Calculate lattice strain from substrate and layer lattice constants, or find the missing constant when strain and one value are known.
Lattice Strain Formula
This calculator reports a geometric lattice mismatch using substrate normalization. It is not a measurement of film strain; its defined convention is:
If you want the answer as a percentage instead of a decimal, convert it with:
Variable Definitions
| Symbol | Meaning | Typical Unit |
|---|---|---|
| LS | Lattice strain | Dimensionless or % |
| As | Substrate lattice constant | Å, nm, µm, cm, m, in, ft |
| Al | Layer lattice constant | Same unit as As |
Because the formula divides one length by another, lattice strain has no physical unit. The calculator converts each lattice constant from its selected unit before comparing them.
Rearranged Equations
If you know any two variables, you can solve for the third. Select the value to solve for; the algebra is:
How to Use the Calculator
- Select the value to solve for, then enter the required lattice constants or signed mismatch.
- Select each length unit; the two input units may differ.
- Click calculate to find the lattice strain.
- Select percent or decimal ratio as the mismatch result unit.
If you are solving for a missing lattice constant instead, select percent or decimal ratio for the known mismatch; the default is percent.
How to Interpret the Result
- LS = 0: the substrate and layer have the same lattice constant in this calculation.
- LS > 0: the layer lattice constant is smaller than the substrate lattice constant.
- LS < 0: the layer lattice constant is larger than the substrate lattice constant.
This sign convention is useful, but not universal. Some textbooks and papers define lattice mismatch with the opposite sign. When comparing published values, always verify which formula was used.
Example
If the substrate lattice constant is 5.65 Ã… and the layer lattice constant is 5.60 Ã…, then:
The geometric mismatch is approximately 0.00884956, which is:
Why Lattice Strain Matters
Lattice strain is an important concept in crystallography, semiconductor fabrication, thin-film growth, and materials engineering. A mismatch between the substrate and the deposited layer can affect:
- epitaxial fit between materials,
- internal stress in the film,
- defect and dislocation formation,
- electronic and optical behavior,
- mechanical stability and long-term reliability.
Even a small strain value can matter when the application requires high crystal quality or tightly controlled material properties.
Practical Notes
- Choose the correct unit for each lattice constant; mixed units are converted automatically.
- A decimal strain of 0.01 is the same as 1% strain.
- If you only care about the magnitude of mismatch, use the absolute value of the result.
- The calculated value is a geometric comparison of lattice constants; real materials may also be influenced by temperature, composition, relaxation, and defects.
Common Mistakes
- Selecting a length unit that does not match the entered value.
- Confusing decimal strain with percent strain.
- Comparing values from different sources without checking the sign convention.
- Assuming the calculated mismatch is identical to the fully measured strain state of a real film.
Common Questions
Is lattice strain the same as lattice mismatch?
This calculator uses the substrate-normalized geometric mismatch shown above. A coherently strained film referenced to its relaxed layer constant instead uses (Aₛ − Aₗ) / Aₗ. Actual strain also depends on orientation, temperature and relaxation.
Do larger numbers always mean worse compatibility?
In general, a larger strain magnitude means the layer and substrate are less closely matched geometrically, which can increase the likelihood of stress-related issues.
Why is the answer unitless?
Strain is a ratio of two lengths, so the units cancel out.
