Valence and oxidation state

IUPAC (Reference McNaught and Wilkinson2019) (the International Union of Pure and Applied Chemistry) defines valence as follows: “The maximum number of univalent atoms (originally hydrogen or chlorine atoms) that may combine with an atom of the element under consideration, or with a fragment, or for which an atom of this element can be substituted”. Two other widely accepted versions are as follows: “…the number of hydrogen atoms that can combine with an element in a binary hydride or twice the number of oxygen atoms combining with an element in its oxide or oxides” (Greenwood and Earnshaw, Reference Greenwood and Earnshaw1997; page 27), and “…the number of electrons that an atom uses in bonding” (Parkin, Reference Parkin2006; page 791). IUPAC (Reference McNaught and Wilkinson2019) defines oxidation state as follows: “the charge on an atom after ionic approximation of its heteronuclear bonds”. Whichever definition one accepts for valence, it is clear that valence does not have a sign whereas oxidation state (number) does have a sign (or is 0).

Electric charge and electroneutrality

A neutral atom has equal numbers of protons and electrons. By extension, a set of neutral atoms has equal numbers of protons and electrons. Where the atoms in this set form a solid, some of the electrons are shared between atoms to form chemical bonds, but the total numbers of electrons and protons remain the same, and this set of bonded atoms, e.g. a crystal, is electroneutral. Electric charge is a continuously differentiable function and hence is subject to a conservation law as required by Noether's first theorem (Quigg, Reference Quigg2013), and the global gauge invariance of the electromagnetic field results in the conservation of electric charge. Thus the electroneutrality of an assemblage of atoms in a crystal structure must be conserved, irrespective of the oxidation states of the constituent ions. Any deviation from conservation of electric charge violates one of the most fundamental laws of Physics.

End-member

An end-member is defined as one end of a range or series made up of similar members; more specifically in Mineralogy, a mineral that occurs at one end in a range of solid solutions (https://en.wiktionary.org/wiki/endmember). At least with respect to minerals, this definition is wrong. As the concept of an end-member is central to Mineralogy and Petrology, and also central to the issues to be considered here, some discussion is desirable.

Let us consider the experimental data on a set of minerals: a set of chemical analyses, a set of unit-cell dimensions, etc. In detail, all of these chemical analyses are different from each other, and all of these unit-cell dimensions are different from each other; how do we know whether these sets of data correspond to the same mineral species?

In order to decide whether objects belong to the same set, one needs a set of properties that are the same for all these objects in this set, and this set of properties constitutes the definition of an abstract object with which we can compare real objects and decide if the real objects belong to that same set, i.e. are the same as each other (or not). Properties of objects may be divided into intrinsic properties and extrinsic properties: an intrinsic property is a property that is characteristic of the object itself and is independent of anything else; an extrinsic property is a property that depends on the relation between the object itself and other things. The chemical analyses (and associated chemical formulae) and the unit-cell dimensions mentioned above are extrinsic properties: they differ from sample to sample, and the unit-cell dimensions also vary with changing temperature and pressure. Hence chemical analyses, chemical formulae and unit-cell dimensions are extrinsic properties and cannot be used to define the mental object with which we can compare our data. Intrinsic properties that may be used to define this mental object, which henceforth I will call an end-member, are as follows: (1) the end-member formula (including Z, the number of formula units in the unit cell); (2) the space group and bond topology of the end-member structure; and (3) the name of the end-member. This set of intrinsic properties constitutes a set of universals that are associated with the archetype of a mineral species. This archetype is an abstract object and all real mineral samples corresponding to this archetype are imperfect copies of that archetype but have the set of intrinsic properties identical to the corresponding set of universals. This correspondence allows us to say that the different mineral samples are the same mineral species (or not). So, to summarise: An end-member is not a mineral, it is an abstract concept that we use to identify minerals and a particular mineral species (Hawthorne et al., Reference Hawthorne, Mills, Hatert and Rumsey2021).

