Proof that a dominant endmember formula can always be written for a mineral or a crystal structure

• Published in: Canadian mineralogist Vol. 59; no. 1; pp. 159 - 167
• Main Author: Hawthorne, Frank C
• Format: Journal Article
• Language: English
• Published: Mineralogical Association of Canada 26-05-2021
• Subjects: amphibole group; chain silicates; concepts; crystal chemistry; crystal structure; end members; formula; garnet group; isomorphism; Mineralogy; nesosilicates; orthosilicates; silicates; valency
• ISSN: 0008-4476; 1499-1276
• DOI: 10.3749/canmin.2000062

An endmember formula must be: (1) conformable with the crystal structure of the mineral, (2) electroneutral (i.e., not carry a net electric charge), and (3) irreducible [i.e., not capable of being factored into components that have the same bond topology (atomic arrangement) as that of the original formula]. The stoichiometry of an endmember formula must match the "stoichiometry" of the sites in the structure; for ease of expression, I denote such a formula here as a chemical endmember. In order for a chemical endmember to be a true endmember, the corresponding structure must obey the valence-sum rule of bond-valence theory. For most minerals, the chemical endmember and the (true) endmember are the same. However, where local order would lead to strong deviation from the valence-sum rule for some local arrangements, such arrangements cannot occur and the (true) endmember differs from the chemical endmember. I present heuristic and algebraic proofs that a specific chemical formula can always be represented by a corresponding dominant endmember formula. That dominant endmember may be derived by calculating the difference between the mineral formula considered and all of the possible endmember compositions; the endmember formula which is closest to the mineral formula considered is the dominant endmember.