Scattering by randomly distributed collinear chains

Jagodzinski, H.; Frey, F.

doi:10.1107/97809553602060000564

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

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International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 425-427 | 1 | 2 |

Section 4.2.4.3.1. Scattering by randomly distributed collinear chains

H. Jagodzinski^a and F. Frey^b

^aInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany, and ^bInstitut für Kristallographie und Mineralogie, Universität, Theresienstrasse 41, D-8000 München 2, Germany

4.2.4.3.1. Scattering by randomly distributed collinear chains

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As found in any elementary textbook of diffraction the simplest result of scattering by a chain with period c $[l({\bf r}) = l(z) = \textstyle\sum\limits_{n_{3}} \delta (z - n_{3}c) \eqno(4.2.4.12)]$ is described by one of the Laue equations: $[G(L) = |L(L)|^{2} = \sin^{2} \pi NL/\sin^{2} \pi L \eqno(4.2.4.13)]$ which gives broadened profiles for small N. In the context of phase transitions the Ornstein–Zernike correlation function is frequently used, i.e. (4.2.4.13) is replaced by a Lorentzian: $[1 / \{\xi^{2} + 4 \pi^{2}(L - l)^{2}\}, \eqno(4.2.4.14)]$ where ξ denotes the correlation length.

In the limiting case $[N \rightarrow \infty]$ , (4.2.4.13) becomes $[\textstyle\sum\limits_{l} \delta (L - l). \eqno(4.2.4.15)]$ The scattering by a real chain a(r) consisting of molecules with structure factor $[F_{M}]$ is therefore determined by $[F_{M}({\bf H}) = \textstyle\sum\limits_{j} f_{j} \exp \{2 \pi i(Hx_{j} + Ky_{j} + Lz_{j})\}. \eqno(4.2.4.16)]$ The Patterson function is: $[\eqalignno{P({\bf r}) &= (1 / c) \textstyle\int \textstyle\int |F_{0}(H, K)|^{2} \cos 2 \pi (Hx + Ky) \;\hbox{d}H \;\hbox{d}K &\cr &\quad + (2 / c) \textstyle\sum\limits_{l} \textstyle\int \textstyle\int |F_{l}|^{2} \exp \{2 \pi i(Hx + Ky)\} &\cr &\quad \times \exp \{- 2 \pi ilz\} \;\hbox{d}H \;\hbox{d}K, &(4.2.4.17)}]$ where the index l denotes the only relevant position [L = l] (the subscript M is omitted).

The intensity is concentrated in diffuse layers perpendicular to $[{\bf c}^{*}]$ from which the structural information may be extracted. Projections are: $[\textstyle\int a({\bf r}) \;\hbox{d}z = \textstyle\int \textstyle\int F_{0}(H, K) \exp \{2 \pi i(Hx + Ky)\} \;\hbox{d}H \;\hbox{d}K \eqno(4.2.4.18)]$ $[\textstyle\int \textstyle\int a({\bf r}) \;\hbox{d}x \;\hbox{d}y = (2 / c) \textstyle\sum\limits_{l} F_{l}(00l) \exp (- 2 \pi ilz). \eqno(4.2.4.19)]$ Obviously the z parameters can be determined by scanning along a meridian (00L) through the diffuse sheets (diffractometer recording). Owing to intersection of the Ewald sphere with the set of planes the meridian cannot be recorded on one photograph; successive equi-inclination photographs are necessary. Only in the case of large c spacings is the meridian well approximated in one photograph.

There are many examples where a tendency to cylindrical symmetry exists: chains with p-fold rotational or screw symmetry around the preferred direction or assemblies of chains (or domains) with statistical orientational distribution around the texture axis. In this context it should be mentioned that symmetry operations with rotational parts belonging to the 1D rod groups actually occur, i.e. not only p = 2, 3, 4, 6.

In all these cases a treatment in the frame of cylindrical coordinates is advantageous (see, e.g., Vainshtein, 1966): $[\let\normalbaselines\relax\openup3pt\matrix{\hbox{Direct space}\hfill &\hbox{Reciprocal space}\hfill\cr x = r \cos \psi \hfill& H = H_{r} \cos \Psi\hfill\cr y = r \sin \psi \hfill & K = H_{r} \sin \Psi\hfill\cr z = z\hfill& L = L\hfill}]$ $[\eqalignno{a(r, \psi, z) &= \textstyle\int \textstyle\int \textstyle\int F({\bf H}) \exp \{- 2 \pi i[H_{r}r \cos (\psi - \Psi) + Lz]\}\cr &\quad \times H_{r} \;\hbox{d}H_{r} \;\hbox{d}\Psi \;\hbox{d}L &(4.2.4.20)}]$ $[\eqalignno{F({\bf H}) &= \textstyle\int \textstyle\int \textstyle\int a(r, \psi, z) \exp \{2 \pi i[H_{r}r \cos (\psi - \Psi) + Lz]\}\cr &\quad \times r \;\hbox{d}r \;\hbox{d}\psi \;\hbox{d}z. &(4.2.4.21)}]$ The integrals may be evaluated by the use of Bessel functions: $[J_{n}(u) = {\textstyle{1\over 2}} \pi i^{n} \textstyle\int \exp \{i(u \cos \varphi + n\varphi)\} \;\hbox{d}\varphi]$ $[(u = 2 \pi rH_{r}\hbox{; } \varphi = \psi - \Psi)]$ .

