dmγ v dt = q E +q v B, (1)

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1 Vacuum I Giuliano Franchetti GSI Darmstadt, D Darmstadt, Germany Abstract This paper is an introduction to the basics of vacuum. It is intended for readers unfamiliar with the topic; more advanced treatments are left to the dedicated CERN Accelerator School on vacuum in accelerators and to the specialized literature. The kinetics of gases, and gas flows through pipes and pumps are reviewed here. The topic of pumps is continued in the paper Vacuum II. 1 Vacuum, mean free path, and beam lifetime In accelerators, the beam dynamics has the general purpose of controlling the beam particles. For charged particles, this happens through the Lorenz force: dmγ v dt = q E +q v B, (1) where q is the charge of the particle and m is the mass. The time and space structure of the electric and magnetic fields E and B in an accelerator has the purpose of guiding beam particles along the design trajectory. However, in a particle accelerator there is also present a jungle of unwanted particles, which creates a background that is damaging to beam operation and experiments. These particles are referred to as residual gas. As the particles that make up the vacuum are not controlled by an electromagnetic field, they are in general treated as a gas in the thermodynamic sense, which is therefore characterized by macroscopic quantities such as the pressure P, temperature T, particle density ñ, and composition. The statistical behaviour of vacuum particles is described by kinetic theory [1, 2]. In accelerators, the aim of all of the vacuum systems is to minimize the interaction of the beam with the residual gas. Table 1 lists some typical values of particle densities for different types of vacuum. Table 1: Examples of particle densities (from Ref. [3]) Particles m 3 Atmosphere Vacuum cleaner Freeze-dryer Light bulb Thermos flask TV tube Low Earth orbit (300 km) H 2 in LHC SRS/Diamond Surface of Moon Interstellar space 10 5 One fundamental process in a gas is particle wall collisions. The velocity of a particle after interaction with a wall depends on the particle wall processes that occur. However, in any case, these collisions are responsible for creating the macroscopic pressure. The SI unit of pressure is the pascal, Pa = N/m 2. Although this is the standard unit, there are also other units in common use, such as the

2 Table 2: Classification of types of vacuum Low vacuum Medium vacuum High vacuum (HV) Ultrahigh vacuum (UHV) Extreme high vacuum (XHV) Atmospheric pressure to 1 mbar 1 to10 3 mbar 10 3 to10 8 mbar 10 8 to10 12 mbar Less than10 12 mbar bar, torr, and atmosphere, which are related to each other by 1 Pa = 10 2 mbar = Torr = atm. Table 2 shows the classification of types of vacuum in terms of pressure. Another important process in a gas is particle particle collisions. This process is the most probable process, as collisions between three or more particles are possible but rather unlikely. In elastic collisions, two fundamental quantities are preserved: energy and momentum. Particle particle collisions are responsible for creating a distribution of velocities in the particles of a gas. The temperature of the gas is related to the mean kinetic energy of a particle: for monatomic particles, 1 2 m v2 = 3 2 k BT, where k B = J K 1 is the Boltzmann constant. For example, for air, assumed to be composed mainly of nitrogen N 2, at T = 20 C, we find v 2 = 511 m/s. This is not the average velocity v a, which is slightly smaller; in fact,v a = v = 0.92 v 2 = 470 m/s. When the gas is in equilibrium, the velocity distribution of the molecules follows the Maxwell Boltzmann distribution 1 dn N dv = 4 ( ) m 3/2 v 2 e mv2 /2k B T. (2) π 2k B T This distribution is shown in Fig. 1. The most probable velocity is v α = 2k B T/m, and the average velocity is v a = (8/π)(k B T/m). As previously discussed, a gas is characterized by its pressure, 1 dn N d(v/v α) 4 π e 1 1 v a v r v α v α v/v α Fig. 1: Maxwell Boltzmann distribution temperature, and volume. When the gas is in equilibrium, i.e., in a stationary state, these quantities are related by the equation of statepv = nr 0 T, wherer 0 = 8.31 N m/(mole K). The pressure is measured in pascals and the volume in m 3, n is the number of moles (1 mole contains a number of particles equal to Avogadro s number N A ),R 0 = N A k B, and T is the absolute temperature, expressed in kelvin. In vacuum physics, the concept of the mean free path plays an important role in determining the behaviour of a gas. Consider a set of particles at rest, and let the particle density of this distribution beñ. Now consider N test particles distributed in a plane of area A. If this plane now moves orthogonally through the remainder of the gas, when the plane is displaced by a distance l, the test particles will have spanned a volumea l (see Fig. 2). On the other hand, these particles have a cross-section with respect to the remainder of the gas of σ = πr 2, where r is the radius of the cross-section. The particles present in the spanned volume will then cover (with respect to interaction with the test particles) the

