Chapter III: Electric Conduction Through Gases (3)

_Calculation of the Mass of the Ions at Low Pressures._--Although at ordinary pressures the ion seems to have a very complex structure and to be the aggregate of many molecules, yet we have evidence that at very low pressures the structure of the ion, and especially of the negative one, becomes very much simpler. This evidence is afforded by determination of the mass of the atom. We can measure the ratio of the mass of an ion to the charge on the ion by observing the deflections produced by magnetic and electric forces on a moving ion. If an ion carrying a charge e is moving with a velocity v, at a point where the magnetic force is H, a mechanical force acts on the ion, whose direction is at right angles both to the direction of motion of the ion and to the magnetic force, and whose magnitude is evH sin [theta], where [theta] is the angle between v and H. Suppose then that we have an ion moving through a gas whose pressure is so low that the free path of the ion is long compared with the distance through which it moves whilst we are experimenting upon it; in this case the motion of the ion will be free, and will not be affected by the presence of the gas.

Since the force is always at right angles to the direction of motion
of the ion, the speed of the ion will not be altered by the action of
this force; and if the ion is projected with a velocity v in a
direction at right angles to the magnetic force, and if the magnetic
force is constant in magnitude and direction, the ion will describe a
curve in a plane at right angles to the magnetic force. If [rho] is
the radius of curvature of this curve, m the mass of the ion,
mv^2/[rho] must equal the normal force acting on the ion, i.e. it must
be equal to Hev, or [rho] = mv/He. Thus the radius of curvature is
constant; the path is therefore a circle, and if we can measure the
radius of this circle we know the value of mv/He. In the case of the
rapidly moving negative ions projected from the cathode in a highly
exhausted tube, which are known as _cathode rays_, the path of the
ions can be readily determined since they make many substances
luminous when they impinge against them. Thus by putting a screen of
such a substance in the path of the rays the shape of the path will be
determined. Let us now suppose that the ion is acted upon by a
vertical electric force X and is free from magnetic force, if it be
projected with a horizontal velocity v, the vertical deflection y
after a time t is 1/2 X et^2/m, or if l is the horizontal distance
travelled over by the ion in this time we have since l = vt,

Xe l^2
y = 1/2 -- ---.
m v^2

Thus if we measure y and l we can deduce e/mv^2. From the effect of the
magnetic force we know e/mv. Combining these results we can find both
e/m and v.

The method by which this determination is carried out in practice is
illustrated in fig. 13. The cathode rays start from the electrode C in
a highly exhausted tube, pass through two small holes in the plugs A
and B, the holes being in the same horizontal line. Thus a pencil of
rays emerging from B is horizontal and produces a bright spot at the
far end of the tube. In the course of their journey to the end of the
tube they pass between the horizontal plates E and D, by connecting
these plates with an electric battery a vertical electric field is
produced between E and D and the phosphorescent spot is deflected. By
measuring this deflection we determine e/mv^2. The tube is now placed
in a uniform magnetic field, the lines of magnetic force being
horizontal and at right angles to the plane of the paper. The magnetic
force makes the rays describe a circle in the plane of the paper, and
by measuring the vertical deflection of the phosphorescent patch at
the end of the tube we can determine the radius of this circle, and
hence the value of e/mv. From the two observations the value of e/m
and v can be calculated.

Another method of finding e/m for the negative ion which is applicable
in many cases to which the preceding one is not suitable, is as
follows: Let us suppose that the ion starts from rest and moves in a
field where the electric and magnetic forces are both uniform, the
electric force X being parallel to the axis of x, and the magnetic
force Z parallel to the axis of z; then if x, y, are the co-ordinates
of the ion at the time t, the equations of motion of the ion are--

d^2x dy
m ---- = Xe - He --,
dt^2 dt

d^2y dx
m ---- = He --.
dt^2 dt

The solution of these equations, if x, y, dx/dt, dy/dt all vanish when
t = 0, is

Xm / / e \ \
x = ---- {1 - cos( -- Ht ) }
eH^2 \ \ m / /

Xm /e / e \ \
y = ---- {-- Ht - sin( -- Ht ) }.
eH^2 \m \ m / /

These equations show that the path of the ion is a cycloid, the
generating circle of which has a diameter equal to 2Xm/eH^2, and rolls
on the line x = 0.

