MATE50005 Diffraction

Lattice

An infinite periodic array of identical lattice points generated by integer combinations of basis vectors a, b, c.

Motif

The group of atoms attached to each lattice point; repeating the motif over the lattice yields the crystal.

Crystal structure

C = L ∗ M — convolution of lattice (L) and motif (M) so the motif repeats at every lattice point.

Unit cell volume

V = a · (b × c)

Primitive unit cell

Contains exactly one lattice point; any lattice point can be formed from integer linear combinations of a, b, c.

Non-primitive unit cell

Contains more than one lattice point; requires fractional u, v, w in t = ua + vb + wc.

Wigner–Seitz cell

Primitive cell formed by bisecting lines from a lattice point to neighbours; region closer to that point than any other.

Lattice vector

t = ua + vb + wc, translation connecting any two lattice points.

Miller indices (h k l)

Reciprocals of intercepts of a plane with the axes, then cleared of fractions to integers h,k,l.

Direction [u v w]

Vector parallel to ua + vb + wc, reduced to smallest integer ratio; overbar for negative.

Family of directions ⟨u v w⟩

All symmetry-equivalent directions in lattice.

Family of planes {h k l}

All symmetry equivalent planes in a crystal

Interplanar spacing (cubic)

d = a / √(h² + k² + l²).

Interplanar spacing (orthorhombic)

1/d² = (h²/a²) + (k²/b²) + (l²/c²).

Interplanar spacing (hexagonal)

1/d² = (4/3)((h² + hk + k²)/a²) + (l²/c²).

Reciprocal lattice & diffraction

A lattice of vectors normal to real-space planes with magnitude 2π/dhkl

Reciprocal basis vectors

a* = (2π/V)(b × c), b* = (2π/V)(c × a), c* = (2π/V)(a × b).

Magnitude of reciprocal vector

|G_hkl| = 2π/d_hkl.

Laue condition

Diffraction occurs when scattering vector Q equals reciprocal lattice vector G.

Wavevector magnitude

|k| = 2π/λ; direction is normal to wavefronts.

Huygens–Fresnel principle

Each point on a wavefront acts as a secondary wave source; explains diffraction & interference.

Fourier transform of lattice

Perfect infinite lattice of Dirac deltas produces reciprocal lattice points.

Crystal FT

C̃(Q) = L̃(Q) × M̃(Q) → diffraction pattern = reciprocal lattice modulated by structure factor.

Convolution theorem

FT(f ∗ g) = FT(f) × FT(g); used to relate crystal structure and diffraction pattern.

Dirac delta

Zero everywhere except one point; integrates to 1; represents an ideal lattice point.

Dirac comb

Infinite series of delta functions spaced periodically; its FT is another comb with reciprocal spacing.

Structure factor definition

F_hkl = Σ_j f_j exp[2πi(hx_j + ky_j + lz_j)], summing over atoms at fractional coordinates (x_j,y_j,z_j).

Systematic absences

Reflections where structure factor sums to zero due to destructive interference; peaks that physically cannot appear.

BCC structure factor rule

Two atoms (0,0,0) and (½,½,½): F = f[1 + exp(iπ(h+k+l))] → reflections allowed only when (h+k+l) even; odd ⇒ absent.

FCC structure factor rule

Atoms at (0,0,0), (0,½,½), (½,0,½), (½,½,0); allowed only when h,k,l are all even or all odd; mixed parity forbidden.

NaCl structure factor

FCC lattice + Na at (0,0,0), Cl at (½,½,½). F = f_Na + f_Cl exp[iπ(h+k+l)]. Even sum ⇒ constructive; odd sum ⇒ partial cancellation → weak or absent peaks.

Glide plane definition

Mirror reflection + fractional translation in the plane; produces destructive interference giving systematic absences.

Glide plane absence rule

If glide translation = ½ along a direction, reflections with odd index in that direction vanish (e.g., h odd ⇒ absent for (h0l)).

Screw axis definition

Rotation + translation along the axis (e.g., 2₁ screw = 180° rotation + ½ translation).

2₁ screw systematic rule

(h00) reflections only allowed when h even, because translation gives exp(iπ h).

Single slit diffraction

Minima occur at θ ≈ (1 +2n)λ/W

Double slit diffraction

Constructive intereference (bright fringes) occurs when dsinθ = nλ

Diffraction grating

Evenly spaced slits, peaks become narrower and more intense due to path interference sinθ = nλ/S

Diffuse vs sharp diffraction

Sharp is in a perfectly infinite crystal lattice, diffuse is in finite crystal (FT of a finite shape)

How does diffraction measure strucutre?

Diffraction pattern is the FT of the electron density (finding F(Q) gives back atomic positions)

Determining lattice parameter (cubic)

a = λ √(h²+k²+l²)/(2 sinθ) from each peak.

Indexing powder patterns

Compute sin²θ or d; match with allowed h²+k²+l² for cubic; first peak identifies lattice type (FCC starts at 111, BCC at 110).

Preferred orientation effect

Certain planes show enhanced intensity if platelets or grains align; peaks differ from ideal powder pattern.

Anisotropic broadening

Peak widths change with hkl due to directional strain or different coherence lengths.

Peak splitting

Occurs when symmetry is lowered (e.g., cubic → tetragonal); previously equivalent reflections separate.

Scherrer equation

D = (K λ)/(β cosθ), K ~ 0.9, β = FWHM (radians). Estimates crystallite size from peak broadening.

Williamson–Hall method

Plot β cosθ vs 4 sinθ/λ → intercept gives size (Scherrer-like), slope gives microstrain.

Is K = 0.9 always correct?

