(a) What are the different types of rotational axes of symmetry present in a crystal ? What are the different types of twinning…

(a) Rotational axes and quartz twinning. A crystal may possess proper rotational axes of symmetry of order 1, 2, 3, 4 and 6 only. A 1-fold axis is the identity operation; a 2-fold axis requires a 180° rotation, a 3-fold axis 120°, a 4-fold axis 90°, and a 6-fold axis 60° for the form to repeat. Five-fold and higher than six-fold axes are absent because of the crystallographic restriction: they cannot be combined with translational periodicity to fill space without gaps or overlaps. Crystals also have rotoinversion axes (1̄, 2̄, 3̄, 4̄, 6̄), which combine rotation with inversion through the centre; 1̄ is a centre of symmetry and 2̄ is equivalent to a mirror plane. Low (α) quartz is trigonal, point group 32, with one 3-fold axis parallel to c and three 2-fold axes perpendicular to it. Twinning in quartz occurs by three laws. Dauphiné-law twins are electrical twins related by a 180° rotation about the c-axis; they combine two crystals of the same handedness, are invisible in polarised light, and form during growth, on inversion from β- to α-quartz, or by mechanical stress. Brazil-law twins are optical twins that combine right- and left-handed quartz across {11-20}, seen as banded or lamellar domains, common in amethyst. Japan-law twins are rare contact twins in which the c-axes of the two individuals meet at 84°33′, giving a flat heart-shaped pair. These twins record growth or deformation history.

(b) Double refraction and birefringence. Double refraction is the splitting of a single incident light ray into two rays when it enters an anisotropic mineral whose refractive index varies with direction. The two rays, ordinary and extraordinary, are plane-polarized with mutually perpendicular vibrations. The ordinary ray obeys Snell's law and has a constant refractive index, nω; the extraordinary ray has a direction-dependent index, nε. Birefringence is the difference between the refractive indices of these rays, written δ = |nε − nω| for a uniaxial mineral, or the maximum minus minimum index for a biaxial mineral. A suitable sketch shows the uniaxial indicatrix as an ellipsoid of revolution with two equal axes nω and a unique axis nε; a ray entering obliquely splits into ordinary and extraordinary rays. If nε > nω the mineral is uniaxial positive, if nε < nω it is uniaxial negative. For example, calcite has δ ≈ 0.172 (a strong double image) and quartz δ ≈ 0.009 (low), while isotropic minerals show none. Birefringence controls interference colours and extinction under crossed polars.

(c) Binary eutectic system and porphyritic basic rock. At 1 atm, a binary eutectic phase diagram has two liquidus curves descending from the melting points of pure components A and B to a minimum eutectic point. Below the liquidus, fields are A + liquid, B + liquid, and, at the eutectic isotherm, liquid transforms into A + B. The horizontal eutectic line and the lever rule give the proportions of phases. To explain a porphyritic basic rock with plagioclase phenocrysts and a plagioclase-clinopyroxene groundmass, treat plagioclase as A and clinopyroxene as B, as in the classic diopside–anorthite system, whose eutectic lies at about 1274 °C and about 42% anorthite (by weight). Because the bulk composition lies on the plagioclase (A) side of the eutectic, the melt first crosses the plagioclase liquidus; plagioclase crystallizes and grows as phenocrysts. The residual liquid becomes enriched in the pyroxene-forming components and moves toward the eutectic composition. At the eutectic temperature, the remaining liquid solidifies simultaneously as interlocking plagioclase and clinopyroxene. If this final crystallization is rapid or quenched, a fine-grained eutectic groundmass is preserved around the earlier phenocrysts, producing a porphyritic basalt or dolerite. The texture reflects two-stage cooling: slow cooling at depth grows the phenocrysts, and rapid cooling after eruption or emplacement freezes the groundmass. The simple binary is an approximation, since natural plagioclase is a solid solution.

(d) Magmatic differentiation. Magmatic differentiation is the change in composition of a cooling magma by physical and chemical processes. Fractional crystallization is central: as temperature falls, minerals crystallize in the order of Bowen's reaction series. Early olivine, pyroxene and calcic plagioclase remove Mg, Fe and Ca, while the residual melt becomes enriched in Si, Al, Na, K and volatiles, evolving from basalt toward andesite, dacite or rhyolite. Crystal settling allows dense crystals to sink, separating them from the melt; in layered mafic intrusions it can produce cumulate layers. Filter pressing occurs when a crystal framework forms and expels interstitial liquid. Gas streaming carries volatile-rich fluids and dissolved elements away from the magma. Liquid immiscibility can separate a silicate melt from an immiscible second liquid, such as a carbonatitic melt. Assimilation happens when country rock is heated, melted and mixed with the magma, modifying its chemistry. Magma mixing occurs when a fresh, hotter magma enters an evolved chamber, producing hybrid compositions, disequilibrium textures and enclaves; thermal diffusion (the Soret effect) can also impose compositional gradients. Together these processes generate the range of igneous rocks from a single parent melt.

(e) Folk's graphic classification of carbonate rocks. Folk's classification separates carbonate rocks by allochems, matrix and textural maturity. Allochems include intraclasts, ooids, bioclasts or fossils, and peloids; matrix is micrite or sparite. The graphic ternary diagram has three end-members: allochems, micrite matrix and sparry calcite cement. Peloids (pellets) are the fourth allochem type used in naming the rock. The main rock names are intrasparite, oosparite, biosparite, pelsparite, intramicrite, oomicrite, biomicrite and pelmicrite, plus micrite, dismicrite and biolithite. The ternary fields are read by plotting the relative proportions of allochems, micrite and spar, giving sparry allochemical, microcrystalline allochemical and micrite rock types. Matrix content and sorting then define Folk's textural spectrum: micrite, fossiliferous micrite, sparse biomicrite, packed biomicrite, poorly washed biosparite, and unsorted, sorted and rounded biosparite; grainstone–mudstone terms belong to Dunham's classification. Well-sorted, rounded, clean grains indicate textural maturity, whereas micrite-rich, poorly sorted grains indicate immaturity. This scheme links grain type, matrix and diagenetic history.