Suppose now that we have a number of ions starting from the plane x =
0, and moving towards the plane x = a. The particles starting from x =
0 describe cycloids, and the greatest distance they can get from the
plane is equal to the diameter of the generating circle of the
cycloid, i.e. to 2Xm/eH^2. (After reaching this distance they begin to
approach the plane.) Hence if a is less than the diameter of the
generating circle, all the particles starting from x = 0 will reach
the plane x = a, if this is unlimited in extent; while if a is greater
than the diameter of the generating circle none of the particles which
start from x = 0 will reach the plane x = a. Thus, if x = 0 is a plane
illuminated by ultra-violet light, and consequently the seat of a
supply of negative ions, and x = a a plane connected with an
electrometer, then if a definite electric intensity is established
between the planes, i.e. if X be fixed, so that the rate of emission
of negative ions from the illuminated plate is given, and if a is less
than 2Xm/eH^2, all the ions which start from x = 0 will reach x = a.
That is, the rate at which this plane receives an electric charge
will be the same whether there is a magnetic field between the plate
or not, but if a is greater than 2Xm/eH^2, then no particle which
starts from the plate x = 0 will reach the plate x = a, and this plate
will receive no charge. Thus the supply of electricity to the plate
has been entirely stopped by the magnetic field. Thus, on this theory,
if the distance between the plates is less than a certain value, the
magnetic force should produce no effect on the rate at which the
electrometer plate receives a charge, while if the distance is greater
than this value the magnetic force would completely stop the supply of
electricity to the plate. The actual phenomena are not so abrupt as
this theory indicates. We find that when the plates are very near
together the magnetic force produces a very slight effect, and this an
increase in the rate of charging of the plate. On increasing the
distance we come to a stage where the magnetic force produces a great
diminution in the rate of charging. It does not, however, stop it
abruptly, there being a considerable range of distance, in which the
magnetic force diminishes but does not destroy the current. At still
greater distances the current to the plate under the magnetic force is
quite inappreciable compared with that when there is no magnetic
force. We should get this gradual instead of abrupt decay of the
current if some of the particles, instead of all starting from rest,
started with a finite velocity; in that case the first particles
stopped would be those which started from rest. This would be when a =
2Xm/eH^2. Thus if we measure the value of a when the magnetic force
first begins to affect the leak to the electrometer we determine
2Xm/eH^2, and as we can easily measure X and H, we can deduce the value
of m/e.

By these methods Thomson determined the value of e/m for the negative ions produced when ultra-violet light falls on a metal plate, as well as for the negative ions produced by an incandescent carbon filament in an atmosphere of hydrogen (_Phil. Mag._ [5], 48, p. 547) as well as for the cathode rays. It was found that the value of e/m for the negative ions was the same in all these cases, and that it was a constant quantity independent of the nature of the gas from which the ions are produced and the means used to produce them. It was found, too, that this value was more than a thousand times the value of e/M, where e is the charge carried by an atom of hydrogen in the electrolysis of solutions, and M the mass of an atom of hydrogen. We have seen that this charge is the same as that carried by the negative ion in gases; thus since e/m is more than a thousand times e/M, it follows that M must be more than a thousand times m. Thus the mass of the negative ion is exceedingly small compared with the mass of the atom of hydrogen, the smallest mass recognized in chemistry. The production of negative ions thus involves the splitting up of the atom, as from a collection of atoms something is detached whose mass is less than that of a single atom. It is important to notice in connexion with this subject that an entirely different line of argument, based on the Zeeman effect (see MAGNETO-OPTICS), leads to the recognition of negatively electrified particles for which e/m is of the same order as that deduced from the consideration of purely electrical phenomena. These small negatively electrified particles are called corpuscles. The latest determinations of e/m for corpuscles available are the following:--

Observer. e/m.