A hypothetical amphibole

Consider the root formula of pargasite (Hawthorne et al., Reference Hawthorne, Oberti, Harlow, Maresch, Martin, Schumacher and Welch2012): NaCa₂(Mg₄Al)(Si₆Al₂)O₂₂(OH)₂. The following substitution maintains the general stoichiometry of the amphibole structure (AB₂C₅T₈O₂₂W₂): ^A□ + ^CMg + ^W□₂ → ^ANa + ^CAl + ^W(OH)₂ → □Ca₂Mg₅(Si₆Al₂)O₂₂□₂; this formula is electroneutral and fits the general stoichiometry of the amphiboles. However, it cannot exist as a crystal structure as it violates several requirements for a stable structure. The occurrence of a vacancy at the W position of the general formula may maintain electroneutrality of the formula, but it means that the O(3) site is occupied by a vacancy which, in turn, means that the M(1) and M(2) sites are [4]-coordinated, in particular imparting a one-sided coordination to the cation(s) at the M(1) site, leading to major violations of the valence-sum rule (Brown, Reference Brown2016; Hawthorne, Reference Hawthorne2012, Reference Hawthorne2015) and non-stability of the amphibole-type arrangement of ions implied by the formula □Ca₂Mg₅(Si₆Al₂)O₂₂□₂. In this regard, it must be emphasised that the algebraic fit of a formula to the general stoichiometry of a mineral structure is a necessary but not sufficient condition for stability. The arrangement of ions implied by the formula must also be a stable structure (Hawthorne, Reference Hawthorne2021a).

A real example: Y-rich hainite

Lyalina et al. (Reference Lyalina, Zolotarev, Selivanova, Savchenko, Zozulya, Krivovichev and Mikhailova2015) refined the crystal structure of Y-rich hainite of approximate chemical formula Na(NaCa)Ca₂(CaY)Ti(Si₂O₇)₂(OF)F₂. This formula may be split into two chemical formulae as follows: NaNa₂Ca₂(CaY)Ti(Si₂O₇)₂F₂F₂ and NaCa₂Ca₂(CaY)Ti(Si₂O₇)₂O₂F₂. Using bond-valence theory, Hawthorne (Reference Hawthorne2021a) showed that these formulae are not structurally stable atomic arrangements and hence cannot be end-members. The formula Na(NaCa)Ca₂(CaY)Ti(Si₂O₇)₂(OF)F₂ has various local (short-range) arrangements of cations that are stable with regard to the valence-sum rule of bond-valence theory. In the formula Na(NaCa)Ca₂(CaY)Ti(Si₂O₇)₂(OF)F₂, there are three sites that contain more than one constituent ion, apparently indicating that the formula is not irreducible. However, there are two valence-sum constraints that remove two of these apparent degrees of freedom, and electroneutrality constrains the third site (Hawthorne, Reference Hawthorne2021a). Thus, the formula Na(NaCa)Ca₂(CaY)Ti(Si₂O₇)₂(OF)F₂ is irreducible, it accords with the valence-sum rule (Brown, Reference Brown2016), and is a true end-member. This point was recognised by Sokolova and Cámara (Reference Sokolova and Cámara2017) who wrote the ideal formula for hainite-(Y) as Na(NaCa)Ca₂(CaY)Ti(Si₂O₇)₂(OF)F₂. Other TS-block minerals have dominant end-member formulae (Sokolova, Reference Sokolova2006; Sokolova and Cámara, Reference Sokolova and Cámara2017) that are likewise constrained by such short-range bond-valence requirements; it is not by accident that these minerals have the empirical formulae that they do. The number of minerals for which this situation occurs is quite small, but mineralogists need to be aware of this issue.

Calculation of the proportions of end-member constituents in minerals

The ostensible reason for not using dominant end-member formulae to define distinct mineral species was invoked by Grew et al. (Reference Grew, Locock, Mills, Galuskina, Galuskin and Hålenius2013) and stated by Bosi et al. (Reference Bosi, Hatert, Hålenius, Pasero, Miyawaki and Mills2019a; page 627) as follows: “As demonstrated for garnet-supergroup minerals, for example, the end-member approach is ambiguous, as end-member proportions strongly depend on the calculation sequence”. This calculation involves the solution of a set of simultaneous equations relating the amount of each end-member constituent to the amount of each ion in the chemical formula of the mineral. The operative word here is ‘simultaneous’; there is no sequence in finding a solution to these equations, they are solved simultaneously. Hawthorne (Reference Hawthorne2021b) examined the data of Rickwood (Reference Rickwood1968) on which these statements are based, and showed that his sequence-dependent results arise because of the use of non-stoichiometric and non-electroneutral formulae, something that Rickwood (Reference Rickwood1968) mentioned in his paper but which seems to have been overlooked by those who have cited his results. If that garnet formula is adjusted slightly such that it exactly fits the general formula of a garnet, ^[8]X₃^[6]Y₂^[4]Z₃O₁₂, and is electroneutral, the simultaneous equations relating its chemical formula to a particular set of end-member constituents have a single unique solution.

