Ion Thermal Velocity Calculator
Find a plasma's ion thermal velocity, v_ti = √(Ti·e/mᵢ), the ion counterpart to electron thermal velocity.
🚶 What is the Ion Thermal Velocity Calculator?
This ion thermal velocity calculator finds the characteristic speed of ions in a plasma from their temperature. Enter the ion temperature and choose a species, and it returns the thermal velocity in metres per second, plus the equivalent percentage of the speed of light.
Ion thermal velocity, v_ti = √(Ti·e/mᵢ), follows the exact same functional form as electron thermal velocity, only with the ion's much larger mass substituted, making ions the slower, heavier counterpart in every plasma process where mass matters.
Because ions are thousands of times more massive than electrons, they move far more sluggishly at the same temperature, setting a much slower characteristic timescale for ion-acoustic waves, ion transport, and ion collision physics compared to the fast electron-scale dynamics.
This calculator is useful for plasma physics and fusion engineering students building up the standard toolkit of plasma parameters, and anyone curious how dramatically slower ions move compared to electrons at the same temperature.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Tokamak plasma, protons
Example 2 - Solar corona, protons
Example 3 - Tokamak plasma, deuterons
❓ Frequently Asked Questions
🔗 Related Calculators
What is ion thermal velocity?
Ion thermal velocity is the characteristic speed of ions in a plasma due to their random thermal motion, the direct ion-species counterpart to electron thermal velocity. It sets the natural speed scale for ion-acoustic waves, ion collision rates, and ion transport processes.
What is the formula for ion thermal velocity?
v_ti = √(Ti·e/mᵢ), where Ti is the ion temperature in electronvolts, e is the elementary charge, and mᵢ is the ion mass. This is the exact same functional form as electron thermal velocity, with the ion mass substituted for the electron mass.
Why is ion thermal velocity so much lower than electron thermal velocity?
Thermal velocity scales as 1/√m at fixed temperature, and since a proton is about 1836 times more massive than an electron, protons move about √1836 ≈ 43 times slower than electrons at the same temperature. This is why electron-scale plasma phenomena are so much faster than the corresponding ion-scale phenomena.
Why do deuterons move slower than protons at the same temperature?
A deuteron (one proton plus one neutron) is almost exactly twice as massive as a bare proton, and since thermal velocity scales as 1/√m, a deuteron moves about 1/√2 ≈ 71% as fast as a proton at the same temperature, a roughly 29% reduction.
Where does ion thermal velocity appear in other plasma formulas?
It sets the characteristic speed scale for ion-acoustic wave speed, ion collision frequency and mean free path, and ion Landau damping. It plays the same structural role for ion-scale physics that electron thermal velocity plays for electron-scale physics.
How fast are ions in a typical fusion plasma?
In a 10 keV tokamak core, protons move at roughly 980 kilometres per second, only about 0.33% of the speed of light, in sharp contrast to electrons at the same temperature, which reach about 14% of light speed. This large gap illustrates just how much lighter and faster electrons are compared to ions.
Does ion thermal velocity depend on plasma density?
No, like electron thermal velocity, it depends only on temperature and the ion's mass, not on how many ions are present. Density instead controls related quantities like collision frequency and plasma frequency.
Why does this calculator include a deuteron preset specifically?
Deuterium is the leading fuel candidate for magnetic confinement fusion, alongside tritium, so its ion mass is a practically useful preset for fusion-relevant ion thermal velocity calculations, alongside ordinary hydrogen (protons).
Is ion thermal velocity the same as ion sound speed?
No, though they are related. Ion thermal velocity describes an individual ion species' own random motion, while ion sound (acoustic) speed describes how a compressional wave propagates through the combined electron-ion plasma, typically dominated by the electron temperature rather than the ion temperature alone.
How does ion thermal velocity relate to electron thermal velocity?
Both follow the identical functional form v_t = √(Te,i · e/m), differing only in which species' temperature and mass are used. The related <a href="/science/plasma-physics/electron-thermal-velocity-calculator/">Electron Thermal Velocity Calculator</a> computes the electron version directly.