3 available area by an amount A = A lñσ. Therefore the number of particles that will pass, without collision, through the portion of gas of length l is N N(A lñσ)/a. We obtain the differential equation dn = N ñσ. (3) dl By integrating this equation, we find the number of surviving test particles N(l) at a distance l. This quantity can be interpreted as the number of particles that did not collide in a distance l, and of these particles, certainly N(l) lñσ will collide with the remaining gas between l and l + l. Therefore the probability that a particle will travel for a distance l and then collide with the remaining gas in the interval betweenlandl+dl isdp(l) = (N(l)/N 0 )ñσdl, wheren 0 is the initial number of particles in the plane considered. The average distance that a test particle will travel between two collisions is then λ = 0 ldp(l) = 1/σñ. Note that in this argument it is assumed that the particles of the gas are at rest, which is not true in a real gas. Maxwell computed the effect of the relative motion of particles of the same species, and this adds a factor to the formula, which becomes λ = 1 2σñ. (4) Fig. 2: Conceptual discussion of the mean free path For example, if we consider air att = 20 C and a pressurep = 1 atm, we find from the equation of state thatñ = P/(k B T) = atoms/m 3. The diameterdof an air molecule is m [2], from which σ = πd 2 = m 2. Therefore the mean free path is λ = m. Table 3 shows the relation between the mean free path and the pressure and viscosity for N 2 att = 20 C, takingd = m (obtained from measurements of viscosity [4]). In synchrotrons and colliders, the circulating beam is disturbed by the presence of residual gas. The importance of the residual gas is of high relevance for circular accelerators, where the beam circulates for a large number of turns. The situation here is of a beam formed by a number of ions of a certain species, which while travelling may collide with residual gas molecules, become ionized, and be lost because they now have the wrong charge state with respect to the machine rigidity. The number of particles surviving a path of length l is N(l+ l) = N(l) N(l)σñ l.

4 Table 3: Mean free path, particle density, and viscosity P ñ ρ η λ (Pa) (m 3 ) (kg m 3 ) (m 2 s 1 ) (m) Atmospheric pressure Primary vacuum High vacuum UHV XHV Assuming that all beam particles have the same longitudinal velocityv 0, we can change the space variable to a time variable, and so we find N(t+ t) = N(t) N(t)σñv 0 t, and therefore dn dt = N(t)σñv 0. As the particle density of the residual gas isñ = P/(k B T), this equation becomes dn dt = N(t)σPv 0 k B T. The time constant of this equation defines the beam lifetime τ = k BT σpv 0. For example, in the LHC the residual gas contains, among other species, molecules of hydrogen H 2 in thermal equilibrium with cold surfaces at a temperature of 5 K. The cross-section for interaction with the energetic protons circulating at 7 TeV sets the above cross-section to σ = m 2. Therefore, for a beam lifetimeτ = 100 h, we find a requirement for the pressure of H 2 inside the vacuum chamber of P = Pa. Clearly, the corresponding requirements for other molecules making up the gas will differ. The pressures foreseen in the LHC for the various components of the gas in the vacuum are shown in Table 4 [5]. Table 4: Pressure of various components of the vacuum in the LHC for a target beam lifetimeτ = 100 h Gas Nuclear scatteringσ (cm 2 ) Gas density (m 3 ) Pressure (Pa) at 5 K H He CH H 2 O CO CO A special note has to be made here about electrons. Electrons can also be considered as part of the residual gas. Owing to their lightness, electrons are strongly influenced by the electric field of the circulating beam in the accelerator. The electrons in the vacuum chamber are often referred to as the

