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:

$N = \dfrac{p\,V}{k_B\,T}$

With the Boltzmann constant k_B=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:

$1\,{\rm sccm} \approx 4.47796\cdot 10^{17}\,\rm s^{-1}\,{\rm [Particles/second]}$

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:

$1\,{\rm A} \approx 13.9397\,{\rm sccm}$

Pumping Speed

Definition

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

$S_\textrm{p} = \dfrac{A\,c}{4}$

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

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

$c = \sqrt{\dfrac{8\,k\,T}{m\,\pi}} = 145.5\dfrac{\rm m}{\rm s}\sqrt{\dfrac{T[\rm K]}{m[\rm u]}}$

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

Species	Relative mass [u]	Mean thermal velocity c [m/s]
Kr	83.8	275
Ar	39.9	399
O*2*	32.0	446
SiH*4*	32.1	445
N*2*	28.0	476
H*2*	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 S_p is usually well below. This can be mended to assign a transmission probability f, which is obtained as follows:

$f = \dfrac{4\,S_\textrm{p}}{A\,c}$

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

Example

A surface sized 16x75 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:

$f = \dfrac{4\,\cdot\,0.5}{0.16\,\cdot\,0.75\,\cdot\,398.5} \approx 0.0418$

Relation between gas flow, pumping speed and pressure

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

$\frac{d}{dt}p\,V=k\,T\frac{d}{dt}N$

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

$\frac{d}{dt}N = F_\textrm{in}f_\textrm{sccm} - \dfrac{c}{4}A_\textrm{p}f_\textrm{p}n\stackrel{!}{=} 0$

with the following parameters:

• F_in = Inflow in sccm (for the sccm unit, see above)
• f_sccm = 4.479 x 10¹⁷ particles/s (see above)
• A_p = Pumping area
• f_p = Transmission probability of pumping surface
• n = Particle density

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

$\dfrac{{k\,T\,F_\textrm{in}f_\textrm{sccm}}}{p} = \dfrac{c}{4}A_\textrm{p}f_\textrm{p} = S_\textrm{p}$

where S_p 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:

$p = \Phi\dfrac{F_\textrm{in}}{S_\textrm{p}}\qquad \Phi = k\,T\,f_\textrm{sccm}$

Typical values for Φ are listed in following.

Temperature [K]	$\Phi$	$\Phi$ 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:

$\lambda = \dfrac{1}{\sqrt{2}\,n\,\sigma} = \dfrac{k_\textrm{B}\,T}{\sqrt{2}\pi\,d^2\,p}$

whereof the variables mean the following:

• n = Density
• p = Pressure
• k_B = 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	$p \times \lambda$ [Pa mm] Atomic diam	ter [nm]
Ar	6.34	0.382
H2	11.06	0.288
N2	5.96	0.394
O2	6.41	0.380

Temperature vs. kinetic energy

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

$E = k_B T_e$.

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

$E = 3/2 k_B T$.

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).