# Paracrystalline

Paracrystalline materials are defined as having short and medium range ordering in their lattice (similar to the liquid crystal phases) but lacking crystal like long-range ordering at least in one direction.[1]

Ordering is the regularity in which atoms appear in a predictable lattice, as measured from one point. In a highly ordered, perfectly crystalline material, or single crystal, the location of every atom in the structure can be described exactly measuring out from a single origin. Conversely, in a disordered structure such as a liquid or amorphous solid, the location of the first and perhaps second nearest neighbors can be described from an origin (with some degree of uncertainty) and the ability to predict locations decreases rapidly from there out. The distance at which atom locations can be predicted is referred to as the correlation length ${\displaystyle \xi }$. A paracrystalline material exhibits correlation somewhere between the fully amorphous and fully crystalline.

The primary, most accessible source of crystallinity information is X-ray diffraction and cryo-electron microscopy,[2] although other techniques may be needed to observe the complex structure of paracrystalline materials, such as fluctuation electron microscopy[3] in combination with Density of states modeling[4] of electronic and vibrational states. Scanning transmission electron microscopy can provide real-space and reciprocal space characterization of paracrystallinity in nanoscale material, such as quantum dot solids.[5]

The scattering of X-rays, neutrons and electrons on paracrystals is quantitatively described by the theories of the ideal[6] and real[7] paracrystal.

Rolf Hosemann’s definition of an ideal paracrystal reads: "The electron density distribution of any material is equivalent to that of a paracrystal when there is for every building block one ideal point so that the distance statistics to other ideal points is identical for all of these points. The electron configuration of each building block around its ideal point is statistically independent of its counterpart in neighboring building blocks. A building block corresponds then to the material content of a cell of this "blurred" space lattice, which is to be considered a paracrystal."[8]

Numerical differences in analyses of diffraction experiments on the basis of either of these two theories of paracrystallinity can often be neglected.[9]

Just like ideal crystals, ideal paracrystals extend theoretically to infinity. Real paracrystals, on the other hand, follow the empirical α*-law,[10] which restricts their size. That size is also indirectly proportional to the components of the tensor of the paracrystalline distortion. Larger solid state aggregates are then composed of micro-paracrystals.[11]

The words "paracrystallinity" and "paracrystal" were coined by the late Friedrich Rinne in the year 1933.[12] Their German equivalents, e.g. "Parakristall", appeared in print one year earlier.[13]

## Paracrystalline model

Structure of silica – an example of a paracrystalline, or partially disordered lattice

The paracrystalline model is a revision of the Continuous Random Network model first proposed by W. H. Zachariasen in 1932.[14] One type of paracrystal model contains highly strained, microcrystalline crystallites surrounded by fully amorphous material.[15] This is a higher energy state than the continuous random network model. The important distinction between this model and the microcrystalline phases is the lack of defined grain boundaries and highly strained lattice parameters, which makes calculations of molecular and lattice dynamics difficult. A general theory of paracrystals has been formulated in a basic textbook,[16] and then further developed/refined by various authors.

## Applications

The paracrystal model has been useful, for example, in describing the state of partially amorphous semiconductor materials after deposition. It has also been successfully applied to synthetic polymers, liquid crystals, biopolymers, quantum dot solids, and biomembranes.[17]