The 2D problem $[a = a(r, \psi)]$ is treated first; an extension to the general case $[a(r, \psi, z)]$ is easily made afterwards.

Along the theory of Fourier series one has: $[\eqalign{a(r, \psi) &= \textstyle\sum\limits_{n} a_{n}(r) \exp \{in \psi\}\cr a_{n}(r) &= {1\over 2 \pi} \int a(r, \psi) \exp \{- in \psi\} \;\hbox{d}\psi\cr} \eqno(4.2.4.22)]$ or with: $[\eqalign{\alpha_{n} &= {1\over 2 \pi} \int a(r, \psi) \cos (n\psi) \;\hbox{d}\psi\cr \beta_{n} &= {1\over 2 \pi} \int a(r, \psi) \sin (n\psi) \;\hbox{d}\psi\cr a_{n} (r) &= |a_{n} (r)| \exp \{- i\psi_{n} (r)\} \cr |a_{n} (r)| &= \sqrt{a_{n}^{2} + \beta_{n}^{2}} \cr \psi_{n} (r) &= \hbox{arctan } \beta_{n}/\alpha_{n}.\cr}]$ If contributions to anomalous scattering are neglected a(r, ψ) is a real function: $[a (r, \psi) = \textstyle\sum\limits_{n} |a_{n} (r)| \cos [n\psi - \psi_{n} (r)]. \eqno(4.2.4.23)]$ Analogously, one has $[F (H_{r}, \Psi) = \textstyle\sum\limits_{n} |F_{n} (H_{r})| \exp (in \Psi). \eqno(4.2.4.24)]$ $[F (H_{r}, \Psi)]$ is a complex function; $[F_{n} (H_{r})]$ are the Fourier coefficients which are to be evaluated from the $[a_{n} (r)]$ : $[\eqalignno{F_{n} (H_{r}) &= {1\over 2\pi} \int F (H_{r}, \Psi) \exp \{- in\Psi\} \;\hbox{d} \Psi &\cr \noalign{\vskip5pt} &= \exp \{in \pi/2\} \int a_{n} (r) J_{n} (2\pi r H_{r}) 2\pi r \;\hbox{d}r &\cr F (H_{r}, \Psi) &= \textstyle\sum\limits_{n} \exp \{in [\Psi + (\pi/2)]\} \textstyle\int a_{n} (r) &\cr &\quad \times J_{n} (2\pi r H_{r}) 2\pi r \;\hbox{d}r &(4.2.4.25)}]$ $[\eqalignno{a (r, \psi) &= \textstyle\sum\limits_{n} \exp \{in [\Psi - (\pi/2)]\} \textstyle\int F_{n} (H_{r}) &\cr &\quad \times J_{n} (2\pi r H_{r}) 2\pi H_{r} \;\hbox{d}H_{r}. &(4.2.4.26)}]$ The formulae may be used for calculation of diffuse intensity distribution within a diffuse sheet, in particular when the chain molecule is projected along the unique axis [cf. equation (4.2.4.18)].

Special cases are:

(a) Complete cylinder symmetry $[\eqalignno{F (H_{r}) &= 2\pi \textstyle\int a (r) J_{0} (2\pi r H_{r}) r \;\hbox{d}r &(4.2.4.27)\cr a (r) &= 2\pi \textstyle\int F (H_{r}) J_{n} (2\pi r H_{r}) H_{r} \;\hbox{d}H_{r}. &(4.2.4.28)}%(4.2.4.28)]$
(b) p-fold symmetry of the projected molecule $[a (r, \psi) = {a [r, \psi + (2\pi/p)]}]$ $[\eqalignno{F_{p} (H_{r}, \Psi) &= \textstyle\sum\limits_{n} \exp \{inp[\Psi + (\pi/2)]\} &\cr \noalign{\vskip5pt} &\quad \times \textstyle\int a_{np} (r) J_{np} (2\pi r H_{r}) 2\pi r \;\hbox{d}r &(4.2.4.29)\cr \noalign{\vskip5pt} a_{p} (r, \psi) &= \textstyle\sum\limits_{n} |a_{np} (r)| \cos [np\psi - \psi_{np} (r)]. &(4.2.4.30)}%(4.2.4.30)]$ Only Bessel functions $[J_{0}, J_{p}, J_{2p}, \ldots]$ occur. In most cases $[J_{2p}]$ and higher orders may be neglected.
(c) Vertical mirror planes

Only cosine terms occur, i.e. all $[\beta_{n} = 0]$ or $[\psi_{n} (r) = 0]$ .

