Ne–Q solid-solution series

Compositions are chosen with some at 10% Q intervals with others having integral numbers of cavity site vacancies. The unit cell with 32 oxygen cations has 24 sites of which 16 are tetrahedral cations in framework sites and 8 are cavity cations and vacancies occupying the interframework (cavity) sites.

Five representative examples for this solid-solution series will be discussed here. Thus the top five rows of data in Table 1 show atomic occupancies for the two end-members 100Ne (Na₈Al₈Si₈O₃₂) and 100Q (□₈Si₁₆O₃₂), and for three intermediate compositions: 87.5Ne12.5Q (□₁Na₇Al₇Si₉O₃₂); 70Ne30Q (□_2.4Na_5.6Al_5.6Si_10.4O₃₂); 50Ne50Q (□₄Na₄Al₄Si₁₂O₃₂). The first column shows the number of vacancies in the cavity sites for each sample; in this system all of the vacancies result from the presence of the excess Si component of the solid solution. The next column is not relevant for this system. The third column shows the number of Na atoms present in the cavity sites for each sample. The fourth column is not relevant for this system. Column 5 shows the atoms of tetrahedral Al for the Ne component and column 6 shows the tetrahedral Si contents for Ne. Column 7 gives the tetrahedral Si contents (Si^xs) for the excess Q^xs component where the excess silica molecule is defined as Si₂O₄ (see Blancher et al., Reference Blancher, D'Arco, Fonteilles and Pascal2010; Hamada et al., Reference Hamada, Akasaka and Ohfuji2019) to keep the same number of oxygens as for the other end-members. Thus, the number of silicons associated with the excess Q^xs molecule is double the number given in column 7 and that molecule has a molecular weight of 120.16 rather than 60.08. Note that the tetrahedral Al + Si always total 16 and that the cavity sites total 8, which includes both Na cations and vacancies associated with the presence of the excess Q’; the total of cations plus vacancies is always 24. Two important features are defined here: the number of vacancies in the cavity site (□^Si) is exactly half of the excess Si^xs value and matches the amount of the Si₂O₄ molecule; no vacancies are associated with the Na component. These features are fixed by the unique nepheline stoichiometry and structure. Column 9 shows the % of Ne component; this is calculated following Barth (Reference Barth1963):

$$\% {\rm Ne} = \lpar {3 \times {\rm atoms}\; {\rm of}\; {\rm Na}} \rpar \times 100/24$$

where stoichiometric Ne has equal numbers of Na, Al and Si atoms so the total cations in Ne can be calculated as 3 × the Na value. As expected the top five bulk compositions are simply 100% Ne, 87.5%, 70%, 50% Ne and 0% Ne; the same result would be obtained by simply taking the number of Na atoms in each sample and relating that to the total number of cavity sites in the structure, i.e. 8.

In columns 10 to 14 the calculations for obtaining values for Barth's An’ and Q’ (cf. CaNe and Q^xs) components are shown; columns 10 and 11 are not relevant to the Ca-free compositions nevertheless several problems would emerge for Ca-bearing samples. Thus, column 10 shows Barth's (Reference Barth1963) equation for calculating the Ca end-member for natural nephelines and it is necessary to explain the protocol here. The atomic proportions for An were based on the stoichiometric formula CaAl₂Si₂O₈ and Barth defined the anorthite proportion as 5 × Ca (i.e. Ca+2Al+2Si). Because Ca is absent in the Ne–Q samples, zero An is reported for all five Ca-free samples, but the wrong An contents would have been reported if Ca had been present. Column 11 gives the correct equation for calculating the An’ (CaNe) component; this equation includes the cavity site vacancy associated with the Ca component.

The Si^xs present in the tetrahedral framework of any nepheline solid solution is most simply calculated as the excess after the Si associated with the other nepheline end-members has been subtracted from the total number of Si atoms in the cell formula; note that one of the Blancher et al. (Reference Blancher, D'Arco, Fonteilles and Pascal2010) calculations defined a value for the excess Si vacancy as □^Si = (Si^total – Al)/2. For recalculating nepheline analyses Barth (Reference Barth1963) appears to have used the calculation Q’ = 24 – 3Na – 3K – 5Ca with the result shown in Table 1, column 12; note that Q’ refers to the total excess Si molecule that takes account of the cavity site vacancies associated with both excess Si and Ca. For the top five rows in Table 1, column 12 shows that the total Si excesses are correctly calculated for all compositions. As expected, this calculation has provided the total Q’ value, which includes both the actual Si^xs, which is a framework component, and the associated cavity cation site vacancies (□^Si). Using the numbers of framework Si atoms and associated Si vacancies defined in Table 1 the separate Si^xs and □^Si values are 8.34 and 4.17% for Ne87.5Q12.5, 20 and 10% for Ne70Q30, 33.33 and 16.66% for Ne50Q50 and 66.66, 33.33% for 100Q. Thus □^Si is exactly half of the excess Si^xs, though that has been defined for ideal nepheline stoichiometry. The Barth calculation only works here because no Ca component is present for this binary solid-solution series. Column 13 in Table 1 gives rigorous definitions of how to calculate excess Si values for Ca-bearing compositions. Thus this equation gives the correct total proportion for Q^xs. Another relationship given in Column 13 defines the vacancy proportion in the cavity cation sites directly: □^Si = ((8 – Na – 2Ca) × 100) / 8 and this also provides the correct value for Q^xs. The most direct equation for calculating the total excess Si component is also shown in Column 13 where the total number of Si atoms minus the Si atoms contributed by the different nepheline components divided by 16 (the total tetrahedral cations) provides the required total Q^’; the fact that the number of excess Si atoms is normalised to 16 rather than 24 means that it is not necessary to consider the presence of any vacancies in the cavity sites. The equation Q’ = ((8 – Na – 2Ca) × 100)/24 gives the proportion of □^Si in the bulk composition equivalent to Q’/3, which leads to a framework excess Si proportion of ⅔Q’, as required for ideal stoichiometry. Finally, a value for the total excess Si content can also be calculated as (Si^total – Al) × 100/16 as Ne is defined to have exact stoichiometry.

All these examples are defined with ideal nepheline stoichiometry, however, analyses of natural minerals would be subject to analytical error and the equation Q^xs = (24 – Si^total – Al – Na – K – 2Ca) × 100/8 would provide a value for excess Q^xs that would include most of the analytical errors (see below).

Solid solutions in the system Ne–An

At this stage to be consistent with published nomenclature the Ca-component will be referred to as An’. The five middle rows of Table 1 give atomic occupancy of compositions for 100Ne together with 90Ne10, 80Ne20, 50Ne50An and 100An. Note that the vacancy associated with the presence of Ca exactly matches the Ca content (columns 2 and 4) and column 5 shows the Al associated with both Ne and An with the former value first. All the solid solutions would contain 8 Si atoms per 32 O (Table 1, column 6); no excess Si is present. The Ne contents are shown correctly in column 9. The data in column 10 shows how Barth (Reference Barth1963) calculated the Ca end-member; however, for the Ne–An samples shown, the An proportions calculated are 0, 8.33, 16.66, 41.67 and 83.33% rather than 0, 10, 20, 50 and 100%, respectively. Even though Barth (Reference Barth1963) stressed that the vacancy contents must be treated as ‘cations’ this was not followed in his calculation (see his tables 1 and 3). In column 11 the correct calculation is shown for the 24 cation basis that uses the term 6 × Ca that includes the appropriate associated Ca vacancy and this gives the correct An’ values; this calculation could also be defined as (5 × Ca + □^Ca) × 100/24. Indeed, the Ca component can also be defined in terms of the cavity cation contents only, giving (Ca + □^Ca) × 100/8 that is the same as 2Ca × 100/8 as by definition for nepheline stoichiometry Ca = □^Ca, and this step defines the naming of the simplest Ca component as □^Ca_0.5Ca_0.5AlSiO₄ (CaNe, unit cell □₄Ca₄Al₈Si₈O₃₂). Data derived for apparent excess Si^xs using the equation defined in column 12 shows that the deficiency for An’ in column 10 is indeed accounted for by Ca vacancies, which would show as excess Si using Barth's calculation. For the Ne–An series column 13 shows correctly that no excess Si is present.

Clearly the doubled Ca term involved in this calculation is related to the divalent character of Ca²⁺ that replaces two sodiums in cavity sites, which in turn is reflected in the fact that the end-member Ca_0.5AlSiO₄ has an atomic formula defined by half a Ca atom so that a given fraction of Ca atoms (in this case on a 4 O basis) leads to a doubled Ca fraction for the CaNe component.

