# Units and conversion factors

## SCCM - Gas flow rate

Sccm stands for Standard cubic centimeters per minute. This rate is proportional to the number of gas molecules per time unit. In this case, the standard conditions are defined as follows:

• Pressure p = 1.01325 bar = 1.01325e+5 Pa
• Temperature T = 0 °C = 273.15 K
• Volume V = 1 cm³ = 1e-6 m³

The number N of particles within one cm³ can be obtained from the ideal gas equation via:

With the Boltzmann constant kB=1.3806504e-23 J/K (see Wikipedia) we obtain N = 2.68675e+19.

Since sccm is the flow rate given per minute but we would like to have the number of particles per second, we have to divide N by 60. The result is:

## Conversion of ampere to sccm

With the electron charge e = 1.60218e-19 you get that one ampere corresponds to N = 6.24150e+18 particles per second. Divided by the above given value of particles per sccm, you get:

## Pumping Speed

### Definition

The pumping speed Sp is usually defined as pumped gas volume / time. In case of an area A with ideal pumping behaviour, the pumping speed is:

• c = Mean thermal velocity of a gas molecule,
• A = pumping area.

The mean thermal velocity can be obtained from the Maxwellian distribution by:

At a temperature of 300 K, the following velocities are obtained:

SpeciesRelative mass [u]Mean thermal velocity c [m/s]
Kr 83.8 275
Ar 39.9 399
O2 32.0 446
SiH4 32.1 445
N2 28.0 476
H2 2.02 1770
H 1.01 2510

### Select transmission probability of a pumping surface

A surface of a given area A can have the maximal pumping speed given above. The pumping speed of a real pump Sp is usually well below. This can be mended to assign a transmission probability f, which is obtained as follows:

• Sp = external pumping speed
• A = surface area
• c = gas velocity

### Example

A surface sized 16×75 cm² should have a real pumping speed of 500 l/s (0.5 m³/s) for Argon at T = 300 K. The according transmission probability calculates as follows:

## Relation between gas flow, pumping speed and pressure

The time derivative of the ideal gas equation can be written as:

In equilibrium, the left hand side is zero. The change of the particle number becomes the inflow minus the outflow:

with the following parameters:

• Fin = Inflow in sccm (for the sccm unit, see above)
• fsccm = 4.479 x 1017 particles/s (see above)
• Ap = Pumping area
• fp = Transmission probability of pumping surface
• n = Particle density

The particle density can again be expressed by using the ideal gas equation, yielding:

where Sp is the pumping speed in [m³/s]. Thus, we end up with a species-independent factor Φ which determines the relation between pressure, gas flow and pumping speed:

Typical values for Φ are listed in following.

Temperature [K] with Sp given in liters/second
300 0.001854 1.854
350 0.002163 2.163
400 0.002472 2.472
450 0.002781 2.781
500 0.003091 3.091

## Mean free path

In following, the main points from http://en.wikipedia.org/wiki/Mean_free_path are summarized.

The mean free path λ is defined as the mean travelling distance of a particle between two collisions. If a particle ensemble has a Maxwellian velocity distribution, the following relations hold:

whereof the variables mean the following:

• n = Density
• p = Pressure
• kB = Boltzman constant
• d = Diameter of the gas particle
• T = Temperature
• λ = Effective collision cross section

The latter formula shows that λ is inversely proportional to the total pressure. From http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_4.html the following numerical values are obtained for a few common gases at T=25 °C:

Species [Pa mm]Atomic diameter [nm]
Ar6.340.382
H211.060.288
N25.960.394
O26.410.380

## Temperature vs. kinetic energy

In many sources the electron temperature Te is defined via the simple relation:

.

However for a Maxwellian velocity distribution that is not correct. Here, any degree of freedom contributes to the mean kinetic energy by 1/2 kB T. Thus, in a 3D Maxwellian distribution, the following relation holds:

.

By this formula, a mean energy of 1.0 eV corresponds to a kinetic temperature of 7736.3 K (and not 11604 K as results from the wrong formula mentioned above).