We tend to treat minor deviations in stoichiometry and electroneutrality in chemical formulae as trivial issues. However, they are not trivial issues. If the number of ions exceeds the number of sites available, a structure of that composition is physically impossible. Furthermore, if the chemical formula has a net electric charge, it is violating the conservation of electric charge, one of the most fundamental laws of Physics. Adjusting the composition of a mineral, normally within the noise of the data, seems a small price to pay for removing such fundamental physical errors from the data (Hawthorne, Reference Hawthorne2021b). Thus the ‘sequence argument’ that has been used to ostensibly justify avoiding use of the dominant end-member in the definition of a mineral species is specious, and the reason given for not using the rule of the dominant constituent is invalid.

The rule of the dominant constituent

The following statement is taken from Nickel and Grice (Reference Nickel and Grice1998; page 917): “In multiple solid-solution series, the 50% rule is interpreted to mean predominant occupancy of a particular structural site. Thus, if there are two types of atom in a structural site, the species is to be defined by the atom comprising at least 50% of that site. If there are more than two substituting atoms in the site, the species is defined by the predominant atom occupying the site”. This statement reinforces the second quote from Nickel (Reference Nickel1992) given above, a procedure which does not necessarily conserve electric charge if the atoms concerned are ions of different oxidation state.

Hatert and Burke (Reference Hatert and Burke2008; page 717) christened the method of Nickel and Grice (Reference Nickel and Grice1998) the “rule of the dominant constituent”, where the word ‘constituent’ “may designate atoms (cations or anions), molecular groups, or vacancies”. As noted above, the problem with the rule of the dominant constituent is that its operation may violate the law of conservation of electric charge. This problem is apparent in many places in Hatert and Burke (Reference Hatert and Burke2008) where one commonly meets the statement “but its idealized end-member formula…is not charge-balanced”. This statement is nonsensical as an end-member formula is electroneutral by definition (Hawthorne, Reference Hawthorne2002). The situation is correctly summarised as follows: in many cases, operation of the rule of the dominant constituent fails to produce an end-member chemical formula.

Valency-imposed double site-occupancy

Hatert and Burke (Reference Hatert and Burke2008; page 719) introduced “valency-imposed double site-occupancy” to deal with the type of issue raised by the occurrence of richterite, end-member formula Na(CaNa)Mg₅Si₈O₂₂(OH)₂ (Hawthorne et al., Reference Hawthorne, Oberti, Harlow, Maresch, Martin, Schumacher and Welch2012), i.e. occupancy of a site by two ions of different oxidation state in the end-member formula. As noted above, the use of the word ‘valency’ here is incorrect; according to the IUPAC definition, valency has no sign (see ‘Definitions’ section above); the correct term is oxidation state. However, to avoid complicating matters any further, I will retain the word ‘valence’ when discussing these various rules. The idea of valency-imposed double site-occupancy is correct, and occurs in many mineral species (Hawthorne, Reference Hawthorne2002). However, it should be realised that valency-imposed double site-occupancy directly contradicts the rule of the dominant constituent, something that has been ignored in the development of these rules.

The dominant-valency rule

Hatert and Burke (Reference Hatert and Burke2008; page 721) also introduced what they called the “dominant-valency rule” to compensate for the fact that in minerals with mixed-valence ions at more than one site, if one of the constituents is substituted by another constituent of the same valence, the valency-imposed double site-occupancy rule fails to derive an end-member formula.

I will consider the example that they give to justify this rule. Consider two sites in a structure: M and N. Hatert and Burke (Reference Hatert and Burke2008, figure 3) label the constituents at these sites in the relevant substitutions by writing the sites M and N as post-subscripts, e.g. Dⁿ⁺_N, which are easily confused with stoichiometric coefficients. Here I have modified their notation for clarity by noting the sites as pre-superscripts, e.g. ^NDⁿ⁺. They consider the following substitution:

$$\it {^M} \rm A^{n + } +{} \it ^N\rm C^{( {n + 1} ) + }\to {}\it ^M \rm B^{( {n + 1} ) + } + {}\it ^N\rm D^{n + }$$