Nephelines in the ternary system Ne–Q–An

The atom occupancy data for three ternary solid solution compositions are given in the lower part of Table 1. The first five columns have similar information to that provided above, though column 6 now shows the Si contents for both Ne and An (in that order). Column 9 again shows that the proportion of Ne can be calculated directly from the Na value either on a 24 cations or an 8 cavity cation basis. Column 10 again gives An’ contents that are ⅙Ca too low whereas column 11 gives the correct An’ values; the An’ deficit shown in column 10 can again be identified as the □^Ca values in column 12. In addition, column 12 now shows that the total excess Si values are too high by that same amount and that the sum of the apparent Si^xs and □^Si now gives the correct Q^xs content. It is clear that the Barth (Reference Barth1963) calculation only provides the total amount for the two vacancy types; Barth (Reference Barth1963) does suggest that the ‘holes’ could be considered as separate components, but this is not considered to be a reasonable suggestion as the An content would always be underestimated and the Q^xs content overestimated. Equations to calculate the correct amounts for Q^xs are shown in column 13 on both 24 and 8 cation bases; correct values would also be obtained using the relationships: (Si–Al) × 100/16; 1.5 × Si^xs × 100/24; 0.5 × Si^xs × 100/8; and □^Si × 100/8.

Dealing with other calculation protocols and nepheline components

The Barth (Reference Barth1963) approach is followed for defining all of the nepheline end-member molecules on a 32 oxygen basis. Based on the section above, it is now clear that the main requirement in calculating molecular/atomic compositional parameters following ideal nepheline stoichiometry are that the vacancies associated with the entry of both Ca and excess Si into the structure must be counted as cavity cations. Naturally occurring nephelines contain significant amounts of kalsilite and the formulae developed so far will simply require the addition of a Ks component. Bannister and Hey (Reference Bannister and Hey1931) and Dollase and Thomas (Reference Dollase and Thomas1978) stressed that high-quality analysed nephelines samples should have the stoichiometry parameters (Si + Al + Fe³⁺) = 16.000 (within error) and (Na + K + 2Ca) = (Al + Fe³⁺) within error (see below). Depending on such considerations it is possible to standardise calculations by assuming exactly 24 total cations and exactly 8 cavity sites as required by symmetry and chemistry; alternatively, the former might be corrected to the analysed cation total for Si, Al, Fe, Ca, Na, K plus vacancies (which should be close to 24 per 32 O) and for the analysed total for Na, K, Ca plus vacancies (which should be close to 8 per 32 O). Application of these rules to recalculating formulae for natural nephelines is assessed in the next section and will be applied to representative published analyses.

Other researchers [Peterson (Reference Peterson1989) and Worley and Cooper (Reference Worley and Cooper1995); Rossi et al. (Reference Rossi, Oberti and Smith1989); and Hamada et al. (Reference Hamada, Akasaka and Ohfuji2019)] have used different oxygen bases and/or end-member components. Thus Peterson (Reference Peterson1989) used an 8 oxygen basis to define the end-member components Ne (Na₂Al₂Si₂O₈), Ks (K₂Al₂Si₂O₈), Nf (Na₂Fe³⁺₂Si₂O₈) (i.e. iron nepheline), An (CaAl₂Si₂O₈), Qz (Si₄O₈), and Cn (Al_16/3O₈); the last component (in effect a theoretical corundum content) was introduced in an attempt to account for analyses so rich in Na and K that deficiencies were reported for both Al and Si. Rossi et al. (Reference Rossi, Oberti and Smith1989) used a 32 O basis and end-members K₂Na₆Al₈Si₈O₃₂, Na₂Na₆Al₈Si₈O₃₂ and Ca□^CaNa₆Al₈Si₈O₃₂ and, more recently, Blancher et al. (Reference Blancher, D'Arco, Fonteilles and Pascal2010) reported nepheline end-members based on 4-oxygen cells as NaAlSiO₄, KAlSiO₄, □^SiSi₂O₄, and □^Ca_0.5Ca_0.5AlSiO₄. Finally, Hamada et al. (Reference Hamada, Akasaka and Ohfuji2019) also reported nepheline end-members based on 4-oxygen cells as NaAlSiO₄, KAlSiO₄, □^SiSi₂O₄, and □^Ca_0.5(Ca,Mg)_0.5AlSiO₄; note that the last component assumes that Mg occupies the cavity sites that is considered to be unlikely (see above). Dealing with the same end-member compositions standardised to any chosen number of oxygens would provide the same results, however application of the Peterson and Hamada approaches could lead to either problems or possibly errors; such problems are discussed in the following section.

Criteria for assessing analysis reliability and end-member calculation protocols

Section S.3 in the supplementary files gives a more detailed treatment of how earlier workers have assessed the reliability of nepheline analyses. In this paper the approaches of Bannister and Hey (Reference Bannister and Hey1931) and Dollase and Thomas (Reference Dollase and Thomas1978) have been extended to show reliable nepheline analyses must have coupled ΔAl^{cavity cation} [Δ({Al+Fe³⁺}–{Na+K+2Ca})] and ΔT^charge [{(Al+Fe³⁺) ×3 + (Si+Ti) ×4} – {16 × measured mean tetrahedral charge}] parameters. For the analyses given by Bannister and Hey (their table 1) ΔAl^cc would have a mean value –0.011 and ΔT^charge a mean value of –0.017. The average ΔAl^cc/ΔT^charge ratio for their samples is 1.17 (range 1.01 to 1.24); the ΔAl^cc/ΔT^charge ratio must reflect a Si/Al ratio > 1 as expected for natural nephelines. For exact nepheline stoichiometry (unit cell 32 oxygens, 16 T atoms, 8 cavity sites, Si = Al) ΔAl^cc and ΔT^charge both equal 0. For departures from these ‘ideal’ values, delta parameters would be coupled with either positive or negative values, though both should have the same sign and have similar values in the absence of significant analytical errors. On the basis of the formulation used to define the delta values, an increase of the T site cation total to > 16.00 would lead to negative ΔT^charge values (coupled to negative ΔAl^cc) and if the T cation site total is < 16.00 the ΔT^charge would be positive (cf. +ve ΔAl^cc).

More than half of the compositions discussed by Bannister and Hey (Reference Bannister and Hey1931) show an excess of cavity cations over trivalent framework components. This excess positive charge could be neutralised by the Si component of the framework (cf. the natural mineral natrosilite, β-Na₂Si₂O₅) but that implies breaking Si–O–Si framework bonds). However, it is also possible that the charge for a small excess alkali content could be neutralised by the presence of large anions (e.g. CO₃^2–, Cl) as in cancrinites (see Supplementary material, section S.3); this is dealt with in more detail below.

The fact that natural nephelines mostly have Si/Al ratios >1.0 has been accounted for by the presence of a feldspar-like component and the same could be said for an anhydrous analcime component (NaAlSi₂O₆); such solid solutions would of course have stuffed tridymite frameworks (see above). Using ‘nepheline’ 32 oxygen unit-cell stoichiometry, ΔAl^cc and ΔT^charge parameters for ideal NaAlSiO₄, NaAlSi₂O₆ and NaAlSi₃O₈ should have zero values for both delta parameters for each of these three end-members with the different Si/Al ratios of 1, 2 and 3, respectively. A ratio for ΔAl^cc/ΔT^charge cannot be calculated for zero delta values, however by allowing a very small degree of non-stoichiometry for the Si, Al or Na components, the estimated ΔAl^cc/ΔT^charge ratios are close to 1.143, 1.091 and 1.067, respectively (Supplementary section S.3); analyses of natural minerals confirm these ΔAl^cc/ΔT^charge ratios (Table S.1). Note that the size of the delta values relative to zero reflects their departure from ideal nepheline stoichiometry. Thus compositional variations within natural nephelines would be expected to show at least this range of ΔAl^cc/ΔT^charge ratios for acceptable nepheline analyses. At this stage, the Dollase and Thomas (Reference Dollase and Thomas1978) approach of defining a range of ±0.25 for both ΔAl^cc and ΔT^charge will be adopted for now, nevertheless the importance of these values being coupled is stressed. In addition, the ΔAl^cc/ΔT^charge range from 1.0 to 1.2 is used here to define acceptable analyses.

On the basis of these criteria, the compositions shown in table 6 of Deer et al. (Reference Deer, Howie and Zussman2004) have 32 of 39 analyses falling within this acceptable range for both ΔAl^cc and ΔT^charge; 25 of the 39 analyses have ΔAl^cc/ΔT^charge ratios within the range 1.0 to 1.2 (22 of these in the narrow range 1.120 to 1.142), and 33 of the analyses have T atom totals within the range 15.9 to 16.1 atoms per 32 O. Any major differences between the two delta parameters must reflect either non-stoichiometry and/or significant analytical errors for one or more of the analytical values.

Table 2 summarises the equations used in this paper to calculate nepheline end-member proportions on the basis of 32 O cell formulae and on the simple formulae: NaAlSiO₄ (Ne), KAlSiO₄ (Ks), □^Ca_0.5Ca_0.5AlSiO₄ (CaNe), and □^SiSi₂O₄ (Q’). Equations 1 and 2 define the excess Si (Si^xs) and, following Barth (Reference Barth1963), equations 3, 4, 5 and 6 define the proportions of Q^xs, Ne, Ks, and CaNe taking account of essential vacancies in the cavity cation sites. Equation 7 provides another estimate of total excess silica (Q^Si) that is based on a 16 T-atom framework; note that the actual per cent of Si in the framework is given by Si^xs = (Si_total – Na – K – 2Ca) × 100/24 and the amount of vacancy associated with the excess Si component □^Si = (8 – Na – K – 2Ca) × 100 / 24. The relative proportions of these Si parameters are 3:2:1, which is fixed by the ideal nepheline stoichiometry and structural symmetry. Equation 8 defines a fourth estimate for the total excess silica (Q^(Si–Al)) and this is also defined on a 16 T-atom framework; this parameter does not depend on the reliability of analyses for the cavity cations.