Classen (_Ber. deut. phys. Ges._ 6, p. 700) 1.7728 X 10^7
Bucherer (_Ann. der Phys._, 28, p. 513) 1.763 X 10^7

It follows from electrical theory that when the corpuscles are moving with a velocity comparable with that of light their masses increase rapidly with their velocity. This effect has been detected by Kauffmann (_Gott. Nach._, Nov. 8, 1901), who used the corpuscles shot out from radium, some of which move with velocities only a few per cent less than that of light. Other experiments on this point have been made by Bucherer (_Ann. der Phys._ 28, p. 513).

_Conductivity Produced by Ultra-Violet Light._--So much use has been made in recent times of ultra-violet light for producing ions that it is desirable to give some account of the electrical effects produced by light. The discovery by Hertz (_Wied. Ann._ 31, p. 983) in 1887, that the incidence of ultra-violet light on a spark gap facilitates the passage of a spark, led to a series of investigations by Hallwachs, Hoor, Righi and Stoletow, on the effect of ultra-violet light on electrified bodies. These researches have shown that a freshly cleaned metal surface, charged with negative electricity, rapidly loses its charge, however small, when exposed to ultra-violet light, and that if the surface is insulated and without charge initially, it acquires a positive charge under the influence of the light. The magnitude of this positive charge may be very much increased by directing a blast of air on the plate. This, as Zeleny (_Phil. Mag._ [5], 45, p. 272) showed, has the effect of blowing from the neighbourhood of the plate negatively electrified gas, which has similar properties to the charged gas obtained by the separation of ions from a gas exposed to Rontgen rays or uranium radiation. If the metal plate is positively electrified, there is no loss of electrification caused by ultra-violet light. This has been questioned, but a very careful examination of the question by Elster and Geitel (_Wied. Ann._ 57, p. 24) has shown that the apparent exceptions are due to the accidental exposure to reflected ultra-violet light of metal surfaces in the neighbourhood of the plate negatively electrified by induction, so that the apparent loss of charge is due to negative electricity coming up to the plate, and not to positive electricity going away from it. The ultra-violet light may be obtained from an arc-lamp, the effectiveness of which is increased if one of the terminals is made of zinc or aluminium, the light from these substances being very rich in ultra-violet rays; it may also be got very conveniently by sparking with an induction coil between zinc or cadmium terminals. Sunlight is not rich in ultra-violet light, and does not produce anything like so great an effect as the arc light. Elster and Geitel, who have investigated with great success the effects of light on electrified bodies, have shown that the more electro-positive metals lose negative charges when exposed to ordinary light, and do not need the presence of the ultra-violet rays. Thus they found that amalgams of sodium or potassium enclosed in a glass vessel lose a negative charge when exposed to daylight, though the glass stops the small amount of ultra-violet light left in sunlight after its passage through the atmosphere. If sodium or potassium be employed, or, what is more convenient, the mercury-like liquid obtained by mixing sodium and potassium in the proportion of their combining weights, they found that negative electricity was discharged by an ordinary petroleum lamp. If the still more electro-positive metal rubidium is used, the discharge can be produced by the light from a glass rod just heated to redness; but there is no discharge till the glass is luminous. Elster and Geitel arrange the metals in the following order for the facility with which negative electrification is discharged by light: rubidium, potassium, alloy of sodium and potassium, sodium, lithium, magnesium, thallium, zinc. With copper, platinum, lead, iron, cadmium, carbon and mercury the effects with ordinary light are too small to be appreciable. The order is the same as that in Volta's electro-chemical series. With ultra-violet light the different metals show much smaller differences in their power of discharging negative electricity than they do with ordinary light. Elster and Geitel found that the ratio of the photo-electric effects of two metals exposed to approximately monochromatic light depended upon the wave-length of the light, different metals showing a maximum sensitiveness in different parts of the spectrum. This is shown by the following table for the alkaline metals. The numbers in the table are the rates of emission of negative electricity under similar circumstances. The rate of emission under the light from a petroleum lamp was taken as unity:--

Blue. Yellow. Orange. Red.
Rb .16 .64 .33 .039
Na .37 .36 .14 .009
K .57 .07 .04 .002