5 electron cloud. Reviews of the complex processes of build-up of electrons and their interplay with the circulating bunches can be found in Refs. [6, 7, 8]. As previously mentioned, the molecules of the vacuum are subject to collisions with one another, generating a Maxwellian distribution of velocities. One relevant feature of a gas is the impingement rate J. This quantity measures the number of molecules that hit a unit area of a surface per unit of time. For a Maxwell Boltzmann velocity distribution, we find [9, 2] J = 1 4ñv a, wherev a = (8/π)(k B T/m) is the average velocity of a gas molecule. 2 Gas flows in pipes The properties of a gas with respect to transport through pipes depend very much on the spatial scale considered, which is set by the size of the pipe or vessel. If the size D of the vessel is much larger than the mean free path λ, then the gas will be dominated by collisions between the gas molecules, and the effect of molecule wall collisions will be negligible. In this regime, the gas will effectively behave as a continuum, as a local change in properties will be propagated as a wave (like sound in air at atmospheric pressure and room temperature). If instead the size of the vessel is much smaller than the mean free path, the molecules will collide mainly with the walls of the vessel. In this case continuum processes are not possible, and the motion of the particles of the gas is dominated by particle wall collisions. This regime is referred to as the molecular regime. The Knudsen number K n characterizes the type of regime in which a gas is found: 2.1 Throughput and conductances K n < 0.01 Continuous regime, K n = λ D 0.01 < K n < 0.5 Transitional regime, K n > 0.5 Molecular regime. The creation of a vacuum requires the extraction of air from a vessel, which is done via a system of pumps connected to vessels through pipes. We now present a short discussion of gas flow in pipes. The flow of a gas through a pipe can be expressed in terms of the number of particles per second dn/dt passing through a reference surface across the pipe. On the other hand, measurements of gas flow are better expressed in terms of macroscopic quantities characterizing the thermodynamic state of the gas. If a gas flows in a pipe at a velocity v across an area A, the rate of particles per second dn/dt is dn dt = ñva = P k B T va = P dv k B T dt = 1 d PV. (5) k B T dt Note that the quantity Q = PV here is a product of a pressure and a volume. Equation (5) can be re-expressed so that the particle flow is dn dt = 1 k B T Q. The quantity Q is called the throughput and is expressed in Pa m 3 /s. In the absence of adsorption or desorption, the rate at which particles pass through a cross-section of a pipe does not change along the pipe, and hence the throughput does not change either. It is now useful to introduce the concept of the conductance of a pipe. If there is a flow of particles in a pipe from a section 1 with pressure P 1 to a section 2 with pressure P 2 (< P 1 ), the relation between

6 the throughput and the difference in pressure between the two sections can be expressed as C = Q P 1 P 2. (6) Here C is the conductance; this is a geometric property of the pipe and of the gas flow. The units of C are m 3 /s. For a composite structure formed from several pipes, it is possible to prove, based on the assumption that the throughput is conserved, that for N pipes each having conductance C i, the conductance C of the composition in series is given by 1 N C = 1. C i For the composition of pipes in parallel, the conductance of the composite structure is i=1 C = N C i. i=1 Note that the law of composition of pipes in series can be used to obtain an indication of how the conductance of a long pipe scales with the length. In fact, if C 1 is the conductance of one pipe, by connecting N of these pipes in series we find a conductance C = C 1 /N. Therefore it is expected that the conductance scales as the inverse of the length of the pipe. Up to this point, the concept of conductance, alias the relation between the throughput and the pressure gradient, has been discussed without considering the physical behaviour of the gas itself. It is therefore to be expected that a pipe will exhibit different conduction properties according to whether the gas is in a molecular or a continuum regime. 2.2 Molecular flow Particle wall collisions characterize the molecular flow of a gas in a pipe. For example, in a vessel of diameterd = 0.1 m, the molecular regimek n > 0.5 is obtained for a pressure ofp < mbar (at room temperature). Owing to imperfections in the surface of the pipe, the gas molecule wall collision becomes a very complex process. However, experimental evidence found by Knudsen [10] suggests that a particle hitting a surface at an arbitrary angle with respect to the normal to the surface will emerge after the interaction with an angle not correlated with the incident direction, with a probability that obeys the cosine law. This law states that the probability that a particle will have a direction within a solid angle dω is dp = 1 π cosθdω. See Fig. 3 for a schematic illustration of the Knudsen law. As a consequence, a particle launched inside a pipe will have a certain probability α of passing through the pipe, and hence a probability 1 α of returning back. Here we also assume that no processes of capture by surfaces take place (i.e., no sticking). If N 1 particles per unit time enter the pipe, then N 1 α will pass through and N 1 (1 α) will return back. Therefore, in the molecular regime, there are always two opposite fluxes of particles coexisting inside a pipe. The simplest example of a conductance is the conductance of an aperture. Consider the situation shown in Fig. 4. Here, two vessels are joined together and connected by an aperture of areaa. In the left vessel, the pressure is P u, and in the right vessel the pressure is P d. The gas has the same temperature T in both vessels. The rate of flow of particles from the vessel at pressure P u to the other vessel is I u = J u A. The gas throughput is obtained as Q u = P u Vu, where V u is the volumetric flow of gas

