Two-dimensional disorder of 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. 425-429 | 1 | 2 |

Section 4.2.4.3. Two-dimensional disorder of 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. Two-dimensional disorder of chains

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In this section disorder phenomena are considered which are related to chain-like structural elements in crystals. This topic includes the so-called `1D crystals' where translational symmetry (in direct space) exists in one direction only – crystals in which highly anisotropic binding forces are responsible for chain-like atomic groups, e.g. compounds which exhibit a well ordered 3D framework structure with tunnels in a unique direction in which atoms, ions or molecules are embedded. Examples are compounds with platinum, iodine or mercury chains, urea inclusion compounds with columnar structures (organic or inorganic), 1D ionic conductors, polymers etc. Diffuse-scattering studies of 1D conductors have been carried out in connection with investigations of stability/instability problems, incommensurate structures, phase transitions, dynamic precursor effects etc. These questions are not treated here. For general reading of diffuse scattering in connection with these topics see, e.g., Comes & Shirane (1979, and references therein). Also excluded are specific problems related to polymers or liquid crystals (mesophases) (see Chapter 4.4 ) and magnetic structures with chain-like spin arrangements.

Trivial diffuse scattering occurs as 1D Bragg scattering (diffuse layers) by internally ordered chains. Diffuse phenomena in reciprocal space are due to `longitudinal' disordering within the chains (along the unique direction) as well as to `transverse' correlations between different chains over a restricted volume. Only static aspects are considered; diffuse scattering resulting from collective excitations or diffusion-like phenomena which are of inelastic or quasielastic origin are not treated here.

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.

4.2.4.3.2. Disorder within randomly distributed collinear chains

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Deviations from strict periodicities in the z direction within one chain may be due to loss of translational symmetry of the centres of the molecules along z and/or due to varying orientations of the molecules with respect to different axes, such as azimuthal misorientation, tilting with respect to the z axis or combinations of both types. As in 3D crystals, there may or may not exist 1D structures in an averaged sense.

4.2.4.3.2.1. General treatment

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All formulae given in this section are only special cases of a 3D treatment (see, e.g., Guinier, 1963). The 1D lattice (4.2.4.12) is replaced by a distribution: $[\eqalignno{d (z) &= \textstyle\sum\limits_{\nu} \delta (z - z_{\nu}) \cr \noalign{\vskip5pt}D (L) &= \textstyle\sum\limits_{\nu} \exp \{2\pi iLz_{\nu}\} &(4.2.4.37)\cr \noalign{\vskip5pt}F ({\bf H}) &= F_{M} ({\bf H}) D (L).}]$ The Patterson function is given by $[P ({\bf r}) = [a_{M} ({\bf r}) * a_{M} - ({\bf r})] * [d (z) * d - (z)]. \eqno(4.2.4.38)]$ Because the autocorrelation function [w = d * d] is centrosymmetric $[{w (z) = N\delta (z) + \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} \delta [z - (z_{\nu} - z_{\mu})]}\ {+ \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} \delta [z + (z_{\nu} - z_{\mu})],} \eqno(4.2.4.39)]$ the interference function [W (L)] $[(= |D (L)|^{2})]$ is given by $[\eqalignno{W (L) &= N + 2 \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} \cos 2\pi [L (z_{\nu} - z_{\mu})] &(4.2.4.40)\cr \noalign{\vskip5pt}I ({\bf H}) &= |F_{M} ({\bf H})|^{2} W (L). &(4.2.4.41)}%(4.2.4.41)]$ Sometimes, e.g. in the following example of orientational disorder, there is an order only within domains. As shown in Section 4.2.3, this may be treated by a box or shape function [b(z) = 1] for $[z \leq z_{N}]$ and 0 elsewhere. $[\eqalignno{d (z) &= d_{\infty} b (z) \cr \noalign{\vskip5pt}a ({\bf r}) &= a_{M} ({\bf r}) * [d_{\infty} b (z)] &(4.2.4.42)\cr \noalign{\vskip5pt}F ({\bf H}) &= F_{M} ({\bf H}) [D_{\infty} * B (L)]}]$ with $[\eqalign{&\ \ b (z) * b - (z) \leftrightarrow |B (L)|^{2}\cr &I = |F_{M} ({\bf H})|^{2}| D_{\infty} * B (L)|^{2}.} \eqno(4.2.4.43)]$ If the order is perfect within one domain one has $[D_{\infty} (L) \simeq \sum \delta (L - l)]$ ; $[(D_{\infty} * B) = \sum D (L - l)]$ ; i.e. each reflection is affected by the shape function.