No. Depends on crystallite shape; spherical≈0.89, plates, needles differ; using 0.9 is an approximation.

BCC vs FCC diffraction

BCC first peak = 110; FCC first peak = 111. Allowed/absent rule identifies structure.

Gold vs Ir FCC alloy test

If alloy: single FCC pattern with peaks shifted between pure Au and pure Ir. If separate phases: two FCC patterns with different a.

X-ray vs neutron scattering

X-rays scatter from electrons, so heavy atoms dominate. Neutrons scatter from nuclei, so light atoms can give strong peaks; intensities differ.

Background noise in XRD

Comes from incoherent scattering, fluorescence, Compton scattering, air scatter, detector noise, and imperfect sample packing; adds broad, low-intensity signal underneath peaks.

Air scatter background

X-rays scatter elastically from air atoms, creating a smooth low-angle background; minimised by shielding, helium purge, or vacuum paths.

Fluorescence background

When incident X-ray energy exceeds absorption edge of the sample, atoms fluoresce and emit broad X-rays, raising background; especially strong for Fe when using Cu sources.

Compton scattering

Inelastic scattering of X-rays causes loss of energy and a broad background hump at low and mid 2θ.

Instrument noise

Electronic noise from detector and readout system contributes random background counts.

Doublet peaks

Cu tube emits Kα1 and Kα2 wavelengths; each reflection produces two slightly separated peaks at predictable spacing, forming a doublet.

Why use a Ni filter with Cu radiation

Ni absorbs Kβ wavelengths, producing a cleaner spectrum with mainly Cu Kα1 and Kα2, reducing extra unwanted peaks.

Why Kα1 and Kα2 are resolved more at high angles

Peak separation in 2θ increases with angle; doublets merge at low angles but split at high angles.

Peak broadening from small crystallites

Finite grain size reduces coherence length; Scherrer broadening produces wide peaks inversely proportional to crystallite size.

Peak broadening from microstrain

Variation of lattice spacings causes angular spread in Bragg condition, giving broad peaks that increase with 2θ.

Instrumental broadening

Finite slit widths, detector resolution and monochromator imperfections produce a baseline broadening added to sample broadening.

Preferred orientation

If grains align, certain hkl planes scatter more strongly, distorting intensities compared to reference powder pattern.

Zero-shift error

If sample is above or below the diffractometer axis, all peaks shift slightly in 2θ; corrected by internal standard or fitting.

Peak asymmetry at low angles

Caused by axial divergence of incident beam; peaks tail toward lower 2θ.

Sample transparency effect

X-rays penetrate substrate causing absorption and refraction, modifying peak shape and positions.

Sample displacement error

If sample surface is not exactly at diffractometer centre, Bragg angles shift by a small constant amount.

Why use monochromators

Reduce Kβ and white radiation; sharpen peaks; reduce fluorescence; improve signal-to-noise ratio.

Why powder samples are rotated during measurement

Ensures random grain orientation, improves statistical averaging, reduces preferred orientation effects.

Why we grind samples to fine powders

Reduces grain size so many crystallites contribute; ensures random orientation; avoids spotty diffraction.

Particle size too large problem

Coarse grains produce spotty (not smooth) powder rings; missing reflections or erratic intensities.

Why peaks get weaker at high 2θ

Atomic form factor decreases with scattering angle, reducing intensity of high-angle reflections.

Scattering from sample holder

If holder is amorphous (glass/plastic), broad humps appear; low-background holders minimise this.

Spurious peaks from Cu anode

Cu Kβ and white radiation produce unwanted extra peaks; removed using Ni filters or monochromators.

Preferred orientation in plate-like samples (e.g. alumina)

Basal planes align, enhancing low-angle reflections like (00l), causing intensity mismatch.

Preferred orientation in needle-like crystals

Preferred alignment enhances certain (h00)/(0k0) peaks depending on habit.

Why peak intensities differ between X-ray and neutron diffraction

X-rays scatter from electron density; neutrons scatter from nuclei with different scattering lengths, so heavy atoms weaken and light atoms strengthen relatively.

Kβ leakage and unwanted peaks

If filter is imperfect, small Kβ intensity remains, causing small peaks at slightly different angles.

Why use internal standards

Correct for zero-shift, lattice parameter accuracy, and instrument drift by comparing known and unknown peaks.

Instrumental drift

X-ray intensity varies over time due to tube ageing or temperature shifts; corrected by normalisation or standards.

Absorption correction

Heavy elements absorb X-rays strongly, changing relative intensities; absorption correction normalises true structure factor intensities.

Why we see noise at low counts

Diffraction follows Poisson statistics; low counts give higher relative statistical scatter.

Lorentz–polarisation factor

Intensity must be corrected for geometric factors of diffractometer and polarisation of X-ray beam.

Wavelength and frequency equation

c = vλ

de Broglie Wavelength

λ = h/p

Bragg’s law

nλ = 2dsinθ

|G_hkl|

2π/d_hkl

Wavevector equation

|k| = 2π/λ

Q

k_f - k_i

Crystal lattice with motif convolution

C = (L * M) x S

Fourier transform formula

FT[ f( r ) x g( r )] = f̃( r* ) * g̃( r* )

What does FT do to the points?

Small features spread out (smear), large features become narrow

Cubic crystal system

a = b = c, α = β = γ = 90°, (h² + k² + l²)/a²

Tetragonal

a = b ≠ c , α = β = γ = 90°, (h² + k²)/a² + l²/c²

Orthorhombic

a ≠ b ≠ c, α = β = γ = 90°, h²/a² + k²/b² + l²/c²