7 θ dω Particle Surface Fig. 3: Description of the cosine law from the vessel at P u. The volumetric flow is obtained as V u = I u /ñ u, and therefore Q u = P u I u /ñ u. The same argument can be repeated for the gas flow from the right vessel to the left. Therefore the total throughput across the aperture is Q = Q u Q u = P u I u /ñ u P d I d /ñ d. Here we also recall that I = AJ = Añv a /4, and thereforei u /ñ u = I d /ñ d = v a A/4. P u,t P d,t I u I d Fig. 4: Schematic illustration of the conductance of an aperture 1 Finally, we obtain Q = A 1 4 v a(p u P d ); comparing this formula with Eq. (6), we find that the conductance of the aperture is 1 R 0 T C a = A 2π M, wherem is the molar mass of the gas. The conductances of all types of pipe are usually referred to the conductance of an aperture C a. For example, for a long pipe the conductance is [10, 2] C L = 4 d 3L C a. Note that C L scales as the inverse of the pipe length L, as already inferred from the law of composition of pipes in series. Observation. The expressions for the conductance in these formulas depend on the geometrical properties of the pipes, and on the type of molecules that the gas is composed of. This property stems from the flow regime, which is molecular here Continuum flow regime For K n < 0.01, there are approximately 100 collisions between 1 particles on average before a particle collides with a wall. In this case perturbations ofñ,p,t propagate through a continuum medium. In this

8 regime, collisions between particles create the phenomenon of viscosity, which affects the behaviour of gases in pipes. If a fluid propagates into a pipe at a small speed, the fluid is in the laminar regime and the motion of the particles is parallel to the pipe axis (see Ref. [11] for extensive discussions of flow in pipes). The velocity of the fluid is zero at the walls of the pipe, and is largest at the centre. If the velocity of the fluid is increased beyond a certain threshold, the fluid changes its properties and its motion becomes turbulent. The quantity that defines the transition from the viscous laminar regime to the turbulent regime is the Reynolds number Re = ρvd h, η where ρ is the density of the fluid in kg/m 3, v is the average velocity of the gas in m/s, and D h is the hydraulic diameter in metres, given by D h = 4A/B, where A is the cross-sectional area and B is the perimeter of the pipe. Finally,η is the viscosity in Pa s. IfRe < 2000, the fluid is in the laminar regime, whereas forre > 3000 the fluid is in the turbulent regime. The Reynolds number can be expressed in terms of the throughput as Re = 4 Q B M R 0 T For air (N 2 ) at T = 20 C, taking η = Pa s, we find Re = k R Q/B, where k R = s/(m 2 Pa). Therefore the transition to turbulent flow (Re 2000) takes place at a transition throughput Q T = 24d, where Q T is in mbar l/s. For a pipe of diameter d = 25 mm, we obtain Q T = 600 mbar l/s, which corresponds at atmospheric pressure to a speed of v = 1.22 m/s. 1 η Laminar regime One source of complexity in characterizing a fluid flow in a pipe is the compressibility of the fluid. However, it is possible to prove that if the velocity of the fluid has a Mach numberma < 0.2, then a fluid in a pipe behaves as if it were incompressible, i.e., the Bernoulli equation takes a form very similar to that of the equation for incompressible fluids. It is important to characterize pipes in terms of the conductance when the flow regime is laminar. The conductance can be given under some assumptions about the laminar flow: (1) the fluid is considered incompressible; (2) the flow is fully developed, that is, it reaches a velocity distribution which does not change along a pipe of constant cross-section (see [2] for further discussion); (3) the particle motion is laminar (Re < 2000); and (4) the velocity of the fluid at the pipe walls is zero. Under these assumptions, we find that the throughput is given by Q = C(P u P d ), where the conductance is now given by C = πd4 128ηL P, where P = (P u + P d )/2. Here P u is the pressure upstream, P d is the pressure downstream, D is the diameter of the pipe,lis its length, andη is the fluid viscosity. This finding can be obtained directly from the Hagen Poiseuille equation [11]. We conclude that in a continuum laminar regime, the conductance depends on the pressure at which the fluid transport occurs Turbulent regime In the turbulent regime, for a long pipe, the expression for the throughput becomes R0 T D h Q = A Pu M f D L 2 Pd 2, where f D is the Darcy friction factor, a quantity which depends nonlinearly on the Reynolds number [12].

