Formula Used
The Einstein relation connects the diffusion coefficient D and mobility μ for charged carriers:
This tool also rearranges the equation to solve for μ, T, or |q| from consistent inputs.
How to Use This Calculator
- Select the quantity you want to compute from Solve for.
- Enter the remaining known values and choose their units.
- Pick the desired output unit when available.
- Press Calculate to show results above the form.
- Use Download CSV or Download PDF to export.
Example Data Table
| Case | Mobility μ | Temperature T | Charge |q| | Computed D |
|---|---|---|---|---|
| Electron-like carrier | 1400 cm²/(V·s) | 300 K | 1 e | 0.0362 m²/s (≈ 362 cm²/s) |
| Ion in solution | 5.0×10⁻⁸ m²/(V·s) | 298 K | 1 e | 1.28×10⁻⁹ m²/s |
| Low-mobility carrier | 10 cm²/(V·s) | 350 K | 1 e | 0.00301 m²/s (≈ 30.1 cm²/s) |
Values are illustrative. Real systems may deviate due to interactions and non-ideal effects.
Professional Guide
1) What the Einstein relation states
The Einstein relation links random thermal spreading to field-driven drift. It connects diffusion D and mobility μ through temperature and charge magnitude. In SI form, D = μ kB T / |q|. This makes the ratio D/μ a direct thermometer of thermal energy.
2) Why it matters in transport modeling
Many drift–diffusion solvers need consistent D and μ. If one value is measured, the other can be estimated quickly. For charge carriers in semiconductors, the relation supports device-scale simulations. In electrolytes, it helps connect ionic mobility to measurable diffusion.
3) Typical mobility ranges you may enter
Electron and hole mobilities vary widely by material and disorder. Silicon at room temperature can be on the order of 10²–10³ cm²/(V·s), while organic semiconductors may sit near 10⁻⁴–10 cm²/(V·s). In liquids, ionic mobilities are often around 10⁻⁸–10⁻⁷ m²/(V·s).
4) Diffusion coefficients: what numbers look like
Diffusion spans many orders of magnitude. Small ions in water commonly show D ≈ 10⁻⁹ m²/s. Neutral molecules can be similar or slightly lower. In solids, diffusion may drop toward 10⁻¹⁵ m²/s or below, especially at lower temperatures.
5) Temperature dependence in practical terms
For fixed mobility, diffusion increases linearly with T. A rise from 300 K to 360 K increases D by 20%. In real materials, mobility itself may change with temperature, so the net trend can be stronger or weaker than linear.
6) Charge magnitude and carrier choice
The calculator uses |q|, so the sign does not affect results. For singly charged carriers, choose 1 e. For doubly charged ions, use 2 e. If a custom charge is known, switch the unit to coulomb.
7) Unit conversions built into this page
Mobility and diffusion are accepted in both SI and cgs-style units. The tool converts inputs to SI internally and then displays your preferred output. For reference, 1 cm² = 10⁻⁴ m². This is why cm²/(V·s) values can look much larger than SI.
8) Interpreting deviations from the relation
Strong interactions, degeneracy, trapping, or correlated motion can break simple Einstein behavior. In such cases, the effective D/μ may differ from kBT/|q|. Use the calculator as a baseline, then compare with experimental trends and model corrections.
FAQs
1) When should I solve for diffusion instead of mobility?
Solve for diffusion when mobility is measured from drift experiments, and you need D for concentration spreading in simulations. This is common in drift–diffusion device models and electrolyte transport work.
2) Does the sign of charge change the computed value?
No. The Einstein relation uses |q|. Electrons and holes with the same magnitude of charge yield the same diffusion–mobility ratio at a given temperature.
3) Why does my result look huge in cm²/s?
The unit cm²/s is 10,000 times larger than m²/s. A value like 3×10⁻³ m²/s becomes 30 cm²/s. The physics is unchanged; only the unit scale differs.
4) Can I use Celsius input for temperature?
Yes. Choose °C and enter the value. The tool converts it to kelvin internally using T(K)=T(°C)+273.15 before computing the target quantity.
5) What charge should I enter for ions with valence z?
Enter |q| = z e. For example, z=2 uses 2 e. If you know the charge in coulomb, switch the unit to C and enter it directly.
6) Why might experiments deviate from Einstein behavior?
Deviations can occur from trapping, carrier degeneracy, strong correlations, or non-equilibrium conditions. These effects alter the effective relationship between drift response and thermal spreading, changing the observed D/μ.
7) Is this calculator valid for neutral particles?
The displayed formula is for charged carriers where mobility is defined against an electric field. For neutral particles, use friction-based Einstein–Smoluchowski forms relating diffusion to drag and thermal energy instead.