By measuring the variation of the current as the distance between the plates is increased, Townsend, to whom we owe much of our knowledge on this subject, determined the values of α for different values of X and for different pressures for air, hydrogen and carbonic acid gas (Phil. Mag. [6], 1, p. 198). Since λ varies inversely as the pressure, we see that α may be written in the form pφ(X/p) or α/X = F(X/p). The following are some of the values of α found by Townsend for air.
| X Volts per cm. | Pressure .17 mm. | Pressure .38 mm. | Pressure 1.10 mm. | Pressure 2.1 mm. | Pressure 4.1 mm. |
| 20 | .24 | ||||
| 40 | .65 | .34 | |||
| 80 | 1.35 | 1.3 | .45 | .13 | |
| 120 | 1.8 | 2.0 | 1.1 | .42 | .13 |
| 160 | 2.1 | 2.8 | 2.0 | .9 | .28 |
| 200 | 3.4 | 2.8 | 1.6 | .5 | |
| 240 | 2.45 | 3.8 | 4.0 | 2.35 | .99 |
| 320 | 2.7 | 4.5 | 5.5 | 4.0 | 2.1 |
| 400 | 5.0 | 6.8 | 6.0 | 3.6 | |
| 480 | 3.15 | 5.4 | 8.0 | 7.8 | 5.3 |
| 560 | 5.8 | 9.3 | 9.4 | 7.1 | |
| 640 | 3.25 | 6.2 | 10.6 | 10.8 | 8.9 |
We see from this table that for a given value of X, α for small pressures increases as the pressure increases; it attains a maximum at a particular pressure, and then diminishes as the pressure increases. The increase in the pressure increases the number of collisions, but diminishes the energy acquired by the corpuscle in the electric field, and thus diminishes the change of any one collision resulting in ionization. If we suppose the field is so strong that at some particular pressure the energy acquired by the corpuscle is well above the value required to ionize at each collision, then it is evident that increasing the number of collisions will increase the amount of ionization, and therefore α, and α cannot begin to diminish until the pressure has increased to such an extent that the mean free path of a corpuscle is so small that the energy acquired by the corpuscle from the electric field falls below the value when each collision results in ionization.
The value of p, when X is given, for which α is a maximum, is proportional to X; this follows at once from the fact that α is of the form X·F(X/p). The value of X/p for which F(X/p) is a maximum is seen from the preceding table to be about 420, when X is expressed in volts per centimetre and p in millimetres of mercury. The maximum value of F(X/p) is about 1⁄60. Since the current passing between two planes at a distance l apart is i0εαl or i0εXlF(X/p), and since the force between the plates is supposed to be uniform, Xl is equal to V, the potential between the plates; hence the current between the plates is i0εV·F(X/p), and the greatest value it can have is i0εV/60. Thus the ratio between the current between the plates when there is ionization and when there is none cannot be greater than εV/60, when V is measured in volts. This result is based on Townsend’s experiments with very weak currents; we must remember, however, that when the collisions are so frequent that the effects of collisions can accumulate, α may have much larger values than when the current is small. In some experiments made by J. J. Thomson with intense currents from cathodes covered with hot lime, the increase in the current when the potential difference was 60 volts, instead of being e times the current when there was no ionization, as the preceding theory indicates, was several hundred times that value, thus indicating a great increase in α with the strength of the current.
Townsend has shown that we can deduce from the values of α the mean free path of a corpuscle. For if the ionization is due to the collisions with the corpuscles, then unless one collision detaches more than one corpuscle the maximum number of corpuscles produced will be equal to the number of collisions. When each collision results in the production of a corpuscle, α = 1/λ and is independent of the strength of the electric field. Hence we see that the value of α, when it is independent of the electric field, is equal to the reciprocal of the free path. Thus from the table we infer that at a pressure of 17 mm. the mean free path is 1⁄325 cm.; hence at 1 mm. the mean free path of a corpuscle is 1⁄19 cm. Townsend has shown that this value of the mean free path agrees well with the value 1⁄21 cm. deduced from the kinetic theory of gases for a corpuscle moving through air. By measuring the values of α for hydrogen and carbonic acid gas Townsend and Kirby (Phil. Mag. [6], 1, p. 630) showed that the mean free paths for corpuscles in these gases are respectively 1⁄11.5 and 1⁄29 cm. at a pressure of 1 mm. These results again agree well with the values given by the kinetic theory of gases.
If the number of positive ions per unit volume is m and v is the velocity, we have nue + mve = i, where i is the current through unit area of the gas. Since nue = i0εnx and i = i0εnl, when l is the distance between the plates, we see that
nu / mv = εnx / (εnl − εnx),
| n | = | v | · | εnx | . |
| m | u | εne − εnx |
Since v/u is a very small quantity we see that n will be less than m except when εnl - εnx is small, i.e. except close to the anode. Thus there will be an excess of positive electricity from the cathode almost up to the anode, while close to the anode there will be an excess of negative. This distribution of electricity will make the electric force diminish from the cathode to the place where there is as much positive as negative electricity, where it will have its minimum value, and then increase up to the anode.
The expression i = i0εαl applies to the case when there is no source of ionization in the gas other than the collisions; if in addition to this there is a source of uniform ionization producing q ions per cubic centimetre, we can easily show that