The table shows that the absorption of light by the metal has great influence on the photo-electric effect, for while potassium is more sensitive in blue light than sodium, the strong absorption of yellow light by sodium makes it more than five times more sensitive to this light than potassium. Stoletow, at an early period, called attention to the connexion between strong absorption and photo-electric effects. He showed that water, which does not absorb to any great extent either the ultra-violet or visible rays, does not show any photo-electric effect, while strongly coloured solutions, and especially solutions of fluorescent substances such as methyl green or methyl violet, do so to a very considerable extent; indeed, a solution of methyl green is more sensitive than zinc. Hallwachs (_Wied. Ann._ 37, p. 666) proved that in liquids showing photo-electric effects there is always strong absorption; we may, however, have absorption without these effects. Phosphorescent substances, such as calcium sulphide show this effect, as also do various specimens of fluor-spar. As phosphorescence and fluorescence are probably accompanied by a very intense absorption by the surface layers, the evidence is strong that to get the photo-electric effects we must have strong absorption of some kind of light, either visible or ultra-violet.

If a conductor A is placed near a conductor B exposed to ultra-violet light, and if B is made the negative electrode and a difference of potential established between A and B, a current of electricity will flow between the conductors. The relation between the magnitude of the current and the difference of potential when A and B are parallel plates has been investigated by Stoletow (_Journal de physique_, 1890, 11, p. 469), von Schweidler (_Wien. Ber._, 1899, 108, p. 273) and Varley (_Phil. Trans. A._, 1904, 202, p. 439). The results of some of Varley's experiments are represented in the curves shown in fig. 14, in which the ordinates are the currents and the abscissae the potentials. It will be seen that when the pressure is exceedingly low the current is independent of the potential difference and is equal to the negative charge carried off in unit time by the corpuscles emitted from the surface exposed to the light. At higher pressures the current rises far above these values and increases rapidly with the potential difference. This is due to the corpuscles emitted by the illuminated surface acquiring under the electric field such high velocities that when they strike against the molecules of the gas through which they are passing they ionize them, producing fresh ions which can carry on additional current. The relation between the current and the potential difference in this case is in accordance with the results of the theory of ionization by collision. The corpuscles emitted from a body under the action of ultra-violet light start from the surface with a finite velocity. The velocity is not the same for all the corpuscles, nor indeed could we expect that it should be: for as Ladenburg has shown (_Ann. der Phys._, 1903, 12, p. 558) the seat of their emission is not confined to the surface layer of the illuminated metal but extends to a layer of finite, though small, thickness. Thus the particles which start deep down will have to force their way through a layer of metal before they reach the surface, and in doing so will have their velocities retarded by an amount depending on the thickness of this layer. The variation in the velocity of the corpuscles is shown in the following table, due to Lenard (_Ann. der Phys._, 1902, 8, p. 149).

+------------------------------------+--------+----------+-----------+
| | Carbon.| Platinum.| Aluminium.|
+------------------------------------+--------+----------+-----------+
| Corpuscles emitted with velocities | | | |
| between 12 and 8 X 10^7 cm sec. | 0.000 | 0.000 | 0.004 |
| between 8 and 4 X 10^7 cm sec. | 0.049 | 0.155 | 0.151 |
| between 4 and 0 X 10^7 cm sec. | 0.67 | 0.65 | 0.49 |
| | | | |
| Corpuscles only emitted with the | | | |
| help of an external electric | 0.28 | 0.21 | 0.35 |
| field. +--------+----------+-----------|
| | 1.00 | 1.00 | 1.00 |
+------------------------------------+--------+----------+-----------+

If the illuminated surface is completely surrounded by an envelope of the same metal insulated from and completely shielded from the light, the emission of the negative corpuscles from the illuminated surface would go on until the potential difference V between this surface and the envelope became so great that the corpuscles with the greatest velocity lost their energy before reaching the envelope, i.e. if m is the mass, e the charge on a corpuscle, v the greatest velocity of projection, until Ve = 1/2mv^2. The values found for V by different observers are not very consistent. Lenard found that V for aluminium was about 3 volts and for platinum 2. Millikan and Winchester (_Phil. Mag._, July 1907) found for aluminium V = .738. The apparatus used by them was so complex that the interpretation of their results is difficult.

