NAN/PHY/MSE 546 (Venables) Sect 2.2

## Notes for NAN/PHY/MSE 546 Sect 2.2 (Venables)

### © Arizona Board of Regents for Arizona State University and John A. Venables

Lecture notes by John A. Venables. Notes revised for Spring 2005. Latest version 26 August 2012, reformatted.

## 2.2 Vacuum Concepts

### System Volumes, Leak Rates and Pumping Speeds

The system to be pumped has a System Volume, V, measured in liters, at pressure p (mbarr or Torr). It is pumped with a pump, with a Pumping Speed, S liter/s. There is a Leak Rate Q into the system, expressed in (mbar or Torr)liters/s. The Pump-down equation, which you can think about, is then:

pS = -Vdp/dt + Q.

The leak rate is composed of two elements: Q = Ql + Qo, where Ql is the true leak rate (i.e. due to a hole in the wall) and Qo is a virtual leak rate. A virtual leak is one which originates inside the system volume; it can be caused by degasssing from the walls, or from trapped volumes, which are to be strongly avoided.

The solution of the pump-down equation has:

i) a short time limit: p = p0exp(-t/τ),

with τ = V/S, where the leak rate is negliglible. This stage will be essentially complete in 10τ. Typical values 10 x 50 liter/ 50 liters/s = 10s. It isn't quite this short in practice, but it is short;

ii) a long time limit: pu, the ultimate pressure = Q/S.

If we don't have any true leaks, Q = Qo, which depends on the surface area, material and the treatment.

For example, if the system volume V = 50 liter, roughly 50x20x50 cm3, then A is roughly 1 m2. Qo = qA, with a typical (good) value for q around 10-8 mbar.liter.m-2.s-1, pu = 2x10-10 mbar. This is a pressure to aim for after bakeout. The bakeout is required to desorb gases, particularly H2O, from the walls.

Note: In doing problems on the pump-down equation, some students used it too uncritically, or deduced solutions which went against their experience in the laboratory. For example, to deduce, via point i) above, that you can get down to 10-6 mbar, say, in a minute or so, is not correct. It is however correct to deduce that in that time the term -Vdp/dt becomes less than +Q; but Q itself varies (decreases) with time, as the walls outgas. This means that for almost all UHV situations we are interested in the long time limit of the equation, but with variable Q, depending on the bakeout and other treatments of the vacuum system.

### The Idea of Conductance

The pumping speed of the pump is reduced by the high impedance, or low conductance, of the pipework between the pump and the vacuum chamber. But, of course, large pipes increase both the system volume and the internal surface area. So, one needs to take care in the design of the system, to avoid obvious pitfalls. Typically pipes have a conductance Ci, and these are in series (draw). Then with the pump speed as S0, we have the effective pumping speed S given by

S-1 = Σ Ci-1 +S0-1,

where Ci are measured in liters/s. In this case, inverse conductances and pumping speeds therefore add as add as resistances in series.

Thus we need to choose Ci large enough so that S is not much less than S0; or equivalently, if S is sufficient, we can economise on the size (S0) of the pump. As with all design problems, we need to have enough in hand so that our solution works routinely and is reliable. On the other hand, over-provision is (very) expensive. We will consider actual values of C in the next section (handout).

Sometimes, if high pumping speed is essential, or if the geometrical aspect ratio is unfavorable (as in the accelerator examples given in sect 2.1), we would use multiple pumps distributed along the length of the apparatus. In this case the conductances are distributed in parallel, and

S = ΣSi, C = ΣCi.

Whether this is a good solution should be clear from geometry. Obviously, a solution involving one UHV pump is simpler, if possible. Sometimes we use more than one pump because different pumps have different characteristics, as described in the next section.

### Measurement of System Pressure

One can measure pressure at various points in the system volume, and these pressures will not necessarily be the same. Manufacturers of complete systems have a tendency, understandable of course, to put the pressure guage right above the pump, where the pressure (p0) is lowest. But the pressure p in the system volume will be worse because of the intervening conductance C. (draw this).

By continuity, we have p0S0 = pS = Q, the flow rate. But the flow rate is also equal to C(p-p0), as this is the definition of the conductance. So, rearranging, we can deduce that

p - p0 = p0S0/C and hence p = p0( 1 + S0/C).

Thus there is a big error in the measurement of p at position p0, if S0 is large and/or C small. One can also use the above relations to prove that S-1 = ΣCi-1 + S0-1.

Note that in general both S and C can be functions of pressure. In the molecular flow regime, at low p where the gas molecules only collide with the walls, and where we are not near the ultimate pressure of the pump, then they are, in fact, both constant.

Continue to section 2.3