Free binary systems are shown not to admit idempotent means. This refutes a conjecture of the author. It is also shown that the extension of Hindman’s theorem to nonassociative binary systems formulated and conjectured by the author is false.

The composition of magmas proposed as parental to the layered suite of the Bushveld Complex, and some models for the manner of their emplacement, are reviewed briefly. Included are some contributions published in South Africa, with which overseas readers might be less familiar. Emphasis is given to the broader features of the cumulates, and the contradictions raised by whole-rock compositional, Sr-isotopic, and trace-element data that cloud their correlation with proposed parental magmas. It is concluded that the Lower, Critical and Main Zones are the derivatives of only two primary magmatic lineages, while a third was added to residual liquids from which the layered rocks above the Pyroxenite Marker were formed. Excessive amounts of olivine and chromium in the cumulates of the Lower and Lower Critical Zones in the northern sector of the Western Limb can seemingly not be accounted for by the composition and volume of the putative magmas. This is attributed to (1) this sector being a proximal facies located close to the original feeder, and/or (2) crystal-charged magma batches, expelled from a lower magma chamber, being periodically injected into and dispersed within the liquids already in place in the Bushveld chamber. Thus, ongoing changes in the bulk composition of the liquids within the chamber would not be reflected in the rinds of earlier, chilled-facies rocks. The expulsion of significant volumes of liquid residua from the chamber during cumulate deposition is not ruled out.

Magmas undergoing shear are prime examples of flows that involve the transport of solids and gases by a separate (silicate melt) carrier phase. Such flows are called multiphase, and have attracted much attention due to their important range of engineering applications. Where the volume fraction of the dispersed phase (crystals) is large, the influence of particles on the fluid motion becomes significant and must be taken into account in any explanation of the bulk behaviour of the mixture. For congested magma deforming well in excess of the dilute limit (particle concentrations >40% by volume), sudden changes in the effective or relative viscosity can be expected. The picture is complicated further by the fact that the melt phase is temperature- and shear-rate-dependent. In the absence of a constitutive law for the flow of congested magma under an applied force, it is far from clear which of the many hundreds of empirical formulae devised to predict the rheology of suspensions as the particle fraction increases with time are best suited. Some of the more commonly used expressions in geology and engineering are reviewed with an aim to home in on those variables key to an improved understanding of magma rheology. These include a temperature, compositional and shear-rate dependency of viscosity of the melt phase with the shear-rate dependency of the crystal (particle) packing arrangement. Building on previous formulations, a new expression for the effective (relative) viscosity of magma is proposed that gives users the option to define a packing fraction range as a function of shear stress. Comparison is drawn between processes (segregation, clustering, jamming), common in industrial slurries, and structures seen preserved in igneous rocks. An equivalence is made such that congested magma, viewed in purely mechanical terms as a high-temperature slurry, is an inherently nonequilibrium material where flow at large Pe´clet numbers may result in shear thinning and spontaneous development of layering.

The common association of mid-crustal migmatites with an upper-level granite pluton could indicate that the migmatites are a feeder zone for the pluton. If magma from a deeper level pervasively intrudes a high temperature metamorphic complex, most of the intruded magma would not freeze because of the prevailing temperature. The interaction between the magma and country rocks, which could include partial melting and crystallisation of the magma passing through, would modify magma to a more granitic composition, as found in the higher-level pluton.

The physical aspect of the magma transport through such a hot feeder zone is modelled by introducing a dimensionless melt transport (MT) number, which is the ratio of the rate of melt movement caused by the bulk flow of the entire mass (melt+solid) to that of porous media flow of melt only through the solid framework. The MT number is strongly dependent on the melt content of the melt-rich zone (MRZ), the diameter of the MRZ and typical particle size in the MRZ.

The ∼300-Ma, diatexitic, Lauterbrunnen migmatites (LM) in the Aar massif, Swiss Alps, may be such a feeder zone for the nearby 303-Ma Gastern granite (GG). The chemical and field evidence indicates that the LM formed by an intrusion of intermediate composition magma, which interacted with country rocks to produce a magma of GG composi

