In the present work the wt.% oxide compositions are calculated for a set of hypothetical end-members and solid-solution, stoichiometric amphibole compositions with fixed vacancies, Fe valences, excess oxygen proportions (O²⁻), and (OH)⁻ in W. Details of the compositions are given in the Supplementary Excel spreadsheet file 1, Folder I. The examples used are chosen to test how calculation protocols can be developed to deal with different types of crystal chemical challenges. Different stages of formula calculation are shown and are intended to be detailed enough for mainstream petrologist readers to follow.

Table 1. Calculation of Fe³⁺ on a 23 O basis for hypothetical vacancy-present and oxo-end-member amphibole molecules.

Table 1 (this work) gives the wt. % oxides of these hypothetical amphiboles, with total Fe calculated as wt.% FeO; structural water is also given with the wt% calculated compositions with the equivalent values for FeO and Fe₂O₃ just below. Table 1 includes both anhydrous 23 O calculations dealing with the full amphibole formula with fixed 16 cation + vacancy formulae, but the availability of values for FeO, Fe₂O₃ and H₂O in Table 1 allows the full 24 anion data to be calculated for these ideal compositions if required. The first example considered is the iron-free ideal glaucophane (□Na₂(Mg₃Al₂)Si₈O₂₂(OH)₂) with a vacancy at A, 2 Na⁺ cations occurring in B, a total of 5 atoms at C (3 Mg²⁺ and 2 Al³⁺), 8 Si⁴⁺ cations at T ; positive charges for the different sites are 0, 2, 12, 32 consecutively, totalling +46. These are balanced electrostatically by 22 O²⁻ in the amphibole double chain sheet framework and 2(OH)⁻ forming the hydroxyl group at W, totalling −46. Note that (OH) is chosen to occupy W in most of the ideal amphiboles considered initially, and this cation charge is fixed by stoichiometry for amphiboles without excess O. Also note that the cation plus vacancy total is 16 which is fixed by the amphibole stoichiometry. In the amphibole spreadsheet used here the first stage is to calculate the atomic proportions of oxygen and then of the cation numbers from Si to H (Na for the amphiboles in Table 1) following ([16]; [84]); in Table 1 (column 1a) the water content is included as the first calculation is on a 24 anion hydrous basis with a zero Fe component. The atomic proportions for the cations and for H as (OH) exactly match the molecular formula shown above; the cation sum (denoted Σ²⁴_cats) equals 15.000 per formula unit (pfu), and as the stoichiometric formula must have 16 possible sites, the vacancy number (□, □^A in this case) must be 1.000. Table 1 column 1b shows the data for the same sample without an H component; thus the stoichiometric proportion of (OH) is assumed. The calculation here is to sum the oxygen for a 23 O formula on an anhydrous basis (Σ²³_O = 2.935pfu) and to incorporate it into the cation correction factor (denoted Y) to calculate the cation proportions on the basis of 23 O, which we define as Y= 23/Σ²³_O; for ideal glaucophane Y = 7.836, but for most amphiboles would lie within an extended range from ~8 - 10. This factor is then used to calculate the numbers for all the cations and thus a value for the total Σ_cats. The 23 O cation values for glaucophane (Table 1, column 1b) match those calculated for the stoichiometric, 24 anion formula (1a); all the components show the stoichiometric values expected for the formula defined, with a cation total (Σ²³_cats) of 15.000. While all the cation data at this stage look complete, they only refer to a 23 O formula and we know that stoichiometric amphibole has 24 anions, 22 in the sheet and 2(OH) at W which shares oxygens with the sheet. However the 24 anions must be related to 16 possible cation sites and the actual cation plus vacancy total has been shown to be 16 so the stoichiometry normalisation factor to convert the 23 O proportions is simply (16^stoic.)/(Σ²³_cats + vacancies) [denoted Q here]; note that this type of correction factor is denoted the ‘ferric normalization factor’ (FNF) by [84].Q = 16/16 = 1 for ideal glaucophane, and the final stoichiometric 24 anion formula for the cations remains the same as for the 23 O formula. The calculation of these values seems straightforward but that reflects the absence of Fe³⁺, indeed for end-member glaucophane there is no Fe at all.

The same approach is followed for ferroglaucophane where Fe²⁺ replaces Mg (□Na₂(Fe²⁺₃Al₂)Si₈O₂₂(OH₂)) as there is no valence difference between Fe²⁺ and the Mg it ‘replaces’. The wt. % oxide values with all iron as FeO and the stoichiometric data for ferroglaucophane are given in Table 1, column 2, although correct oxide amounts for Fe₂O₃ and FeO are given in italics at the bottom of the oxide data; for this amphibole only FeO is present. The calculated oxide data with separate Fe₂O₃ and FeO values, including H₂O, would give exact stoichiometric atomic values where calculated on a 24 anion basis with cation sums of 16.000 atoms, as long as vacancy values are included as essential empty stoichiometric cation sites;note that for the 24 anion calculation the [21] formula used to model the Fe vacancy values must use a 24 O value.

The 23 O formula calculation gives exactly the same cation proportions (Table 1, column 2) as the 23 O formula for glaucophane (and to the 24 anion calculation), as long as all Fe is present as Fe²⁺, but now the [21] formula basis is defined by requiring a 23 O value. Thus following Droop, the calculation to obtain the Fe³⁺ in the 24 anion formula is shown as Equation (1) below:

F= 2X(1 –T/S)(1)

where Droop defined F as the number of Fe³⁺ ions per Xoxygens, T as the ideal number of cations per formula and Sas the observed cation total perX oxygens; Tis denoted Σ^stoic here (usually either 16 or 13 total cations) and Sis denoted Σ²³_cats; for strict stoichiometry both Σ^stoic and Σ²³_cats should include vacancy values but we will see later that it may be convenient (simpler) to ignore a vacancy altogether as long as it occurs at A, which is normally the case for amphiboles (see above). The next stage in producing a full 24 anion formulaformula would be to convert the 23 O cation proportions to the full formula on the basis of either 16 cation or 13 cations proportions. Thus, for ferroglaucophane, 1 vacancy is required to be added to each calculation leading to the calculated apparent Fe³⁺ values of 2 × 23 × (1 − 16/16) (Table 1, column 2, row 2b) and 2 × 23 × (1 − 13/13), respectively. In each case the apparent Fe³⁺ is zero, as required by the ideal formula. The final 24 correct anion proportions would therefore be the same as for the first stage 23 O values. However, another second stage approach might be to accept that ferroglaucophane has one vacancy and therefore the cation sum ratios could be defined simply as 15/15 which gives the correct calculation for Fe³⁺ of 0.0 (Table 1, column 2, row 2c). Indeed, the presence of a vacancy could be ignored with the 23 O cation sum alone totalling 15 atoms (Σ²³_cats) to be corrected by subtracting the stoichiometric 23 O, B category content (2Na^B ignoring the vacancy), to also give Fe³⁺ = 0.0 for the 13/13 correction (Table 1, column 2d); thus both the 16 and 13 cation corrections give exactly the same zero Fe³⁺ as required by the strict stoichiometry of the input data. The 23 O cation values for all the elements multiplied by the stoichiometry correction factor (denoted Q) which for this sample has the value Q = 13/13 = 1.000 (Table 1, column 2 row e); the final formula values would now match all the stoichiometric cation proportions with 3 Fe²⁺ and zero Fe³⁺ pfu (see formula given above and in Table 1 footnote).

