2.2.1 MOLECULAR VELOCITIES

The well-known kinetic theory of gases provides us with an atomistic picture of the state of affairs in a confined rarefied gas (Refs. 3, 4). A fundamental assumption is that the large number of atoms or molecules making up the gas are in a continuous state of random motion that is intimately dependent on their temperature. As they move, the gas particles collide with each other as well as with the walls of the confining vessel. Just how many molecule–molecule or molecule–wall impacts occur depends on the concentration or pressure of the gas. In the perfect or ideal gas approximation there are no attractive or repulsive forces between molecules. Rather, they may be considered to behave like independent elastic spheres separated from each other by distances that are large compared to their size. The net result of the continual elastic collisions and exchange of kinetic energy is that a steady-state distribution of molecular velocities emerges given by the celebrated Maxwell–Boltzmann formula

     (2-1)

This centerpiece of the kinetic theory of gases states that the fractional number of molecules f(ν), where ν is the number per unit volume in the velocity range ν to ν + dv, is related to their molecular weight (M) and absolute temperature (T). In this formula the units of the gas constant R are on a per-mole basis. Among the important implications of Eq. 2-1, which is shown plotted in Fig. 2-1, is that molecules can have neither zero nor infinite velocity. Rather, the most probable molecular velocity of the distribution is realized at the maximum value of f(ν) and can be calculated from the condition that df(ν)/dv = 0. Since the net velocity is always the resultant of three rectilinear components νx, νy, and νz, one (or even two) but, of course, not all three of these may be zero simultaneously. Therefore, a similar distribution function of molecular velocities in each of the componentdirections can be defined; i.e.,

     (2-2)

and similarly for the y and z components.

A number of important results emerge as a consequences of the foregoing equations. For example, the most probable (νm), average and mean square velocities are given, respectively, by

     (2-3a)

     (2-3b)

     (2-3c)

These velocities, which are noted in Fig. 2-1, simply depend on the molecular weight of the gas and the temperature. In air at 300 K, for example, the average molecular velocity is 4.6 × 104 cm/s, which is almost 1030 miles per hour. However, the kinetic energy of any collection of gas molecules is solely dependent on temperature. For a mole it is given by with ½RT partitioned in each of the three coordinate directions.

2.2.2 PRESSURE

Momentum transfer from the gas molecules to the container walls gives rise to the forces that sustain the pressure in the system. Kinetic theory shows that the gas pressure, P, is related to the mean-square velocity of the molecules and thus, alternately to their kinetic energy or temperature. Thus,

     (2-4)

where NA is Avogadro’s number. From the definition of n, n/NA is the number of moles per unit volume and therefore, Eq. 2-4 is an expression of the perfect gas law.

Pressure is the most widely quoted system variable in vacuum technology and this fact has generated a large number of units that have been used to define it under various circumstances. Basically, two broad types of pressure units have arisen in practice. In what we shall call the scientific system or coherent unit system (Ref. 4), pressure is defined as the rate of change of the normal component of momentum of impinging molecules per unit area of surface. Thus, the pressure is defined as a force per unit area, and examples of these units are dynes/cm2 (CGS) or newtons/meter2 (N/m2) (MKS). Vacuum levels are now commonly reported in SI units or pascals; 1 pascal (Pa) = 1 N/m2. Historically, however, pressure was, and still is, measured by the height of a column of liquid, e.g., Hg or H2O. This has led to a set of what we shall call practical or noncoherent units, such as millimeters and microns of Hg, torr, and atmospheres, which are still widely employed by practitioners as well as by equipment manufacturers. Definitions of some units together with important conversions include

The mean distance traveled by molecules between successive collisions, called the mean-free path, λmfp, is an important property of the gas that is dependent on the pressure. To calculate λmfp we note that each molecule presents a target area πd2c to others where dc is its collision diameter. A binary collision occurs each time the center of one molecule approaches within a distance dc of the other. If we imagine the diameter of one molecule increased to 2dc while the other molecules are reduced to points, then in traveling a distance λmfp the former sweeps out a cylindrical volume πd2c λmfp. One collision will occur under the condition that πd2c λmfp n = 1. For air at room temperature and atmospheric pressure, λmfp ∼ 500 Å, assuming dc 5 Å.

A molecule collides in a time given by λmfp/ν, and under the above conditions, air molecules make about 1010 collisions per second. This is why gases mix together rather slowly even though the individual molecules are moving at great speeds. The gas particles do not travel in uninterrupted linear trajectories. As a result of collisions, they are continually knocked to and fro, executing a zigzag motion and accomplishing little net movement. Since n is directly proportional to P, a simple relation for ambient air is

     (2-5)

with λmfp given in centimeters and P in torr. At pressures below 10−3 torr, λmfp is so large that molecules effectively collide only with the walls of the vacuum chamber.

2.2.3 GAS IMPINGEMENT ON SURFACES

A most important quantity that plays a role in both vacuum science and vapor deposition is the gas impingement flux Φ. It is a measure of the frequency with which molecules impinge on or collide with a surface, and should be distinguished from the previously discussed molecular collisions in the gas phase. The number of molecules that strike an element of...