Table 2. Equations for calculations based on a 32 oxygen nepheline structure unit cell.

Of course, all of these end-member parameters can also be defined from the occupancy of the cavity cation sites alone (equations 9, 11, 12 and 13) and these provide the same numbers as equations 3, 4, 5 and 6. Equation 10 provides an additional estimate of the total excess silica proportion (Q^cav2) based on the total cations; this value uses the proportion of vacancies in the cavity site corrected for the presence of divalent Ca. For ideal stoichiometry all calculated total excess silica values should be identical; different values will depend on the scale of non-stoichiometry and/or the presence of analytical errors. In addition the second value defined for Q^cav would be influenced by any analytical error for Al (equation 10). Blancher et al. (Reference Blancher, D'Arco, Fonteilles and Pascal2010) calculated excess Q simply as 0.5(Si^tot – Al) × 100/8 (cf. Table 2, equation 8) whereas their most complex method involved using least squares fitting to minimise the function

$$R^2 = \lpar {{\rm Na} - {\rm Ne}} \rpar ^2 + \lpar {{\rm K} - {\rm Ks}} \rpar ^2 + \lpar {{\rm Ca} - {\rm An}/2} \rpar ^2 + \lpar {{\rm Al} - {\rm Na} - {\rm Ks} - {\rm An}} \rpar ^2 + \lpar {{\rm Si} - {\rm Na} - {\rm Ks}- {\rm An} - 2{\rm Q}^{\prime}} \rpar $$

where Na, K, Ca, Al, and Si are numbers of atoms per 4 oxygens and Ne, Ks, An and Q’ are mole fractions of the chosen end-members per 4 O.

The values calculated here for the chosen nepheline end-members CaNe (An’) and total excess Si (Q^xs) for the 39 analyses quoted by Deer et al. (Reference Deer, Howie and Zussman2004, their table 6) are given in the Supplementary material (Table S.2; these are referred to here as ‘DHZ values’) and show significant differences to those given in the original source. For Ca-free nepheline analyses the CaNe and Q^xs values calculated here match those in the original source, but for Ca-bearing samples the new calculations generally have higher CaNe and lower excess Q values than the DHZ values. Thus, it is probable that the DHZ values were calculated using the Barth (Reference Barth1963) procedure where their An content determination did not take account of the vacancy associated with the presence of Ca; this results in An contents that are ⅙Ca too low and excess Q^xs values that are ⅙Ca too high. Tables S.2 (Supplementary file) and Table 4 (main paper) also show values for a new nepheline end-member defined by its content of a divalent tetrahedral component of formula K₈M ²⁺₄Si₁₂O₃₂; note that M = Mg+Fe²⁺+Mn (see above) where Mg tends to be the largest divalent component reported in natural nephelines. On the basis of the simplest formula for this component of KM ²⁺_0.5Si_1.5O₄ (8 per equivalent nepheline unit cell) the proportion of the KsM molecule would be based on 0.5 atoms of M ²⁺ and calculated from % KsM = 6M × 100/24 (Table 2, equation 14). It is probable that the most reliable data for this component comes from electron microprobe analyses (e.g. Dawson et al., Reference Dawson, Smith and Steele1995; analysis # 34) as the older ‘wet’ chemical analyses of mineral separates might have such components reflecting the presence of impurity mineral phases. Calculation of the excess Q^xs is not affected by incorporation of this KsM component as the total K, Si-equivalent is subtracted from the analysed Si.

The end-member molecules calculated for natural nephelines by Peterson (Reference Peterson1989) give values for both excess and deficient Si and Al molecules which, based on the 32 O cell used here, have the ideal formulae □₈Si₁₆O₃₂ and □_2.666Al_21.333O₃₂; the percentage values for these are simply calculated as [(total Si – Na – K – 2Ca) × 100] / 16 and [(Al – Na – K – 2Ca) × 100] / 21.333, respectively. The concept of an excess or deficient content of Al (+ Fe³⁺) compared with that associated with Na, K and Ca is not straightforward crystal chemically. Analytical error is probably an important factor with an excess of Al related to alkali ‘loss’ in the electron microprobe if a defocussed beam or raster scan is not used (Morgan and London, Reference Morgan and London2005; Henderson and Pierozynski, Reference Henderson and Pierozynski2012); a lower-than-normal beam current is also essential. If the analysis conditions are optimal, an excess end-member alumina (Al₂O₃) content implies that this component has a framework structure with both tetrahedral and octahedral Al sites, perhaps similar to an anhydrous, hexagonal kappa- or chi-alumina phase (Okumiya and Yamaguchi, Reference Okumiya and Yamaguchi1971; Levin and Brandon, Reference Levin and Brandon1998). Though the presence of a small amount of such a molecule in solid solution might be possible, the occurrence of significant interstitial octahedral Al cations is more problematical. The presence of some CaAl₂O₄ in solid solution might be possible (Goldsmith, Reference Goldsmith1949; Donnay et al., Reference Donnay, Schairer and Donnay1959) but that would not explain a deficiency in the Δ(Al – cations) parameter. Perhaps it is possible that a small amount of an alkali-rich and Al- and Si-poor alteration phase is present in such natural nephelines. The most probable alteration mineral would be cancrinite as this has a hexagonal structure with six-rings of tetrahedra similar to those of nepheline, although the linkages of these units are different in the two mineral groups. The general formula for the cancrinite group is (Na,K)₆Ca₂[(Al,Si)₁₂O₂₄](CO₃,SO₄,Cl,OH)₂⋅nH₂O, which on a 32 O basis would be (Na,K)₈Ca_2.66[(Al,Si)₁₆O₃₂](CO₃,SO₄,Cl,OH)_2.66⋅nH₂O. This is compositionally similar to nepheline but is structurally different with the extra Na, Ca and large anions occupying ‘cancrinite cages’ and large continuous channels (Deer et al., Reference Deer, Howie and Zussman2004; their figure 245); note that entry of any extra Ca with its anion would not be associated with entry of an equivalent vacancy in a cavity site. Some cancrinite species are possibly stable at high temperatures (Larsen and Foshag, Reference Larsen and Foshag1926) and Edgar (Reference Edgar1964). Indeed, Sirbescu and Jenkins (Reference Sirbescu and Jenkins1999) have investigated the system Na₂O–CaO–Al₂O₃–SiO₂–CO₂–H₂O at 2 kbar and reported a triple point involving nepheline, cancrinite and silicate melt at ~950°C. It appears that no-one has studied the possibility of some solid solution between nepheline and cancrinite.

A new database of published nepheline analyses

For this paper a database of 310 published nepheline analyses as up-to-date as possible has been assembled. Table 3 shows a list of source publications and the following treatment is based on these data, together with other papers referred to in the text. The summary comments made in Table 3 are supplemented in the following text. The representative published analyses shown in Table 4 were chosen to reflect nepheline compositions from a range of magmatic rock types where the original authors might have used different protocols for calculating CaNe (An’) and excess Q’. Thus, Table 4 shows the calculations of the formula units based on the 32 oxygen cell; the CaNe component was defined by multiplying the number of Ca atoms by 6 and the total Q^xs is calculated from equation 3 of Table 2; an excess Q’ value based on the total Si atoms (Q^Si) is calculated using equation 7 of Table 2. Table 4 also gives two further estimates for excess Q’, the third defined above as Q^cav (equation 9 or 10) and the fourth which is based on the excess of Si over that for Al alone (Q^(Si–Al); Table 2 equation 8). All four excess Q estimates should be similar for acceptable analyses and would have identical values for ideal nepheline stoichiometry. Fourteen of the 16 analyses shown in Table 4 have ΔAl^cc and ΔT^charge within ±0.25 and represent ‘acceptable’ analyses, whereas compositions 15 and 17 fall outside the acceptable range. In addition, 13 have ΔAl^cc/ΔT^charge within the narrow range 1.07 to 1.14 with analyses 13, 15 and 18 having significantly lower values of 0.65, 0.73 and 0.75, respectively (see below). Significant differences between the four Q’ values mark a departure from ideal stoichiometry although analytical errors could be partly responsible; as a ‘rule of thumb’, a ΔAl^cc value of 0.25 would lead to a difference of at least 10% (relative) between the values for Q^xs and Q^Si.

Table 3. Summary of papers dealing with nepheline compositions in natural rocks.

ΔAl = atoms (Al + Fe³⁺) minus (Na + K + 2Ca) per 32 O; Δ^atoms = 24 minus (total all other atoms) per 32 O. D&T and H&G refer to references Dollase and Thomas (Reference Dollase and Thomas1978) and Henderson and Gibb (Reference Henderson and Gibb1983), respectively.

Table 4. Recalculation of nepheline atomic and molecular formulae for published analyses based on ideal nepheline stoichiometry.