The next example is an end-member with a fixed Fe³⁺ content together with a single vacancy, e.g., magnesio-ferri-hornblende ((□Ca₂(Mg₄Fe³⁺)[Si₇Al]O₂₂(OH)₂); for a 23 O calculation the Fe₂O₃ content of that end-member (9.47 wt.%) is input to the amphibole spreadsheet as an anhydrous amphibole (0% H₂O) with 8.52 wt% FeO. However, as this is the first of our examples to have ferric Fe, the first calculation considered is for the full 24 anion formula with Fe₂O₃ and H₂O included (Table 1, column 3a). The 24 anion cation values shown match those for the stoichiometric formula (see above and Table 1 footnote), with 15 cations and 1 vacancy giving Si = 7.000, Fe³⁺ = 1.000 and (OH) = 2.000. The 23 O calculation is shown in column 3b with Fe input as FeO and without the water value. The Σ²³_O cation values now show higher cation proportions than for those shown for the 24 anion cations (column 3a) and with those in the molecular formula; for example, Si = 7.1555 pfu compared with the required 24 anion value of 7.000 pfu (see above formula); each element, including the bulk Fe value, shows the same excess proportion with a total cation sum for Σ²³_cats = 15.3335 pfu. These values are too high because not enough O is present in the calculation due to the total iron being reported only as FeO. The Σ_cats = 15.3335 is used to convert a proportion of the input Fe²⁺ to the known stoichiometric value for Fe³⁺ following the Droop method, as explained above with Equation (1). The ‘16’ cation calculation must include the vacancy of 1.0 pfu so that Fe³⁺ = 2 × 23 × (1 − 16/16.3335) = 0.939 pfu (Table 1, column 3, row b); that value is slightly less than the known Fe³⁺ amount of 1.000 atom pfu. The deficiency reflects the fact that the value calculated for 16 cations has used cation data calculated on a 23 O basis with a cation total of 15 cations. However, the presence of one vacant site increases the effective totalsite occupancy to 16 leading to a correction factor of 16/15 increasing the Fe³⁺ content to 1.0 (0.939 × 16/15 = 1.00), the known Fe³⁺ content. Note that the equation adds a known vacancy to obtain a 16 cation plus vacancy total for a stoichiometric amphibole, then subtracts this vacancy value to convert to a 13 cation basis! Thus, a numerically correct and simpler approach is to ignore the vacancy at A, to use the Σ²³_cats of 15.3335 without the vacancy value, and to subtract the Σ²³_Ca value (2.045) to give Q= 13/13.2886 and thus Fe³⁺ = 1.000; this formulation now defines step (d) in the amphibole spreadsheet for the Fe³⁺ calculation. Thus step (b) described above depends on knowing the exact vacancy contents and onlyworks if the vacancy is restricted to A. However, the final 13 cation (d) correction factor could be used to deliver the preferred Fe³⁺ cation values for the full formula for any stoichiometric amphiboles containing vacancies at A; indeed, it wouldwork for any vacancy content without the requirement of the value being known exactly (see later). In addition, if the cation count is taken to be exactly 15 for the ‘16’ cation calculation and 13 for the 13 cation calculation exactly the same Fe³⁺ is given reflecting the exact stoichiometry of the input data (see above for ferroglaucophane). In effect, ignoring the known vacancy at A follows the usual practice of researchers who choose the 13 cation calculation for amphiboles (rather than a 16 cation calculation) to give a more reliable Fe³⁺ estimate; it seems that although ignoring a known vacancy content is counter-intuitive it is valid for stoichiometric amphiboles. However, if the essential vacancy is ignored, the basic amphibole spreadsheet for a 16 cation formula would calculate an apparent Fe³⁺ for magnesio-ferri-hornblende as 2 × 23 × (1 − 16/15.3335), i.e., giving the non-physical value of minus1.999 Fe³⁺ in a phase known to have one atom of Fe²⁺; clearly that calculation is not valid (cf. [21]). The [55] and [77] approaches are to set such a non-physical negative result for the model fitting parameters to zero, leading to possible mismatches with charge balance! Thus, it is crucial to match all the correction factors in terms of whether the vacancy values are included, or not. Further examples below will test how reliability for this approach.

The end-member ferri-tschermakite (Table 1, column 4) contains a vacant A and only Fe³⁺ i.e. twice that of the hornblende matched by a higher substitution of Al for Si in the Tsite; hence the name. As expected, the 23 O cation values are higher than the stoichiometric values (Si = 6.273 rather than 6, total cations + vacancy = 16.682) and the calculated Fe³⁺ estimates show the expected characteristics (Table 1, column 4 row b 1.880 × 16/15 = 2.004, and column 4 rows c ans d = 2.000); the Q factors to correct the 23 O cation data to the full 24 anion formula are similar for ferri-hornblende and ferri-tschermakite at 0.978 and 0.957, respectively (Table 1 column 3 row e and column 4 row e).

Ideal riebeckite (□Na₂(Fe²⁺₃Fe³⁺₂)Si₈O₂₂(OH)⁻₂) has a vacancy at A while C contains both iron Fe oxidation states (Table 1, column 5); this is the first example of our ideal amphiboles with this feature. Table 1, column 5a shows the complete 24 anion data for the hydrous phase with both Fe₂O₃ and FeO values declared. The 24 anion cation proportions match the stoichiometric formula with Si = 8.000, Fe³⁺ = 2.000, Fe²⁺ = 3.000, Σ²³_cats = 15.000. The anhydrous 23 O formula (Table 1, column 5b) is calculated with all Fe as FeO (30.96 wt.%) giving cation values larger than those in 5a with Si = 8.364, Fe = 5.227 and Σ²³_cats = 15.681 plus 1 vacancy. Calculated Fe³⁺ values are 1.879 (Table 1 5a, row b) and 2.000 (Table 1 column 5b, rows d and e). The multiplication of all the 23 O cation values by the 24 anion correction factor (Q) of 0.956 (row e) now gives the correct stoichiometric formula values with Si = 8.000, Fe³⁺ = 2.000 and Fe²⁺ = 3.000. Thus, stoichiometric amphiboles with vacant A groups and with only Fe³⁺ or with mixed Fe valence components have the same calculation characteristics.

