Lecture notes by John A. Venables. Notes revised for Spring 2005.

The leak rate is composed of two elements: Q = Q_{l} + Q_{o},
where Q_{l} is the true leak rate (i.e. due to a hole in the wall)
and Q_{o} 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:

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;

If we don't have any true leaks, Q = Q

For example, if the system volume V = 50 liter, roughly 50x20x50 cm^{3},
then A is roughly 1 m^{2}. Q_{o} = qA, with a typical (good) value for q
around 10^{-8} mbar.liter.m^{-2}.s^{-1}, p_{u} =
2x10^{-10} mbar. This is a pressure to aim for after bakeout. The bakeout is
required to desorb gases, particularly H_{2}O, 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.

where C_{i} 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 C_{i} large enough so that S is not much less than S_{0};
or equivalently, if S is sufficient, we can economise on the size (S_{0}) 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

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.

By continuity, we have p_{0}S_{0} = pS = Q, the flow rate. But the flow rate is also equal
to C(p-p_{0}), as this is the definition of the conductance. So, rearranging, we can deduce that

Thus there is a big error in the measurement of p at position p_{0}, if S_{0}
is large and/or C small. One can also use the above relations to prove that
S^{-1} = ΣC_{i}^{-1} + S_{0}^{-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.