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:
The number N of particles within one cm³ can be obtained from the ideal gas equation via:
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:
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:
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:
The mean thermal velocity can be obtained from the Maxwellian distribution by:
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 |
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:
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:
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:
The particle density can again be expressed by using the ideal gas equation, yielding:
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:
Typical values for Φ are listed in following.
Temperature [K] | with S_{p} 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 |
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:
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] |
---|---|---|
Ar | 6.34 | 0.382 |
H_{2} | 11.06 | 0.288 |
N_{2} | 5.96 | 0.394 |
O_{2} | 6.41 | 0.380 |
In many sources the electron temperature T_{e} 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 k_{B} 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).