The following examples are for more complex hypothetical amphibole compositions including solid solutions with variable vacancy contents, Ti-bearing, and oxo-amphibole compositions. Table 1 (continued), columns 6 and 7 show data for solid solutions which show the presence of 0.5 vacancies at A. The binary solid solutions chosen are 0.5 ideal riebeckite: 0.5 ferri-taramite (□_0.5Na_0.5(Na_1.5Ca_0.5)(Mg_1.5Fe²⁺_1.5Fe³⁺₂)[Si₇Al]O₂₂(OH)⁻₂) (column 6) and 0.5 Mg-ferrihornblende: 0.5 ferro-taramite (□_0.5Na_0.5(Na_0.5Ca_1.5)(Mg_2.0Fe²⁺_1.5Fe³⁺_0.5Al) [Si_6.5Al_1.5]O₂₂(OH)⁻₂) (columns 7). As expected, both have 23 O atom cation totals greater than the stoichiometric values with total cation values of 16.205 + □_0.5 and 15.670 + □_0.5, Fe³⁺ contents of 1.940, 2.000 (column 6 row b and 6 row d) compared with 0.484, 0.500 (column 7 row b and 7 row d), and Q correction factors of 0.956 and 0.989 (columns 6 row e and 7 row e); application of those Q values results in 24 anion formulas that match the stoichiometric formulae. However, the data for row (c) are invalid (highlighted in the Table) because the parameter was developed specifically for compositions with 1 vacant site. The main conclusion is that theFe³⁺estimations described can be used for any vacancy content as long as it can berestrictedtoRomanA. The amphiboles in Table 1 (continued), columns 8 and 9 will be dealt with within the context of some natural amphibole compositions.

Calculation of formulae for end-member oxo amphiboles is considered for ideal oxo-ferro-ferri-sadanagaite NaCa₂(Fe²⁺Fe³⁺₂Al₂)[Si₅Al₃]O₂₂O₂ (Table 1 continued, column 10a). A 23 O calculation for the analytical data with all Fe as FeO gives the correct cation proportions (column 10a) but that cannot be valid because the whole basis of the 23 O model assumes a stoichiometric amphibole with 22 O atoms and 2(OH) per 24 anion formula and this amphibole is (OH) free. Thus the standard Fe³⁺ calculations give values of 0, 3.0 and 0 (column 10 rows a, c, d), none of which are correct. End-member oxo-amphiboles all require a 24 O calculation (i.e., 23 + ^wO²⁻/2) to obtain the meaningful cation contents shown in column 10b. All of these values are higher than the correct values and correction by the factor 0.9584 (column 10b row c) reduces the cation values shown for Si to Na provides the correct 24 anion values for all elements including Fe³⁺ and Fe²⁺ values. Thus, both 16 cation (column 10b, row (b) and 13 atom calculations (row (c) provide correct values for Fe³⁺ of 2.000 pfu but the value of 4.874 in column 10b, row (c) is nonsense and has no validity for a formula with no vacancies and no (OH). It was mentioned above that many natural amphiboles show the presence of oxo-amphibole components; thus, calculation of meaningful formulae requires quantification of the ^WO²⁻, correction factor.

The composition given in Table 1, continued (column 11) is for a hypothetical amphibole solid solution with 25% ferri-taramite, 25% riebeckite and 50% oxo-ferro-sadanagaite (i.e., with 0.25 A vacancies and 1 ^WO²⁻ and 1 (OH)⁻ pfu (ideal formula □_0.25Na_0.75(Na_0.75Ca_1.25)(Mg_0.75Fe²⁺_1.25Fe³⁺₂Al)[Si₆Al₂]O₂₂(OH)O. In Table 1 examples 1 to 8 are all stoichiometric amphiboles having the required formulae with 2 (OH)⁻ pfu; however, examples 10 and 11 involve amphibole formulae with oxo-components requiring the first stage in the recalculation to be carried out on a recalculation defined by (23 + ^WO²⁻/2) rather than 23 O. Thus, the presence of 50% oxo molecule (e.g., example 11) would require the formula to be corrected on a 23.5 O basis, which delivers cation proportions larger than the stoichiometric values. Calculation of the Fe³⁺ values using the standard calculations gives the values 1.968 for column 11a row (b) (note that 1.968 × 16/15.75 = 2.000), 4.78 for row c, and 2.000 for row d. It is clear that the calculation for column 11, row c is not valid for such samples; it was developed to deal with amphiboles having a fixed vacancy total of 1.000 atoms pfu. Thus, that calculation basis involves normalisation to 15 cations, but the correct normalising values should be 16 minus the known vacancy count, a total of 15.75 in this case. The 13 atom calculation defines the value of Q = 0.9575 for recalculating the correct cation atomic proportions for 24 anions and the corrected values are shown in Column 11b. Based on all of the examples discussed here,the most reliable calculation for assessing the Fe valence proportions is that for row(d)inTable 1,which gives the data for a13cation count for23O calculations for normal amphiboles with2(OH)and for amphiboles with a proportion of oxo-amphibole components in solid solution using a(23 + ^WO²⁻/2)calculation.

The main requirement for dealing with EMP analyses of amphiboles is to quantify the oxy-amphibole component; [77] and [55] provide a better basis for this approach. Those databases include data of well analysed amphiboles to provide reference ‘standards’ with reliable EMP data, H/(OH) values estimated with SIMS/vibrational spectroscopy, and separate values for Fe²⁺ and Fe³⁺ obtained by chemical, Mössbauer, XANES, or from single crystal refined X-ray data with Fe³⁺ based on charge balance (SREF). The compositional, crystal chemical and structural information for key samples in the [77] database were analysed using multivariate regression analysis, while [55] applied a machine-learning method based on principal component regression analysis. Note that we only use those papers to estimate ^WO²⁻ for new and published analyses that lack data for H₂O. In their databases, we have checked if the regressions have significantly modified the original atomic proportions of the reference samples (e.g., [24], 2016) and are satisfied that the calculated model ^WO²⁻ values are generally within ~5% - 10% of the original ‘analysed’ values. Thus we will apply our own crystal chemical calculations to obtain atomic proportions constrained by strict stoichiometry, with the possibility of a small uncertainty in the estimated Fe³⁺/ΣFe at that level.