Abbreviations in top row: b.d.l. below detection limit; n.a. not analysed; Syn = synthetic; Crinan. = crinanite’ Theral / Ther = theralite; Ne Sy = nepheline syenite; Nephel = nephelinite. *Calculation for Q^(Si–Al) values; ^& -ve Q recalculated Ne and Ks

References and sample numbers: [1] Barth (Reference Barth1963), table 2, sample (#) 1; [4] Dollase and Thomas (Reference Dollase and Thomas1978), table 2, # 1; [5] Henderson and Gibb (Reference Henderson and Gibb1983), table 1, # g; [6] Gibb and Henderson (Reference Gibb and Henderson1978), table 1, # AC490; [7] Wilkinson and Hensel (Reference Wilkinson and Hensel1994), table 2, # 1; 8. Melluso et al. (Reference Melluso, Srivastava, Petrone, Guarino and Sinha2012), table 6, # J56, includes Sr 0.064 atoms; [9] Blancher et al. (Reference Blancher, D'Arco, Fonteilles and Pascal2010), table 1, # 5; [10]. Dollase and Thomas (Reference Dollase and Thomas1978), table 2, # 4; [11] Henderson and Gibb (Reference Henderson and Gibb1972), table 2, Average of 5; [12] Mitchell and Platt (Reference Mitchell and Platt1979a), table 1, # 14; [13] Zhu et al. (Reference Zhu, Yang, Sun, Zhang and Wu2016), table 11, # 11SM17, includes Ti 0.009, Mn 0.005 atoms; [14] Andersen et al. (Reference Andersen, Elburg and Erambert2017), supplementary appendix 1 Nepheline folder Analysis 17/1; [15] Dawson et al. (Reference Dawson, Smith and Steele1995), table 2, # 16; [16] Dawson and Hill (Reference Dawson and Hill1998), table 1, # 5; [17] Peterson (Reference Peterson1989), table 3, # SH43; [18] Hamada et al. (Reference Hamada, Akasaka and Ohfuji2019), table 1; includes Ti 0.009, Cr 0.005 atoms.

In Table 4, column 1 shows data from Barth (Reference Barth1963) for his Sample 1 (his tables 2 and 4); calculations of atoms per 32 oxygens only differ slightly from his, presumably reflecting the different atomic weights used. Note that Barth recorded the proportion of ‘holes’ (cavity site vacancies), which includes components from both the presence of Ca and excess Si. Data in column 2 show the new calculations of molecular end-members for the total CaNe and Q^xs, which both include their vacancy contributions. It is clear that the total for CaNe and Q^xs matches Barth's total for An’+ Q’+‘holes’. The next column shows data for the hypothetical 70Ne–20Q–10CaNe composition; as expected the ideal nepheline stoichiometry for this composition and the reliability of the equations used for calculating composition parameters deliver zero values for the ΔAl^cc and ΔT^charge terms and identical values for the four different ways of calculating excess Q values. The cation total is 22 atoms leaving 2 of the 24 sites vacant consisting of 0.4 vacancies associated with Ca and 1.6 with the excess Q’. These vacancies together with the cavity cations Na and Ca are located in the 8 cavity sites and give occupancies for CaNe of (0.4Ca+0.4□^Ca) × 100/8 = 10% and for Q^xs of 1.5□^Si × 100/8 = 20%, which are identical to the values shown for the same parameters in Table 4 reflecting the ideal stoichiometry for this composition.

Analysis 4 is for a disequilibrium synthetic nepheline (Dollase and Thomas, Reference Dollase and Thomas1978); analyses 5, 6 and 7 are for nephelines in alkaline basaltic rocks (respectively: Henderson and Gibb, Reference Henderson and Gibb1983, Gibb and Henderson, Reference Gibb and Henderson1978, Wilkinson and Hensel, Reference Wilkinson and Hensel1994), and analysis 11 is from a nepheline syenite (Dollase and Thomas, Reference Dollase and Thomas1978). For these five samples calculations all show high Q^xs contents but these are ~⅓ smaller than those quoted in the original papers; in addition the CaNe content given here for analysis 5 is approximately double the An value quoted by Henderson and Gibb (Reference Henderson and Gibb1983) (see earlier comments). Clearly the present calculations use a different procedure and it seems that the higher Q^xs and lower CaNe values obtained in the original publications were most probably obtained using the following procedures:

$$\%\; {\rm excess}\; {\rm Qz} = \lpar {{\rm S}{\rm i}_{\rm total}- {\rm Na} - {\rm K} - 2{\rm Ca}} \rpar \times 100 / [ {{\rm Na} + {\rm K} + {\rm Ca} + \lpar {{\rm S}{\rm i}_{\rm total}- {\rm Na} - {\rm K} - 2{\rm Ca}} \rpar } ] $$

$$\%\; {\rm An}\; \lpar {{\rm CaNe}} \rpar = {\rm Ca} \times 100 / \lpar {{\rm S}{\rm i}_{{\rm total}}- {\rm Na} - {\rm K} - 2{\rm Ca}} \rpar $$

The original An value calculated is much smaller than that calculated here for CaNe because the vacancy associated with Ca was not included in the numerator whereas the denominators in both equations are too small because the total excess Si was subtracted instead of half of this value, which defines the vacancy in the cavity sites related to the excess Si present (i.e. □^Si = Si^xs/2). Note that for small Ca contents the denominator is little different from the value for total Si atoms per 32 O. By following a procedure similar to the early stages of calculating a CIPW norm, and taking the residual mole proportion of SiO₂ after subtracting the Si components contained in Ne, Ks and An, and combining this with moles of Ne (2 moles from Na₂O), Ks (2 moles from K₂O), and An (1 mole from CaO), provides very similar molar Ne–Ks–Q^xs values to those reported by Dollase and Thomas (Reference Dollase and Thomas1978). Using the new calculations, the effect of departure from nepheline stoichiometry using the data in columns 4 and 5 can be considered as follows; for sample 4 ΔAl^cc = 0.041; Q^xs = 22.1; Q^Si = 21.9. Equivalent values for column 5 are: 0.19, 17.33 and 16.5, respectively; the very small ΔAl^cc for sample 4 is matched by the almost identical excess Q values, whereas the larger ΔAl^cc for sample 5 leads to a Q^xs vs. Q^Si difference of ~5.5% of the excess Si present reflecting a combination of analytical error and/or minor non-stoichiometry in the latter sample.

The nepheline compositions shown in columns 7 and 8–14 include analyses from rock types varying from pyroxenites (8) to theralites (7, 9) to nepheline syenites (10–14). The published end-member data for two samples (columns 8 and 12) were given as wt.% values but the mol.% values given here are directly comparable to these values and simply reflect the different molecular masses of the end-members (assuming a molecular weight of 120 for the excess Si component). For nepheline analyses published by the Mitchell and Melluso groups the data presented here show similar nepheline end-member compositions to the published values suggesting that the calculation protocols are similar and reliable. However, differences for other samples, particularly for the Q^xs values, are difficult to reconcile with the analytical data published, but excess Q’ values from Worley and Cooper (Reference Worley and Cooper1995), Wittke and Holm (Reference Wittke and Holm1996), Brotzu et al. (Reference Brotzu, Gomes, Melluso, Morbidelli, Morra and Ruberti1997) and Zhu et al. (Reference Zhu, Yang, Sun, Zhang and Wu2016) match Q^Si better than Q^xs values. For this range of natural rock data CaNe and Q^xs values vary from 0 to 12.9% and ~3 to 20% though there does not seem to be a clear relationship with the rock type other than the indications are that the highest Q^xs values are shown by nepheline crystallising metastably as late-crystallising groundmass phases in basaltic differentiated sill rocks and that the higher Ks contents reflect the co-precipitation with alkali feldspars (cf. Gibb and Henderson, Reference Gibb and Henderson1978; Henderson and Gibb, Reference Henderson and Gibb1983, Reference Henderson and Gibb1987; Wilkinson and Hensel, Reference Wilkinson and Hensel1994). In some papers (e.g. Balassone et al., Reference Balassone, Kahlenberg, Altomsre, Mormone, Rizzi, Saviano and Mondillo2014; Valentin et al., Reference Valentin, Botelho and Dantas2020) it is clear that the Ca content and its associated cavity site value were not used together in the calculation of A-site vacancies, which in turn would lead to a Q’ estimate that is too high. In another very recent paper (Vrublevskii et al., Reference Vrublevskii, Nikiforov, Sugorakova and Kozulina2020), although the microprobe analyses look reliable, the calculated Ne, Ks, An and Q’ values make no sense and, although some errors could be recognised, the calculation method followed could not be deduced.

