Scherrer Equation: Calculating Crystallite Size from XRD

By Aniket Bhardwaj · 6 November 2026 · Quantum & structure

X-ray powder diffraction (XRD) is a primary characterisation method in solid-state chemistry and materials science. While peak positions reveal unit cell geometry through Bragg's law, diffraction peak breadths reflect crystallite microstructure. For nanoscale crystalline domains, finite size prevents complete destructive interference, broadening the diffraction reflections. The Scherrer equation translates this peak broadening into the mean size of coherent scattering domains. This calculation is a recurring topic in advanced competitive exams including IIT-JAM, GATE, CSIR-NET, and CUET-PG.

Origin of Peak Broadening and the Scherrer Formula

In large crystals, diffraction peaks are very sharp and limited only by instrument geometry. When crystallite dimensions fall below roughly 100 nm, incomplete destructive interference at angles close to the Bragg condition produces noticeable peak broadening. Paul Scherrer formulated the mathematical relationship linking domain size to diffraction line width:

The Scherrer Equation:

D = (K · λ) / (β · cos θ)

Variables, definitions, and units:

D: Mean crystallite dimension along the direction perpendicular to reflecting lattice planes, in nanometres (nm) or Ångströms (Å).
K: Dimensionless shape factor, typically 0.89 to 0.94 (standard value 0.90 for spherical crystallites with FWHM definition).
λ: X-ray wavelength in nm or Å (for laboratory Cu Kα radiation, λ = 0.15406 nm or 1.5406 Å).
β: Sample-induced peak broadening at full width at half maximum (FWHM) in radians, corrected for instrumental broadening.
θ: Bragg angle in degrees or radians, equal to half of the detector angle 2θ. The cosine is evaluated at θ, not 2θ.

Instrumental Broadening Correction

A diffractometer introduces non-sample broadening due to slit dimensions, beam divergence, and detector resolution. This instrumental breadth (β_inst) must be subtracted from the measured full width at half maximum (β_obs) using an appropriate peak shape model:

Gaussian model: β = √((β_obs)² - (β_inst)²)
Lorentzian (Cauchy) model: β = β_obs - β_inst

Worked Numerical Problems

The following numerical problems demonstrate crystallite size calculations, angular unit conversions, and instrumental corrections. Each solution shows every arithmetic step and is verified by a second route.

Exam details

Source: practice problem written for this article (not a past-paper question)
Relevant exams: IIT-JAM, GATE Chemistry, CSIR-NET Chemical Sciences, CUET-PG
Topic: Solid State Chemistry — Scherrer Equation and XRD
Difficulty: Advanced undergraduate to postgraduate entrance level

Question: An XRD pattern of anatase TiO2 nanoparticles recorded with Cu Kα radiation (λ = 0.15406 nm) displays a (101) peak centered at 2θ = 25.30°. The measured FWHM is 0.450°. A standard reference material under identical geometry gives an instrumental broadening of 0.090°. Assuming Gaussian peak profiles and shape factor K = 0.90, calculate: (a) the corrected sample broadening β in radians, and (b) the mean crystallite size D in nanometres.

Step-by-step solution:

Step 1: Determine Bragg angle θ and cos θ.

θ = 25.30° / 2 = 12.65°

cos θ = cos(12.65°) = 0.97573

Step 2: Correct for instrumental broadening using the Gaussian profile.

(β_obs)² = (0.450°)² = 0.20250 deg²

(β_inst)² = (0.090°)² = 0.00810 deg²

β_deg = √(0.20250 - 0.00810) = √(0.19440) = 0.44091°

Step 3: Convert corrected broadening to radians.

β = 0.44091° × (π / 180°) = 0.44091 × (3.14159265 / 180) = 0.0076953 rad = 7.70 × 10⁻³ rad

Step 4: Compute mean crystallite size D.

D = (K · λ) / (β · cos θ)

Numerator = 0.90 × 0.15406 nm = 0.138654 nm

Denominator = 0.0076953 rad × 0.97573 = 0.0075085 rad

D = 0.138654 / 0.0075085 = 18.47 nm

Step 5: Double-check by second route converting to radians first.