Comb-layered quartz is a type of unidirectional solidification texture found at the roofs of shallow silicic intrusions that are often associated spatially with Mo and W mineralisation. The texture consists of multiple layers of euhedral, prismatic quartz crystals (Type I) that have grown on subplanar aplite substrates. The layers are separated by porphyritic aplite containing equant phenocrysts of quartz (Type II), which resemble quartz typical of volcanic rocks and porphyry intrusions. At Logtung, Type I quartz within comb layers is zoned with respect to a number of trace elements, including Al and K. Concentrations of these elements as well as Mn, Ti, Ge, Rb and H are anomalous and much higher than found in Type II quartz from Logtung or in igneous quartz reported elsewhere. The two populations appear to have formed under different conditions. The Type II quartz phenocrysts almost certainly grew from a high-silica melt between 600 and 800°C (as β-quartz); in contrast, the morphology of Type I quartz is consistent with precipitation from a hydrothermal solution, possibly as α-quartz grown below 600°C. The bulk compositions of comb-layered rocks, as well as the aplite interlayers, are consistent with the hypothesis that these textures did not precipitate solely from a crystallising silicate melt. Instead, Type I quartz may have grown from pockets of exsolved magmatic fluid located between the magma and its crystallised border. The Type II quartz represents pre-existing phenocrysts in the underlying magma; this magma was quenched to aplite during fracturing/degassing events. Renewed and repeated formation and disruption of the pockets of exsolved aqueous fluid accounts for the rhythmic banding of the rocks.

Blocks of medium-grained granodiorite to 4 m, and minor diabase, quartz diorite, granite, aplite and granophyre, are common in ejecta of the ∼6,900 yrBP calderaforming eruption of Mount Mazama. The blocks show degrees of melting from 0–50 vol%. Because very few have adhering juvenile magma, it is thought that the blocks are fragments of the Holocene magma chamber's walls. Primary crystallisation of granodiorite produced phenocrystic pl + hyp + aug + mt + il + ap + zc, followed by qz + hb + bt + alkali feldspar (af). Presence of fluid inclusions in all samples implies complete crystallisation before melting. Subsolidus exchange with meteoric hydrothermal fluids before melting is evident in δ18O values of −3·4+4·9‰ for quartz and plagioclase in partially melted granodiorites (fresh lavas from the region have δ18O values of +5·8−+7·0‰); δ18O values of unmelted granodiorites from preclimatic eruptive units suggest hydrothermal exchange began between ∼70 and 24 ka. Before eruption, the granitic rocks equilibrated at temperatures, estimated from Fe-Ti oxide compositions, of up to ∼1000°C for c. 102–104 years at a minimum pressure of 100-180 MPa. Heating caused progressive breakdown or dissolution of hb, af, bt, and qz, so that samples with the highest melt fractions have residual pl + qz and new or re-equilibrated af + hyp + aug + mt + il in high-silica rhyolitic glass (75-77% SiO2). Mineral compositions vary systematically with increasing temperature. Hornblende is absent in rocks with Fe-Ti oxide temperatures >870°C, and bt above 970°C. Oxygen isotope fractionation between qz, pl, and glass in partially fused granodiorite also is consistent with equilibration at T≥900°C (Δ18Oqz.pl = +0·7 ± 0·5‰). Element partitioning between glass and crystals reflects the large fraction of refractory pl, re-equilibration of af and isolation or incomplete dissolution of accessory phases. Ba and REE contents of analysed glass separates can be successfully modelled by observed degrees of partial melting of granodiorite, but Rb, Sr and Sc concentrations cannot. Several samples have veins of microlite-free glass 1–5 mm thick that are compositionally and physically continuous with intergranular melt and which apparently formed after the climactic eruption began. Whole-rock H2O content, microprobe glass analysis sums near 100% and evidence for high temperature suggest liquids in the hotter samples were nearly anhydrous. The occurrence of similar granodiorite blocks at all azimuths around the 8 × 10 km caldera implies derivation from one pluton. Compositional similarity between granodiorite and pre-Mazama rhyodacites suggests that the pluton may have crystallised as recently as 0·4 Ma; compositional data preclude crystallisation from the Holocene chamber. The history of crystallisation, hydrothermal alteration, and remelting of the granitic rocks may be characteristic of shallow igneous systems in which the balance between hydrothermal cooling and magmatic input changes repeatedly over intervals of 104-106 years.

S-type granites have properties that are a result of their derivation from sedimentary source rocks. Slightly more than half of the granites exposed in the Lachlan Fold Belt of southeastern Australia are of this type. These S-type rocks occur in all environments ranging from an association with migmatites and high grade regional metamorphic rocks, through an occurrence as large batholiths, to those occurring as related volcanic rocks. The association with high grade metamorphic rocks is uncommon. Most of the S-type granites were derived from deeper parts of the crust and emplaced at higher levels; hence their study provides insights into the nature of that deeper crust. Only source rocks that contain enough of the granite-forming elements (Si, Al, Na and K) to provide substantial quantities of melt can produce magmas and there is therefore a fertile window in the composition of these sedimentary rocks corresponding to feldspathic greywacke, from which granite magmas may be formed.

