(a) What are the symmetry elements present in the normal class of an isometric system ? Write the Hermann-Mauguin notation of the…

(a) Isometric normal class. The normal or holohedral isometric class has the maximum symmetry of the system: thirteen axes—three four-fold, four three-fold and six two-fold—nine mirror planes and one centre of symmetry, i.e. 23 symmetry elements. Its Hermann–Mauguin notation is 4/m 3̄ 2/m. To plot a general face (hkl), place its pole in the positive octant of the stereogram. The nine mirror planes appear as great-circle traces and divide the upper stereogram into 24 fundamental spherical triangles. The three-fold axes and mirror planes generate the six positive permutations in that octant: (hkl), (hlk), (khl), (klh), (lkh) and (lhk); the four-fold and two-fold axes, with the centre, produce the remaining sign-changed poles. Thus the complete form has 48 equivalent poles, distributed as 6 per octant; the upper-hemisphere stereogram displays 24 projected poles, with closed and open symbols for opposite faces. The generated form is the hexoctahedron {hkl}. Special positions reduce the form: {h0l} gives a tetrahexahedron, {hhl} a trisoctahedron, {100} a cube, {110} a dodecahedron and {111} an octahedron.

(b) Mica structure. Micas are 2:1 phyllosilicates with a T-O-T layer structure. In a sketch, in each of two tetrahedral sheets, tetrahedra share basal oxygens to form a (Si2O5)2− layer; the inward-pointing apical oxygens of both sheets link to a central octahedral sheet in which Al, Mg or Fe occupy octahedral sites with OH or O. An interlayer of K+ (or Ca, Na, H2O) balances charge and separates the T-O-T packages, producing perfect {001} cleavage. The weak interlayer bonding explains the easy cleavage and the low hardness parallel to {001}. Dioctahedral micas have only two of three octahedral sites occupied; muscovite, KAl2(AlSi3O10)(OH)2, is Al-rich and usually colourless or pale. Trioctahedral micas have all three sites occupied; biotite, K(Mg,Fe)3(AlSi3O10)(OH)2, and phlogopite, KMg3(AlSi3O10)(OH)2, are Fe-Mg rich and brown to green. A diagnostic physical property is that mica flakes are elastic (they spring back when bent), unlike the merely flexible chlorite. Optically, micas are biaxial, usually negative, with small to moderate 2V (about 0–25° in biotite, about 30–47° in muscovite), moderate relief, low hardness and strong perfect basal cleavage. Muscovite shows weak pleochroism, while biotite shows strong brown pleochroism and pleochroic haloes around radioactive inclusions. Birefringence is high (muscovite 0.036–0.042, up to third-order interference colours; biotite 0.03–0.06), often masked by the strong body colour. Important Indian occurrences include Koderma in Jharkhand and Nellore in Andhra Pradesh.

(c) Polymorphism. Polymorphism is the occurrence of one chemical composition in two or more different crystal structures. Polymorphic transitions are changes from one polymorph to another with temperature or pressure. Reconstructive transitions require breaking and reforming bonds, have high activation energy and are sluggish; examples include quartz–tridymite–cristobalite and kyanite–andalusite–sillimanite. Displacive transitions involve small shifts or rotations of atoms without bond breaking, have low activation energy and are rapid; α–β quartz at 573°C and α–β cristobalite are typical. Order–disorder transitions involve a change in the ordering of atoms over crystallographic sites, as in high- to low-temperature feldspars. Transitions are also classed as enantiotropic (reversible, each form stable over a T–P range, e.g., quartz ⇌ tridymite) or monotropic (one form metastable throughout, so the change is irreversible, e.g., marcasite → pyrite); aragonite at surface conditions is metastable and slowly inverts to calcite. The SiO₂ polymorphs are quartz, tridymite, cristobalite, coesite and stishovite; the α and β varieties of quartz, tridymite and cristobalite are low- and high-temperature forms. Quartz is stable at low pressure and low to moderate temperature; tridymite at high temperature and low pressure (about 870–1470 °C at 1 atm); cristobalite at still higher temperature (about 1470–1713 °C, the melting point); coesite at high pressure and moderate temperature; stishovite at very high pressure, so it is an impact indicator. The Al₂SiO₅ polymorphs are kyanite, andalusite and sillimanite. Kyanite is stable at high pressure and lower temperature, andalusite at low pressure and lower temperature, and sillimanite at high temperature. Their triple point, near 500°C and 3.8 kbar (Holdaway, 1971), is a key metamorphic index. In Barrovian higher-pressure sequences, kyanite occurs before sillimanite; in Buchan low-pressure sequences, andalusite occurs before sillimanite. In Indian Precambrian terrains, sillimanite at Sonapahar in Meghalaya and Pipra in Madhya Pradesh records high-grade metamorphism.