9 3 Sources of vacuum degradation After the air has been pumped out from a chamber, the vacuum can degrade because new molecules may enter the vessel. One phenomenon that contributes to spoiling the vacuum is the evaporation and condensation of liquids present on surfaces (because those surfaces have not been cleaned). If there is a spot of liquid on a surface in a high-vacuum vessel, particles evaporate from the surface of the liquid, increasing the pressure in the vessel. At the same time, vapour particles in the vessel impinge on the liquid surface, bringing molecules into the liquid. There are therefore two fluxes of particles: one of evaporation and the other of condensation. The process of condensation depends on the pressure in the vessel, whereas the evaporation process depends only on the temperature. If one waits long enough, the pressure in the vessel will stabilize to the saturated vapour pressurep E, the value of which is given by the Clausius Clapeyron equation [13]. Therefore we conclude that the fluxes of evaporation J E and condensation J C are equal. Hence the evaporation flux is 1 J E = P E N a 2πR0 MT. A spot of liquid in a vacuum chamber will therefore emit a flux of particles J E. This is equivalent to a throughput into the vessel equal to Q E = AJ E k B T, where A is the surface area of liquid exposed to the vacuum in the vessel. Another source of vacuum degradation is the phenomenon of outgassing. Outgassing is the passage of gas from the walls of a vessel or pipe into the vacuum; it is the release of gas molecules previously adsorbed on surfaces. The locations where gas molecules are captured (and later released) are called adsorption sites. When molecules hit one of these sites, there is a certain probability of capture (see the discussion of sticking coefficients in Ref. [2]). Clearly, when an adsorption site is occupied, it cannot accommodate another molecule, and the amount of gas adsorbed by a surface is proportional to the fraction of adsorption sites occupied, Θ. Typically, a molecule captured on an adsorption site remains there for a time called the mean stay time τ d before being released. Therefore, in a time intervaldt, a fraction Table 5: Mean stay time at T = 20 K E d (kcal/mole) Case τ d (s) 0.1 Helium H 2 physisorption Ar, CO, N 2, CO 2 physisorption Weak chemisorption H 2 chemisorption ( half a week) 30 CO/Ni chemisorption ( 50 years) ( half the age of the Earth) 150 O/W chemisorption (larger than the age of the Universe) Θdt/τ d of the total number of sites present on the surface will become free and release gas. The time evolution of Θ, neglecting readsorption, then follows the equation dθ dt = Θ τ d. The mean stay time is strongly related to the surface temperature and the binding energy of the gas molecule. As shown in [14], we find that τ d = τ 0 exp[e d /(R 0 T)], where τ 0 = s. Table 5 shows the mean stay times of molecules with different binding energiese d. The number of adsorption sitesn s is equal to A , where A is the surface area in cm 2. The outgassing, in terms of throughput,

10 then becomes Q G = k B T N sθ τ d. A third source of vacuum degradation is the presence of leaks. If a small hole (i.e., a small channel) is created in a vessel containing a high vacuum, then the throughput of gas from the outside to the inside of the vessel is given by Q C a P 0, where P 0 is the atmospheric pressure (or, more generally, the outside pressure). In the case of air, composed mainly of N 2, at room temperature T = 293 K, for a small hole of diameter d = m we find a conductance C a = m 3 /s. Therefore the throughput is Q L = Pa m 3 /s = mbar l/s. For a hole with d = 10 9 m, Q L mbar l/s, and therefore 1 cm 3 of air needs 317 years to enter the vessel. Leaks in a vacuum system can be distinguished into the following classes according to the throughput [15]: very tight if Q L < 10 6 mbar l/s, tight if 10 6 < Q L < 10 5 mbar l/s, and with leaks if 10 5 < Q L < 10 4 mbar l/s. Finally, among the sources of vacuum degradation, we should mention the phenomenon of permeation: this process happens when gases are adsorbed on the material of the walls, diffuse through the walls, and are later desorbed into the vessel. The throughput depends on the surface temperature, the thickness and composition of the walls, and the composition of the gas. A discussion of this effect has been given in Ref. [2]. 4 Pipes and pumps The creation of a vacuum is achieved via pumps that are connected to vessels via pipes. An ideal pump behaves in such a way that all of the gas particles that enter the pump inlet never return. The pumping speed of a pump is referred to the volumetric speed S of the gas through the inlet, and is expressed in units of m 3 /s. If the inlet of a pump has diameter D, then the gas flow through the ideal pump aperture is given by I = JD 2 π/4, where I is the number of molecules per second passing through the inlet surface. The volumetric pumping speed is then S = dv/dt = I/ñ = v a D 2 π/16. For example, for N 2 att = 20 C, ifd = 0.1 m we find a volumetric pumping speeds 0 = 0.92 m 3 /s. Of course, this pumping speed is an ideal value and the effective pumping speed of a real pump will be smaller. The situation is summarized in Fig. 5. The throughput in the section AA is Q = PS; it is the same throughout the pipe, Q = PS Q 0 = P 0 S 0 A A Vessel Ideal Pump Fig. 5: Effective pumping speed S and its relation to the pumping speed S 0 of an ideal pump and is equal to the throughput of gas entering the pump, Q 0 = P 0 S 0. At the same time, Q = C(P P 0 ), wherec is the conductance of the pipe connecting the vessel to the pump. From these relations, we find 1 S = 1 C + 1 S 0,