However, in a more reliable paper, Blancher et al. (Reference Blancher, D'Arco, Fonteilles and Pascal2010) described a detailed study of nepheline compositions from the differentiated sub-volcanic ring-complex of Messum, Namibia. Nephelines from the rock series theralite–diorite–monzonite–syenite–nephelinite and peralkaline syenite were assessed in terms of their compositional variations within the quaternary system Ne–Ks–An–Qz (all defined on a 4 oxygen basis), on projections into the Barth ‘composition plane of natural nepheline compositions’, and their crystal–liquid fractionation trends projected onto the experimental Ne–Ks–Qz system at 1 kbar water vapour pressure (Blancher et al., Reference Blancher, D'Arco, Fonteilles and Pascal2010). Thus the trend of decreasing Ca and increasing excess Si in nepheline with magmatic differentiation is well established for this well-characterised complex. This type of interpretation requires access to high quality analyses and rigorous nepheline end-member calculation. The data published by Blancher et al. (Reference Blancher, D'Arco, Fonteilles and Pascal2010) includes 40 samples from representative rock types and all analyses fall within the recommended ΔAl^cc monitor ranging –0.25 to +0.25. All 40 samples are included in the new database and 37 of these fall within the range Δ(T^charge) –0.25 to +0.09 with the other three having Δ(T^charge) values of –0.33, –0.30 and –0.27. All 40 samples define a linear equation y (Δ(T^charge) = 0.82x (ΔAl^cc) – 0.102 (R ² = 0.703). In addition, note that for the 40 analyses the average ΔAl^cc/ΔT^charge ratio is 1.136 ±0.003; this very tight ratio for such a range of nepheline compositions from the same magmatic complex is a good example of a high quality dataset. The mol.% proportions calculated here for Ne and Ks are very close to those reported by Blancher et al. (Reference Blancher, D'Arco, Fonteilles and Pascal2010) but the agreement for the CaNe and Q’ values fluctuates, total CaNe + Q generally being slightly larger than the Blancher et al. values. The fluctuation of CaNe values reflects whether the Blancher factor to convert atoms Ca per to mole fraction CaNe is greater or less than the standard equivalent value used by us (i.e. 6 for 32 O atoms atomic proportions).

The parent rock types for nepheline analyses in columns 4–12 are all metaluminous (Al > Na+K+2Ca) but have nephelines with positive or very small negative ΔAl^cc parameters and have ΔAl^cc/ΔT^charge ratios within the narrow range 1.10 to 1.13 pointing to the charge distribution of the cavity cations matching that for the framework composition. The positive ΔAl^cc parameters are accompanied by small positive excess alumina parameters quantified here as % Cn values (‘normative’ corundum values; Table 4, cf. Peterson, Reference Peterson1989). Analyses 14 to 17 are from peralkaline parent rocks and are distinguished by having significant negative ΔAl^cc parameters (range –0.15 to –0.61); the column 13 nepheline analysis is from a slightly peralkaline rock type and is intermediate with a negative ΔAl^cc parameter of –0.044. These negative ΔAl^cc values point to the presence of excess Na (Na^xs) in nephelines from these rocks. It seems that the nepheline compositions in both the metaluminous rocks and the peralkaline silica-undersaturated syenites reflect the chemical affinity of the parental magmas and to the best of knowledge this has not been recognised before. Columns 15 and 16 show nepheline analyses from rocks occurring in the Oldoinyo-Lengai nephelinite–natrocarbonatite volcano, Tanzania (Dawson et al., Reference Dawson, Smith and Steele1995; Dawson and Hill, Reference Dawson and Hill1998); these microprobe analyses are characterised by high Si+Al contents. In addition analysis 15 has a higher Mg content than most natural nephelines and this is equivalent to 2.45 mol.% KsM nepheline end-member (KMg_0.5Si_1.5O₄). Microprobe Mg analyses are usually much smaller than this value (<0.1 wt.%) and might usually be ignored nevertheless Mg and Mn should be included in the list of elements analysed by microprobe. The different Q’ values for the nephelines from an Oldoinyo Lengai ijolite (Table 4, column 15) vary in the range 2.02 to 5.59 reflecting non-stoichiometry; however, the full nepheline dataset given in Dawson et al. (Reference Dawson, Smith and Steele1995) shows a much wider variation with ΔAl^cc in the range –0.35 to +0.29 (20 of the 27 having negative values) with Q^xs ranging from –0.84 to 7.5 (3 negative); Q^(Si–Al) values are generally higher varying from 1.0 to 6.7 mol.%. The fact that the 20 peralkaline compositions all give positive Q^(Si–Al) concentrations suggests that these estimates are more reliable than the Q^xs values.

The analyses shown in columns 15 to 18 provide some extra insight about dealing with apparent nepheline non-stoichiometry. Peterson (Reference Peterson1989) reported nepheline phenocrysts in nephelinites from Shombole (five rocks), Kenya, and Oldoinyo-Lengai (four rocks), Tanzania; all have negative Cn contents from –0.7 to –3.7 mol.% and some had small negative Q^xs values (–0.6 to –1.6 mol.%) whereas others ranged from 0 to 4.7 (mean 1.3 ± 1.0 mol.%). Analysis 17 from Shombole volcano (Peterson, Reference Peterson1989) has high delta monitor values (~ –0.6) and high negative Cn and Q’. The high negative ΔAl^cc is due to Na+K+Ca totalling 8.23, larger than the 8 cavity sites available in the nepheline structure. For nepheline analyses returning negative Q values it might be preferable to define the excess Q’ parameter as (Si^total – Al)/16; clearly this estimate does not depend on Na, K and Ca analyses. Thus, Table 4 includes calculated excess Q^(Si–Al) for all the analyses; for most of them the agreement between the different Q’ values is reasonably good. In particular, the Q^(Si–Al) values for the nephelinite nephelines are now sensible, small and positive, rather than non-physical negative values. The final analysis in Table 4 (column 18, Hamada et al., Reference Hamada, Akasaka and Ohfuji2019) shows a similar level of Al deficiency; note that the small KsM component quoted for analysis 18 reflects the assignment of M ²⁺ in that analysis to a framework site rather than to a cavity cation site together with Ca, which also accounts for the smaller CaNe (An) given here compared with the Hamada et al. analysis.

The significance of the ΔAl^cc/ΔT^charge ratio

The full nepheline database of 310 analyses has been used to explore the significance of variation for the coupled ΔAl^cc and ΔT^charge parameters and this variation is plotted in Fig. 1 using the same symbol for each separate analysis; most of the analyses plot on the same linear trend described by the linear fit ΔT^charge = 0.8928 × ΔAl^cc – 0.0072 (R ² = 0.992). The slope of this line is equivalent to a mean ΔAl^cc/ΔT^charge ratio of 1.12. The full dataset has some analyses with ΔAl^cc/ΔT^charge ratios differing widely from this mean and removal of ~35 analyses by only including those with ratios from 1.0 to 1.2 gives a trend with the equation ΔT^charge = (0.8896 × ΔAl^cc) – 0.0020 (R ² = 0.9987) and a mean ΔAl^cc/ΔT^charge ratio of 1.130 ± 0.022 that is very close to the average of 1.136 ± 0.003 for the 40 nepheline analyses reported by Blancher et al. (Reference Blancher, D'Arco, Fonteilles and Pascal2010) for the Messum magmatic complex. It appears that this approach could provide another reliable method for assessing the quality of analyses however the fundamental significance of the value provided by this ratio must be assessed. For example, in a strict stoichiometric composition both individual delta parameters are zero and the ratio has no mathematical meaning. However a tiny variation from exact end-member stoichiometry (e.g. Na₈Al₈Si_8.000001O₃₂) provides a ratio that is close to a theoretical value of 1.42857 whereas Na₈Al₈Si_8.0001O₃₂ and Na₈Al₈Si_8.01O₃₂ are little different at 1.42856 and 1.142755.

Fig. 1. Atomic formula units calculated to 32 oxygens are used to plot delta parameters Δ(Al – cavity cations) [ΔAl^cc] vs. Δ (T-site charge) [ΔT^charge]. The whole database of 310 analyses is plotted with the same symbol (small blue diamond). These points define a linear trend with the statistics of the fit displayed. A large proportion of points lie close to the line over the range ±0.5 for both ΔAl^cc and ΔT^charge showing that these parameters are coupled closely for reliable analyses. Points that fall at delta values outside that range and falling further away from the linear trend are marked with symbols that are different for each literature source. Wittke = Wittke and Holm (Reference Wittke and Holm1996); Melluso Jasra = Melluso et al. (Reference Melluso, Srivastava, Petrone, Guarino and Sinha2012); Trupia = Trupia and Nicholls (Reference Trupia and Nicholls1996); Dawson = Dawson et al. (Reference Dawson, Smith and Steele1995), Dawson and Hill (Reference Dawson and Hill1998); Chakra. = Chakrabarty et al. (Reference Chakrabarty, Mitchell, Ren, Saha, Pal, Pruseth and Sen2016); Concei. = Conceição et al. (Reference Conceição, Rosa, Moura, Buenano, Macambira and Glazara2009); Melluso VUV = Melluso et al. (Reference Melluso, Morra and Girolamo1996); Brotzu = Brotzu et al. (Reference Brotzu, Gomes, Melluso, Morbidelli, Morra and Ruberti1997); and Andersen = Andersen et al. (Reference Andersen, Elburg and Erambert2017). Also shown are the compositions declared for samples that have been used to determine crystal structures by X-ray diffraction; points falling clear of the linear trend are considered to have unreliable compositions. See text for further detail.

On the basis of the concept of the end-member stoichiometric nepheline having a stuffed tridymite unit cell with 32 oxygens, 16 T atoms and exactly 8 interframework cavity sites for Na, it is clear that once one quadrivalent Si atom is replaced with one trivalent Al atom that must be accompanied by the entry of one Na atom into a cavity site. For all atomic substitutions that retain strict nepheline stoichiometry the crystal chemical ‘die is cast’ (“Alea iacta est” Julius Caesar, January 10, 49 BC). Thus it should be possible to define the delta parameter ratio in terms of the Al(+Fe³⁺) and Si atoms alone (Kevin Knight pers. comm., July 4, 2020).