Our calculation protocol provides atomic formulae for conventional EMP amphibole analyses with all Fe reported as FeO, and which lack H₂O/(OH) data. We prefer to assess the reliability of Fe³⁺ estimates against reference samples with Fe valence values determined by chemical analysis, or by Mössbauer / XAS methods rather than single crystal XRD refinement (SREF). Indeed, least squares regression analyses (SREF) for chemically-complex amphiboles involve assessing how valence, size, molecular mass, and electron density properties for key elements (e.g., Mn, Fe, Mg, Al, Ti, Cr, Ni, Zn, Li) might be used to predict cation-site occupancies, especially ordering between the 4 distinct sites containing C and B group cations. This is difficult enough for pairs of competing elements in multi-site crystal structures, but multi-element structural interactions are even more difficult to predict as discussed by [35] and [71]. Thus, it is difficult to assess the results summarised in the [55] spreadsheet, but it is our objective to use a simple model aimed at obtaining reliable Fe³⁺/total Fe ratios (Fe³⁺/ΣFe) based on the overall charge distribution over the whole structure on the basis of calculating Fe³⁺ for 16 (T + C + B + A) and 13 (T + C) cation + vacancy formulae. Fortunately in most amphiboles, any vacancies occur in A (see above).

The ^WO²⁻ values calculated for the amphibole database assembled here using both the [77] and [55] spreadsheets provide similar sets of results. However, the [77] paper also provides a formula (their equation 3) for calculating a value for all elements from Si to K, and another formula (equation 4a) approximates a ^WO²⁻ value based on the atomic proportions of Ti and Fe³⁺ in C and of (Na+K) in A. The coefficients for the dependent variables in those equations are based on the standard amphibole dataset some of which provide high positive or even high non-physical negative ^WO²⁻ values; however, atomic proportions for the latter are returned as zero values and, thus, we do not use either of those equations. In addition, some end-member and hypothetical amphibole compositions with vacancies (e.g., tremolite and glaucophane) return very small, or even negative, ^WO²⁻ values; indeed estimated ^WO²⁻ values show significant differences to the known values for some Al-rich compositions (see bottommost rows in Table 1). Indeed, care is needed in considering the reliability of such estimates for certain amphibole compositions and related model Fe³⁺ and Σ¹⁶_cats data, but small ^WO²⁻ values will have little effect on a 23 O formula calculation (see below).

In this work, the ^WO²⁻ values incorporated used are those obtained using the [55] spreadsheet with folder ‘Dataset of Li-free AMPH’; the quality of Li-et al. estimated ^WO²⁻ values are checked against the values obtained for high quality, fully-analysed samples (e.g., [24], [83]) and are found to be within 5% - 10% relative (see above). Our approach is to obtain amphibole compositions from the published literature, and to test whether these follow rigorous stoichiometry with reliable 16-cation plus vacancy, and 24 anion formulae. We have shown above that using the 13 cation calculation provides reliable Fe³⁺ values irrespective of vacancy content as long as they only occupy A (see above). Thus, we normalize cation proportions using the factor (23 + ^WO²⁻/2)/ΣO; note that half of the determined‘extra oxygen’is required to supply the extra charge provided by O²⁻compared to(OH)⁻.

The hypothetical amphibole examples used have had exact stoichiometries and vary depending on the presence or absence of vacancies and on the proportions of extra oxygen values. As long as such information is available the correct amount of Fe³⁺ can be calculated with the same value being returned on the basis of calculation to either16 or13 cations pfu. Well-analysed natural amphiboles would be expected to show some departure from this ideal situation reflecting the level of analytical error on individual chemical components, especially for the reliable estimation of the ^WO²⁻ value; in addition the analyses must include all the essential amphibole major elements and possible minor elements including, for example Cr, V, Ni, Sr, Ba, Zr and Li. If such information is not available, it would be even more difficult to deal with the presence of vacancies when the amount of Fe³⁺ is being estimated for conventional EMP analyses.

The first samples discussed are Ti-rich ‘ferri-kaersutites’ from northern Tanzania ([101]); which were renamed as ‘oxo-magnesio-hastingsites’ (end-member NaCa₂Mg₂Fe³⁺₃[Si₆Al₂]O₂₂O₂) by [101]. Zaitsev et al. report EMP analyses for 8 amphiboles; three have Fe-valence values determined by Mössbauer spectroscopy and all have water values reported. The three amphiboles with measured Fe valence show formulae calculated to 24 anions apfu as shown in Table 2 in [101]. We deal with the three samples analysed by Mössbauer (OL 22, OL 313 and OL 246), together with another sample from the same locality as OL 22 (DT -/11), and an average for the four other samples (LL 1/5, LL -/4, LL 3/4 and OL 165) (Table 2 this work). For all of those samples the wt.% values with all Fe as FeO have been recalculated as anhydrous formulae on the basis of (23 + ^WO²⁻/2) with ^wO²⁻ obtained from the [55] spreadsheet. The initial cation totals with all Fe as Fe²⁺ are shown at the top of the data columns (denoted Σ¹⁶; Table 2 row (i)) All show values with totals > 16 atom pfu reflecting the presence of Fe³⁺. The calculated Fe³⁺/ΣFe ratios are given in Table 2 row (ii); these values might be reliable as the Σ¹⁶ are so high that the proportions of any vacant cation sites could be small. Row (iii) of Table 2 gives the total cation values for the 13 cation formulae defined as Σ¹⁶ – (Ca + Na + K) in B and A (here denoted Σ¹³_noCNK). We have shown earlier (Section 2) that stoichiometric amphiboles should give very similar values for estimated model Fe³⁺ without vacant sites having any effect. Indeed, the first two estimates (Σ¹⁶ and Σ¹³_noCNK; rows (ii) and (iv) for all 5 samples show good agreement suggesting high Fe³⁺. Except for OL 22, calculated estimates are much more oxidising than Mössbauer values for OL 313 and OL246 (Table 2, columns 1b and 1c). These analyses show significant amounts of C-type atoms greater than the limit of 5.0 atoms pfu with the excess over 5.00 termed ∆C following [77]; we define this excess as ∆C = Σ¹³_noCNK – 13; however, that value still has the enhanced values for all the T and C components due to the presence of all Fe as Fe²⁺ coupled to the absence of the extra O defining the amount of Fe³⁺ present. For OL 22 that initial excess ∆C = 0.169 and that amount is firstly distributed based on the stoichiometric amphibole T and C (8 and 5, respectively). Thus, an initial value is calculated as ∆C* = (Σ¹³_noCNK – 13) × 5/13 (row (vi)) and the estimated Fe³⁺/ΣFe all show lower ratios reflecting the different calculated ∆C* values. The estimated oxidation states for OL 313 and OL 246 are similar to the Mössbauer analyses suggesting that this ∆C calculation might be reliable. However, when dealing with other published data sources it soon became clear that this method for modelling the value for ∆C* tended to give estimated Fe³⁺ that appeared to be too high so it was decided to define the composition for a hypothetical amphibole with a chosen proportion of component that might occur in either C or B. Thus a fixed proportion of the Ca in the ideal amphibole 0.5 Mg-ferri-hornblende – 0.5 ferrotaramite of composition □_0.5Na_0.5(Na_0.5Ca_1.5) (Mg₂Fe²⁺_1.5Fe³⁺_0.5Al)[Si_6.5Al_1.5]O₂₂(OH)₂) (see our Table 1, column 7) is replaced with 0.15 atoms Mn to give the composition □_0.5Na_0.5(Na_0.5Ca_1.35Mn_0.15) (Mg₂Fe²⁺_1.5Fe³⁺_0.5Al)[Si_6.5Al_1.5]O₂₂(OH)₂ (Table 1, column 8); note that both compositions have the same Fe³⁺ - Fe²⁺ contents. Recalculation of the new composition with the spreadsheet (Folders I, III) with the Mn component gives a Σ¹³_noCNK (i.e. all Mn included with Mg and Fe) estimated Fe³⁺ of 0.632 rather than the known value of 0.500 pfu with all the element proportions being slightly smaller than the known values (e.g. Si 6.481 instead of 6.500 pfu!). Thus the excess C calculation has been varied by trial and error to provide the new calculation defined as ∆C^corrected = (Σ¹³_noCNK – 13) × 6.7/13. This calculation now gives Fe³⁺ = 0.500 pfu for the Mn amphibole chosen to calibrate a feasible ‘correction’ method; note that the final formula data (Table 1, column 8 and attached spreadsheet Folder I) show a Σ¹³_noCNK value of 13.15 withoutany Mn; thus Mn = 0.15 pfu clearly belongs to B as shown in its ideal formula. Thus, the new values for the Zaitsev amphiboles using the ∆C^corredted factor are shown in Table 2 and columns 1a to 1e.