In this paper, three contrasting S-type granite suites of the Lachlan Fold Belt are discussed. Firstly, the Cooma Granodiorite occurs within a regional metamorphic complex and is associated with migmatites. It has isotopic and chemical features matching those of the widespread Ordovician sediments that occur in the fold belt. Secondly, the S-type granites of the Bullenbalong Suite are found as voluminous contact-aureole and subvolcanic granites, with volcanic equivalents. These granites are all cordierite-bearing and have low Na2O, CaO and Sr, high Ni, strongly negative εNd and high 87Sr/86Sr, all indicative of S-type character. However, the values of these parameters are not as extreme as for the Cooma Granodiorite. Evidence is discussed to show that these granites were derived from a less mature, unexposed, deeper and older sedimentary source. Other hypotheses such as basalt mixing are discussed and can be ruled out. The Strathbogie Suite granites are more felsic but all are cordierite-bearing and have chemical and other features indicative of an immature sedimentary source. They are closely associated with cordierite-bearing volcanic rocks. The more felsic nature of the suite results in part from crystal fractionation. It is suggested that the magma may have entered this “crystal fractionation” stage of evolution because it was a slightly higher temperature magma produced from an even less mature sediment than the Bullenbalong Suite. The production of these S-type magmas is discussed in terms of vapour-absent melting of metagreywackes involving both muscovite and biotite. The production of a magma in this way is consistent with the low H2O contents and geological setting of S-type granites and volcanic rocks in the Lachlan Fold Belt.

Major and trace element, including REE, analytical data are used as bases for interpreting the petrogenesis of the major igneous components of the northern part of the Ballantrae complex which occurs in the southwestern part of the Midland Valley of Scotland. Most of the peridotite, now serpentinised, is similar to ultramafic rocks in other ophiolite complexes. Mean crystallisation conditions, determined on the basis of co-existing orthopy-roxenes and clinopyroxenes for the dominant peridotite and minor pyroxenite were 1060 (±60)°C—20 (±2) kb and 1240 (±89)°C—25 (±25) kb, respectively. These rocks, of mantle provenance, have compositions consistent with being residues after the extraction of 20–30% of tholeiitic material from the mantle. The presence among them of a rock whose REE contents indicate that it is a plagioclase peridotite, point to the tectonic incorporation of the products of a high level magma chamber.

The mafic parts of the complex have tholeiitic characteristics and developed between 1300° and 1100°C. They do not represent primary mantle melt but fractionated material. Clinopyroxene was the main fractionating phase and more than 10% fractional crystallisation is indicated with increase from gabbros, through beerbachites (metadolerites) of a sheeted dyke complex and pillow lavas, to microgabbros and pyroxene diorites. Biotite diorites and trondhjemites represent the most fractionated products, the latter having affinities with ophiolitic plagiogranites.

The beerbachites of the sheeted dyke complex do not all represent the same stage of fractionation. The pillow lavas have REE patterns similar to rocks found in marginal basins but are markedly different from pillow lavas from the Highland Border Complex in Arran, near the northern margin of the Midland Valley.

The lead isotope compositions of 61 galenas from central and southern Scotland vary markedly between different regions. Most galenas from the southern Grampian Highlands yield isotope ratios (206Pb/204Pb 17·77 ± 0·25, 207Pb/204Pb 15·47 ± 0·05, 208Pb/204Pb 37·63 ± 0·26) less radiogenic than those from Midland Valley galenas (18·22 ± 0·12, 15·55 ± 0·05, 38·13 ± 0·14) whilst galena lead from the Southern Uplands (18·28 ± 0·12, 15·56 ± 0·03, 38·21 ± 0·18) is more radiogenic than that from the southern Midland Valley (18·12 ± 0·06, 15·52 ± 0·02, 38·06 ±0·10). The change in isotopie composition across the Highland Boundary fault reflects the presence or absence of Dalradian rocks which included a magmatic component of lead. Galenas from the Dalradian sequence in Islay, where igneous rocks are lacking, have a composition (18·14±0·04, 15·51±0·01, 37·90±0·02) more like Midland Valley galenas. In the Southern Uplands, galenas yield lead isotope ratios similar to those of feldspars from Caledonian granite (18·30 ± 0·14, 15·57 ± 0·04, 37·96 ± 0·15) analysed by Blaxland et al. (1979). The similar ratios reflect the incorporation of Lower Palaeozoic sedimentary rocks into the granite magma, rather than a granitic source for the mineralisation. The granites were then thermal-structural foci for later mineralising fluids which leached metals from the surrounding rocks. Within the Midland Valley, galenas hosted in Lower Devonian-Lower Carboniferous lavas are notably more radiogenic (18·31 ±0·12, 15·58 ± 0·06, 38·20 ± 0·16) than sediment-hosted galenas (18·14 ± 0·07, 15·52 ± 0·02, 38·08 ± 0·10). The Devonian lavas at least may have inherited lead from subducted (? Lower Palaeozoic) rock incorporated in the primary magma.