An extremely interesting fact discovered by Lenard is that the velocity with which the corpuscles are emitted from the metal is independent of the intensity of the incident light. The quantity of corpuscles increases with the intensity, but the velocity of the individual corpuscles does not. It is worthy of notice that in other cases when negative corpuscles are emitted from metals, as for example when the metals are exposed to cathode rays, Canal-strahlen, or Rontgen rays, the velocity of the emitted corpuscles is independent of the intensity of the primary radiation which excites them. The velocity is not, however, independent of the nature of the primary rays. Thus when light is used to produce the emission of corpuscles the velocity, as Ladenburg has shown, depends on the wave length of the light, increasing as the wave length diminishes. The velocity of corpuscles emitted under the action of cathode rays is greater than that of those ejected by light, while the incidence of Rontgen rays produces the emission of corpuscles moving much more rapidly than those in the cases already mentioned, and the harder the primary rays the greater is the velocity of the corpuscles.

The importance of the fact that the velocity and therefore the energy of the corpuscles emitted from the metal is independent of the intensity of the incident light can hardly be overestimated. It raises the most fundamental questions as to the nature of light and the constitution of the molecules. What is the source of the energy possessed by these corpuscles? Is it the light, or in the stores of internal energy possessed by the molecule? Let us follow the consequences of supposing that the energy comes from the light. Then, since the energy is independent of the intensity of the light, the electric forces which liberate the corpuscles must also be independent of that intensity. But this cannot be the case if, as is usually assumed in the electromagnetic theory, the wave front consists of a uniform distribution of electric force without structure, for in this case the magnitude of the electric force is proportional to the square root of the intensity. On the emission theory of light a difficulty of this kind would not arise, for on that theory the energy in a luminiferous particle remains constant as the particle pursues its flight through space. Thus any process which a single particle is able to effect by virtue of its energy will be done just as well a thousand miles away from the source of light as at the source itself, though of course in a given space there will not be nearly so many particles to do this process far from the source as there are close in. Thus, if one of the particles when it struck against a piece of metal caused the ejection of a corpuscle with a given velocity, the velocity of emission would not depend on the intensity of the light. There does not seem any reason for believing that the electromagnetic theory is inconsistent with the idea that on this theory, as on the emission theory, the energy in the light wave may instead of being uniformly distributed through space be concentrated in bundles which occupy only a small fraction of the volume traversed by the light, and that as the wave travels out the bundles get farther apart, the energy in each remaining undiminished. Some such view of the structure of light seems to be required to account for the fact that when a plate of metal is struck by a wave of ultra-violet light, it would take years before the corpuscles emitted from the metal would equal in number the molecules on the surface of the metal plate, and yet on the ordinary theory of light each one of these is without interruption exposed to the action of the light. The fact discovered by E. Ladenburg (_Verh. d. deutsch. physik. Ges._ 9, p. 504) that the velocity with which the corpuscles are emitted depends on the wave length of the light suggests that the energy in each bundle depends upon the wave length and increases as the wave length diminishes.