The following stages show how such a relationship could be developed: let N, K, C, A and S represent the numbers of atoms of Na, K, Ca, Al (or Al + Fe³⁺) and Si, per 32 O unit cell, respectively. Charge balance requires the constraint N + K + 2C + 3A +4S = 64. It has been shown above that ΔAl^cc = A – 2C – N – K, from the charge balance constraint ΔA = –64 + 4S + 4A and this definition is used here as it contains the same variables as ΔT^charge. Total charge = 4S + 3A; total charge per T-site unit = (4S + 3A)/(S + A); ideal charge per T-site unit (4S + 3A)/16. ΔT^charge is defined above as the difference in total charge from the ideal charge and the total charge per T-site unit. Thus,

$${\Delta}{\rm T}^{{\rm charge}} = {\rm }16\left( {\left( {4{\rm S}{\rm } + {\rm }3{\rm A}} \right)/16{\rm }- {\rm }\left( {4{\rm S}{\rm } + {\rm }3{\rm A}} \right)/\left( {{\rm S}{\rm } + {\rm }{\rm A}} \right)} \right){\rm } = {\rm }\left( {4{\rm S}{\rm } + {\rm }3{\rm A}} \right)\left( {1{\rm }- {\rm }16/\left( {{\rm S}{\rm } + {\rm }{\rm A}} \right)} \right){\rm } = {\rm }\left( {\left( {4{\rm S}{\rm } + {\rm }3{\rm A}} \right)\left( {\left( {{\rm S}{\rm } + {\rm }{\rm A}} \right){\rm }- {\rm }16} \right)} \right)/\left( {{\rm S}{\rm } + {\rm }{\rm A}} \right)$$

The next step is to calculate the ratio (ΔA/ΔT) (defined above as ΔAl^cc/ΔT^charge):

$$\lpar \Delta {\rm A}/\Delta {\rm T}\rpar = \lpar {- 64 + 4{\rm S} + 4{\rm A}} \rpar /\lpar {\lpar {\lpar {4{\rm S} + 3{\rm A}} \rpar \lpar {\lpar {{\rm S} + {\rm A}} \rpar - 16} \rpar } \rpar /\lpar {{\rm S} + {\rm A}} \rpar } \rpar $$

$$\lpar \Delta {\rm A}/\Delta {\rm T}\rpar = \lpar {\lpar {- 64 + 4{\rm S} + 4{\rm A}} \rpar \lpar {{\rm S} + {\rm A}} \rpar } \rpar /\lpar {\lpar {4{\rm S} + 3{\rm A}} \rpar \lpar {{\rm S} + {\rm A} - 16} \rpar } \rpar $$

$$ \eqalign{ ( \Delta {\rm A}/\Delta {\rm T} ) =\; &( {- 64{\rm S} - 64{\rm A} + 4{\rm S}^2 + 4{\rm SA} + 4{\rm SA} + 4{\rm A}^2} ) / \cr& ( {4{\rm S}^2 + 4{\rm SA} - 64{\rm S} + 3{\rm SA} + 3{\rm A}^2- 48A} ) $$

$$ \eqalign{ \( \Delta {\rm A}/\Delta {\rm T}\) =\; &( {- 64{\rm S} - 64{\rm A} + 4{\rm S}^2 + 8{\rm SA} + 4{\rm A}^2} ) / \cr& {{( {- 64{\rm S} - 48{\rm A} + 4{\rm S}^2 + 7{\rm SA} + 3{\rm A}^2} ) $$

and this leads to a key equation:

$$ \eqalign{ ( \Delta {\rm Al}^{\rm cc}/\Delta {\rm T}^{{\rm charge}} ) =\; &( {64{\rm S} + 64{\rm A} - 4{\rm S}^2- 8{\rm SA} - 4{\rm A}^2} ) / \cr& ( {64{\rm S} + 48{\rm A} - 4{\rm S}^2- 7{\rm SA} - 3{\rm A}^2} ) $$

This expression can be recast in terms of S and the difference between S and A. Let S – A = δ, therefore A = S – δ’ Thus,

$$\eqalign{\lpar \Delta {\rm A}/\Delta {\rm T}\rpar =\;& \lpar 64{\rm S} + 64 \lpar {\rm S} - {\rm \delta} \rpar - 4{\rm S}^2- 8{\rm S}\lpar {\rm S} - {\rm \delta} \rpar - 4\lpar {\rm S} - {\rm \delta} \rpar ^2\rpar / \cr&\lpar 64{\rm S} + 48\lpar {\rm S} - {\rm \delta} \rpar - 4{\rm S}^2- 7S\lpar {\rm S} - {\rm \delta} \rpar - 3\lpar {\rm S} - {\rm \delta} \rpar ^2\rpar} $$

$$\lpar \Delta {\rm A}/\Delta {\rm T}\rpar = \lpar 128{\rm S} - 16{\rm S}^2- 4{\rm \delta }^2 + 16{\rm S\delta }- 64{\rm \delta }\rpar /\lpar 112{\rm S} - 14{\rm S}^2- 3{\rm \delta} ^2 + 13{\rm S}{\rm \delta} - 48{\rm \delta }\rpar $$

leading to another key equation:

$$\Delta {\rm Al}^{\rm cc}/\Delta {\rm T}^{{\rm charge}} = \lpar 128{\rm S} - 16{\rm S}^2 + 16{\rm S\delta }- 64{\rm \delta }- 4{\rm \delta }^2\rpar / \lpar 112{\rm S} - 14{\rm S}^2 + 13{\rm S\delta }- 48{\rm \delta }- 3{\rm \delta }^2\rpar $$

For the relationships: δ is small, S ≈ A, S ≈ 8, 128S ≈ 16S² and 112 ≈ 14S², δ² is small enough to be ignored and δ can be cancelled in the numerator and denominator:

Thus, ΔAl^cc/ΔT^charge ≈ (128–64) / (104–48) = (64 / 56) = 8/7 = 1.1429

Even though the individual ΔAl^cc and ΔT^charge parameters are each effectively zero, the strict nepheline stoichiometry still delivers a ΔAl^cc/ΔT^charge ratio of 1.1429 for end-member NaAlSiO₄.

In the context of a strict stuffed-tridymite stoichiometry defining the nepheline unit cell, the integers 64 and 8 can be rationalised as referring to the ideal tridymite cell that has a total charge of 64 for 16 quadrivalent silicons (total S = 16) in the 32 O unit cell. The unit cell has 8 □^SiSi₂O₄ ‘nepheline’ molecules and thus 64/8 = 8 per unit molecule. The integers 56 and 7 refer to the Na₈Al₈Si₈O₃₂ species (8 NaAlSiO₄ per unit cell) where the T site total is 56 (8 × 4 + 8 × 3) and 56/8 = 7. Following the same logic, an albite molecule expressed as having 32 O cell would have the formula Na₄Al₄Si₁₂O₃₂ (total T charge (12 × 4 + 4 × 3) =60) and 60/8 = 7.5; an anhydrous analcime would have the formula Na_5.333Al_5.333Si_10.667O₃₂ (total T charge (10.667 × 4 + 5.333 × 3) = 58.667) and 58.667/8 = 7.3334. These values lead to ΔAl^cc/ΔT^charge ratios of 8/7.5 (1.0667) and 8/7.333 (1.0909), respectively; see the values given earlier. The three end-members all have ideal stoichiometries and would have zero values for each delta parameter so the presence of excess silica in nephelines is not a sign of non-stoichiometry. The equivalent ΔAl^cc/ΔT^charge ratios for tridymite and the extreme alumina ‘nepheline’ end-member Cn molecule would be 1.000 and 1.3333. Departures from such ideal nepheline stoichiometry would allow delta ratios to deviate from these ‘standard’ values as would the presence of analytical errors. However, it is clear that the ΔAl^cc/ΔT^charge ratio is a direct monitor of how much a nepheline analysis deviates from the model, stuffed-tridymite unit cell and we will exploit this concept in dealing with the nepheline database. Indeed using the ΔAl^cc/ΔT^charge values for albite, analcime and leucite gives the determinative equation for calculating a Si/Al ratio from the ΔAl^cc/ΔT^charge obtained from a nepheline analysis or from any average nepheline value as follows:

$${\rm Si}/{\rm Al} = 289.89\lpar \Delta {\rm A}{\rm l}^{{\rm cc}}/\Delta {\rm T}^{{\rm charge}}\rpar ^{2}- 66.84\lpar \Delta {\rm A}{\rm l}^{{\rm cc}}/\Delta {\rm T}^{{\rm charge}}\rpar + 384.44 \;\, \lpar {{ R}^2 = 1} \rpar $$

Deviation of nepheline compositions from ideal stoichiometry

The ΔAl^cc vs. ΔT^charge values for all the analyses in the new database (310 microprobe analyses) are plotted in Fig. 1. The individual analyses nephelines have ΔAl^cc and ΔT^charge ranging from ~ +1.7 to –1.9, far exceeding the values assumed to define acceptable analyses (±0.25). Of these analyses ~80% fall within ±0.5 and ~46% have negative ratios. Whether or not a new extended range for the ΔAl^cc/ΔT^charge ratio could be adopted to select reliable microprobe data as long as care was taken to avoid Na loss (i.e. low sample current, defocussed beam) must be assessed.