4.2.4.3.2.2. Orientational disorder

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A misorientation of the chain molecules with respect to one another is taken into account by different structure factors $[F_{M}]$ . $[I ({\bf H}) = \textstyle\sum\limits_{\nu} \textstyle\sum\limits_{\mu} F_{\nu} ({\bf H}) F_{\mu} ({\bf H})^{+} \exp \{2\pi iL (z_{\nu} - z_{\mu})\}. \eqno(4.2.4.44)]$ A further discussion follows the same arguments outlined in Section 4.2.3. For example, a very simple result is found in the case of uncorrelated orientations. Averaging over all pairs $[F_{\nu} F_{\mu}^{+}]$ yields $[I ({\bf H}) = N (\langle |F|^{2}\rangle - \langle |F|\rangle^{2}) + \langle |F|\rangle^{2} L (L), \eqno(4.2.4.44a)]$ where $[\eqalign{\langle |F|\rangle^{2} &= 1/N^{2} \langle F_{\nu} F_{\mu}^{+}\rangle \cr \noalign{\vskip5pt}&= \textstyle\sum\limits_{\nu} \alpha_{\nu} F_{\nu} ({\bf H}) \textstyle\sum\limits_{\mu} \alpha_{\mu} F_{\mu}^{+} ({\bf H}) \quad (\nu \neq \mu) \cr \noalign{\vskip5pt}\langle |F|^{2}\rangle &= 1/N \langle F_{\nu} F_{\nu}^{+}\rangle = \textstyle\sum\limits_{\nu} \alpha_{\nu} |F_{\nu} ({\bf H})|^{2}.}]$ Besides the diffuse layer system there is a diffuse background modulated by the H dependence of $[[\langle |F ({\bf H})|^{2}\rangle - \langle |F ({\bf H})|\rangle^{2}]]$ .

4.2.4.3.2.3. Longitudinal disorder

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In this context the structure factor of a chain molecule is neglected. Irregular distances between the molecules within a chain occur owing to the shape of the molecules, intrachain interactions and/or interaction forces via a surrounding matrix. A general discussion is given by Guinier (1963). It is convenient to reformulate the discrete Patterson function, i.e. the correlation function (4.2.4.39). $[\eqalignno{w (z) &= N\delta (z) + \textstyle\sum\limits_{\nu} \delta [z \pm (z_{\nu} - z_{\nu + 1})] &\cr &\quad + \textstyle\sum\limits_{\nu} \delta [z \pm (z_{\nu} - z_{\nu + 2})] + \ldots &(4.2.4.39a)}]$ in terms of continuous functions $[a_{\mu} (z)]$ which describe the probability of finding the µth neighbour within an arbitrary distance $[\eqalignno{w' (z) &= w (z)/N = \delta (z) + a_{1} (z) + a_{-1} (z) + \ldots \cr &\quad + a_{\mu} (z) a_{-\mu} (z) + \ldots &(4.2.4.45)}]$ $[[\textstyle\int a_{\mu} (z) \;\hbox{d}z = 1, a_{\mu} (z) = a_{-\mu} (-z)].]$