These considerations illustrate the evidence afforded by photo-electric effects on the nature of light; these effects may also have a deep significance with regard to the structure of matter. The fact that the energy of the individual corpuscles is independent of the intensity of the light might be explained by the hypothesis that the energy of the corpuscles does not come from the light but from the energy stored up in the molecules of the metal exposed to the light. We may suppose that under the action of the light some of the molecules are thrown into an unstable state and explode, ejecting corpuscles; the light in this case acts only as a trigger to liberate the energy in the atom, and it is this energy and not that of the light which goes into the corpuscles. In this way the velocity of the corpuscles would be independent of the intensity of the light. But it may be asked, is this view consistent with the result obtained by Ladenburg that the velocity of the corpuscles depends upon the nature of the light? If light of a definite wave length expelled corpuscles with a definite and uniform velocity, it would be very improbable that the emission of the corpuscles is due to an explosion of the atoms. The experimental facts as far as they are known at present do not allow us to say that the connexion between the velocity of the corpuscles and the wave length of the light is of this definite character, and a connexion such as a gradual increase of average velocity as the wave length of the light diminishes, would be quite consistent with the view that the corpuscles are ejected by the explosion of the atom. For in a complex thing like an atom there may be more than one system which becomes unstable when exposed to light. Let us suppose that there are two such systems, A and B, of which B ejects the corpuscles with the greater velocity. If B is more sensitive to the short waves, and A to the long ones, then as the wave length of the light diminishes the proportion of the corpuscles which come from B will increase, and as these are the faster, the average velocity of the corpuscles emitted will also increase. And although the potential acquired by a perfectly insulated piece of metal when exposed to ultra-violet light would depend only on the velocity of the fastest corpuscles and not upon their number, in practice perfect insulation is unattainable, and the potential actually acquired is determined by the condition that the gain of negative electricity by the metal through lack of insulation, is equal to the loss by the emission of negatively electrified corpuscles. The potential acquired will fall below that corresponding to perfect insulation by an amount depending on the number of the faster corpuscles emitted, and the potential will rise if the proportion of the rapidly moving corpuscles is increased, even though there is no increase in their velocity. It is interesting to compare other cases in which corpuscles are emitted with the case of ultra-violet light. When a metal or gas is bombarded by cathode rays it emits corpuscles and the velocity of these is found to be independent of the velocity of the cathode rays which excite them; the velocity is greater than for corpuscles emitted under ultra-violet light. Again, when bodies are exposed to Rontgen rays they emit corpuscles moving with a much greater velocity than those excited by cathode rays, but again the velocity does not depend upon the intensity of the rays although it does to some extent on their hardness. In the case of cathode and Rontgen rays, the velocity with which the corpuscles are emitted seems, as far as we know at present, to vary slightly, but only slightly, with the nature of the substance on which the rays fall. May not this indicate that the first effect of the primary rays is to detach a neutral doublet, consisting of a positive and negative charge, this doublet being the same from whatever system it is detached? And that the doublet is unstable and explodes, expelling the negative charge with a high velocity, and the positive one, having a much larger charge, with a much smaller velocity, the momentum of the negative charge being equal to that of the positive.

Up to now we have been considering the effects produced when light is incident on metals. Lenard found (and the result has been confirmed by the experiments of J. J. Thomson and Lyman) that certain kinds of ultra-violet light ionize a gas when they pass through. The type of ultra-violet light which produces this effect is so easily absorbed that it is stopped by a layer a few millimetres thick of air at atmospheric pressure.

_Ionization by Collision._--When the ionization of the gas is produced by external agents such as Rontgen rays or ultra-violet light, the electric field produces a current by setting the positive ions moving in one direction, and the negative ones in the opposite; it makes use of ions already made and does not itself give rise to ionization. In many cases, however, such as in electric sparks, there are no external agents to produce ionization and the electric field has to produce the ions as well as set them in motion. When the ionization is produced by external means the smallest electric field is able to produce a current through the gas; when, however, these external means are absent no current is produced unless the strength of the electric field exceeds a certain critical value, which depends not merely upon the nature of the gas but also upon the pressure and the dimensions of the vessel in which it is contained. The variation of the electric field required to produce discharge can be completely explained if we suppose that the ionization of the gas is produced by the impact with its molecules of corpuscles, and in certain cases of positive ions, which under the influence of the electric field have acquired considerable kinetic energy. We have direct evidence that rapidly moving corpuscles are able to ionize molecules against which they strike, for the cathode rays consist of such corpuscles, and these when they pass through a gas produce large amounts of ionization. Suppose then that we have in a gas exposed to an electric field a few corpuscles. These will be set in motion by the field and will acquire an amount of energy in proportion to the product of the electric force, their charge, and the distance travelled in the direction of the electric field between two collisions with the molecules of the gas. If this energy is sufficient to give them the ionizing property possessed by cathode rays, then when a corpuscle strikes against a molecule it will detach another corpuscle; this under the action of the electric field will acquire enough energy to produce corpuscles on its own account, and so as the corpuscles move through the gas their number will increase in geometrical progression. Thus, though there were but few corpuscles to begin with, there may be great ionization after these have been driven some distance through the gas by the electric field.

