(a) How does the Bragg equation explain X-ray diffraction from a crystal? (20 marks) (b) How does Si-O polymerism help to…

(a) The Bragg equation and X-ray diffraction

X-rays have wavelengths (about 0.5–2.5 Å) of the order of the spacing of atomic planes in a crystal, so a crystal acts as a three-dimensional diffraction grating. W. H. Bragg and W. L. Bragg showed in 1913 that diffraction can be treated as reflection from sets of parallel lattice planes of spacing d.

Derivation. A monochromatic beam falls on the planes at a glancing angle θ. Rays reflected from two successive planes differ in path by 2d sinθ. They reinforce only if this difference is a whole number of wavelengths:

nλ = 2d sinθ

where n is the order of reflection. Since sinθ cannot exceed 1, the equation requires λ ≤ 2d. At other angles the reflected waves cancel, so a crystal gives sharp peaks at certain angles only.

Use. For a cubic crystal, d = a/√(h²+k²+l²). For quartz, d(101) = 3.343 Å; with Cu Kα (λ = 1.5406 Å) and n = 1, sinθ = 1.5406/(2 × 3.343) = 0.2304, so θ = 13.32° and the peak is at 2θ = 26.64°.

(i) Powder method. A powdered sample has crystallites in every orientation, so each set of planes reflects at its own angle, and a diffractometer records peaks of 2θ and intensity. The d-values and intensities identify a mineral by comparison with a database.

(ii) Single crystals (Laue, rotation and four-circle methods) give the unit cell and the crystal structure.

(iii) Clay minerals are identified by the basal spacing, before and after glycol treatment and heating.

The peak positions give cell dimensions, peak widths give crystallite size and strain, and intensities depend on the atoms in the cell. Mixtures can be analysed quantitatively.

(b) Si–O polymerisation and the classification of silicates

The basic unit is the SiO₄ tetrahedron. Tetrahedra may share corner (bridging) oxygens, and the degree of sharing, which is the polymerisation, raises the Si:O ratio and lowers the negative charge per Si, which has to be balanced by cations. This gives the classes:

(i) Nesosilicates have isolated tetrahedra (Si:O = 1:4): olivine.

(ii) Sorosilicates have paired tetrahedra sharing one oxygen (Si₂O₇, 1:3.5): epidote.

(iii) Cyclosilicates have rings (Si₆O₁₈, 1:3): beryl.

(iv) Inosilicates have single chains (1:3): pyroxene, such as diopside. Double chains (4:11) give amphibole, such as hornblende.

(v) Phyllosilicates have sheets sharing three oxygens (2:5): muscovite, biotite and the clay minerals.

(vi) Tectosilicates share all four oxygens in a framework (1:2): quartz and feldspar.

With increased polymerisation, the silicates are more stable at the surface. Olivine and pyroxene weather first and quartz last, which is the Goldich weathering order. Polymerisation also controls cleavage: chains give two cleavages, sheets one, and frameworks none or two.

(c) Kaolinite, smectite and illite compared

(i) Structure. Kaolinite is 1:1, a tetrahedral sheet joined to an octahedral sheet. Smectite and illite are 2:1, with an octahedral sheet between two tetrahedral sheets.

(ii) Layer charge. Kaolinite has almost none; smectite has a low charge of 0.2–0.6 per formula unit, and illite a higher charge of 0.6–0.9.

(iii) Interlayer. Kaolinite layers are joined by hydrogen bonds. Smectite holds hydrated exchangeable cations (Na, Ca) in its interlayer. Illite holds fixed potassium, as in mica.

(iv) Basal spacing. About 7.2 Å for kaolinite; 10–15 Å for smectite, expanding to about 17 Å with glycol; and 10 Å for illite, which is unchanged.

(v) Swelling and cation exchange capacity. Kaolinite does not swell and has a capacity of about 3–15 meq/100 g; smectite swells strongly and has about 80–150; illite swells little and has about 10–40.

(vi) Formation. Kaolinite forms from feldspar by intense leaching in humid, warm climates. Smectite forms from volcanic ash and basalt in poorly drained, alkaline settings. Illite forms from micas and feldspars in cooler or arid weathering and during diagenesis.

(vii) Uses and Indian examples. Kaolinite is china clay in ceramics and paper, as in Kerala and Rajasthan. Smectite (bentonite) is used in drilling mud, as in Kutch, and forms the swelling black cotton soil on the Deccan Traps. Illite is found in shales and brick clays.