How this instrument works
Number density counts discrete things — molecules, ions, electrons, dopant atoms — per unit of volume, and nothing else. Divide however many particles occupy a region, N, by the volume that holds them, V, and the units fall out on their own: particles per cubic metre. It carries no mass and no charge information; a reading of 10²³ per cubic metre says only that a cubic metre of space contains that many countable objects, whatever they happen to be.
The formula n = N ⁄ V is deliberately unweighted: it returns one average value for the whole volume, treating every corner of the container as equally populated. Real gases and plasmas are rarely that tidy — a candle flame's ion population is far denser near the wick than at its edge, and Earth's atmosphere thins with altitude. Where the local value matters, physicists reach for the differential form n = dN ⁄ dV, sampling a region small enough that the count inside it is locally uniform; this instrument returns the bulk average, which is exactly what a total count and a container volume can honestly support.
This is the n in the kinetic-theory form of the ideal gas law, p = nkT, and it is the figure printed on a semiconductor datasheet as a doping concentration, usually quoted per cubic centimetre and converted here to per cubic metre. It is easy to confuse with mass density, kilograms per cubic metre — the two agree only after multiplying by one particle's mass, ρ = nm. Treating a doping figure given in atoms per cubic centimetre as if it were already atoms per cubic metre is a common slip, and it is off by a factor of a million, not ten.
- Enter the Number of particles — the total count occupying the region, from a gauge reading, a dopant dose, or a simulation output.
- Enter the Volume those particles occupy; switch its unit to litres or cubic centimetres if that matches your source data.
- Read off Number density, particles ⁄ m³ — the instrument converts any volume unit to cubic metres before dividing.
- Compare the result against a known reference, such as air at standard conditions, to sanity-check the order of magnitude.
Worked example — 10²⁰ particles in a one-litre chamber
A vacuum engineer pumps down a one-litre plasma-etch chamber and an ion gauge reports 1×10²⁰ residual gas molecules still inside — thin, but far from empty. Entering particleCount = 1×10²⁰ and volume = 0.001 m³ (one litre) into n = N ⁄ V gives n = 1×10²⁰ ⁄ 0.001 = 1×10²³ particles per cubic metre.
That is roughly 265 times more dilute than the room air surrounding the chamber, whose number density sits near 2.65×10²⁵ particles per cubic metre — the NIST Loschmidt constant for air at 0 degrees Celsius and 100 kPa. It is a useful sanity check: the etch process only behaves predictably once the chamber sits well below atmospheric density, and the same division applies unchanged to a doping figure or a stellar number density, only the labels on N and V change.
Questions
What does number density actually count?
Whatever discrete objects are specified — molecules, ions, electrons, dopant atoms — per cubic metre of space. It is a raw count divided by volume, with no reference to mass, charge, or the particle's identity; a gauge reading, a simulation, or a doping-dose calculation can each supply the N that goes into n = N ⁄ V.
How is number density different from mass density?
Mass density, in kilograms per cubic metre, weighs each particle; number density, in particles per cubic metre, only counts them. The two connect through one particle's mass, m, as ρ = nm. Helium and neon at the same number density have very different mass densities, because a neon atom outweighs a helium atom by roughly five times.
Why does converting cubic centimetres to cubic metres trip people up?
Because the factor is a million, not a thousand. One cubic metre holds 10⁶ cubic centimetres, so a dopant concentration given as atoms per cubic centimetre must be multiplied by 1×10⁶ before it is atoms per cubic metre. Mixing up the two units is one of the most common doping-density errors, and it changes the answer by six orders of magnitude.
Is this the same n used in the ideal gas law?
Yes — in its kinetic-theory form, p = nkT, n is exactly this number density, with k the Boltzmann constant and T the temperature. The more familiar pV = nRT works in moles instead of a raw particle count; multiply this instrument's result by the volume and divide by Avogadro's constant, 6.02214076×10²³ per mole, to get moles.
Does number density stay the same throughout a real container?
Usually not. The formula n = N ⁄ V returns a single average for the whole volume, but a flame, a plasma discharge, or a planetary atmosphere is denser in some regions than others. Capturing that variation needs the differential form n = dN ⁄ dV evaluated over a small local volume, rather than one bulk figure for the whole container.
What if I only have a concentration in moles per litre?
Convert it first: multiply the molar concentration by Avogadro's constant, 6.02214076×10²³ per mole, then by 1000 to turn litres into cubic metres, and the result is particles per cubic metre. Treat that value as N ⁄ V from there — it is what feeds directly into kinetic-theory formulas such as p = nkT.