The end-member albite, anhydrous analcime and nepheline compositions calculated on an ideal nepheline stoichiometry all plot at the 0,0 point on Fig. 1 and the overall trend of points passes through this zero point; 55 of the full dataset of 310 analyses have delta values within a ΔAl^cc and ΔT^charge range of ±0.05. Positive delta values have excess alumina over cavity cations and T sites totalling > 16.000 atoms per 32 O; negative delta values have excess cavity cations (mainly Na and K) over alumina and T sites totalling <16.000 atoms per 32 O, in effect these nephelines are ‘peralkaline’ in bulk composition. Some analyses plot off this main trend or at surprisingly high or low points on the trend; all of such data points for each of those data sources are plotted over the main set of points with clearly different data symbols. Thus, the groundmass nepheline analyses from Trupia and Nicholls (Reference Trupia and Nicholls1996) (blue open triangles in Fig. 4) all fall below the main trend suggesting that multiphase groundmass grains were analysed in that work. Four of the analyses from Brotzu et al. (Reference Brotzu, Gomes, Melluso, Morbidelli, Morra and Ruberti1997) are deficient in cavity cations with ΔAl^cc values higher than +0.5; this might be due to Na loss in the electron beam. Analyses with excess alkali are more common (ΔAl^cc more negative than –0.5) as shown by some of the data points from Melluso et al. (Reference Melluso, Morra and Girolamo1996, Reference Melluso, Srivastava, Petrone, Guarino and Sinha2012) and Chakrabarty et al. (Reference Chakrabarty, Mitchell, Ren, Saha, Pal, Pruseth and Sen2016). Although most analyses appear to pass the tests for acceptability it should be remembered that calculating nepheline analyses on an ideal 32 oxygen basis alongside expecting Si+Al+Fe³⁺ to total close to 16.000 with fixed numbers of cavity cation sites (8) and total cation sites (24) and fixed end-member stoichiometry, all place constraints on the end data calculated. Indeed, the excess Q’ proportions in particular are essentialy dependent on a fixed cavity site of 8 or a total cation sum of 24. Small analytical errors for Si, Al or Na would have the largest effect on the calculations and it seems that, in general, such errors cause the calculated points to move very slightly along or very close to the main linear trend with the main difference showing up for the excess Q’ values. Because of this particular result, it is necessary to choose the best possible protocol for calculating end-member mineral proportions, in particular excess silica (Q’). Nevertheless, based on the fact that a very large proportion of the analyses plot in a concentrated band defining the linear trend of coupled delta values it seems that a range ±0.6 for both ΔAl^cc and ΔAl^charge could be used as a criterion to accept analyses.

The compositions for nephelines that have been used for X-ray single-crystal or powder structure determination (large orange open triangles) are also shown on Fig. 1. The synthetic samples are reported to have exact nepheline stoichiometry or to have very small delta parameters (Gregorkiewitz, Reference Gregorkiewitz1984; Hippler and Böhm, Reference Hippler and Böhm1989; Dollase and Thomas, Reference Dollase and Thomas1978), whereas natural samples usually have very small delta parameters (Dollase, Reference Dollase1970; Dollase and Peacor, Reference Dollase and Peacor1971; Hassan et al., Reference Hassan, Antao and Hersi2003; Tait et al., Reference Tait, Sokolova, Hawthorne and Khomyakov2003; Antao and Hassan, Reference Antao and Hassan2010). However, the sample from Kola (Tait et al., Reference Tait, Sokolova, Hawthorne and Khomyakov2003) has excess cavity cations and delta parameters close to –0.5 but plots on the main trend (Fig. 1). The formulae given for samples studied by Foreman and Peacor (Reference Foreman and Peacor1970) and Simmons and Peacor (Reference Simmons and Peacor1972) appear to have been recalculated on the basis of Si+Al = 16.000 and give zero ΔT^charge values but non-zero ΔAl^cc values of +0.25 and +0.72 respectively; such values would suggest a mismatch of charges between cavity cations and the framework. It is clear that the analytical data for these samples are not complete or are in error as it is essential that acceptable nepheline analyses should show closely coupled ΔAl^cc and ΔT^charge values.

It is clear that the Si/Al ratios of the analysed nephelines influences the values of the delta parameters though the tight stoichiometric controls imposed for the ideal nepheline cell also requires that both parameters are zero for analyses with T atoms totalling 16.000 irrespective of the Si/Al ratio (cf. end members NaAlSiO₄, NaAlSi₂O₆, and NaAlSi₃O₈ all have zero ΔAl^cc and ΔT^charge values, see above). Thus Fig. 2 is used to show how the ΔAl^cc/ΔT^charge ratio varies as a function of Si/Al ratio; the full database of analyses shows that the large majority define a linear trend showing only a small variation from ΔAl^cc/ΔT^charge ≈1.10 over the Si/Al range from 1.0 to 1.6. Approximately 30 analyses scatter around this trend, mainly with lower delta ratios. Some analyses are anomalously far-removed from the main trend and these are mainly due to analyses with close to 16.000 atom totals; for these analyses the ΔT^charge values tend to be smaller than for analyses with lower Si+Al totals, and thus have less reliable delta ratios. Most of the ‘scattered’ analyses are from a few of the data sources and all of those are identified with different symbols in Fig. 2a. Figure 2b shows the analyses which all fall within the ΔAl^cc/ΔT^charge range from 1.0 to 1.25 and all of these are considered to be acceptable analyses. The main trend is defined by ~250 analyses of which ~200 are concentrated between ΔAl^cc/ΔT^charge values of 1.144 and 1.125 over the Si/Al range 1.0 to 1.15. The Messum igneous complex nephelines (Blancher et al., Reference Blancher, D'Arco, Fonteilles and Pascal2010) show the clear dependence of how the delta ratio decreases with increasing Si/Al. The points falling off this main trend tend to be from a few of the data sources and these are identified with different symbols in Fig. 2b. All of the analyses shown fall in the acceptable category though the ones showing the largest departures from ideal nepheline stoichiometry are easily identified (Fig. 2b) from those defining the main linear trend.

Fig. 2. Plot of atomic Si/Al vs ΔAl^cc/ΔT^charge. (a) All of the database analyses are plotted with the same symbol and most analyses are seen to fall on a slightly falling delta ratio over the range Si/Al from 1.0 to 1.55. As in Fig. 1, points falling away from the main trend are identified with different symbols. One of the set of the Moreau analyses and three of Trupia's four analyses plot far from the main trend; all of those analyses and those having ΔAl^cc/ΔT^charge ratios between 0 and 1.0 and those which have the delta ratio > 1.25 are considered to be unreliable (see text). (b) Si/Al vs. ΔAl^cc/ΔT^charge shows a large concentration of points define the trend of decreasing ΔAl^cc/ΔT^charge ratios with increasing Si/Al; that trend is well-displayed by the data for nephelines from the genetically related Messum magmatic complex (Blancher et al., Reference Blancher, D'Arco, Fonteilles and Pascal2010). Other samples labelled separately tend to be from peralkaline rock types. All of these points are believed to have reliable compositions. Andersen = Andersen et al. (Reference Andersen, Elburg and Erambert2017); Blancher = Blancher (Reference Blancher, D'Arco, Fonteilles and Pascal2010); Concei = Conceição et al. (Reference Conceição, Rosa, Moura, Buenano, Macambira and Glazara2009); Dawson 1995 = Dawson et al. (Reference Dawson, Smith and Steele1995); Hamada = Hamada et al. (Reference Hamada, Akasaka and Ohfuji2019); Moreau = Moreau et al. (Reference Moreau, Ohnenstetter, Demaiffe and Robineau1996); Paslick = Paslick et al. (Reference Paslick, Halliday, Lange, James and Dawson1996); Trupia = Trupia and Nicholls (Reference Trupia and Nicholls1996); Zhu = Zhu et al. (Reference Zhu, Yang, Sun, Zhang and Wu2016).