The general 3D expressions valid for extended chains with period c [equation (4.2.4.12)] are found in an analogous way: $[a (r, \psi, z) = a_{M} (r, \psi, z) * l(z)]$ $[\eqalignno{F ({\bf H}) &= F_{l} (H_{r}, \Psi, L) = F_{M} ({\bf H}) L (L) &\cr &= \textstyle\int \textstyle\int \textstyle\int a_{M} (r, \psi, z) \exp \{2\pi i [H_{r} r \cos (\psi - \Psi) + Lz]\} &\cr &\quad \times 2\pi r \;\hbox{d}r\;\hbox{d}\psi\;\hbox{d}z &(4.2.4.31)}]$ using a series expansion analogous to (4.2.4.23) and (4.2.4.24): $[a_{nl} (r) = {1\over 2\pi} \int \int a_{M} \exp \{- i (n\psi - 2\pi lz)\} \;\hbox{d}\psi \;\hbox{d}z \eqno(4.2.4.32)]$ $[F_{nl} (H_{r}) = \exp \{in \pi/2\} \textstyle\int a_{nl} (r) J_{n} (2\pi H_{r} r) 2\pi r \;\hbox{d}r \eqno(4.2.4.33)]$ one has: $[{F_{l} ({\bf H}) = \textstyle\sum\limits_{n} \exp \{in [\Psi + (\pi/2)]\} \textstyle\int a_{nl} (r) J_{n} (2\pi H_{r} r) 2\pi r \;\hbox{d}r.} \eqno(4.2.4.34)]$ In practice the integrals are often replaced by discrete summation of j atoms at positions: $[r = r_{j}]$ , $[\psi = \psi_{j}]$ , $[z = z_{j}]$ $[(0 \leq z_{j} \lt c)]$ : $[\eqalignno{F_{l} ({\bf H}) &= \textstyle\sum\limits_{j} \textstyle\sum\limits_{n} f_{j} J_{n} (2\pi H_{r} r_{j}) \exp \{-in \psi_{j}\} &\cr &\quad \times \exp (2\pi ilz_{j}) \exp \{in [\Psi + (\pi/2)]\} &(4.2.4.35)}]$ or $[F_{l} ({\bf H}) = \textstyle\sum\limits_{n} (\alpha_{n} + i\beta_{n}) \exp \{in \Psi\}]$ $[\eqalign{\alpha_{n} &= \textstyle\sum\limits_{j} f_{j} J_{n} (2\pi H_{r} r_{j}) \cos \{n [(\pi/2) - \psi_{j}] + 2\pi lz_{j}\} \cr \beta_{n} &= \textstyle\sum\limits_{j} f_{j} J_{n} (2\pi H_{r} r_{j}) \sin \{n [(\pi/2) - \psi_{j}] + 2\pi lz_{j}\}.}]$ Intensity in the lth diffuse layer is given by $[\eqalignno{I_{l} &= \textstyle\sum\limits_{n} \textstyle\sum\limits_{n'} [(\alpha_{n} \alpha_{n'} + \beta_{n} \beta_{n'}) + i (\alpha_{n'} \beta_{n} - \alpha_{n} \beta_{n'})] &\cr &\quad \times \exp \{i (n - n') \Psi\}. &(4.2.4.36)}]$

(a) Cylinder symmetry (free rotating molecules around the chain axis or statistical averaging with respect to ψ over an assembly of chains). Only component $[F_{0l}]$ occurs: $[F_{0l} (H_{r}, L) = 2\pi \textstyle\int \textstyle\int \langle a_{M} \rangle J_{0} (2\pi H_{r} r) \exp \{2\pi ilz\} r \;\hbox{d}r \;\hbox{d}z]$ or $[F_{0l} (H_{r}, L) = \textstyle\sum\limits_{j} f_{j} J_{0} (2\pi H_{r} r_{j}) \exp \{2\pi ilz_{j}\}.]$ In particular, $[F_{00} (H_{r})]$ determines the radial component of the molecule projected along z: $[F_{00} (H_{r}) = \textstyle\sum\limits_{j} f_{j} J_{0} (2\pi H_{r} r_{j}).]$
(b) p-fold symmetry of a plane molecule (or projected molecule) as outlined previously: only components np instead of n occur. Bessel functions $[J_{0}]$ and $[J_{p}]$ are sufficient in most cases.
(c) Vertical mirror plane: see above.
(d) Horizontal mirror plane (perpendicular to the chain): Exponentials $[\exp \{2\pi ilz\}]$ in equation (4.2.4.32) may be replaced by $[\cos 2\pi lz]$ .
(e) Twofold symmetry axis perpendicular to the chain axis (at positions $[\psi = 0]$ , $[2\pi/p, \ldots]$ ). Exponentials in equation (4.2.4.32), $[\exp \{- i (np\psi - 2\pi lz)\}]$ , are replaced by the corresponding cosine term $[\cos (np\psi + 2\pi lz)]$ .

Formulae concerning the reverse method (Fourier synthesis) are not given here (see, e.g., Vainshtein, 1966). Usually there is no practical use in diffuse-scattering work because it is very difficult to separate out a single component $[F_{nl}]$ . Every diffuse layer is affected by all components $[F_{nl}]$ . There is a chance if one diffuse layer corresponds predominantly to one Bessel function.

References

Vainshtein, B. K. (1966). Diffraction of X-rays by chain molecules. Amsterdam: Elsevier.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 425-427