DBPapers
DOI: 10.5593/sgem2017/61/S24.040

SOLID SOLUTIONS – A HUME-ROTHERY CONDITION IN THE CONTEXT OF THE SET-THEORETIC NOTION OF BINARY RELATION

M. Michalak, G. Bytomski
Tuesday 12 September 2017 by Libadmin2017

References: 17th International Multidisciplinary Scientific GeoConference SGEM 2017, www.sgem.org, SGEM2017 Conference Proceedings, ISBN 978-619-7408-12-6 / ISSN 1314-2704, 29 June - 5 July, 2017, Vol. 17, Issue 61, 301-306 pp, DOI: 10.5593/sgem2017/61/S24.040

ABSTRACT

In this paper, the authors discussed one of the Hume-Rothery conditions for occurring substitutional solid solutions in the context of binary relations. The analyzed condition says that substitutional solid solutions may form if the ionic radius of the solute and solvent ions differ by no more than 15%. Because the problem concerns a mathematical connection between two objects, the authors suggested describing this problem using the set-theoretic notion of binary relation. In this context, the possibility of forming solid solutions is reduced to meeting a mathematical condition accompanying the introduced binary relation. From a mathematical point of view, it seems to be natural to check whether the introduced relation can be treated as an equivalence relation. The authors presented a proof that such a relation is not an equivalence relation as it does not meet all the necessary conditions (reflexive, symmetric and transitive) for an equivalence relation. There is presented the main consequence of this result – nonexistence of equivalence classes that in terms of the described problem represent disjoint sets containing ions that form solid solutions.

Keywords: Hume-Rothery conditions, solid solutions, ionic radii, mineralogy, binary relation