There are two principal ways to define $[a_{\mu} (z)]$ . The first is the case of a well defined one-dimensional lattice with positional fluctuations of the molecules around the lattice points, i.e. long-range order is retained: $[a_{\mu} (z) = \mu c_{0} + z_{\mu}]$ , where $[z_{\mu}]$ denotes the displacement of the µth molecule in the chain. Frequently used are Gaussian distributions: $[c' \exp \{- (z - \mu c_{0})^{2}/2\Delta^{2}\}]$ ( [c'] = normalizing constant; $[\Delta ]$ = standard deviation). Fourier transformation [equation (4.2.4.45)] gives the well known result $[I_{d} \sim (1 - \exp \{- L^{2} \Delta^{2}\}),]$ i.e. a monotonically increasing intensity with L (modulation due to a molecular structure factor neglected). This result is quite analogous to the treatment of the scattering of independently vibrating atoms. If (short-range) correlations exist between the molecules the Gaussian distribution is replaced by a multivariate normal distribution where correlation coefficients $[\kappa^{\mu}]$ $[(0 \lt \kappa \lt 1)]$ between a molecule and its µth neighbour are incorporated. $[\kappa^{\mu}]$ is defined by the second moment: $[\langle z_{0} z_{\mu}\rangle/\Delta^{2}]$ . $[a_{\mu} (z) = c'' \exp \{- (z - \mu c_{0})^{2}/2\Delta^{2} (1 - \kappa^{\mu})\}.]$ Obviously the variance increases if the correlation diminishes and reaches an upper bound of twice the single site variance. Fourier transformation gives an expression for diffuse intensity (Welberry, 1985): $[\eqalignno{I_{d} (L) &\sim \exp \{- L^{2} \Delta^{2}\} \textstyle\sum\limits_{j} (- L^{2} \Delta^{2})\hskip 2pt^{j}/j! &\cr &\quad \times (1 - \kappa^{2j})/(1 + \kappa^{2j} - 2\kappa\hskip 2pt^{j} \cos 2\pi Lc_{0}). &(4.2.4.46)}]$ For small Δ, terms with $[j \gt 1]$ are mostly neglected. The terms become increasingly important with higher values of L. On the other hand, $[\kappa\hskip 2pt^{j}]$ becomes smaller with increasing j, each additional term in equation (4.2.4.46) becomes broader and, as a consequence, the diffuse planes in reciprocal space become broader with higher L.

In a different way – in the paracrystal method – the position of the second and subsequent molecules with respect to some reference zero point depends on the actual position of the predecessor. The variance of the position of the µth molecule relative to the first becomes unlimited. There is a continuous transition to a fluid-like behaviour of the chain molecules. This 1D paracrystal (sometimes called distortions of second kind) is only a special case of the 3D paracrystal concept (see Hosemann & Bagchi, 1962; Wilke, 1983). Despite some difficulties with this concept (Brämer, 1975; Brämer & Ruland, 1976) it is widely used as a theoretical model for describing diffraction of highly distorted lattices. One essential development is to limit the size of a paracrystalline grain so that fluctuations never become too large (Hosemann, 1975).

If this concept is used for the 1D case, $[a_{\mu}(z)]$ is defined by convolution products of $[a_{1}(z)]$ . For example, the probability of finding the next-nearest molecule is given by $[a_{2}(z) = \textstyle\int a_{1}(z')a_{1}(z - z') \;\hbox{d}z' = a_{1}(z) * a_{1}(z)]$ and, generally: $[a_{\mu}(z) = a_{1}(z) * a_{1}(z) * a_{1}(z) * \ldots * a_{1}(z)]$ (µ-fold convolution).

The mean distance between next-nearest neighbours is $[\langle c \rangle = \textstyle\int z' a_{1}(z') \;\hbox{d}z']$ and between neighbours of the µth order: $[\mu \langle c \rangle]$ . The average value of $[a_{\mu} = 1/\langle c \rangle]$ , which is also the value of w(z) for $[z > z_{k}]$ , where the distribution function is completely smeared out. The general expression for the interference function G(L) is $[G(L) = 1 + \textstyle\sum\limits_{\mu} \{F^{\mu} + F^{+\mu}\} = \hbox{Re} \{(1 + F)/(1 - F)\} \eqno(4.2.4.47)]$ with $[F(L) \leftrightarrow a_{1}(z), F^{\mu}(L) \leftrightarrow a_{\mu}(z)]$ .