## References

1. ^ Voyles, P. M.; Zotov, N.; Nakhmanson, S. M.; Drabold, D. A.; Gibson, J. M.; Treacy, M. M. J.; Keblinski, P. (2001). "Structure and physical properties of paracrystalline atomistic models of amorphous silicon". Journal of Applied Physics. 90 (9): 4437. Bibcode:2001JAP....90.4437V. doi:10.1063/1.1407319.
2. ^ Berriman, J. A.; Li, S.; Hewlett, L. J.; Wasilewski, S.; Kiskin, F. N.; Carter, T.; Hannah, M. J.; Rosenthal, P. B. (29 September 2009). "Structural organization of Weibel-Palade bodies revealed by cryo-EM of vitrified endothelial cells". Proceedings of the National Academy of Sciences. 106 (41): 17407–17412. Bibcode:2009PNAS..10617407B. doi:10.1073/pnas.0902977106.
3. ^ Biswas, Parthapratim; Atta-Fynn, Raymond; Chakraborty, S; Drabold, D A (2007). "Real space information from fluctuation electron microscopy: Applications to amorphous silicon". Journal of Physics: Condensed Matter. 19 (45): 455202. arXiv:. Bibcode:2007JPCM...19S5202B. doi:10.1088/0953-8984/19/45/455202.
4. ^ Nakhmanson, S.; Voyles, P.; Mousseau, Normand; Barkema, G.; Drabold, D. (2001). "Realistic models of paracrystalline silicon". Physical Review B. 63 (23). Bibcode:2001PhRvB..63w5207N. doi:10.1103/PhysRevB.63.235207.
5. ^ B. Savitzky, R. Hovden, K. Whitham, J. Yang, F. Wise, T. Hanrath, and L.F. Kourkoutis (2016). "Propagation of Structural Disorder in Epitaxially Connected Quantum Dot Solids from Atomic to Micron Scale". 16: 5714−5718. doi:10.1021/acs.nanolett.6b02382.
6. ^ R. Hosemann: Röntgeninterferenzen an Stoffen mit flüssigkeitsstatistischen Gitterstörungen, Zeitschrift für Physik. Band 128, Nr. 1, 1950, S. 1–35, doi:10.1007/BF01339555
7. ^ R. Hosemann: Grundlagen der Theorie des Parakristalls und ihre Anwendungensmöglichkeiten bei der Untersuchung der Realstruktur kristalliner Stoffe, Kristall und Technik, Band 11, 1976, S. 1139–1151
8. ^ R. Hosemann, Der ideale Parakristall und die von ihm gestreute kohaerente Roentgenstrahlung, Zeitschrift für Physik 128 (1950) 465-492
9. ^ R. Hosemann, W. Vogel, D. Weick, F. J. Baltá-Calleja: Novel aspects of the real paracrystal. In: Acta Crystallographica Section A. Band 37, Nr. 1, January 1981, pp. 85–91, doi:10.1107/S0567739481000156
10. ^ R. Hosemann, M. P. Hentschel, F. J. Balta-Calleja, E. Lopez Cabarcos, A. M. Hindeleh, The α*-constant, equilibrium state and bearing netplanes in polymers, biopolymers and catalysts, Journal of Physics C: Solid State Physics. Band 18, Nr. 5, June 2001, pp. 249–254
11. ^ A. M. Hindeleh and R. Hosemann: Microparacrystals: The intermediate stage between crystalline and amorphous, J. Materials Sci. Band 26, 1991, pp. 5127–5133, doi:10.1007/BF01143202
12. ^ F. Rinne, Investigations and considerations concerning paracrystallinity, Transactions of the Faraday Society 29 (1933) 1016-1032
13. ^ Friedrich Rinne: Investigations and considerations concerning paracrystallinity, Transactions of the Faraday Society. Band 29, Nr. 140, January 1933, pp. 1016–1032, ISSN 0014-7672, doi:10.1039/TF9332901016
14. ^ Zachariasen, W.H. (1932). "The Atomic Arrangement in Glass". J. Am. Chem. Soc. 54 (10): 3841. doi:10.1021/ja01349a006.
15. ^ Cowley, J.M. (1981). I. Hargittai; W. J. Orville Thomas, eds. Diffraction Studies on Non-Crystalline Substances. Budapest: Akademia Kiado. p. 13. ISBN 9780444997524.
16. ^ Hosemann R.; Bagchi R.N. (1962). Direct analysis of diffraction by matter. Amsterdam; New York: North-Holland. OCLC 594302398.
17. ^ Baianu I.C. (1978). "X-ray scattering by partially disordered membrane systems". Acta Crystallogr. A. 34 (5): 751–753. Bibcode:1978AcCrA..34..751B. doi:10.1107/S0567739478001540.