where Aⁿ⁺, B⁽ⁿ⁺¹⁾⁺, C⁽ⁿ⁺¹⁾⁺ and Dⁿ⁺ are different constituents. They focus on the specific composition ^M(Aⁿ⁺_0.4 B⁽ⁿ⁺¹⁾⁺_0.6) ^N(C⁽ⁿ⁺¹⁾⁺_0.4 Dⁿ⁺_0.6) and allow the substitution ^NEⁿ⁺ → ^NDⁿ⁺ to replace half of ^NDⁿ⁺; the result is the composition ^M(Aⁿ⁺_0.4 B⁽ⁿ⁺¹⁾⁺_0.6)^N(C⁽ⁿ⁺¹⁾⁺_0.4 Dⁿ⁺_0.3 Eⁿ⁺_0.3). According to Hatert and Burke (Reference Hatert and Burke2008; page 721): “The strict application of the dominant-constituent rule would indicate that this composition corresponds to a new species with C⁽ⁿ⁺¹⁾⁺ instead of Dⁿ⁺ as the dominant constituent at the N site. However, the end-member formula for this supposedly new species, [B⁽ⁿ⁺¹⁾⁺]_M [C⁽ⁿ⁺¹⁾⁺]_M is impossible because it is not charge-balanced. The valency-nomenclature problem can be solved by considering the elements of the homovalent substitution Eⁿ⁺_N → Dⁿ⁺_N as a whole, so that the group of cations with n+ valency are still dominant at the N site, in spite of the majority C⁽ⁿ⁺¹⁾⁺. Consequently, species with such coupled heterovalent–homovalent substitutions must be defined by the most abundant amongst the cations with the same valency state at this site, here n+. This rule is called the dominant-valency rule, as it is necessary to preserve charge balance in any end-member formula”. Hatert and Burke (Reference Hatert and Burke2008) fail to recognise that end-member formulae are electroneutral by definition (Hawthorne, Reference Hawthorne2002) and need no such arbitrary rules to ensure this property. In this example, the three end-member constituents may be read directly from the composition ^M(Aⁿ⁺_0.4 B⁽ⁿ⁺¹⁾⁺_0.6)^N(C⁽ⁿ⁺¹⁾⁺_0.4 Dⁿ⁺_0.3 Eⁿ⁺_0.3):

and the dominant end-member formula is ^M(Aⁿ⁺) ^N(C⁽ⁿ⁺¹⁾⁺). Again, the situation is correctly summarised by the following statement: in many cases, operation of the rule of the dominant valency fails to produce an end-member chemical formula.

Specific examples of the dominant-valency rule

Hatert and Burke (Reference Hatert and Burke2008) and Bosi et al. (Reference Bosi, Hatert, Hålenius, Pasero, Miyawaki and Mills2019a) have given several examples of the operation of the dominant-valency rule to justify its use. I will consider the validity of (some of) these examples below.

Ternary feldspars

Hatert and Burke (Reference Hatert and Burke2008; page 722) considered a feldspar with the chemical formula (Ca_0.4Na_0.35K_0.25)(Al_1.4Si_2.6)O₈ and noted that “According to the current dominant-constituent rule, this mineral is Ca-dominant and would thus be anorthite. But its idealized end-member formula, CaAlSi₃O₈ is not charge-balanced. Application of the dominant-valency rule, however, clearly shows that the monovalent cations are dominant at the large crystallographic site, not Ca. Amongst these monovalent cations, Na is the dominant one, and this sample is thus simply a Ca- and K-rich albite”. What is wrong with this statement?

1. (1) A crystallographic site is a point, and a (zero-dimensional) point does not have a size.

2. (2) The formula CaAlSi₃O₈ is not an end-member formula (idealised or not), it is merely the formula given by the dominant-constituent rule.

