) = \textstyle\sum\limits_{n}\displaystyle \hbox{{\bf G}}_{n, \, l} (R) \exp [in(\psi + \pi /2)], \eqno(19.5.3.3)]$ where the Fourier–Bessel structure factor $[\hbox{{\bf G}}_{n, \, l} (R)]$ is independent of ψ and is given by $[\hbox{{\bf G}}_{n, \, l} (R) = \textstyle\sum\limits_{j}\displaystyle f_{j} J_{n} (2\pi Rr_{j}) \exp [i(-n\varphi_{j} + 2\pi lz_{j}/c)]. \eqno(19.5.3.4)]$ $[J_{n} (0)]$ is 1 when [n = 0] and 0 otherwise. For this reason, the structure factors on the meridian [(R = 0)] are nonzero only on layer lines for which l is an integral multiple of u. Hence, a visual inspection of the diffraction pattern often helps to determine u.

References

Cochran, W., Crick, F. H. & Vand, V. (1952). The structure of synthetic polypeptides. I. The transform of atoms in a helix. Acta Cryst. 5, 581–586.Google Scholar

Klug, A., Crick, F. H. C. & Wyckoff, H. W. (1958). Diffraction from helical structures. Acta Cryst. 11, 199–212.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 19.5, p. 445