The same crystal chemical reasoning has been used for the other samples in Table 2 (rows (vii – x)), columns 1b-1e and in all cases the formulae show that the 13 cation formulas with the recalibrated ∆C^corrected component added to the B + A term have much less oxidised Fe valence ratios. All have similar calculated values for the Fe³⁺/ΣFe ratios (range 0.23 - 0.32) which are slightly lower than the analysed Fe³⁺/ΣFe Mössbauer values for samples OL 313 and OL 246 (0.33 and 0.34; [101]). However, the calculated crystal chemical Fe³⁺/ΣFe value for OL 22 is completely different to that from the Mössbauer study results which showed only peak pairs for Fe³⁺ ([101]). It is possible that the sample used for that study was more altered than the others but it is clear that the estimated ^WO²⁻ values for amphibole are dependent on the whole mineral composition, especially the coupled Si-Al-Ti relationships which are much more difficult to reset than a simple and much more rapid Fe²⁺ oxidation which can occur very rapidly by an electron shuttle mechanism for micas, amphiboles and spinels (cf. [24]; [78]; [43]; [19]; [4]; [41]). At the bottom of Table 2 the mg#, total cation numbers and Fe³⁺/ΣFe ratios are summarised for 13 and 16 cation formulae with data from this work, [101], and for data obtained using the [55] and [77] spreadsheets.

The work of Dyar and colleagues on Ti- bearing kaersutite amphiboles is very relevant to this paper, the element specific Mössbauer ([24]) and X-ray absorption spectroscopic (XAS, [23]) techniques were used to determine Fe-valences directly. Table 2 (column 2) shows the average composition for 20 samples from their Table 5 ([24]) with all Fe input as FeO and on an anhydrous basis. In addition, another publication repeats the data for some of 1993 samples as well as giving XAS pre-edge and XANES K-edge data for new samples in good agreement with Mössbauer Fe³⁺/ΣFe results ([23]). The average data reported here (our Table 2, column 2) give the data in the same format as for the 5 [101] samples. The average 16 cation formula gives a total cation sum of 16.165 equivalent to a Fe³⁺/ΣFe ratio of 0.32 [Table 2, (ii) (range 0.15 - 0.46 for the 20 individual samples], while the uncorrected 13 cation formula (only Ca, Na, and K at B and A) totals 13.388 with a Fe³⁺/ΣFe ratio of 0.94 (row (iv)) (range 0.8 - 1.9!). The first estimate is only slightly smaller than the [24] average (0.41 ± 0.14) while the second is clearly incorrect because the assumed A + B occupancy by Ca + Na + K is only 2.755 which could reflect the presence of unanalysed components such as Sr or Ba. Note that a cation shortfall cannot be explained by the presence of vacancies (see above), however, the [24] samples show that large proportions of C-type cations occupy B. Thus, the first calculation we have used to estimate the ∆C content gives a value of 0.149 pfu (row (vi)) leading to an A + B occupancy of 2.93, a 13 total cation sum of 13.239 (row (v)) and an Fe³⁺/ΣFe ratio of 0.58 (range 0.42 - 0.96) for the average composition (Table 2, column 2, row (vii)). The preferred ∆C^corrected − Σ¹³ cation calculation (row (viii-x)) now gives an average Fe³⁺/ΣFe of 0.45, within error of the [24] average Mössbauer composition (0.41 (16)). Our data for 9 representative samples from the [23] paper shows a similar level of agreement but note that two heated samples with very high proportions of Fe³⁺ (>0.9) both show Fe³⁺/ΣFe estimates < 0.2; this confirms our earlier suggestion that the overall crystal chemistry estimate of ^WO²⁻ appears to be coupled to the overall Si-Al-Ti composition which leads to lower Fe³⁺/ΣFe ratios consistent with primary amphibole crystallization conditions, or mantle heterogeneities, rather than those of later dehydroxylation and rapid Fe-oxidation reactions ([30]; [24]; cf. Ba-Ti-rich micas, [78]; [41]).

[93] reported analysis, with Fe valence determined by Mössbauer techniques, of hornblendes and hastingsites. The average composition for 7 samples (Table 2, column 3) shows atomic proportions for a 13 cation formula. The ‘16 cation’ total is < 16.0 pfu (row (i)) so negative Fe³⁺ contents are implied; however an uncorrected 13 cation formula gives 13.151 cations pfu giving a model Fe³⁺/ΣFe = 0.23 (rows (iii-iv)). The initial calculated ∆C value is 0.055 (row (vi)) giving a corrected B + A occupancy of 2.413 and a lower 13 cation total of 13.088 leading to a Fe³⁺/ΣFe = 0.14 (rows (v, vii)); the equivalent values for the ∆C^corrected − Σ¹³ cation formula are 2.48, 13.069 and 0.11 (rows (viii, x); both the Σ¹³ cation formula estimates are close to the spectroscopically determined value of 0.15 (2) ([93]). The final dataset (our Table 2, column 4) is for a single sample of a synthetically prepared richterite ([20]). For this sample our calculated data are reported mainly for a 13 cation formula but the full 16 cation total is 16.182 atoms pfu (Fe³⁺/ΣFe = 0.10) and the uncorrected 13 cation total is 13.253 pfu with a higher Fe³⁺/ΣFe = 0.17. The calculated ∆C* value is 0.097 leading to a recalculated Fe³⁺/ΣFe = 0.11 (column 3, rows (vi, vii)) and the equivalent calculated ∆C^corrected values are 0.130 and 0.083 (Table 2, column 4, rows (ix, x)).