3. (3) There are three end-member formulae in this simple feldspar system: KAlSi₃O₈, NaAlSi₃O₈ and CaAl₂Si₂O₈. The composition (Ca_0.4Na_0.35K_0.25)(Al_1.4Si_2.6)O₈ is denoted by the red circle labelled X on the ternary-feldspar composition diagram (Fig. 1). According to the dominant-valency rule, the alkali feldspars lie in the compositional field KNCD and composition X lies in the compositional field of albite: NCFB, and the compositional field of anorthite is ADC. However, dividing the diagram on the basis of the dominant end-member, the field is divided into three equal areas: KBED, NCEB and ADEC for orthoclase, albite and anorthite. The difference in the two schemes involves the yellow-shaded triangle CED in Fig. 1: this area is classed as anorthite according to the dominant end-member, and classed as albite and orthoclase according to the dominant-valency rule. The problem with the latter scheme is that the compositions CEF and DEF are called albite and orthoclase, but the highest component activity (according to Raoult's Law; Spear, Reference Spear1993; Anderson, Reference Anderson2005) is that of anorthite. Although the feldspar system is strongly non-ideal from a thermodynamic perspective, it does not seem logical to name a mineral albite where the highest (ideal) component activity is that of anorthite.

Fig. 1. The compositional field for the system orthoclase–albite–anorthite. Dashed lines BE, CE and DE are the compositional boundaries using the dominant end-member formulae to define the species. The dotted lines CFD and BEF are the compositional boundaries using the dominant-valency rule to define the species. X denotes a composition that is considered as anorthite using the dominant end-member formulae to define the species, and considered as albite using the dominant-valency rule to define the species.

Li,Fe²⁺MgMn²⁺Al-bearing tourmaline

Hatert and Burke (Reference Hatert and Burke2008; page 722) state the following: “The Y-site composition (Fe²⁺_1.5Li_0.75Al_0.75) is the boundary between schorl and elbaite series in their solid-solution series (see above). A composition (Fe²⁺_1.60Li_0.70Al_0.70) represents thus schorl, but what about the composition (Fe²⁺_0.60Mg_0.50Mn_0.50Li_0.70Al_0.70) caused by a multiple homovalent substitution? Application of the current dominant-constituent rule would lead to the name elbaite (as Li and Al are now the dominant elements at the site). But this is erroneous: the divalent ions (Fe + Mg + Mn) are still dominant (S = 1.60), with Fe²⁺ as the dominant ion, and the composition corresponds to schorl”.

One cannot say that the name elbaite is erroneous based on the dominant-valency rule as it is the dominant-valency rule that is erroneous, i.e. it has no scientific basis. Using the end-members listed in Table 1, we may write the relation between the end-member constituents and the formula as a set of simultaneous equations as shown in Table 2. We may solve for a, b, c and d, and the resultant values are listed in the bottom row of Table 2: elbaite is the dominant end-member.

Table 1. End-member formulae for the Y-site of a tourmaline-supergroup group mineral of composition Na[Fe²⁺_0.6Mg_0.5Mn²⁺_0.5Li_0.7Al_0.7]Al₆[Si₆O₁₈][BO₃]₃O₃(OH).

Table 2. Simultaneous equations expressing the Y-site composition [Fe²⁺_0.6Mg_0.5Mn_0.5Li_0.7Al_0.7] in a tourmaline.

a = 0.60 / 3 = 0.20; b = 0.70 × 2 / 3 = 0.46; c = 0.50 / 3 = 0.17; d = 0.50 / 3 = 0.17.

Epidote

Hatert and Burke (Reference Hatert and Burke2008; page 722) state the following: “Also, the many coupled heterovalent–homovalent substitutions in the epidote-group minerals require the application of the dominant-valency rule in the solid-solution series. This is necessary because strict adherence to the rule based on the dominant ionic species leads to inconsistencies and unbalanced formulae (Armbruster et al., Reference Armbruster, Bonazzi, Akasaka, Bermanec, Chopin, Gieré, Heuss-Assbichler, Liebscher, Menchetti, Pan and Pasero2006)”. However, the examples they gave are mainly concerned with assigning names, and as noted in the Introduction, assigning end-member formulae is independent of naming the resulting species. In the example that follows, I write the site constituents as elements or ions with the amounts indicated by post-subscripts (as in Hatert and Burke, Reference Hatert and Burke2008).

Armbruster et al. (Reference Armbruster, Bonazzi, Akasaka, Bermanec, Chopin, Gieré, Heuss-Assbichler, Liebscher, Menchetti, Pan and Pasero2006; page 560) stated the following (including section in italics): “Example: An allanite-subgroup mineral where M3 is not dominated by a single divalent cation but by several, so that a trivalent cation is the most abundant one: e.g. Ca(La_0.6Ca_0.4)Al₂(Fe²⁺_0.3Mg_0.2Mn²⁺_0.1Al_0.4)[Si₂O₇][SiO₄]O (OH) [One might be tempted to write a meaningless, non-charge-balanced end-member CaLaAl₃[Si₂O₇][SiO₄]O(OH)]”.

