Abstract
Disperse systems represent extreme situations of total absence of correlation between the scattering units. There are internal correlations only. At the other extreme are the crystalline systems where the correlations between the scattering units are constant, resulting in understanding the long-range order in macroscopic scales. Between these two extreme situations are the systems with arbitrary correlations, which are complex systems where there are correlations between the scattering units, but these correlations may vary along the physical length of the systems.
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Notes
- 1.
In a gas, the scattering units are molecules and the internal correlations producing interference effects are the interatomic distances r ab , e.g. (2.7). In the case of monatomic gas, the atoms are the scattering units and the internal correlations are the atomic orbitals contributing to the atomic scattering factor, (1.46).
- 2.
Recalling that p(u) = u 2 c(u) and \(\sum _{a}\vert f_{a}(Q)\vert ^{2} = N_{at}\langle f^{2}(Q)\rangle = N_{at}f_{\mathrm{m}}^{2}(Q)\).
- 3.
The crystalline structure of Au is face-centered cubic (fcc).
- 4.
- 5.
Mean square deviation: \(\langle \mathrm{d}x^{2}\rangle =\sum _{ i=1}^{N}(x_{i} -\bar{ x})^{2}/(N - 1)\).
- 6.
The validity of this approximation for different weight functions can be checked by a numerical comparison similar to that performed for verifying (3.21).
- 7.
Results obtained by the expansion in Taylor series of \(\exp (-Q^{2}\langle r_{jk}^{2}\rangle /6)\) and using that \(\sum _{j=1,\,k>j}^{N_{m}}(k - j)^{\,n} \simeq N_{m}^{n+2}/[(n + 2)(n + 1)]\) for N m → ∞. See also Lindner and Zemb (2002, p. 268).
- 8.
This phenomenon is analogous to that observed in particles with a crystalline structure, e.g. Exercise 3.3. The larger the particle, the more definite are the diffraction peaks.
- 9.
The convolution of two FTs is done in a variable \({\boldsymbol q}\), (1.32), and when one of them is a delta function, the other FT remains unchanged.
- 10.
In disperse systems, the structural function S(Q) is associated with the internal structure of the particles, such as in (2.8). In literature, it is common to represent the interference function arising from the mutual interference between particles also with \(S({\boldsymbol Q})\), which is the same symbol, a similar meaning, but in slightly different context.
- 11.
Although the general aspects of the RDF of liquid water were taken from Hura et al. (2003), the interference function S(Q), especially in the interval Q < 2 Å−1, is very sensitive to the accuracy of function g(u). Therefore, the scattering simulated in this exercise should not be considered an exact reproduction of the water’s experimental scattering at NTP.
- 12.
In isotropic systems, \(g({\boldsymbol u})\,\mathrm{d}V _{u}\; \rightarrow \; 4\pi u^{2}g(u)\,\mathrm{d}u\).
- 13.
Hard sphere model: interaction potential is zero for u ≥ d and infinite for u < d (Hansen and McDonald 1990).
- 14.
For more complex systems there are other methods for extracting the RDFs g α β (u) from the experimental data. See, for example, Egami and Billinge (2003, Ch. 3, p. 64).
Bibliography
Als-Nielsen, J., McMorrow, D.: Elements of Modern X-Ray Physics. Wiley, New York (2001)
Barbosa, L.R.S.: Small angle X-ray scattering study of biological relevant systems. Ph.D. Thesis, Institute of Physics, University of São Paulo (2008). http://www.teses.usp.br/teses/disponiveis/43/43134/tde-20032009-101046/pt-br.php
Egami, T., Billinge, S.J.L. (eds.): Underneath the Bragg Peaks. Pergamon, Oxford (2003)
Giacovazzo, C. (ed.): Fundamentals of Crystallography, 2nd edn. Oxford University, Oxford (2002)
Glatter, O., Kratky, O. (eds.): Small Angle Scattering of X-Rays. Academic, London (1982)
Guinier, A., Fournet, G.: Small-Angle Scattering of X-Rays. Wiley, New York (1955)
Hansen, J.-P., McDonald, I.R.: Theory of Simple Liquids, 2nd edn. Academic, London (1990)
Hura, G., Russo, D., Glaeser, R.M., Head-Gordon, T., Krack, M., Parrinello, M.: Water structure as a function of temperature from x-ray scattering experiments and ab initio molecular dynamics. Phys. Chem. Chem. Phys. 5, 1981–1991 (2003)
Lindner, P., Zemb, T. (eds.): Neutrons, X-Rays and Light: Scattering Methods Applied to Soft Condensed Matter. Elsevier, Amsterdam (2002)
Vocadlo, D.J., Davies, G.J., Laine, R., Withers, S.G.: Catalysis by hen egg-white lysozyme proceeds via a covalent intermediate. Nature 412, 835–838 (2001)
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Morelhão, S.L. (2016). Complex Systems. In: Computer Simulation Tools for X-ray Analysis. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-19554-4_3
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DOI: https://doi.org/10.1007/978-3-319-19554-4_3
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