Inorganic chemistry

Crystal structures and X-ray diffraction

A crystal is periodic order on the atomic scale. X-rays scattered by that periodic array interfere constructively only in specific directions, turning a diffraction pattern into evidence about the unit cell and atomic arrangement.

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IntuitionIntuition: periodic planes act like a diffraction grating

When the path difference between X-rays scattered from adjacent lattice planes equals an integer number of wavelengths, waves reinforce. Changing crystal spacing or X-ray wavelength moves the diffraction peaks.

Switch simple cubic, bcc, fcc, NaCl or diamond structures to compare peak positions and systematic absences.

SchoolSchool level: cells and Bragg’s law

Definition: Unit cell

A unit cell is a smallest repeating volume that, by translation in three dimensions, generates the crystal lattice. Atoms on faces, edges and corners are shared with neighbouring cells, so their contribution to a cell is fractional.

n\lambda = 2d\sin\theta

In Bragg’s law, d is the spacing between a family of lattice planes, θ is the glancing angle and n is an integer order. A powder diffractometer commonly reports 2θ, the angle between incident and diffracted beams.

Cubic lattices and allowed reflections
Lattice	Condition	Examples
Simple cubic	All hkl (basis may alter intensity)	α-Po
Body-centred cubic	h+k+l even	α-Fe
Face-centred cubic	h,k,l all odd or all even	Cu, Ni

Example: First peak of fcc copper

Copper is fcc with a ≈ 3.615 Å. For the (111) planes, use Cu Kα radiation λ = 1.5406 Å to find 2θ.

Solution

For cubic crystals d111 = a/√3 ≈ 2.087 Å. First-order Bragg diffraction gives θ = asin[λ/(2d)] ≈ 21.65°, so 2θ ≈ 43.3°.

UndergraduateUndergraduate: reciprocal space and structure factors

d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}}\quad(\text{cubic});\qquad I_{hkl}\propto |F_{hkl}|^2

The structure factor Fhkl is a coherent sum over atoms in the unit cell, each weighted by its scattering factor and phase. Lattice centring and multi-atom bases cause systematic absences or intensity contrasts; peak positions determine metric spacings, while intensities carry basis information.

F_{hkl}=\sum_j f_j\exp\!\left[2\pi i(hx_j+ky_j+lz_j)\right]

Powder averaging superposes reflections from crystallites in all orientations. Peak widths can reflect instrument resolution, finite coherent domain size and microstrain; background, preferred orientation and absorption complicate quantitative refinement. Rietveld refinement fits the whole profile rather than isolated peak heights.

Example: Index a cubic reflection

A cubic powder peak occurs at 2θ = 29.4° with λ = 1.5406 Å and is assigned to (200). Estimate the lattice parameter a.

Solution

θ = 14.7°. Bragg gives d = λ/(2 sin θ) ≈ 3.03 Å. For (200), d = a/2, therefore a ≈ 6.06 Å.

AdvancedReal crystals, defects and reciprocal-space tools

A finite crystallite broadens reciprocal-lattice points; strain shifts and broadens peaks anisotropically. Total scattering and pair-distribution-function analysis can reveal local order even where Bragg diffraction sees only average periodic structure. Single-crystal diffraction additionally resolves directions and anisotropic displacement.

Diffraction is not chemically unique: similar patterns can arise from related phases, and nanoscale or amorphous material may yield broad features. Reliable identification compares calibrated data with reference patterns and checks composition and complementary measurements.

References

Elements of X-Ray Diffraction · B. D. Cullity, S. R. Stock, 2001
The Structure of Some Crystals as Indicated by their Diffraction of X-rays · W. L. Bragg, 1913