First, an end-member formula is neutral by definition; the supposed ‘end-member formula’ produced above just shows that the procedure “one might be tempted” to use does not work.

Second, one may write the following end-member formulae involving the cations at the A2 and M3 sites (and omitting ^{A 1}Ca^{M 1}Al^{M 2}Al[Si₂O₇][SiO₄]O(OH), the fixed part of the formulae) as shown in Table 3. Using the numbered end-members listed in Table 4, we may write the relation between the end-member constituents and the formula of a mineral as a set of simultaneous equations as shown in Table 5. One may solve this set of equations and the resultant amount of each constituent is shown on the bottom line of Table 5. Armbruster et al. (Reference Armbruster, Bonazzi, Akasaka, Bermanec, Chopin, Gieré, Heuss-Assbichler, Liebscher, Menchetti, Pan and Pasero2006) assign the end-member formula as CaLaAl₂Fe²⁺[Si₂O₇][SiO₄]O(OH) whereas the dominant end-member formula is CaCaAl₂Al[Si₂O₇][SiO₄]O(OH). These conflicting results arise because the former formula conflates classification and calculation of end-member formulae whereas the latter formula results solely from determining the dominant end-member.

Table 3. Constituents of the A2 and M3 sites in epidote.

Table 4. End-member formulae* for an epidote-group mineral of composition Ca(La_0.6Ca_0.4)Al₂(Fe²⁺_0.3Mg_0.2Mn²⁺_0.1Al_0.4)[Si₂O₇][SiO₄]O(OH).

*Omitting the fixed ^{A 1}Ca^{M 1}Al^{M 2}Al[Si₂O₇][SiO₄]O(OH) part of the formula.

**Labels from Table 3.

Table 5. Simultaneous equations expressing the formula for the A2 and M3 sites in terms of the end-members 1–4 for the epidote-group mineral of Table 1.

The fallacy of the dominant-valency rule

As noted above, the dominant-valency rule was introduced to compensate for the fact that in minerals with mixed-valence ions at more than one site, if one of the constituents is substituted by another constituent of the same valence, the valency-imposed double site-occupancy rule fails to derive an end-member formula. It must be emphasised the dominant-valency rule has no basis in Physics or Chemistry; it is an arbitrary 'rule' introduced in an attempt to compensate for the failure of the rule of the dominant constituent and the valency-imposed double site-occupancy rule to always accord with the conservation of electric charge (Hawthorne, Reference Hawthorne2021a). Moreover, the dominant-valency rule leads to some compositions within particular mineral groups being assigned a name that is different from the component in the mineral that has the highest concentration and the highest (ideal) thermodynamic activity, e.g. as in the ternary feldspar system, hardly an ideal situation.

The site-total-charge approach:

Bosi et al. (Reference Bosi, Hatert, Hålenius, Pasero, Miyawaki and Mills2019a; page 629) define the charge constraint as follows: “The charge constraint can be defined as an integer number close (or next) to the observed site total charge (STC)”. There are three problems with this ‘definition’:

1. (1) This statement is not a definition. A number is not a constraint until it is used in some fashion in an argument, and then the constraint is the kernel of that argument rather than the number itself.

2. (2) Wherever the integer defined in this process is not that closest to the observed site-total-charge but just a number close to it, the site-total-charge approach violates the rule of the dominant constituent on which all of the IMA procedures are based.

3. (3) There is a degree of arbitrariness in picking the integer number close or next to the observed site total. For example, Bosi et al. (Reference Bosi, Biagioni and Oberti2019b) consider the composition K ^M(Li_1.49Mn³⁺_1.02Al_0.49) Si₄O₁₀^A(O_1.02F_0.98) intermediate between norrishite, ideally K ^M(LiMn³⁺₂)Si₄O₁₀^AO₂, and polylithionite, ideally K ^M(Li₂Al)Si₄O₁₀^AF₂. The relevant sums of the STC are ^M6.02⁺ and ^A3.02^–, and Bosi et al. (Reference Bosi, Biagioni and Oberti2019b) arbitrarily pick the alternative values ^M7⁺ and ^A4^– and assign the end-member as norrishite. However, according to the explanation of the STC method that they provide, they could also have picked the STC pair ^M5⁺ and ^A2^– to give the end-member as polylithionite. The STC values of the empirical formula are slightly closer to those of norrishite than those of polylithionite, but this is not given as a criterion in the rule (Bosi et al., Reference Bosi, Hatert, Hålenius, Pasero, Miyawaki and Mills2019a).

