Inputs
Example data
| n (m⁻³) | d (nm) | T (K) | Species | ν (s⁻¹) | λ (m) | τ (s) |
|---|---|---|---|---|---|---|
| 2.5e25 | 0.30 | 300 | N₂–N₂ | 3.37e9 | 1.41e-7 | 2.97e-10 |
| 1.0e20 | 0.20 | 2 | e⁻–Ar | ~1e6 | ~8e-3 | ~1e-6 |
Formula used
- ν = n σ v where n is number density, σ is collision cross-section, and v is relative speed.
- Hard-sphere approximation: σ = π d² using collision diameter d.
- Thermal mode uses mean relative speed: ⟨v_rel⟩ = √(8 k T / (π μ)).
- Reduced mass: μ = m₁ m₂ / (m₁ + m₂).
- Mean free path: λ = 1 / (n σ) and mean time: τ = 1 / ν.
How to use this calculator
- Enter number density and select its unit.
- Choose whether to input σ directly or compute from diameter.
- Select thermal mode (enter temperature) or provide relative speed.
- Pick two species masses to set the reduced mass in thermal mode.
- Press Calculate to see ν, λ, and τ above the form.
- Use the download buttons to export CSV or PDF.
Professional notes on collision frequency
1) Why collision frequency matters
Collision frequency, ν, is a compact way to describe how often particles exchange momentum and energy. In gases it controls viscosity, diffusion, and thermal conduction; in plasmas it helps predict conductivity, resistivity, and how quickly distributions relax toward equilibrium.
2) Core inputs and physical meaning
The calculator combines number density (n), collision cross‑section (σ), and relative speed (v). For dilute, binary collisions the rate scales linearly with each input: doubling density or cross‑section doubles ν, while higher speed increases how many targets are encountered per second.
3) Typical density scales you can sanity‑check
At room conditions, air has n ≈ 2.5×1025 m−3. A low‑pressure vacuum chamber at 1 Pa is roughly 10−5 of atmospheric pressure, so n drops near 1020 m−3. Many laboratory plasmas span 1015–1020 m−3.
4) Cross‑section choices and common magnitudes
If you use a hard‑sphere diameter, σ = πd². For molecular diameters around d = 0.3 nm, σ is about 2.8×10−19 m². Electron‑neutral momentum‑transfer cross‑sections are often 10−20–10−19 m² depending on energy and species.
5) Thermal mode and reduced mass
Thermal mode estimates the mean relative speed ⟨vrel⟩ = √(8kT/(πμ)). The reduced mass μ = m₁m₂/(m₁+m₂) captures two‑body kinematics. When one partner is much heavier, μ approaches the lighter mass, so the lighter particle largely sets the thermal speed.
6) Mean free path and time between collisions
The calculator also reports λ = 1/(nσ) and τ = 1/ν. For air at n = 2.5×1025 m−3 and σ ≈ 3×10−19 m², λ is roughly 1×10−7 m (about 100 nm). If v is a few hundred m/s, ν lands near 109–1010 s−1.
7) Interpreting results for transport and modeling
High ν means frequent scattering, short λ, and rapid equilibration—useful for continuum assumptions. Low ν implies long λ and non‑local transport, where kinetic descriptions become important. In plasma modeling, comparing ν to the gyrofrequency or wave frequency helps judge whether collisions dominate the dynamics.
8) Practical workflow for reliable inputs
Start with n from an equation of state, pressure gauge, or plasma diagnostics. Choose σ from literature tables at the relevant energy or use a diameter estimate for order‑of‑magnitude work. Use thermal mode for equilibrium gases; use speed mode for beams, drift, or directed flows.
FAQs
1) What does collision frequency represent?
It is the expected number of collisions a particle experiences per second under the chosen density, cross‑section, and relative speed, using the dilute binary‑collision model.
2) Which speed should I use?
Use thermal mode for near‑equilibrium gases or plasmas. Use a specified relative speed for drifting plasmas, directed beams, or flows where a known vrel better represents encounters.
3) How do I pick a cross‑section?
For quick estimates, compute σ from an effective diameter. For higher accuracy, use published momentum‑transfer or scattering cross‑sections at the particle energy and target species you are modeling.
4) Why does reduced mass matter in thermal mode?
Thermal relative motion depends on two‑body kinematics. Reduced mass converts two moving particles into an equivalent single‑mass problem, giving the correct mean relative speed for Maxwellian distributions.
5) Is mean free path independent of temperature here?
In this simple model λ = 1/(nσ) does not include temperature directly. Temperature can still affect σ through energy dependence, and it affects ν through the relative speed term.
6) Can I use this for liquids or dense gases?
Not reliably. Dense media have correlations, many‑body effects, and different transport physics. The ν = nσv estimate is best for dilute gases, low‑density plasmas, and particle beams.
7) What unit system do the results use?
All results are reported in SI: ν in s−1, λ in meters, τ in seconds, σ in m², and speed in m/s. Inputs can be entered in common alternative units.