Lattice Strain Calculator
Find micro-strain from X-ray diffraction peak broadening, from a change in lattice parameter, or from epitaxial film and substrate mismatch.
📏 What is Lattice Strain?
Lattice strain is the fractional change in the spacing of a crystal's atomic planes compared with their relaxed, unstrained value. Because it is a ratio of two lengths it has no units, so it is quoted either as a plain fraction such as 0.004269, as a percentage such as 0.4269%, or in units of 10⁻³. A positive strain means the lattice has been stretched (tensile), and a negative strain means it has been squeezed (compressive). Strain is the quantity that links a crystal's measured geometry to the mechanical stress acting on it, through Hooke's law and the material's elastic constants.
Crystallographers meet lattice strain in three very different measurement situations, and this calculator covers all three. In powder X-ray diffraction, a distribution of local strains inside the grains makes every diffraction peak wider without moving its centre, so the strain has to be extracted from the peak's width using the Stokes-Wilson relation ε = β / (4 tanθ). In a refinement or a simple peak-position measurement, a uniform macro-strain shifts the whole pattern, so the strain follows directly from the refined lattice parameter compared with a reference value. In thin-film epitaxy, a film grown coherently on a mismatched substrate is forced into a known in-plane strain set by the two lattice parameters, and Poisson's ratio then fixes the out-of-plane response.
A common misconception is that any broad diffraction peak means high strain. Peak width also grows when crystallites are small (the Scherrer effect), and the diffractometer itself contributes a fixed instrumental width of roughly 0.05 to 0.12 degrees FWHM. Strain broadening scales as 4ε tanθ while size broadening scales as 1/cosθ, so the two can only be separated by measuring several peaks across a wide angular range, which is exactly what a Williamson-Hall analysis does. Treating all broadening as strain, as this calculator's first mode deliberately does, gives an upper bound on the strain rather than a definitive value.
This calculator is useful for materials scientists quantifying residual strain in cold-worked or ball-milled powders, for thin-film engineers checking whether a heterostructure is still coherently strained, and for students learning why peak width and peak position carry two independent pieces of physical information. Every mode shows the full working, so the numbers can be checked by hand or carried straight into a lab notebook.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Ball-Milled Powder, Peak Broadening
Measured FWHM 0.35°, instrumental 0.08°, peak at 2θ = 38.4°, Gaussian correction
Example 2 — Gold Thin Film, Lattice Parameter Shift
Bulk gold a₀ = 4.0782 Å, refined film value a = 4.0863 Å
Example 3 — SiGe Film on a Silicon Substrate
Relaxed film afilm = 5.5 Å, silicon substrate asub = 5.431 Å, Poisson's ratio ν = 0.28
❓ Frequently Asked Questions
🔗 Related Calculators
What is lattice strain in crystallography?
Lattice strain is the fractional change in the spacing of atomic planes relative to their unstrained value, written as e = (d - d0)/d0 or (a - a0)/a0. It is dimensionless, usually quoted as a percentage or in units of 10 to the minus 3. Positive strain means the lattice is stretched (tensile), negative means compressed.
How do you calculate lattice strain from XRD peak broadening?
Use the Stokes-Wilson relation e = beta / (4 tan theta), where beta is the strain-only integral breadth in radians after instrumental broadening has been removed, and theta is half the measured 2theta peak position. For a corrected FWHM of 0.3407 degrees at 2theta = 38.4 degrees, e = 0.004269, or 0.4269 percent.
Why must instrumental broadening be subtracted first?
Every diffractometer adds width to a peak through its optics, slits, and detector, typically 0.05 to 0.12 degrees FWHM. If that width is not removed, the calculated strain is far too large. Subtract it in quadrature for Gaussian profiles, beta = sqrt(beta_obs squared minus beta_inst squared), or linearly for Lorentzian profiles.
What is the difference between micro-strain and macro-strain?
Macro-strain is a uniform change in lattice parameter across the whole sample, so it shifts the diffraction peak to a new 2theta position. Micro-strain is a distribution of local strains around the average, so it leaves the peak position alone and broadens the peak instead. This calculator handles both, in different modes.
How is strain broadening different from size broadening?
Strain broadening scales as 4 e tan theta and therefore grows quickly at high angles, while crystallite size broadening scales as K lambda / (D cos theta) and grows more slowly. Because their angular dependences differ, plotting beta cos theta against sin theta for several peaks (a Williamson-Hall plot) separates the two contributions.
What is epitaxial misfit strain?
When a thin film grows coherently on a substrate with a different lattice parameter, the film is forced to adopt the substrate spacing in the plane of the interface. The resulting in-plane strain is e = (a_sub - a_film) / a_film. A film with a larger natural spacing than the substrate is compressed, giving a negative in-plane strain.
How does Poisson's ratio affect out-of-plane strain?
A film squeezed in the plane must expand along the growth direction. For a cubic film on a (001) substrate the relation is e_perp = -(2 nu / (1 - nu)) e_par, where nu is Poisson's ratio. For nu = 0.28 the factor is 0.7778, so an in-plane strain of -1.2545 percent produces an out-of-plane strain of +0.9758 percent.
What units does lattice strain use?
Strain is dimensionless because it is a ratio of two lengths. It is reported as a plain fraction (0.004269), as a percentage (0.4269 percent), in units of 10 to the minus 3, or as micro-strain in parts per million. This calculator shows the fraction, the percentage, and the value in units of 10 to the minus 3 together.
Is a strain of 0.4 percent large for a real material?
Yes, for a bulk crystal that is a substantial micro-strain, typical of heavily cold-worked or ball-milled metals and nanocrystalline powders. Well annealed bulk crystals usually show less than 0.05 percent. Epitaxial films can hold 1 to 4 percent coherent strain below the critical thickness before dislocations form.
Can lattice strain be negative?
Yes. Negative strain means compression, the measured spacing is smaller than the reference spacing. This calculator reports the sign in every mode and labels the result as tensile or compressive, since sign carries real physical meaning for residual stress and for epitaxial film design.
What reference lattice parameter should I use?
Use a bulk, fully relaxed, unstrained value for the same composition and temperature, ideally from a standards database such as ICDD PDF or a NIST standard reference material. Using a value for a different composition is the most common source of error, since alloying often changes the lattice parameter more than the strain being measured.
How does this relate to the Scherrer and Williamson-Hall calculators?
The Scherrer equation attributes all broadening to crystallite size, this calculator attributes it all to strain. Real samples contain both, so the Williamson-Hall method fits several peaks at once to split the total broadening into a size term and a strain term. A strain-only treatment is appropriate when strain clearly dominates, or when the size contribution has already been determined independently.