We have shown that calculation of Fe valences depends on correctly dealing with the possibility of significant vacancy proportions. Amongst others, [35], [24], and [37] have pointed out that there might not be any significant correlation between calculated and measured amounts, however, it is possible that other workers might not have developed sufficiently robust of ways of quantifying, or approximating, the vacancy count. We have attempted to do that and we conclude that, depending on the amphibole species, the most reliable calculation for the atomic proportions is for a 13 cation calculation with a robust treatment of the possible presence of a C-group excess over 5 for the small divalent cations species (mainly Mn and Fe²⁺). This preferred approach is denoted here as the (∆C^corrected, Σ¹³) cation calculation. Other workers have published Fe valence data which shows mixed agreement with our approach as demonstrated in the next section. Indeed, the most useful calculation mode found for amphiboles in the work is that defined by Σ¹³ – (Ca + Na + K) − ∆C^corrected with ∆C^corrected having the empirical 6.7/13 factor and the numerical details are shown in the attached spreadsheet, Folder III. Calculation details for Σ¹⁶ formulas are shown in Folder II as the first set of labelled final formula data and for Σ¹³ – (Ca + Na + K) (∆C = 0) as the second labelled set of final data.

Mössbauer spectroscopy Fe values are preferred ([87]; [26]; [74]; [48]; [1]); followed by chemical analysis for Fe valence ([83]); Fe-X-ray Absorption Spectroscopy data ([8]; [23]); and indirect Fe³⁺ estimation from single crystal diffraction methods (e.g., [39]; [29]). Particular attention was paid to Ti-rich amphiboles (e.g., [73]; [39]; [8]). Our approach deals with averaged data for each source paper, but additional samples are included in the amphibole spreadsheet (Folders II to IV). Thus, Table 3 summarises the best approaches for calculating formulae for Ca-rich or Na-rich amphiboles for 24 anion (23O + ^WO²⁻/2) formulae, followed by (i) 13 cation Σ¹³ = Σ¹⁶ – Σ(Ca + Na + K) (denoted ∆C = 0) or (ii) as Σ¹³ = Σ¹⁶ – ΣCa + Na + K − ∆C^corr (correction parameter 6.7/13). Ca-rich amphiboles are best fitted using (ii) and Na-rich by method (i). Ca-rich compositions in columns 1 - 5 show how maximum and minimum Fe³⁺ contents could be estimated.

Hawthorne and colleagues used two chemically analysed data sets ([6]; [16]) to discuss how amphibole analyses could be modelled based on different cation totals to estimate the lower and higher possible Fe³⁺ contents. The first two columns in Table 3 show the source chemical data for those amphiboles; method (ii, above) column 1 shows that with ∆C^corr = 0.176 our calculated Fe³⁺/ΣFe of 0.20 is in excellent agreement with the [6] chemical analysis (0.21); while that for column 2 is only slightly higher than for the [16] analysis (0.36 and 0.30, respectively). For further comparison, an early reliable chemical analysis of a Glenelg hornblende with H₂O+ = 2.05 wt. % (equivalent to

Table 3. Amphibole formula checks for published analyses with Fe-valence determined or estimated.

0.02 ^WO²⁻ pfu) and Fe³⁺/ΣFe = 0.21 ([89]) gives a slightly higher calculated 13 cation Fe³⁺/ΣFe = 0.29 (∆C^corr = 0.124, this work) (see Table 3, column 3). We also show the recommended estimated minimum and maximum Fe³⁺/ΣFe from the source references. Columns 4 and 5 show similar data for key amphibole papers ([85]; [86]). For all 5 published Ca-amphibole examples we suggest that our estimated Fe oxidation ratios are more useful for petrologists than those recommended previously based on model calculations for maximum and minimum Fe³⁺ contents (e.g., [84]; [38]) and for those obtained with [57] corrected by subtracting 2Ti from the atomic O values to obtain a corrected value without any estimated ^WO²⁻ component. The same format is adopted to test how the spreadsheet deals will Ti-rich kaersutite compositions (see Folder III, rows 363 and 281-286); Table 3 column 6 shows our data for a chemically analysed sample ([17]) and column 7 an EMP analysis with Fe-valence determined using Mössbauer spectrocscopy ([67]).

[87] [Table 3] studied upper mantle Fe- and Ti-bearing Ca amphiboles, with 13 samples with Mössbauer Fe valence data. We calculated Fe³⁺ for 15 analyses based on formulae calculated using Li et al. ^WO²⁻ with 13 cations and ∆C^corr = 0.116 (Table 3, column 8). The model Fe³⁺/ΣFe ratio is 0.29 slightly smaller than the average Mössbauer value for 12 samples of 0.38 (range 0.13 - 1.0). However, two of the samples show very high Fe³⁺ ratios (1.0 and 0.54) which are not matched by high calculated ^WO²⁻, in line with our earlier suggestion that it is much easier to oxidise Fe during later sample alteration than to equilibrate the bulk composition of the Ti and other components coupled with a high extra-O content (see above). Without the two higher-Fe³⁺ samples the average Fe³⁺/ΣFe ratio for 10 samples is 0.30(8), in excellent agreement with our model value. Both [74] and [29] studied the crystal chemistry of kaersutite from the same Greenland locality with H determined by manometry (Popp) or SIMS (Gatta). Mössbauer (⁵⁷Fe, Popp) and stoichiometry plus SC X-ray and neutron diffraction (Gatta) were used to determine the Fe-valence data giving published Fe³⁺/ΣFe of 0.22 ([74]) and 0.28 ([29]). The coupled high ^WO²⁻ and Ti-Ca-rich composition has ∆C^corrected = 0.090 leading to a model Fe³⁺ content of 0.302 apfu and Fe³⁺/ΣFe = 0.25 matching the average of the Popp and Gatta estimates. [83] studied Ti-Fe-rich calcic amphiboles (edenite to kaersutite) with Fe²⁺ determined chemically, and H and O data by mass spectrometry. We report atomic proportions for the average of 7 samples (Table 3, column 10) with ^WO²⁻ values from the source paper (0.698 pfu). These samples are Ca-Ti-rich with high ^WO²⁻ leading to cation totals higher than 16, and very high 13 cation totals refection C >> 5. The atomic proportions are therefore calculated on the basis of Σ¹³_noCNS − ∆C^corrected (Table 3, column 10). The calculated Fe³⁺/ΣFe for the sample average composition is 0.14 (range for 7 samples 0.09 - 0.23) in excellent agreement with the chemically-determined ratio of 0.13 (range 0.07 - 0.17).