β_obs = 0.450 × (π / 180) = 0.0078540 rad; β_inst = 0.090 × (π / 180) = 0.0015708 rad

β = √((0.0078540)² - (0.0015708)²) = √(5.9218 × 10⁻⁵) = 0.0076953 rad

D = (0.90 × 0.15406 × 180) / (0.44091 × π × 0.97573) = 18.47 nm

Both calculation paths confirm D = 18.47 nm.

Answer: Corrected broadening β = 7.70 × 10⁻³ radians; mean crystallite size D = 18.47 nm

Question: A ceria (CeO2) powder sample produces a (111) diffraction reflection at 2θ = 28.55° with an observed FWHM of 0.780° using Cu Kα radiation (λ = 0.15406 nm). Diffractometer instrumental broadening is 0.120°. Assuming Lorentzian peak profiles (β = β_obs - β_inst) and taking K = 0.94, determine: (a) corrected broadening in radians, (b) mean crystallite size D in nanometres, and (c) the percentage error if instrumental broadening is neglected.

Step-by-step solution:

Step 1: Find Bragg angle θ and cos θ.

θ = 28.55° / 2 = 14.275°

cos(14.275°) = 0.96914

Step 2: Calculate corrected broadening β.

β_deg = 0.780° - 0.120° = 0.660°

β = 0.660° × (π / 180°) = 0.0115192 rad = 1.152 × 10⁻² rad

Step 3: Calculate corrected crystallite size D.

Numerator = K · λ = 0.94 × 0.15406 nm = 0.144816 nm

Denominator = β · cos θ = 0.0115192 rad × 0.96914 = 0.0111637 rad

D = 0.144816 / 0.0111637 = 12.97 nm

Step 4: Compute uncorrected size and error.

β_uncorr = 0.780° × (π / 180°) = 0.0136136 rad

D_uncorr = 0.144816 / (0.0136136 × 0.96914) = 10.98 nm

Percentage underestimation = ((12.97 - 10.98) / 12.97) × 100% = 15.34%

Step 5: Double-check by second route using single-line arithmetic.

D = (0.94 × 0.15406 × 180) / (0.660 × 3.14159265 × 0.96914) = 26.0669 / 2.00947 = 12.97 nm

Both methods yield identical results.

Answer: Corrected broadening β = 0.0115 radians; mean crystallite size D = 12.97 nm; error = 15.3% underestimation

Common Mistakes in Scherrer Calculations

Inserting β in degrees rather than radians: Substituting degrees directly into the denominator underestimates D by a factor of 57.3. Always multiply degrees by π / 180 before calculating.
Using 2θ instead of θ in cos θ: Powder diffractograms show 2θ along the x-axis. You must divide the peak position by 2 before evaluating cos θ.
Confusing crystallite size with particle size: Crystallite size measures coherent scattering single-crystal domains. Particles are often polycrystalline aggregates of multiple crystallites.
Neglecting instrumental broadening: As crystallites exceed 40–50 nm, sample broadening narrows and becomes comparable to instrument width. Omitting β_inst produces severe errors.
Applying Scherrer beyond its physical limits: The formula is reliable only between ~2 nm and ~100 nm. Above 100 nm, instrumental breadth dominates; below 2 nm, diffraction merges into diffuse halos.

Exam Relevance and Question Patterns

The Scherrer equation is frequently tested in graduate-level physical chemistry and solid-state papers:

Examination	Question Format	Typical Problem Focus
IIT-JAM Chemistry	Numerical Answer Type (NAT) and MCQs	Direct calculations of D from given λ, 2θ, and peak FWHM; converting degrees to radians.
GATE Chemistry	Numerical Answer Type (NAT)	Problems requiring instrumental broadening subtraction and comparing reflections across planes.
CSIR-NET Chemical Sciences	Multiple-choice conceptual and numericals	Differentiating crystallite from particle size, and analyzing how changing radiation wavelength alters peak width.
CUET-PG Chemistry	Objective MCQs	Direct parameter relationships, shape factor definitions, and rapid one-step estimations.