This may be written :
Xe\
XcX
,dx.
where F
Thus if a is the chance that an electron may produce one elec- tron per unit path, since X for the same gas is inversely propor- tioned to the pressure p, a will be of the form nf( ' J : and since n is proportional to the number of molecules per unit volume, a may be written as pf ( ) . When the spheres described round the elec- trons with radius d do not overlap, n will also be proportional to the number of electrons in the molecule. The greatest value of d is e 2 / 2 Q; hence if D, the distance between two electrons, is greater than e 2 / 2 Q> there can be no overlapping; if D is less than this quan- tity there may be overlapping; since the value of d diminishes as the kinetic energy of the electron increases, n for very fast elec- trons will be proportional to the number of electrons in the molecule.
Some of the electrons will by adhesion to a neutral molecule become negative ions. Let the chance of an electron doing so while passing over i centimetre be yp. If N be the number of electrons
per c.c. at a place fixed by the coordinate x, then T. +^(NU) =rate
of increase of number of ions per c.c., where U is the velocity of the electron parallel to x.
The number of electrons passing through the unit of area in unit time is NU. The new electrons produced by the passage of them through the unit volume is NUa, while NVyp will disappear; hence :
^+j x (NU)=NU(a- 7 )+2 (19),
where q is the ionization due to external sources; when things are in a steady state dN/dt = o, and the solution of the equation, when the electric field may be taken as constant from one electrode to another, is:
NU-Cet^* ^-T.
a-jp
Most of the experiments on this subject have been made with- out external ionization; a supply of electrons has been obtained from the cathode, either by raising it to incandescence or by expos- ing it to ultra-violet light. In such cases g = o, and
NU = . ' (a Y * ) * (20),
where to is the number of electrons emitted in unit time from the cathode. Townsend, and Townsend and Kirkby have determined the value of a yp for various gases and over a considerable range of pressure. A series of these values for air are given in the follow- ing table:
X= volts per cm.
Pressure
(mm.)
17
38
I-IO
2-1
4-1
20
24
40
65
34
80
i-35
i'3
45
13
120
1-8
2-0
i-i
42
13
160
2-1
2-8
2-O
9
28
2OO
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
4OO
5-o
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-i
640
3-25
6-2
10-6
10-8
8-9
It will be seen that, when X is given, the increase in the number of electrons reaches a maximum for a particular pressure. From gen- eral reasoning this must be so, for if p o there will be no collisions to make fresh electrons, and if p is infinite the free path of the elec- trons will be so small that they cannot acquire sufficient energy to
(X\ ) , and y does not depend
upon p, ayp will be a maximum when
(X\ /X\ X
) y=f'{ -r )r- This equation determines
hence the critical pressure will be proportional to the electric force.
At this critical pressure XeX bears to Q a ratio which depends upon the way in which the chance of an electron ionizing by a colli- sion depends upon the energy of the electron. If, for example, the chance were independent of this energy, provided the energy were greater than Q, the maximum current would be when XeX = Q; this relation would not hold for other and more probable laws con- necting ionizing power with the energy, but we should expect that for any such law the ratio of XeX to Q would neither be very large nor very small.
Since the electrons cannot begin to ionize until their energy is equal to Q, and to attain this energy they must pass through a distance Q/Xe, it is cjear that we ought in such an equation as (19) to write x Q/Xe in place of x. If V is the potential difference between the plates, X = VJd, so that x-Q/Xe = xdQ/V if Q is measured in volts. Thus in finding the current between two elec- trodes we must, if we use equation (19), write dl i Hj instead of d.
Partz (Verh. d. Deutsch. Phys. Gesell. xiy, p. 60) has shown that theory and experiment agree better by this change.
Spark Discharge. The production of ions by moving electrons will not by itself explain why a current of electricity can be main- tained through a gas by an electric field when all other sources of ionization are excluded. The electrons are continually being driven towards the anode, and unless there is some source of supply near the cathode the ionization and therefore the current will rapidly come to an end. One way in which the electrons could be supplied by the action of the electric field would be by the positive ions which strike against the cathode communicating so much energy to the anode that it is raised to incandescence. Since an incandescent metal gives out large quantities of electrons there will be a continuous supply of electrons from the cathode, which will ionize the gas and produce fresh positive ions to strike against the cathode and keep it hot. This is what happens in the arc discharge when the cathode is kept in a state of incan- descence by the discharge. In this case there is a large amount of energy put into the arc. There are, however, other forms of continuous discharge where the cathode does not become incan- descent, so that there must be other ways in which the supply of electrons is maintained. From what we know about ions there are several ways in which this might occur.
It has been found by experiment (Fiichtbauer, Ann. der Phys. 23, p. 301 (1907); Saxen, Ann. der Phys. 38, p. 319 (1912); Baerwald, Ann. der Phys. 41, p. 643 (1913); 42, p. 1207 (1913) that electrons are emitted from metals when these are bombarded by high- speed positive ions even though the metal is not raised to incan- descence. According to Baerwald the emissions of electrons from metals bombarded by positive hydrogen atoms does not become appreciable until these have an amount of energy exceeding that represented by 900 volts. We know too that, when the electric discharge passes through a gas, radiation capable of ionizing a gas through which it passes, or of ejecting electrons from a metal on which it falls, is an accompaniment of the discharge. Again posi- tive ions ionize a gas through which they pass. This was shown by McClelland, who found that the relation between the potential difference and the current from a hot wire anode surrounded by gas at low pressure was represented 'by a curve like that shown in
40
160 200
FIG. 6
320 360
fig. 6. The hot wire furnishes positive ions as well as negative ones, and the curve shows that fresh ions are formed when the potential difference is greater than about 200 volts. This is a much greater potential difference than that needed to produce ionization by electrons, but it is smaller than would be expected by the consid- erations given above. As it requires less work to eject an electron from a metal than from a molecule, we should expect that if 200 volts ions could eject electrons from a gas through which they pass they would be able to do so from a metal against which they strike, but from Baerwald's experiments much more energy than 200 volts