The next 4 examples (Table 3, columns 11 to 14) are for Na-rich amphiboles which are best calculated using method (i) described above (i.e. ∆C = 0). [26] studied Na-rich amphiboles (variable Fe and low Ca and Ti), all with Fe Mössbauer spectroscopy, and microchemical analysis for lithium and H₂O (1.9 - 2.7 wt.%). We report average data for the 12 samples (Table 3 column 13) with 24 anion formulae (Li et al. ^WO²⁻ values), all of which have very low values typical of sodic amphiboles. (low ^WO²⁻, low Ti and with cation totals (Σ¹⁶) < 16.000 requiring the Fe valence values to be calculated on a 13 cation formula simply defined as Σ¹⁶ – Ca + Na + K (denoted Σ¹³_noCNS) (see attached spreadsheet Folder II). The atomic proportions have a model Fe³⁺/ΣFe value of 0.32 (range 0.07 - 0.69) matching measured Mössbauer values of 0.31 (range 0.07 - 0.52). Our calculation method confirms the stoichiometric approach of [26] where their Mössbauer results show good agreement with those obtained using an EMP high-resolution, microanalytical flank method ([47]).

A similar approach was used to calculate formulae and Fe valence data for amphiboles in metasomatised sandstone xenoliths ([39]). In that work, EMP and H-ion probe analyses of 11 samples were studied with Fe³⁺ values estimated from single-crystal X-ray data. [39] showed that Ti and ^WO²⁻ values are very closely correlated (2Ti = ^WO²); this type of relationship is commonly found in amphiboles and biotite micas ([41]). We divided the samples into Na-Ca-rich and Na-rich groups and used ^WO²⁻ values from [39] rather than Li et al. ^WO²⁻ values, which are much smaller for the Ti rich samples. However, the [77]^WO²⁻ values agree with the Hawthorne data, except for the most sodic sample. Our 13 cation formula results are shown in Table 3, (columns 11 and 12) where the atomic proportions and Fe valence values are calculated relative to Σ¹⁶ – (Ca + Na + K) (i.e., Σ_noCNK; attached spreadsheet II). These results support the conclusion of [39] that the formula are best for calculated with ∆C = 0. The Fe³⁺/ΣFe ratio for the Ca-Na average is 0.10 (range 0.06 - 0.11) and that for the Na-amphibole is 0.24 (range 0.18 - 0.26). These values are only slightly different to the SREF estimates reported by Hawthorne et al., i.e 0 - 0.11 and 0.13 - 0.24, respectively.

Column 14 gives data for a Ti-rich ferro-ferri-kataphorite ([11]) with Fe-valence determined with Mössbauer methods and extra-O being modelled on SREF Fe-valence and on a Ti – O3 relationship similar to that discussed above (extra-O = 2Ti). The Folder II calculation (Folder II, row 275) uses the Li et al. estimated ^WO²⁻ and the full formula is very similar to that reported in the first column of Table 2 published in ([11]) with our Fe³⁺/ ΣFe = 0.33, only slightly smaller than the Mössbauer value of 0.38 (our Table 3, column 14) which, in turn, is lower than the value of 0.43 obtained using SREF ([11]).

Table 3 column 15 data gives calculated Fe³⁺/ΣFe ratios significantly larger than those of the published values which are difficult to reproduce using our adopted calculation protocols, but we will draw attention to possible problems. Thus, [8] studied Ti-rich calcic amphiboles within metasomatised lherzolites. Analytical results are reported for 6 samples, including SIMS for H, single crystal XRD, together with pre-edge X-ray absorption spectroscopy (XAS) for two samples to estimate Fe³⁺/ΣFe values of 0.21(5) and 0.16(5). Formulae for these samples (Table 3, column 15) are reported for a 24 O formula with Fe³⁺/ΣFe varying from 0.14 to 0.24 (average 0.17(4)). We have recalculated those analyses with all Fe input as FeO using ^WO²⁻ values from [8], ranging 0.499 - 0.994 (average 0.866). Our formula proportions for these Ca-rich amphiboles are calculated on the basis of Σ¹³_noCNS − ∆C^corrected (average ∆^corrected = 0.138); we report an average Fe³⁺/ΣFe of 0.63 (range 0.5 - 0.8) much more oxidising than the [8] values. Model ^WO²⁻ values obtained from the Li et al. spreadsheet are smaller ranging 0.372 - 0.756 (average 0.623) which provide lower Fe³⁺/ΣFe ratios of 0.39 - 0.61 (average 0.47). We will consider below the possibility that the XAS-determined values for Fe³⁺/ΣFe values could be overestimates.

[8] calibrated the Fe³⁺/ΣFe values for their samples using the positions of the pre-edge feature in the XANES spectrum (e.g., [98]). The peaks resolved at the absorption edge show some support for low oxidation values but post-edge features are much more similar to more oxidised compositions. Indeed, amphibole XANES spectra in [37] and [23] have amphiboles with Fe³⁺/ΣFe ratios as high as 0.4 - 0.5 show XANES features consistent with those for the Bonadiman samples (cf. data for a Ba-Ti rich mica, [42]). Indeed, it has been shown that using particular XANES energies to deduce Fe³⁺/ΣFe values could provide more robust data than PE refinements (e.g., amphiboles ([23]); garnets ([5]; [22])). We have tested the suggestion that some Ti content might involve a coupled Ti-vacancy in the A site (^A□^Ti) by using a hypothetical amphibole molecule based on the solid solution 0.5-ferrihornblende – 0.5ferrotaramite (our Table 1, column 7: □_0.5Na_0.5(Na_0.5Ca_1.5) (Mg₂Fe²⁺_1.5Fe³⁺_0.5Al)[Si_6.5Al_1.5]O₂₂(OH)₂). The proportions of Fe²⁺ and Fe³⁺ are kept constant, as are those of Si and Al, but 0.5 atoms Ti are substituted for 0.5 atoms Mg at C leaving C full with a Mg_1.5 content; in addition the original vacant 0.5 apfu at A is substituted with an A vacancy related to Ti entry (□^Ti_0.5). The result is that there is a cation and anion valence imbalance which is restored by substituting the Na content at B with Ca and with one oxygen replacing hydroxyl to cope with the molecule now having 47 positive charges. This amphibole now has 50% amphibole in solid solution with the overall formula □_0.5□^Ti_0.5(Ca₂)(Mg_1.5Ti_0.5Fe²⁺_1.5Fe³⁺_0.5Al)[Si_6.5Al_1.5]O₂₃(OH)) requiring the formula to be recalculated to 23.5 oxygens pfu (cf. our Table 1, column 9). With this approach the phase has a cation total of 15.1574 not including the known vacancy account of 1.0 apfu; as shown above in this case the Fe³⁺ could be calculated on the basis of a 15 cation calculation and this gives a value of 0.488 pfu for Fe³⁺, close to the known input value. The 23.5O Σ¹³ cation formula calculated from Σ¹⁶_novacancy – ΣCaNaK gives the value 15.1574 − 2.019 = 12.633, with Fe³⁺ = 12/12.63 = 2.52 apfu, 5 times too high. Calculation of a possible value for Fe³⁺ can only be made using a ‘cation’ sum of 12.5 with the value for the □^Ti of 0.5 apfu counted as a coupled, essential virtual Ti cation rather than a real vacancy, thus Fe³⁺ = 12.5/12.633 = 0.495 pfu, the known value within error. It is clear that this approach could be used for dealing crystal-chemical problems with Ti-rich amphiboles.