Significance of ‘nepheline’ analyses with high excess Na contents

Although small amounts of an excess alumina component might occur in nepheline it is more probable that low Na microprobe analyses are the main explanation of positive ΔAl^cc values (Morgan and London, Reference Morgan and London2005). However, the occurrence of apparently reliable nepheline analyses showing significant cavity cation excesses over Al+Fe³⁺ and the extension of the trend towards high negative delta values must be considered further. The presence of a small amount of an alteration phase containing low Si and Al and high alkalis (particularly Na) would produce the small negative ΔAl^cc values shown by some analyses. Cancrinite-group minerals commonly occur as alteration phases of nepheline and sodalite groups of minerals in nepheline syenites as well as in hydrothermal veins associated with such rocks (e.g. Deer et al., Reference Deer, Howie and Zussman2004, Moreau et al., Reference Moreau, Ohnenstetter, Demaiffe and Robineau1996). The cancrinite-group mineral vishnevite is characterised by having lower Si and Al than nepheline with significantly higher Na contents in sodic varieties, and comparable Na contents and higher Ca occurs in cancrinite itself (Grundy and Hassan, Reference Grundy and Hassan1982; Hassan and Grundy, Reference Hassan and Grundy1984, Reference Hassan and Grundy1991). Those authors have shown that Na and H₂O are located within three-fold symmetry cancrinite ‘cages’ defined by top and bottom 6-rings of ordered SiO₄ and AlO₄ tetrahedra, whereas Na, Ca, OH^–, Cl^–, CO₃^2– (or SO₄^2–) occur in large, continuous pseudo-hexagonal channels bounded by puckered 12-rings of tetrahedra; the two sites occupied by Na are structurally distinct. Note that end-member hydroxy-cancrinite (Na₈[Al₆Si₆O₂₄](OH)₂⋅2H₂O) has only hydroxyl as the anion in the large channel.

The overall effect of the compositional differences shown by Na-rich, low-Ca nephelines (e.g. Peterson, Reference Peterson1989) would be that ~5% of a sodic cancrinite impurity phase, perhaps concentrated in cleavage cracks or occurring as inconspicuous fine alteration products, would produce the level of non-stoichiometry discussed above for the nephelines occurring in peralkaline rock types and some nephelinites (Table 4). However, the absence of any features associated with such an alteration process together with the database having many recent, apparently reliable, microprobe analyses of primary nepheline with coupled ΔAl^cc and ΔT^charge parameters significantly more negative than –0.25 supports the possibility that such nepheline composition samples might show a small degree of solid solution of an alkali-rich molecule with cancrinite-like characteristics. Perhaps some embryo cage-like modifications might occur in the large hexagonal channels within the six-ring nepheline structure that might contain the excess alkalis in some nephelines? Indeed it is well known that many natural nephelines show features such as satellite diffraction peaks indicative of an incommensurate structure (Sahama, Reference Sahama1958; McConnell, Reference McConnell1962, Reference McConnell1981; Parker, Reference Parker1972; Hayward et al., Reference Hayward, Pryde, de Dombal, Carpenter and Dove2000; Hassan et al., Reference Hassan, Antao and Hersi2003; Antao and Hassan, Reference Antao and Hassan2010). Friese et al. (Reference Friese, Grzechnik, Petřīč, Schönleber, van Smaalen and Morgenroth2011) have re-determined the incommensurately modulated structure of one of the original nephelines studied by McConnell (Reference McConnell1962) (K_0.54Na_3.24Ca_0.03Al_3.84Si_4.16O₁₆) using superspace crystallography that was developed to study periodic structures that show long-range order but which lack translational symmetry. They found that all atoms are displacively modulated with amplitudes <0.1 Å and that Na fills the smaller (oval) channels, whereas K, Na and Ca and vacancies occupy the larger (hexagonal) channels in a highly disordered manner. A large proportion of the framework oxygens show split-atom modulations and these effects are coupled to the occupational modulations of the cations (and vacancies) in the large channels. Perhaps this disordered local geometry could allow access of ‘additional’ small cations (Na and Ca) into the large nepheline channels to extend the effects of non-stoichiometry in natural nephelines.

The presence of a cancrinite-like molecule in nepheline would imply the presence of large anions and hydrous species associated with the extra Na. Small amounts of structural ‘H₂O’ (up to ~0.5 wt.%) have been reported in nepheline for many years (Barth, Reference Barth1963; Beran, Reference Beran1974; Balassone and Beran, Reference Balassone and Beran1995). Beran and Rossman (Reference Beran and Rossman1989) studied doubly polished slices of clear (non-turbid) single-crystal nephelines using controlled temperature infrared (IR) spectroscopy and reported the presence of 2 to 3 absorption features over the range ~3500 to 3630 cm^–1. These were identified as fundamental OH-stretching bands due to the presence of molecular water occupying the large, K-rich cavity sites; Belassone and Beran (Reference Balassone and Beran1995) confirmed these IR results using nephelines from Monte Somma, Italy, and reported that the amount of water present was related to the proportion of vacancies present in the large B site in nepheline that contains all the K, Ca, vacancies and some of the Na. Yesinowski et al. (Reference Yesinowski, Eckert and Rossman1988) reported similar IR features and in addition used MAS proton nuclear magnetic resonance (NMR) to show a sharp main central peak close to 0 kHz with a small feature on the flank nearest to the 0 kHz point; much less intense spinning side bands cover a frequency range of 50–60 kHz. Yesinowski et al. (Reference Yesinowski, Eckert and Rossman1988) concluded that IR and ¹H-NMR are consistent with two distinct water molecules in nepheline (cf. Beran, Reference Beran1974), however, the NMR spectrum for analcime is distinctly different with a sharp central band close to 0 kHz with two pairs of spinning side bands, of similar intensities to the central band flanking it, and other weaker ones which cover a frequency range of ~100kHz. Simakin et al. (Reference Simakin, Salova and Zavel'skii2008) found similar IR results for clear nepheline single crystals with 0.20 ±0.05 wt.% H₂O (SIMS) but then also used ¹H NMR pulse spectrometry at 200 MHz that showed a central peak at 0 kHz with a PWHH of ~20 kHz and broad shoulders on the flanks at ~ +20 and –20 kHz. It is possible that the OH-stretching bands in nepheline suggest that Na could be associated with both OH and water molecules; the Na-cancrinite vishnevite ((Na,Ca,K)_7–8[Al₆Si₆O₂₄](SO₄CO₃,Cl₂)⋅2H₂O has also been reported to show this structural environment for species in the large channel (Ventura et al., Reference Ventura, Bellatreccia, Parodi, Camara and Piccinini2007). In addition, there might be some similarities for H₂O/OH speciation in nepheline to those in hydrous silicate glasses. For example, McMillan et al. (Reference McMillan, Jakobsson, Holloway and Silver1983) studied clear hydrous albite glasses using Raman spectroscopy and reported the presence of a broad absorption band from ~3400 to 3600 cm^–1 due to O–H stretching and Stolper (Reference Stolper1982), based on IR and NIR spectroscopy of volcanic and synthetic hydrous silicate glasses, concluded that hydroxyl was the main species up to ~0.5 wt.% water content, though above that amount molecular H₂O became more important. Kohn et al. (Reference Kohn, Dupree and Smith1989) studied hydrous albite glasses by multinuclear NMR (²⁹Si, ²⁷Al, ²³Na and ¹H) and suggested that Na(OH) complexes may be present with the polar hydrogen of the molecular water – Na complex linked to the residual charge of a framework oxygen (their fig. 8). It seems that there might be some prima facie evidence for a small amount of a cancrinite-like Na–H₂O complex in nepheline.

The high-Na content of nephelines from peralkaline rocks could suggest the presence of ~5% of a cancrinite-like phase. Nepheline phenocrysts from Shombole and Oldoinyo Lengai carbonatite volcanoes (E. Africa; Peterson et al., Reference Peterson1989) have high negative delta parameters; for example, analysis 17 (Table 4) has ΔAl^cc–0.61 and ΔT^charge–0.54; the excess Na must be of the order of 0.5 to 0.6 atoms per 32 O, which is ~10% of the total Na content. Assuming that this nepheline has the typical water content of ~0.4 wt.% H₂O this would be equivalent to ~0.52 molecules of OH per 32 O; the similarity of the Na and OH concentrations could be significant! In addition, Dawson et al. (Reference Dawson, Smith and Steele1995) reported the presence of vishnevite in the Oldinyo-Lengai host rock BD875 of nepheline analysis 15 (Table 4), while Donaldson et al. (Reference Donaldson1987) gave analyses for coexisting phenocrysts of nepheline and vishnevite from an Oldinyo Lengai nephelinite (BD51; see their fig 1); that nepheline analysis has delta values close to zero (0.08 and 0.08, respectively). If the coexisting nepheline and vishnevite in BD51 are in equilibrium it is reasonable to assume that the nepheline would be saturated in a vishnevite-like molecule. However, in a very recent investigation of Oldoinyo-Lengai samples, Berkesi et al. (Reference Berkesi, Bali, Bodnar, Szabó and Guzmics2020) describe the compositions of melt inclusions found within nepheline phenocrysts that point to the coexistence of quenched, Na-rich immiscible, silicate- and carbonate-melt phases. It is suggested that these immiscible melt phases also coexisted with a separate peralkaline, fluorine-rich aqueous fluid phase. Note that an analysis of a parent nepheline grain that is contaminated by the presence of associated Na-rich silicate glass and/or trapped peralkaline fluid would have a composition similar to a cancrinite phase. Other researchers may wish to further consider this possibility.

The possible relationship between nepheline and coexisting cancrinite is considered in the next section of this paper assuming either a primary solid-solution origin or a late-magmatic/post-magmatic, hydrothermal alteration origin. Analyses used are for coexisting nepheline and cancrinite in undersaturated dyke-rock phonolites from the Marangudzi ring complex, Zimbabwe, a protocol will be developed for dealing with the compositions of cancrinite-bearing mixtures by calculating all compositions on the basis of a 32 O nepheline unit cell.