11 which gives the effective pumping speed of an ideal pump. For example, taking a long pipe, C L = 4D/(3L)C a, and noting that the volumetric speed of the ideal pump iss 0 = C a, we find S = S 0 1+3L/(4D). Therefore pumps should be placed as close as possible to vessels in order to exploit the nominal pumping capability of the pump. 5 Making the vacuum: pump-down time and ultimate pressure We now put together all the different sources of vacuum degradation, which are summarized in Fig. 6. The quantity of gas inside a vesselqvaries according to Hole P, V Q L Pump C Q S Q G Q E S S 0 Fig. 6: Summary of sources of vacuum degradation Q = Q T Q S, where Q S is the throughput removed by the pump, and Q T is the total throughput entering the vessel because of sources of vacuum degradation. This equation becomes dp dt V = Q T Q S, (7) where V is the volume of the vessel. If the sources of vacuum degradation and the pumping speed are independent of the pressure in the vessel, the evolution of the pressure, i.e., the solution of Eq. (7), is P = P u +(P 0 P u )e (S/V)t, where P 0 is the initial pressure at t = 0, and P u = Q T /S. The time constant of the process τ pd = V/S is the pump-down time. From this equation, we see that for t the pressure in the vessel converges to P u, which is called the ultimate pressure. This pressure is determined by the equilibrium between the throughput of the sources of degradation and the throughput removed by the pump. The vacuum requirements vary from project to project. Table 6 lists examples of the vacuum requirements in the SNS, LHC, and FAIR [16, 5, 17]. 6 Pumps The control and creation of a vacuum is performed via a system of pumps. Pumps are classified according to their principle of operation.

12 Table 6: Examples of vacuum pressures in some accelerators SNS LHC FAIR Front end DTL CTL SCL HEBT Ring RTBR HEBT SIS to Torr Torr Torr < 10 9 Torr < Torr 10 8 Torr 10 7 Torr mbar 10 9 mbar mbar 6.1 Positive-displacement pumps The principle of this type of pump is based on the displacement of a volume V of gas from the vessel to the outside. This process implies that the action of the pump is to seal a volume V and then to open it outside the vessel. Clearly, when the volumev of gas is displaced and opened, in order for the gas to flow out, the pressure in the volume has to be larger than the outside pressure P outlet. Therefore it is always necessary that the pump should compress the volumev so that the initial pressure of the gas, equal to the pressure P inlet in the vessel, becomes larger than P outlet. If the gas pressure in the vessel is so low that the compression performed by the pump is not enough to reach a pressure larger than P outlet, then gas will be transported from the outlet to the inlet! This reasoning shows that a positive-displacement pump will be able to extract gas only if the ratio P outlet /P inlet can be reached by the compression process Piston pump In this type of pump, a piston moves in a cylinder, creating a volume that varies from V min to V max. The ideal volumetric pumping speed is S 0 = V max N c, where N c is the number of cycles per second of the piston. When the piston creates the minimum volume V min, the pressure inside the chamber is equal to P outlet. Next, the piston expands the volume and, for an isothermal process, when the volume is V i = P outlet V min /P inlet, the gas will start flowing from the vessel into the piston chamber. The volumetric amount of gas which enters the piston chamber is V max V i, and this is the amount of gas later expelled. The effective pumping speed is therefore S = N c (V max V i ), that is, ( S = S 0 1 P ) outlet V min. P inlet V max Figure 7 shows a plot of the volumetric pumping speed versus the ratio of inlet to outlet pressure. The above discussion shows that the pump stops pumping at a limiting inlet pressure that depends on the compression performed by the pump. The higher the compression, the lower the limiting pressure. If the thermodynamic process characterizing the compression is of a different nature, the dependence of S on P inlet /P outlet will be different: for example, for an isentropic compression we find S = S 0 [1 (P outlet /P inlet ) 1/γ V min /V max ]. The result is, however, that in any case there exists a limiting pressure at which the pump stops functioning. Note that this feature determines an ultimate pressure in a vessel + pump system independently of the presence of sources of vacuum degradation. In general, from the point of view of the gas flow, a pump is an object that absorbs a throughput P inlet S 0 through the inlet but, owing to other uncontrolled processes, is also subject to a back-flow Q b. The total throughput is then Q = P inlet S 0 Q b. In order to characterize the back-flow, it is useful to define the zero-load compression ratio. This quantity is obtained as follows: the inlet is closed and the pressure P i0 is measured at the entrance of the pump. This is the inlet pressure at zero load. The