Bosi et al. (Reference Bosi, Hatert, Hålenius, Pasero, Miyawaki and Mills2019a; page 629) noted that Hatert and Burke “… have not considered the case … of coupled heterovalent substitutions at two sites along with the heterovalent substitution at a single site. This more complex case is now clarified (Fig. 2)” and give the following cases of a hypothetical feldspar composition: the Sc-bearing pyroxene jervisite, and a garnet from Magnet Cove as examples. I will consider each of these examples below.

Magnet Cove garnet

Bosi et al. (Reference Bosi, Hatert, Hålenius, Pasero, Miyawaki and Mills2019a; page 631) also considered a garnet from Magnet Cove (Table 6) and stated that “The dominant-valency rule is able to identify this garnet if the concepts of site total charges and charge constraints, dictated by the mineral composition and the electroneutrality principle, are taken into account”. However, it is straightforward to rigorously assign the dominant end-member composition to this garnet by examining all possible charge arrangements (Hawthorne, Reference Hawthorne2021a) and subsequently assigning the cations of the formula to the dominant root-charge arrangements without any complicated arguments. I define site total charge as the sum of the charges of the ions at a site and constituent site charge as the charges of the individual ions at a site.

Table 6. Possible charge arrangements and corresponding end-members for a garnet from Magnet Cove*. ^X[Ca²⁺_2.907Fe²⁺_0.043Mn²⁺_0.035Na⁺_0.015] ^Y[Ti⁴⁺_1.069Zr⁴⁺_0.055Fe³⁺_0.517Fe²⁺_0.204Mg²⁺_0.155] ^Z[Si⁴⁺_2.250Fe³⁺_0.588Al³⁺_0.162]O₁₂

*From Chakhmouradian and McCammon (Reference Chakhmouradian and McCammon2005).

**Site-total-charge combinations in italics cannot be end-members as they involve more than one charge species at more than one site.

All possible site-total-charge arrangements for a garnet structure with this composition are listed in Table 6; thus (2₂1₁)⁵⁺ denotes two cations of charge 2⁺ and one cation of charge 1⁺ in the X group (denoted as 2 and 1, respectively) with a site total charge of 5⁺. The sum of the charges of the various arrangements must sum to 24⁺ to satisfy electroneutrality, giving the possible combinations of charges at the various sites in Table 6. There are six site-total-charge arrangements that satisfy this criterion, but three of these do not conform to the end-member requirement of a maximum of one site occupied by more than one charge (ignoring valence-sum restrictions associated with short-range order, see above discussion of Y-rich hainite and Hawthorne, Reference Hawthorne2021a); these arrangements are italicised in Table 6. Only 6⁺8⁺10⁺, 6⁺6⁺12⁺ and 4⁺8⁺12⁺ (Table 6) are end-member charge arrangements. We may use the site total charge at each site in the Magnet Cove garnet to solve for the amounts of these three end-member site-total-charge arrangements. This is done in Table 7 which lists the relevant simultaneous equations and their solution in terms of a, b and c, the amounts of each end-member site-total-charge arrangement corresponding to the composition of Magnet Cove garnet (Table 5).

Table 7. Simultaneous equations for the site total charges in terms of the possible arrangements of site total charges.

*Solution, giving the amounts a, b and c of the distinct site-total-charges at each site in arrangements [1], [2] and [3] (see Table 6).

