NAN/PHY/MSE 546 (Venables) Sect 2.1

Notes for NAN/PHY/MSE 546 Sect 2.1 (Venables)

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

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Lecture notes by John A. Venables. Notes revised for Spring 2005: Latest version 25 August 2012, reformatted.

2. Surfaces in Vacuum: Ultrahigh Vacuum Techniques

Refs: Prutton, Introduction, p5-10 and Luth, Panel I, p6-17 give an introduction to the 'feel' of surface science apparatus and experiments. Woodruff and Delchar devote Chap 1 to this topic amongst others. It is probably not possible to do justice to this topic in a book: it would take too much space, but Luth has made a brave attempt within the allotted number of pages. More detailed books exist, which you may wish to look at, including A. Roth, Vacuum Technology (3rd Edn, 1990), and J.F. O'Hanlon, A Users Guide to Vacuum Technology (1989); I'd be interested to know which book of this type you found actually useful, except for reference.

Manufacturers’ catalogues are useful, assuming that you know that they are attempting to get you to buy something (in the long run). Although all such catalogues provide detailed information about the products, the Leybold-Heraeus catalogue has traditionally included a tutorial section which helps one understand what the products are doing, and what choices the purchaser needs to make. Relatively small performance improvements in vacuum components can cause quite a commercial stir. So one always needs to consider what the latest model, or flavor of the month, is really doing.

I will emphasize the physical principles on which these devices are based, in the hope that these do not change too fast. Also, you may not have to buy anything. Surface Science is now a fairly mature discipline, so there will be kit lying around. But you need to know what it can, and cannot, do. In practice there is no substitute for visiting, and then working in, a Surface Science laboratory, each of which will have its own practices and recipes. So we will arrange a lab visit for those of you not already working in this environment, before too long.

2.1 Kinetic Theory Concepts

  • Arrival Rate of Atoms at a Surface
  • This arrival rate R, or Flux F, is related to the the molecular density n, the mean speed of the molecules (v or c-bar) and the pressure p, via the standard kinetic theory formulae
    R or F = nv/4 per unit area = p/(2πmkT)1/2;
    you may also need n = p/kT and v (c-bar) = (8kT/πm)1/2, which are required to connect the two versions of the above formula.

    So let us work through an example to find the molecular density n, the mean free path λ, and the monolayer arrival time, τ. Take the residual gas in a vacuum system, which is often a mixture of CO, H2 and H2O. Settle on Carbon Monoxide, CO, with molecular weight 28. Then

    m = 28 x 1.6605 x 10-27 kg; k = 1.3807 x 10-23 J/K;
    and T = 293 K (UK, if you're lucky), or 300K (Arizona, ditto).
    We shall also need the molecular diameter of CO, 0.316 nm (approx).

    The main problem with all this is the question of units, especially of pressure, where the SI unit is the Pascal (Nm-2). 1 bar = 105 Pascal, and Gauges are now calibrated in millibar: 1 mbar = 100 Pa. The older unit Torr (mm Hg), after the founder of Vacuum, Signor Torricelli. 1 Torr = 1.333 mbar (760 Torr = 1013 mbar = 1 atmosphere).

  • The Molecular Density
  • Note added 13 may '02: There is a typographic error (in the first printing of the book) in the second equation for n (below), which should have (p/T) on the right hand side, not /T. Thanks to Karsten Pohl for spotting this mistake.

    At low pressures n = Ap. With n per cm3 and p in mbar, we have the constant A = (100)/(kT x 106). This gives n = 7.2464 x 1018 (p/T). Don't forget T is in degrees (K), you wouldn't would you? Roth has suitable diagrams (diagram 41) and tables (diagrams 42 and 43) which spell this out. A typical number to get hold of is that at 10-6 mbar there are 2.42 x 1010 molecules/cm3 in Arizona and 2.47 x 1010 in the UK. Just checking: its the temperature, stupid. There are still lots of molecules around, even in the best vacuum.

  • The Mean Free Path
  • The mean free path between molecular collisions in the gas phase is inversely related to the density n and the molecular cross section proportional to σ2. The proportionality constant f in the equation λ = f/nσ2 has been much discussed, and is given by Roth as 1/π√2 = 0.225. Thus for CO, λ = 2.25 x 1012/n and the mean free path is 10-6 mbar is 9.1-9.3 m at room temperature.

    This is much greater than the typical dimensions of an Ultra-high Vacuum chamber, operating, say, below 10-9 mbar. Thus the gas molecules will travel from wall to wall. or from wall to sample, without intermediate collisions. Higher pressure gas reactors, operating at 10-3 mbar and above, start to run into gas collison and diffusion effects, but the UHV community largely ignore this, except for particle accelerators. At a large installation, such as CERN or FermiLab, the accelerated particles are constrained to miss the walls, but of course they hit the gas molecules.One of the challenging aspects of the (late) Superconducting Supercollider was how to design a toroidal pipe some 80 km long and say 15 cm in diameter with a vacuum everywhere better than 10-12 mbar. The LEP ring at CERN (only 20 km in circumference) has a similar specification. The vacuum design has to be taken rather seriously!

  • The Monolayer Arrival Time
  • If N0 is the number of atoms in a monolayer (the ML unit) then Rτ = N0. We have already had that R = Cp with C = (2πmkT)-1/2. Now we need the conversion from mbar to Pascal, T and other constants. For CO, R in atoms.m-2.s-1, and p in mbar, C is then 2.876 x 1024 for T = 300K, or R = 2.876 x 1018 atoms.m-2.s-1 at 10-6 mbar, i.e. of order 3 x 1014 at 10-10 mbar, a typically (good) UHV pressure.

    The definition of N0 requires above all consistency. It can be defined in terms of the substrate, the deposit or the gas molecules, but it must be done consistently, and the ML unit needs definition, essentially in each paper or description: there is no accepted standard.

    For example, consider condensation on Ag(111), with a (1x1) structure. It is perfectly reasonable to define N0 as the number of Ag atoms per unit area. With the bulk lattice parameter a0 = 0.4086 nm, the surface mesh area (draw) is (√3/2)a2, where the surface lattice constant a = a0/√2. Thus N0 = 1.383 x 1019 atoms.m-2. With this definition, τ = 4.81 s at 10-6 mbar (CO) and 13.4 hours at 10-10 mbar. This is, of course, the reason for doing experiments in UHV conditions; only at low pressures can one maintain a clean surface for long enough to do the experiment.

    However, the above definition of the monolayer arrival time only makes sense if we have a well defined substrate. If the substrate N0 is ill-defined or irrelevant (e.g. the inside of a stainless steel vacuum chamber, or an incommensurate deposit), then a definition in terms of the deposit makes more sense. In our case we might use a close-packed monolayer of condensed CO; with a = 0.316 nm, the corresponding value of τ = 4.02 s at 10-6 mbar (CO) and 11.2 hours at 10-10 mbar. Although these values are of the same order, they are not the same. Thus for quantitative work, either define the ML unit explicitly, or work with a value of R expressed in atoms.m-2.s-1, rather than in ML/s. Note also that had the deposit been something other than CO, and we wanted to track the result in terms of pressure, then we have to use the correct m and T in the constant C.

    To summarize: 1) the density n is still high even in UHV; 2) the mean free path λ >> apparatus dimensions; 3) the monolayer arrival time τ is greater than 1 hour only for p < 10-9 mbar, and for good measure 4) the monolayer (ML) unit, if used, has to be defined consistently.

    Continue to section 2.2

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