13 S/S Vmin V max V min 1 V max P inlet P outlet Fig. 7: Pumping speed of a piston pump zero-load compression ratio is therefore defined as K 0 = P 0 /P i0, which is a quantity measurable as a function of the outlet pressurep 0 and the pumping speed. As the throughput at the inlet Q is zero in this special case, we find that the back-flow throughput Q b is equal tos 0 P 0 /K 0. For technical reasons, the compression ratio of a pump cannot be made arbitrarily large. Hence other techniques have been developed to improve the lower limit on pressure Rotary pumps These pumps are characterized by a pumping speed S = m 3 /h. The lowest pressure achievable reaches mbar for one-stage pumps, and 10 3 mbar for two-stage pumps. An illustration of this type of pump is shown in Fig. 8. These pumps perform the compression via the rotation of Fig. 8: Illustration of a rotary pump a rotor, which moves a vane that compresses the gas to a high compression ratio. However, during the compression, there may be a component G of the gas for which the partial pressurep G becomes too high, resulting in condensation of that component. This problem is avoided by injecting a non-condensable gas during the compression phase. By doing so, the maximum partial pressurep G can be lowered below the condensation point. This process is called gas ballast [18].

14 6.1.3 Liquid-ring pumps Liquid-ring pumps contain a rotating impeller, the axis of which is off centre relative to the casing (see the illustration in Fig. 9). Centrifugal force pushes the liquid against the casing, creating a liquid ring, which seals the impeller, creating vanes that enclose a variable volume. The typical pumping speed is m 3 /h, and gas can be pumped at pressures from 1000 mbar to 33 mbar. Fig. 9: Illustration of a liquid-ring pump A problem related to this type of pump arises when the gas expands in the cavities and the pressure becomes lower than the vapour pressure P s of the liquid (typically water). At that point the liquid boils, but the motion of the impeller then compresses the gas, causing the vapour bubbles to implode, creating shock waves in the liquid medium. This phenomenon is called cavitation, and it causes anomalous functioning of the pump, setting a limit on the pressure at which the pump can work. At T = 15 C, the vapour pressure of water is 33 mbar, which is a typical limiting pressure for this type of pump because of the need to avoid cavitation. For a review of liquid-ring pumps, see Ref. [19] Dry vacuum pumps: the Roots pump These pumps contain two rotating elements which are separated from the case and from each other by approximately 1 mm. The elements rotate in opposite directions, and as a result of their motion they create a moving vane, which brings a gas volume from the inlet to the outlet. The pumping speed is m 3 /h and the operating pressure is mbar. An illustration of this type of pump is shown in Fig. 10. For a general reference, see Ref. [20]. Fig. 10: Illustration of a Roots pump

15 7 Kinetic vacuum pumps A different class of pumps, based on a different principle, is that of kinetic vacuum pumps. These pumps give a momentum to gas particles so that they are moved from the inlet to the outlet. 7.1 Molecular-drag pump This pump is based on the molecular-drag effect. If a gas molecule hits a surface, it will emerge from it in some direction with a probability determined by the cosine law. However, if the surface is in motion with a tangential velocity, the molecule will emerge from reflection from the moving surface with an additional velocity component equal to the speed of the surface. This effect is called the drag effect, and it can be used to create a pump [21]. Consider an long, open channel of transverse cross-section h w closed by a surface moving with a velocity U (Fig. 11 (top)). A gas molecule hitting the moving Fig. 11: Principle of the drag pump (top), and schematic illustration of gas flow in the channel of a drag pump (bottom) surface will acquire a velocity U owing to the drag effect. When the molecule hits any of the other walls of the channel, it will be reflected on average orthogonally to that surface, by the cosine law (see Fig. 11 (bottom)). Therefore a volumetric flow S 0 = whu/2 into the channel is established. Taking the back-flow into account, the pumping speed at the inlet is S i = S 0 K K 0 1 K 0, wherek 0 = P outlet /P inlet,0 is the zero-load compression ratio, andk = P outlet /P inlet. It can be shown that the compression ratio at zero load takes the form K 0 = exp(s 0 /C), where C is the conductance of the channel. For a long tube, S 0 /C = 3UL/(4hv a ), where v a is the average thermal velocity and L is the length of the channel. For example, for L = 250 mm and h = 3 mm, we have S 0 /C > 10 and K 0 1, so that we retrieve the form ( S = S 0 1 K ). K 0 An illustration of the functioning of a molecular-drag pump is shown in Fig. 12. The typical pumping speed of a molecular-drag pump is l/s, at an operating pressure of Pa. The ultimate pressure reachable is Pa. For references, see Refs. [9, 22]. 7.2 Turbomolecular pump The turbomolecular pump is based on the rotation of a set of blades at a velocityu. The blades are tilted at an angle φ. The gas molecules enter the inlet, and the motion of the blades imparts a momentum to