With $[F = |F|e^{i\chi}\ (\chi = L\langle c \rangle)]$ , equation (4.2.4.47) is written: $[G(L) = [1 - |F(L)|^{2}]/[1 - 2|F(L)| \cos \chi + |F(L)|^{2}]. \eqno(4.2.4.47a)]$ [Note the close similarity to the diffuse part of equation (4.2.4.5), which is valid for 1D disorder problems.]

This function has maxima of height [(1 + |F|)/(1 - |F|)] and minima of height [(1 - |F|)/(1 + |F|)] at positions $[lc^{*}]$ and $[\left(l + {1\over 2}\right)c^{*}]$ , respectively. With decreasing [|F|] the oscillations vanish; a critical L value (corresponding to $[z_{k}]$ ) may be defined by G_max/G_min ≲ 1.2. Actual values depend strongly on $[F({\bf H})]$ .

The paracrystal method is substantiated by the choice $[a_{1}(z)]$ , i.e. the disorder model. Again, frequently used is a Gaussian distribution: $[\eqalign{a_{1}(z) &= 1/\sqrt{2\pi} \Delta \cdot \exp \{- (z - \langle c \rangle)^{2}/2\Delta^{2}\}\cr a_{\mu}(z) &= 1/\sqrt{\mu} \cdot 1/\sqrt{2\pi} \Delta \cdot \exp \{- (z - \mu \langle c \rangle)^{2}/2\mu \Delta^{2}\}} \eqno(4.2.4.48)]$ with the two parameters $[\langle c \rangle, \Delta]$ .

There are peaks of height $[1/[\pi^{2} L^{2} (\Delta/\langle c \rangle)^{2}]]$ which obviously decrease with $[L^{2}]$ and $[(\Delta/\langle c \rangle)^{2}]$ . The oscillations vanish for $[|F| \simeq 0.1]$ , i.e. $[1/\langle c \rangle \simeq 0.25/\Delta]$ . The width of the mth peak is $[\Delta_{m} = \sqrt{m} \Delta]$ . The integral reflectivity is approximately $[1/\langle c \rangle [1 - \pi^{2} L^{2} (\Delta/\langle c \rangle)^{2}]]$ and the integral width (defined by integral reflectivity divided by peak reflectivity) (background subtracted!) $[1/\langle c \rangle \pi^{2} L^{2} (\Delta/\langle c \rangle)^{2}]$ which, therefore, increases with $[L^{2}]$ . In principle the same results are given by Zernike & Prins (1927). In practice a single Gaussian distribution is not fully adequate and modified functions must be used (Rosshirt et al., 1985).

A final remark concerns the normalization [equation (4.2.4.39)]. Going from (4.2.4.39) to (4.2.4.45) it is assumed that N is a large number so that the correct normalization factors $[(N - |\mu|)]$ for each $[a_{\mu}(z)]$ may be approximated by a uniform N. If this is not true then $[\eqalignno{G(L) &= N + \textstyle\sum\limits_{\mu} (N - |\mu|)(F^{\mu} + F^{+\mu}) \cr &= N \;\hbox{Re}\; \{(1 + |F|)/(1 - |F|)\} &(4.2.4.49)\cr &\quad -2 \;\hbox{Re}\; \{|F|(1 - |F|^{N})/(1 - |F|^{2})\}.}]$ The correction term may be important in the case of relatively small (1D) domains. As mentioned above, the structure factor of a chain molecule was neglected. The H dependence of $[F_{m}]$ , of course, obscures the intensity variation of the diffuse layers as described by (4.2.4.47a).