Calc-alkaline lamprophyres are porphyritic dyke-rocks, richer in amphibole, biotite, ultramark elements (Mg, Cr, Ni) and incompatible elements (K, F, P, Rb, Sr, Zn, Nb, Ba, REE, Th, U) than other rocks of comparable colour index (35–67) or % SiO2 (46–57). Field and petrological criteria are suggested for identifying them uniquely. New average compositions, based on some 600 screened analyses, confirm the chemical variability but possible heteromorphism of vogesites, kersantites, spessartites and some minettes. Calc-alkaline lamprophyres probably crystallise from volatile-rich, crystal-laden fluids. Microdioritic ‘porphyrites’, K-rich syenites and quartz-feldspar porphyries commonly form from these fluids by acidic hybridisation, and more rarely by in situ differentiation. Calc-alkaline lamprophyres have close compositional equivalents among, for example, shonkinites and absarokites, but not among kimberlites or common plutonie or volcanic rocks. They are compositionally more ‘crustal’ than lamproites and leucitites, despite some overlap. They are far more widespread than other K-rich or alkaline rocks. Three genetic petrological associations are confirmed: with calc-alkaline granitoid plutons (A), with shoshonitic volcanic and subvolcanic suites (B), and with appinite—breccia-pipe complexes (C). Most calc-alkaline lamprophyres, from association A, perhaps form by crustal modification of primary lamproitic or leucititic magmas; a very few, carrying rare mantle-type xenoliths, may represent relatively unmodified, but otherwise similar, primary magmas. Those of association B may form merely by volatile enrichment of shoshonitic magmas during subvolcanic crystallisation. Different origins for minettes in these associations are suggested by compositional differences, revealed by discriminant analysis.

Studies of the chemistry of Lower Carboniferous igneous rocks in the Midland Valley of Scotland throw light on the nature and magmatic evolution of transitional basaltic suites. Major oxide and trace element chemistry of least altered lavas of the Campsie Fells reveals a lower group of basalts, basaltic hawaiites and hawaiites, and an upper, more variable group, ranging from basalt to trachyte. Chemical composition of associated vent intrusions confirms their association with the lower group of flows. The magmatically evolved condition of the erupted material indicates high-pressure fractionation of clinopyroxene at depth followed by fractional crystallisation of olivine, plagioclase and Ti-Fe ores at lower pressures. In the case of the upper lavas, further differentiation appears to have taken place in a shallow magma chamber under a central volcano.

The observed chemical variations in the Campsie Fells and other areas of Lower Carboniferousvolcanic activity can be attributed to a number of different factors during the evolution of the magmas. Detailed studies of the cumulative phases of the lavas may throw further light on the various stages involved.

Pb- and Sr-isotope, major element and trace element analyses are presented for 38 Tertiary igneous rocks and older country rocks from Arran. Tertiary igneous rocks define silica-undersaturated and -oversaturated differentiation trends, but acidic and basic examples of the latter trend, exemplified in composite dykes, are separated by a wide Daly gap. In contrast, Pb-, and to some extent Sr-isotope compositions of these rocks display continuous variation, almost independent of major element composition.

Isotopic variations are interpreted as resulting principally from crustal contamination of the differentiation products of mantle-derived basic magma. A dominant rôle by mantle heterogeneity or crustal anatexis in producing the isotopic variation is ruled out. Contamination is believed to have occurred primarily after differentiation, but some rocks, such as the fine unit of the Northern Granite, display evidence of post-contamination differentiation.

Tertiary igneous rocks define an array on a Pb/Pb isochron diagram whose slope corresponds to an apparent age of 770±180 Ma (2σ). This points to the existence of a late Proterozoic basement complex under Arran.

Acidic and basic members of composite dykes are interpreted as basaltic differentiates which have been variably affected by contamination, yielding wide ranges of isotopic composition. Composite intrusion is ascribed to the simultaneous presence of acidic and basic magma in the crust.