We conclude that our calculation method is robust for stoichiometric amphiboles; however, if the [8] amphiboles are indeed reduced, we can only suggest that some of the Ti in C must be present associated with a matching vacancy at A. Thus, if 19 % of the Ti is present coupled to a matching vacancy (see earlier), that would reduce the average Fe³⁺/ΣFe from 0.63 to 0.18, to match the published values ([8])! See the attached spreadsheet Folder III for the numerical details. The last dataset in Table 3, column 16 is for a synthetic Al-rich,Fe-free kaersutite (sample (K(1) [88]) with water, F and Cl measured using SIMS. The calculated 24 anion formula (Supplementary file 1, Column I) has a moderate ^WO²⁻ value of 0.62 apfu mainly reflecting the high Ti and Al contents (cf, a Li et al. spreadsheet vaue of 0.55 apfu). The formula shown in Table 3 is calculated based on an O factor of (23 + ^WO²⁻/2) and 16 atoms (Supplementary file 1, Folder II). Those values and the equivalent data shown in Folders III and IV are all in close agreement with the 24 anion data (Folder I) with slightly smaller values for the anhydrous data calulations (e.g., ~1% for Si and ~0.2% for Mg (relative) for the column III data. The strcture determined by [88] has Ti ordered at M(1) and Al at M(2).

Finally, the high quality analyses for Ca-amphiboles discussed above, with EMP TiO₂^, Mössbauer or chemical Fe valence, and heat-extracted water reduced with U or SIMS H values (e.g., [12]; [24]; [83]; [74]) all show a clear positive relationship between analytically estimated ^WO²⁻ and Ti content up to ~8% TiO₂ consistent with data for Coyote samples ([39]; see above). Indeed, the common practice is to assume the relationship 2Ti = ^WO²⁻ apfu^, in effect with Ti as a proxy for ^WO² ([38]; [71]). Indeed, [73] have also suggested that that relationship is site-specific for Ti occupying the M(1) octahedral site in the C cation group; that site is coordinated with 2 O(3) sites in the W position. Most crystallographers working on low Ti amphiboles have assigned Ti to M(2) together with trivalent cations (mainly Fe³⁺ with Cr if present) rather than (see above); higher Ti contents are ordered at M(1) (see above). However, most of that work has been done using single-crystal X-ray diffraction which shows poor scattering contrast for Ti with dominant Mg and Fe components but it is generally assumed that occupancy of M(3) is minor while that at M(2) is very low. However, [49] studied a Ti-rich kaersutite by single-crystal neutron diffraction which shows high scattering contrast with (Ti cross section b_c = −3.4 fm), compared with Fe (9.45) and Mg (5.4); Kitamura found a strong preference for Ti in M(1) with 0.27 apfu and low occupancies at M(2) 0.02 and M(3) 0.04 apfu. [29] used X-ray and neutron scattering to study a Greenland kaersutite and reported 0.38 apfu Ti in O(1) with 0.16 apfu in bothO(2) and O(3); thus 2Ti^O(1) = 0.76 which is lower than measured ^WO²⁻ of 0.935 apfu ([29]). However, M(3) also has 2O(3) atoms as nearest neighbours so for this sample 2 Ti^O(1+3) = 1.08 apfu; however, [39] point out that the local geometry of the M(3) site in richterites is not favourable for the presence of Ti. Nevertheless, it appears that for Ti-rich, Ca amphiboles, the overall Ti distribution in the C group might indeed not follow a strict ^M(1)Ti vs ^WO²⁻ but may involve a contribution from some Ti in M(3). Indeed, the more general relationship might be Ti⁴⁺ + 2 O²⁻ = (Mg, Fe²⁺) + 2(OH)⁻, as suggested in [73] and [39].

This is the approach we will follow for the remainder of this paper. However, robust Fe³⁺ contents can only be obtained on a 16 cation basis if it is known that the vacancy count is very small. However, if Σ¹⁶ is >16.2 with Σ¹³ ~13.2 it is worth trying a 16 cation total calculation (see spreadsheet in Folder II of the attached EXCEL file). Nevertheless, if a ‘13 cation’ total (Σ¹³) at this stage is larger than ~13.20 to 13.25, a 13 cation calculation might be preferred calculated as Σ¹³ = Σ¹⁶ − (Ca + Na + K) − ∆C^corr, where the ∆ term includes the correction factor 6.7/13 (see above and supplementary file, Folder III, this work). However, many of the analyses shown here have Σ¹⁶ total less than, or only slightly larger, than 16; and may have ∆C^corr values < 0.05 - 0.1; thus, in this work these amphiboles have 13 cation atomic proportions on the basis of ∆C = 0. However, if some particularly Ti-rich samples cannot be reasonably treated using the above approaches, the possibility of some Ti being present coupled to essential ‘virtual-Ti’ vacancies should be considered. However, if the vacancy content and Fe valence ratio is not known from reliable analyses, that approach would not particularly useful for assessing robust Fe valence estimates.

In Table 2 and Table 3 we have compared our adopted estimates for Fe³⁺/Fe with the analysed values (Mössbauer or chemical analyses) and reported generally good agreement for both Ca- and Na-rich amphiboles. We have also calculated the root-mean-square error (RMSE) for 16 samples and can report an average error of 0. 011 apfu for an average sample with Fe³⁺/ΣFe = 0.25 (i.e., 0.25 ± 0.011 apfu); this relative error is equivalent to a total error of 18% for Fe ³⁺/ΣFe values which is equivalent to the errors reported by [77] and [55]. However, the relative values and general trends for Fe³⁺/ΣFe ratios found here for different samples will be more robust than calculated Fe³⁺ atomic proportions.