The site-total-charge arrangement 6⁺6⁺12⁺ corresponds to the two end-member constituent-site-charge arrangements (2₃1₀)⁶⁺(4₁3₀2₁)⁶⁺(4₃3₀)¹²⁺ and (2₃1₀)⁶⁺(4₀3₃2₀)⁶⁺(4₃3₀)¹²⁺ (Table 8) and the amounts of these two arrangements can be derived by dividing the amount of that site-total-charge arrangement (0.6175, Table 9) into two parts according to the ratio ^Y[Ti⁴⁺_0.359Fe²⁺_0.204Mg_0.155]: ^Y[Fe³⁺_0.517]; ^Y[Ti⁴⁺_0.359Fe²⁺_0.204Mg_0.155] = 0.6175 × 0.718 / (0.718 + 0.517) = 0.3590 and ^Y[Fe³⁺_0.517] = 0.6175 × 0.517 / (0.718 + 0.517) = 0.2585 (Table 9, Amount of constituent-site-charge arrangements). There is (Fe²⁺,Mg) occupancy of the Y site and (Fe³⁺,Al) occupancy of the Z site, and the corresponding end-members may be derived by partitioning the amounts of the constituent-site-charge arrangements according to the amounts of the homovalent substituents. This is done in Table 9 (‘Amounts of end-members’) where it can be seen that the dominant end-member is Ca₃(Ti⁴⁺₂)(SiFe³⁺₂)O₁₂: schorlomite. This contrasts with the end-member assignment of Bosi et al. (Reference Bosi, Hatert, Hålenius, Pasero, Miyawaki and Mills2019a) which assigns the end-member formula as Ca₃(Ti⁴⁺Fe²⁺)Si₃O₁₂: morimotoite.

Table 8. End-member site-total-charge arrangements and possible constituent-site-charge arrangements in Magnet Cove garnet.

*M²⁺ = Fe²⁺ and Mg; M³⁺ = Fe³⁺ and Al.

Table 9. Amounts of end-members in Magnet Cove garnet.

Which is the correct end-member, schorlomite or morimotoite? Morimotoite is derived by a procedure (the rule of the dominant constituent and its additional rules) that has been shown above to be scientifically flawed. Schorlomite has been rigorously shown to be the dominant end-member constituent, and it is difficult to logically justify giving this garnet any other name.

End-member names and root names

Some supergroups of common minerals, particularly those involving more complicated minerals such as amphiboles (Hawthorne et al., Reference Hawthorne, Oberti, Harlow, Maresch, Martin, Schumacher and Welch2012) and micas (Rieder et al., Reference Rieder, Cavazzini, D'yakonov, Frank-Kamenetskii, Gottardi, Guggenheim, Müller, Neiva, Radoslovich, Robert and Sassi1999), are not organised strictly according to the rule of the dominant constituent and are unlikely ever to be organised rigorously on the basis of end-member formulae. This situation arises because the names of many common minerals occur extensively in the scientific literature and have done so for many (sometimes hundreds of) years, and many of these minerals are involved in rock nomenclature and as petrogenetic indicators. For example, the root composition of magnesio-hornblende: □Ca₂(Mg₄Al)(Si₇Al)O₂₂(OH)₂ is intermediate between the end-member compositions of tremolite: □Ca₂Mg₅Si₈O₂₂(OH)₂ and tschermakite: □Ca₂(Mg₃Al₂)(Si₆Al₂)O₂₂(OH)₂, and the root composition of pargasite: NaCa₂(Mg₄Al)(Si₆Al₂)O₂₂(OH)₂ is intermediate between the end-member compositions of edenite: NaCa₂Mg₅(Si₇Al)O₂₂(OH)₂ and sadanagaite: NaCa₂(Mg₃Al₂)(Si₅Al₃)O₂₂(OH)₂ (Hawthorne et al., Reference Hawthorne, Oberti, Harlow, Maresch, Martin, Schumacher and Welch2012).

The classification of the amphiboles is based on the following rules:

1. (1) All distinct arrangements of integral charges over the amphibole formula are considered as root-charge arrangements.

2. (2) For a given root-charge arrangement, specific ions of appropriate charge are associated with sites in the structure, and each distinct chemical formula is a root formula.

Root-charge arrangements and root formulae should not be confused with end-member charge arrangements and end-member formulae. In the amphibole examples given above, tremolite, magnesio-hornblende and tschermakite are all root names but only tremolite and tschermakite are (also) end-member names. Deleting magnesio-hornblende, pargasite and the associated prefixed names (e.g. ferro-pargasite) would not be acceptable to the Petrology community. Moreover, the loss of such intermediate species would decrease the petrological utility of amphibole names; for example, tremolite would become much more common and would occur in a far wider spectrum of rocks than it does at present.

Nickel (Reference Nickel1992) states that mineral names or definitions already in the literature that contravene recommendations should not be changed unless there are compelling reasons to do so. This seems an eminently sensible approach to the issue of rigorously applying principles of mineral nomenclature to all minerals and mineral groups/supergroups without consideration of the utility of the nomenclature. However, this does not void the requirement that the method of defining mineral species be rigorous and free of scientific error.