Inputs
Enter temperature, choose a model, and adjust effective mass for materials where electrons behave heavier or lighter.
Formula Used
Effective mass: The calculator uses m = factor × me. This is useful in semiconductors where electrons respond as if they have a different inertial mass.
Relativistic estimate (optional): It approximates mean kinetic energy as E ≈ (3/2)kT and uses E = (γ − 1)mc² to estimate v = βc. This is a practical check when speeds become a noticeable fraction of c.
How to Use This Calculator
- Enter the temperature value in your preferred unit.
- Select the unit, including kT in eV when working with plasmas.
- Set the effective mass factor to match your material model.
- Choose a thermal speed metric or compute all values.
- Press Calculate to display results above the form.
- Use Download CSV for spreadsheets and records.
- Use Download PDF to print or save results.
Example Data Table
Sample values below assume effective mass factor equals 1.
| Temperature (K) | vmp (m/s) | v̄ (m/s) | vrms (m/s) |
|---|---|---|---|
| 300 | 9.53 × 104 | 1.08 × 105 | 1.17 × 105 |
| 1000 | 1.74 × 105 | 1.97 × 105 | 2.13 × 105 |
| 10000 | 5.50 × 105 | 6.21 × 105 | 6.74 × 105 |
| 1.0 (kT in eV) | 5.93 × 105 | 6.70 × 105 | 7.27 × 105 |
Technical Article
1) What “thermal velocity” means
Electron thermal velocity describes the typical random speed of electrons caused by temperature. It is not a drift speed from an electric field. In many problems, you need a representative value to estimate collision rates, transport time scales, diffusion, or diagnostic thresholds in plasma and semiconductor work.
2) Why there are three common speed choices
The calculator reports most probable, mean, and RMS speeds because the Maxwell–Boltzmann distribution is broad. The most probable speed is where the distribution peaks, the mean speed averages over all speeds, and the RMS speed connects directly to average kinetic energy via ⟨E⟩ = (3/2)kT.
3) Practical numbers at room temperature
At 300 K with free-electron mass, typical thermal speeds are on the order of 105 m/s (hundreds of km/s). These values are far below the speed of light but large enough that microscopic processes can occur extremely quickly, which is why time steps and measurement bandwidth matter in electron systems.
4) Temperature in eV for plasma calculations
Plasma physics often expresses temperature as kT in electron-volts. One eV corresponds to about 11,605 K, so even “a few eV” indicates very hot electrons. Selecting kT (eV) in the unit menu converts your value to Kelvin internally and keeps the same kinetic-theory formulas.
5) Effective mass in materials
In many semiconductors, electrons respond as if their inertia differs from the free-electron mass. This is modeled with an effective mass factor. A smaller factor increases predicted thermal speeds, while a larger factor decreases them. Use literature values for your band structure and direction when accuracy matters.
6) When to care about relativity
As temperature rises, speeds can become a noticeable fraction of c. The calculator includes a simple relativistic estimate based on E ≈ (3/2)kT and E = (γ−1)mc². Treat it as a sanity check; accurate high‑energy modeling can require relativistic distributions.
7) Interpreting results for engineering decisions
Thermal speed helps estimate mean free path and collision frequency when combined with density and cross sections. In devices, it supports quick order‑of‑magnitude checks: whether electrons traverse a feature length within a given time, and whether transport is likely ballistic or scattering‑dominated under your conditions.
8) Reporting and exporting results
For documentation, export CSV to preserve numeric outputs and units for spreadsheets, or export PDF for print-ready records. Always record the temperature unit and the effective mass factor used. If you compare runs, keep the same speed metric (most probable, mean, or RMS) to avoid inconsistent conclusions.
FAQs
1) Which speed should I use for calculations?
Use RMS speed when linking to average kinetic energy. Use mean speed for average transport estimates. Use most probable speed when you care about the distribution peak, such as qualitative comparisons or simplified models.
2) Is this the same as drift velocity?
No. Thermal velocity is random motion from temperature. Drift velocity is the net motion caused by an electric field and is usually much smaller in conductors and semiconductors.
3) Why does effective mass change the result?
Thermal speed scales as 1/√m. If the effective mass is smaller than the free-electron mass, electrons accelerate more for the same thermal energy, increasing the representative speed.
4) What does “kT (eV)” mean?
It means you are entering thermal energy in electron-volts. The calculator converts kT from eV to Kelvin internally and then applies the same formulas consistently.
5) Do these formulas assume a specific distribution?
Yes. The non‑relativistic results assume a Maxwell–Boltzmann speed distribution. In strongly degenerate electron gases or very high‑energy plasmas, other distributions may be more appropriate.
6) Are results valid near the speed of light?
The Maxwell–Boltzmann formulas are not. Use the relativistic estimate as a quick check. For precise work, use a relativistic thermal distribution and energy-dependent modeling.
7) How can I verify my inputs quickly?
Convert temperature to Kelvin and confirm it is positive. For free electrons at 300 K, speeds should be around 105 m/s. Large deviations often indicate unit or mass‑factor mistakes.