The matrix method developed for the case of planar disorder was adapted to 2D disorder by Scaringe & Ibers (1979). Other models and corresponding expressions for diffuse scattering are developed from specific microscopic models (potentials), e.g. in the case of Hg_3−δAsF₆ (Emery & Axe, 1978; Radons et al., 1983), hollandites (Beyeler et al., 1980; Ishii, 1983), iodine chain compounds (Endres et al., 1982) or urea inclusion compounds (Forst et al., 1987).

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

Beyeler, H. U., Pietronero, L. & Strässler, S. (1980). Configurational model for a one-dimensional ionic conductor. Phys. Rev. B, 22, 2988–3000.Google Scholar

Brämer, R. (1975). Statistische Probleme der Theorie des Parakristalls. Acta Cryst. A31, 551–560.Google Scholar

Brämer, R. & Ruland, W. (1976). The limitations of the paracrystalline model of disorder. Macromol. Chem. 177, 3601–3617.Google Scholar

Comes, R. & Shirane, G. (1979). X-ray and neutron scattering from one-dimensional conductors. In Highly conducting one dimensional solids, ch. 2, edited by J. T. Devreese, R. P. Evrard & V. E. Van Doren. New York: Plenum.Google Scholar

Emery, V. J. & Axe, J. D. (1978). One-dimensional fluctuations and the chain-ordering transformation in Hg_{3 − δ}AsF₆. Phys. Rev. Lett. 40, 1507–1511.Google Scholar

Endres, H., Pouget, J. P. & Comes, R. (1982). Diffuse X-ray scattering and order–disorder effects in the iodine chain compounds N,N′-diethyl-N,N′-dihydrophenazinium iodide, E₂PI_1.6 and N,N′-dibenzyl-N,N′-dihydrophenazinium iodide, B₂PI_1.6. J. Phys. Chem. Solids, 43, 739–748.Google Scholar

Forst, R., Jagodzinski, H., Boysen, H. & Frey, F. (1987). Diffuse scattering and disorder in urea inclusion compounds OC(NH₂)₂ + C_nH_{2n + 2}. Acta Cryst. B43, 187–197.Google Scholar

Guinier, A. (1963). X-ray diffraction in crystals, imperfect solids and amorphous bodies. San Francisco: Freeman.Google Scholar

Hosemann, R. (1975). Micro paracrystallites and paracrystalline superstructures. Macromol. Chem. Suppl. 1, pp. 559–577.Google Scholar

Hosemann, R. & Bagchi, S. N. (1962). Direct analysis of diffraction by matter. Amsterdam: North-Holland.Google Scholar

Ishii, T. (1983). Static structure factor of Frenkel–Kontorova-systems at high temperatures. Application to K-hollandite. J. Phys. Soc. Jpn, 52, 4066–4073.Google Scholar

Radons, W., Keller, J. & Geisel, T. (1983). Dynamical structure factor of a 1-d harmonic liquid: comparison of different approximation methods. Z. Phys. B, 50, 289–296.Google Scholar

Rosshirt, E., Frey, F., Boysen, H. & Jagodzinski, H. (1985). Chain ordering in E₂PI_1.6 (5,10-diethylphenazinium iodide). Acta Cryst. B41, 66–76.Google Scholar

Scaringe, P. R. & Ibers, J. A. (1979). Application of the matrix method to the calculation of diffuse scattering in linearly disordered crystals. Acta Cryst. A35, 803–810.Google Scholar

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

Welberry, T. R. (1985). Diffuse X-ray scattering and models of disorder. Rep. Prog. Phys. 48, 1543–1593.Google Scholar

Wilke, W. (1983). General lattice factor of the ideal paracrystal. Acta Cryst. A39, 864–867.Google Scholar

Zernike, F. & Prins, J. A. (1927). Die Beugung von Röntgenstrahlen in Flüssigkeiten als Effekt der Molekülanordnung. Z. Phys. 41, 184–194.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 4.2, pp. 425-429