16 Fig. 12: Illustration of the functioning of a molecular-drag pump the gas. The situation is illustrated in Fig. 13. When particles leave the set of rotating blades, the gas has acquired a rotational velocity, which makes it difficult to use a subsequent series of rotating blades (as they would move at a similar velocity to the gas). For this reason, a stator consisting of blades at rest is placed after the rotating blades so as to remove the rotational component of the particle velocity. By using this strategy, several rotor + stator blocks can be arranged in a multistage pumping structure. Fig. 13: Principle of the turbomolecular pump The probability that molecules entering the pump will be pushed out is W = Ṅ/(J ia), where J i is the rate of impingement of molecules on the inlet surface. The maximum probability W max is found whenp outlet = P inlet. Analogously to the case of the molecular-drag pump, the pumping probabilityw is given by W = W max K 0 K K 0 1, where K 0 is the compression at zero load. In Ref. [23, 9], it is shown that K 0 g(φ)exp(u/v a ), and

17 therefore K0 exp( M ), where M is the molar mass of the gas. This means that different gas species have different pumping probabilities. In addition, the maximum probability Wmax is proportional to U/va M. Therefore the maximum pumping speed Smax = Wmax J is independent of the molecular mass of the gas, and S K0 K =. Smax K0 1 See Ref. [22, 2] for more details. Figure 14 (left) shows an example of the maximum compression as a function of the foreline pressure for several gas species [2]. The typical pumping speed of these types of Fig. 14: Maximum compression rate as a function of the foreline pressure (left), and pumping speed (right) (Balzers Pfeiffer) pump is l/s (Fig. 14 (right)), and the ultimate pressure reachable is Pa. Acknowledgements The author thanks Maria Cristina Bellachioma for comments on this paper and corrections. References [1] E.H. Kennard, Kinetic Theory of Gases, with an Introduction to Statistical Mechanics (McGrawHill, New York, 1938). [2] J.M. Lafferty, Foundations of Vacuum Science and Technology (Wiley, New York, 1988). [3] R.J. Reid, Vacuum science and technology in accelerators, Cockcroft Institute Lectures, [4] R.C. Weast, Ed., Handbook of Chemistry and Physics (CRC Press, Boca Raton, FL, 1975). [5] LHC, LHC Design Report, Technical report, [6] G. Rumolo, F. Ruggiero, and F. Zimmermann, Phys. Rev. ST Accel. Beams 4(1) (2001) [7] F. Zimmermann, Phys. Rev. ST Accel. Beams 7 (2004)

18 [8] M.A. Furman and M.T.F. Pivi, Phys. Rev. ST Accel. Beams 5 (2002) [9] A. Chambers, Modern Vacuum Physics (CRC Press, Boca Raton, FL, 2004). [10] M. Knudsen, Ann. Phys. 28 (1909) 75. [11] R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena (Wiley, New York, 2007). [12] S.E. Haaland, J. Fluids Eng. 105 (1983) 89. [13] C.H. Collie, Kinetic Theory and Entropy (Longman, London, 1982). [14] P.A. Redhead, J. Vac. Sci. Technol. A 13 (1995) [15] K. Zapfe, Leak detection, CERN Accelerator School, CERN (2007), p [16] J.Y. Tang, SNS vacuum instrumentation and control system, Proc. 8th International Conf. on Accelerator and Large Experimental Physics Control Systems, San Jose, CA, 2001, p [17] A. Kraemer, The vacuum system of FAIR accelerator facility, Proc. EPAC 2006, Edinburgh, 2006, p [18] W. Gaede, Z. Naturforsch. 2a (1947) [19] H. Bannwarth, Liquid Ring Vacuum Pumps, Compressors and Systems (Wiley-VCH, Weinheim, 2005). [20] A.P. Troup and N.T.M. Dennis, J. Vac. Sci. Technol. A 9 (1991) [21] W. Gaede, Ann. Phys. 346(7) (1913) [22] J.F. O Hanlon, A User s Guide to Vacuum Technology (Wiley, Hoboken, NJ, 2003). [23] C.H. Kruger and A.H. Shapiro, in Rarefied Gas Dynamics, Ed. L. Talbot (Academic Press, New York, 1960), p. 117.