An extension is proposed of the Shannon entropy-based structural complexity measure introduced by Krivovichev, taking into account the geometric coordinational degrees of freedom a crystal structure has. This allows a discrimination to be made between crystal structures which share the same number of atoms in their reduced cells, yet differ in the number of their free parameters with respect to their fractional atomic coordinates. The strong additivity property of the Shannon entropy is used to shed light on the complexity measure of Krivovichev and how it gains complexity contributions due to single Wyckoff positions. Using the same property allows for combining the proposed coordinational complexity measure with Krivovichev's combinatorial one to give a unique quantitative descriptor of a crystal structure's configurational complexity. An additional contribution of chemical degrees of freedom is discussed, yielding an even more refined scheme of complexity measures which can be obtained from a crystal structure's description: the six C's of complexity.

Institute of Physics of the Academy of Sciences of the Czech Republic Na Slovance 2 182 21 Praha 8 Czech Republic

See more in PubMed

Aczél, J. & Daróczy, Z. (1975). On Measures of Information and Their Characterizations. Mathematics in Science and Engineering, Vol. 115. New York: Academic Press.

Aczél, J., Forte, B. & Ng, C. T. (1974). Adv. Appl. Probab. 6, 131–146.

Aroyo, M. I. (2016). International Tables for Crystallography, Vol. A, Space-group symmetry. Chichester: Wiley.

Baez, J. C., Fritz, T. & Leinster, T. (2011). Entropy, 13, 1945–1957.

Bertz, S. H. (1981). J. Am. Chem. Soc. 103, 3599–3601.

Bertz, S. H. (1983). Bull. Math. Biol. 45, 849–855.

Blaikie, N. (2003). Analyzing Quantitative Data – From Description to Explanation. London: SAGE Publications.

Csiszár, I. (2008). Entropy 10, 261–273.

Dshemuchadse, J. & Steurer, W. (2015). Inorg. Chem. 54, 1120–1128. PubMed

Flores-Gallegos, N. (2019). Chem. Phys. Lett. 720, 1–6.

Hornfeck, W. (2012). Acta Cryst. A68, 167–180. PubMed

Hornfeck, W. & Hoch, C. (2015). Acta Cryst. B71, 752–767. PubMed

Krivovichev, S. (2012a). Acta Cryst. A68, 393–398. PubMed

Krivovichev, S. V. (2012b). Struct. Chem. 23, 1045–1052.

Krivovichev, S. V. (2013a). Microporous Mesoporous Mater. 171, 223–229.

Krivovichev, S. V. (2013b). Miner. Mag. 77, 275–326.

Krivovichev, S. V. (2014a). Miner. Mag. 78, 415–435.

Krivovichev, S. V. (2014b). Angew. Chem. Int. Ed. 53, 654–661. PubMed

Krivovichev, S. V. (2016). Acta Cryst. B72, 274–276. PubMed

Krivovichev, S. V. (2017). Crystallogr. Rev. 23, 2–71.

Krivovichev, S. V., Hawthorne, F. C. & Williams, P. A. (2017). Struct. Chem. 28, 153–159.

Krivovichev, S. V. & Krivovichev, V. G. (2020). Acta Cryst. A76, 429–431. PubMed PMC

Krivovichev, S. V., Krivovichev, V. G. & Hazen, R. M. (2018). Eur. J. Mineral. 30, 231–236.

Mackay, A. L. (1984). Croat. Chem. Acta, 57, 725–736.

Mackay, A. L. (2001). Crystallogr. Rep. 46, 524–526.

Parthé, E., Cenzual, K. & Gladyshevskii, R. E. (1993). J. Alloys Compd. 197, 291–301.

Parthé, E. & Gelato, L. M. (1984). Acta Cryst. A40, 169–183.

Pettifor, D. G. (1984). Solid State Commun. 51, 31–34.

Pettifor, D. G. (1986). J. Phys. C Solid State Phys. 19, 285–313.

Rashevsky, N. (1955). Bull. Math. Biophys. 17, 229–235.

Shannon, C. E. (1948a). Bell Syst. Tech. J. 27, 379–423.

Shannon, C. E. (1948b). Bell Syst. Tech. J. 27, 623–656.

Siidra, O. I., Zenko, D. S. & Krivovichev, S. V. (2014). Am. Mineral. 99, 817–823.

Steurer, W. & Dshemuchadse, J. (2016). Intermetallics – Structures, Properties and Statistics. IUCr Monographs on Crystallography, No. 26. Oxford University Press.

Tambornino, F., Sappl, J. & Hoch, C. (2015). J. Alloys Compd. 618, 326–335.

Taneja, I. J. (2001). Generalized Information Measures and Their Applications. Ebook, http://www.mtm.ufsc.br/~taneja/book/book.html. Retrieved 20/06/2019.

Villars, P. & Cenzual, K. (2013). Pearson’s Crystal Data Crystal Structure Database for Inorganic Compounds. ASM International, Materials Park, Ohio, USA. http://www.crystalimpact.com/pcd/Default.htm.

Yodogawa, E. (1982). Percept. Psychophys. 32, 230–240. PubMed

On the combinatorics of crystal structures. II. Number of Wyckoff sequences of a given subdivision complexity