Correlations between different almost 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, p. 429 | 1 | 2 |

Section 4.2.4.3.3. Correlations between different almost 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

4.2.4.3.3. Correlations between different almost collinear chains

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In real cases there are more or less strong correlations between different chains at least within small domains. Deviations from a strict (3D) order of chain-like structural elements are due to several reasons: shape and structure of the chains, varying binding forces, thermodynamical or kinetic considerations.

Many types of disorder occur. (1) Relative shifts parallel to the common axis while projections along this axis give a perfect 2D ordered net (`axial disorder'). (2) Relative fluctuations of the distances between the chains (perpendicular to the unique axis) with short-range order along the transverse a and/or b directions. The net of projected chains down to the ab plane is distorted (`net distortions'). Disorder of types (1) and (2) is sometimes correlated owing to non-uniform cross sections of the chains. (3) Turns, twists and torsions of chains or parts of chains. This azimuthal type of disorder may be treated similarly to the case of azimuthal disorder of single-chain molecules. Correlations between axial shifts and torsions produce `screw shifts' (helical structures). Torsion of chain parts may be of dynamic origin (rotational vibrations). (4) Tilting or bending of the chains in a uniform or non-uniform way (`conforming/non-conforming'). Many of these types and a variety of combinations between them are found in polymer and liquid crystals and are treated therefore separately. Only some simple basic ideas are discussed here in brief.

For the sake of simplicity the paracrystal concept in combination with Gaussians is used again. Distribution functions are given by convolution products of next-nearest-neighbour distribution functions. As long as averaged lattice directions and lattice constants in a plane perpendicular to the chain axis exist, only two functions $[a_{100} = a_{1}(xyz)]$ and $[a_{010} = a_{2}(xyz)]$ are needed to describe the arrangement of next-nearest chains. Longitudinal disorder is treated as before by a third distribution function $[a_{001} = a_{3}(xyz)]$ . The phenomena of chain bending or tilting may be incorporated by an x and y dependence of $[a_{3}]$ . Any general fluctuation in the spatial arrangement of chains is given by $[a_{mpq} = a_{1} * \ldots * a_{1} * a_{2} * \ldots * a_{2} * a_{3} * \ldots * a_{3}. \eqno(4.2.4.50)]$ (m-fold, p-fold, q-fold self-convolution of $[a_{1}, a_{2}, a_{3}]$ , respectively.) $[w({\bf r}) = \delta ({\bf r}) \textstyle\sum\limits_{m} \textstyle\sum\limits_{p} \textstyle\sum\limits_{q} [a_{mpq} ({\bf r}) + a_{-mpq} ({\bf r})]. \eqno(4.2.4.51)]$ $[a_{\nu}\ (\nu = 1, 2, 3)]$ are called fundamental functions. If an averaged lattice cannot be defined, more fundamental functions $[a_{\nu}]$ are needed to account for correlations between them.

By Fourier transformations the interference function is given by $[\eqalign{G({\bf H}) &= \textstyle\sum\limits_{m} \textstyle\sum\limits_{p} \textstyle\sum\limits_{q} F_{1}^{m} F_{2}^{p} F_{3}^{q} = G_{1} G_{2} G_{3}\hbox{;}\cr G_{\nu} &= \hbox{Re} \{(1 + |F_{\nu}|)/(1 - |F_{\nu}|)\}.} \eqno(4.2.4.52)]$ If Gaussian functions are assumed, simple pictures are derived. For example: $[\eqalignno{a_{1}({\bf r} + \langle {\bf a} \rangle) &= 1/(2\pi)^{3/2} \cdot 1/(\Delta_{11} \Delta_{12} \Delta_{13}) &\cr &\quad \times \exp \left\{- {\textstyle{1\over 2}}[(x^{2}/\Delta_{11}^{2}) + (y^{2}/\Delta_{12}^{2}) + (z^{2}/\Delta_{13}^{2})]\right\} &&\cr&&(4.2.4.53)}]$ describes the distribution of neighbours in the x direction (mean distance $[\langle a \rangle]$ ). Parameter $[\Delta_{13}]$ concerns axial, $[\Delta_{11}]$ and $[\Delta_{12}]$ radial and tangential fluctuations, respectively. Pure axial distribution along c is given by projection of $[a_{1}]$ on the z axis, pure net distortions by projection on the [x - y] plane. If the chain-like structure is neglected the interference function $[G_{1}({\bf H}) = \exp \{-2 \pi^{2}(\Delta_{11}^{2} H^{2} + \Delta_{12}^{2} K^{2} + \Delta_{13}^{2} L^{2})\} \eqno(4.2.4.54)]$ describes a set of diffuse planes perpendicular to $[{\bf a}^{*}]$ with mean distance $[1/\langle a \rangle]$ . These diffuse layers broaden along H with $[m\Delta_{11}]$ and decrease in intensity along K and L monotonically. There is an ellipsoidal-shaped region in reciprocal space defined by main axes of length $[1/\Delta_{11}, 1/\Delta_{12}, 1/\Delta_{13}]$ with a limiting surface given by $[|F| \simeq 0.1]$ , beyond which the diffuse intensity is completely smeared out. The influence of $[a_{2}]$ may be discussed in an analogous way.

If the chain-like arrangement parallel to c [equation (4.2.4.12)] is taken into consideration, $[l(z) = \textstyle\sum\limits_{n_{3}} \delta (z - n_{3} c)\hbox{;}]$ the set of planes perpendicular to $[{\bf a}^{*}]$ (and/or $[{\bf b}^{*}]$ ) is subdivided in the L direction by a set of planes located at $[l \cdot 1/c]$ [equation (4.2.4.15)].

Longitudinal disorder is given by $[a_{3}(z)]$ [equation (4.2.4.48), $[\Delta_{33} = \Delta]$ ] and leads to two intersecting sets of broadened diffuse layer systems.

Particular cases like pure axial distributions $[(\Delta_{11}, \Delta_{12} \sim 0)]$ , pure tangential distributions (net distortions: $[\Delta_{11}, \Delta_{13} \sim 0]$ ), uniform bending of chains or combinations of these effects are discussed in the monograph of Vainshtein (1966).

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, p. 429