Disorder within 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. 427-429 | 1 | 2 |

Section 4.2.4.3.2. Disorder within 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

4.2.4.3.2. 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).

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

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

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. 427-429