Mineralogy & Crystallography

Crystallography & X-ray Diffraction

Seven crystal systems, Miller indices and Bragg's law — how symmetry and X-rays fix the atomic architecture of minerals.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Crystals are periodic atomic lattices classified by symmetry: 7 crystal systems, 14 Bravais lattices, 32 point groups (crystal classes) and 230 space groups. Lattice planes are labelled by Miller indices

(hk l)

. X-ray diffraction probes the spacing between these planes: constructive interference occurs when Bragg's law

nλ = 2 d sin θ

is satisfied, and for a cubic crystal the interplanar spacing is

d_{hk l} = a / h^{2} + k^{2} + l^{2}

The Core Formula Matrix

Cubic interplanar spacing:

d_{hk l} = \frac{a}{h ^{2} + k ^{2} + l ^{2}}

Bragg's law:

nλ = 2 d sin θ

Unit-cell density:

ρ = \frac{Z M}{N _{A} a ^{3}}

(

Z

formula units,

M

molar mass).

Symmetry hierarchy: 7 systems · 14 Bravais · 32 classes · 230 space groups.

The ‘IIT Traps’

⚠
** $sin θ$ , not $θ$ .** Solve Bragg's law for $sin θ$ first, then take the inverse sine.
⚠
Sum of squares. $d$ depends on $h^{2} + k^{2} + l^{2}$ — square each index before adding.
⚠
Å vs nm. Keep wavelength and cell edge in the same unit (Å).

📚 Standard references

Manual of Mineral Science — Klein & Dutrow (Dana)
Introduction to Mineralogy — William D. Nesse

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

The number of crystal systems and Bravais lattices are, respectively,

Q2MEDIUM2 Marks · NAT

Halite is cubic with

a = 5.64

Å. The

d_{200}

spacing is _____ Å.

Q3HARD2 Marks · NAT

X-rays (

λ = 1.54

Å) reflect (first order) from planes with

d = 3.08

Å. The Bragg angle

θ

is _____ degrees.