The number of ions produced by collisions can be calculated by the
following method. Let the electric force be parallel to the axis of x,
and let n be the number of corpuscles per unit volume at a place fixed
by the co-ordinate x; then in unit time these corpuscles will make
nu/[lambda] collisions with the molecules, if u is the velocity of a
corpuscle and [lambda] the mean free path of a corpuscle. When the
corpuscles are moving fast enough to produce ions by collision their
velocities are very much greater than those they would possess at the
same temperature if they were not acted on by electrical force, and so
we may regard the velocities as being parallel to the axis of x and
determined by the electric force and the mean free path of the
corpuscles. We have to consider how many of the nu/[lambda] collisions
which take place per second will produce ions. We should expect that
the ionization of a molecule would require a certain amount of energy,
so that if the energy of the corpuscle fell below this amount no
ionization would take place, while if the energy of the corpuscle were
exceedingly large, every collision would result in ionization. We
shall suppose that a certain fraction of the number of collisions
result in ionization and that this fraction is a function of the
energy possessed by the corpuscle when it collides against the
molecules. This energy is proportional to Xe[lambda] when X is the
electric force, e the charge on the corpuscle, and [lambda] the mean
free path. If the fraction of collisions which produce ionization is
[int](Xe[lambda]), then the number of ions produced per cubic
centimetre per second is [int](Xe[lambda])nu/[lambda]. If the
collisions follow each other with great rapidity so that a molecule
has not had time to recover from one collision before it is struck
again, the effect of collisions might be cumulative, so that a
succession of collisions might give rise to ionization, though none of
the collisions would produce an ion by itself. In this case [int]
would involve the frequency of the collisions as well as the energy of
the corpuscle; in other words, it might depend on the current through
the gas as well as upon the intensity of the electric field. We
shall, however, to begin with, assume that the current is so small
that this cumulative effect may be neglected.

Let us now consider the rate of increase, dn/dt, in the number of
corpuscles per unit volume. In consequence of the collisions,
[int](Xe[lambda])nu/[lambda] corpuscles are produced per second; in
consequence of the motion of the corpuscles, the number which leave
unit volume per second is greater than those which enter it by
(d/dx)(nu); while in a certain number of collisions a corpuscle will
stick to the molecule and will thus cease to be a free corpuscle. Let
the fraction of the number of collisions in which this occurs be
[beta]. Thus the gain in the number of corpuscles is
[int](Xe[lambda])nu/[lambda], while the loss is (d/dx)(nu) +
[beta](nu)/[lambda]; hence

dn nu d [beta]nu
-- = [int](Xe[lambda]) -------- - --(nu) - --------.
dt [lambda] dx [lambda]

When things are in a steady state dn/dt = 0, and we have

d 1 / \
--(nu) = --------( [int](Xe[lambda]) - [beta] )nu.
dx [lambda] \ /

If the current is so small that the electrical charges in the gas are
not able to produce any appreciable variations in the field, X will be
constant and we get nu = C[epsilon]^{[alpha]x}, where [alpha] =
{[int](Xe[lambda]) - [beta]}/[lambda]. If we take the origin from
which we measure x at the cathode, C is the value of nu at the
cathode, i.e. it is the number of corpuscles emitted per unit area of
the cathode per unit time; this is equal to i0/e if i0 is the quantity
of negative electricity coming from unit area of the cathode per
second, and e the electric charge carried by a corpuscle. Hence we
have nue = i0[epsilon]^{[alpha]x}. If l is the distance between the
anode and the cathode, the value of nue, when x = l, is the current
passing through unit area of the gas, if we neglect the electricity
carried by negatively electrified carriers other than corpuscles.
Hence i = i0[epsilon]^{[alpha]l}. Thus the current between the plates
increases in geometrical progression with the distance between the
plates.

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 [alpha] for different values of
X and for different pressures for air, hydrogen and carbonic acid gas
(_Phil. Mag._ [6], 1, p. 198). Since [lambda] varies inversely as the
pressure, we see that [alpha] may be written in the form p[phi](X/p)
or [alpha]/X = F(X/p). The following are some of the values of [alpha]
found by Townsend for air.

+---------+----------+----------+----------+----------+----------+
| X Volts | Pressure | Pressure | Pressure | Pressure | Pressure |
| per cm. | .17 mm. | .38 mm. | 1.10 mm. | 2.1 mm. | 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, [alpha] 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 [alpha], and [alpha] 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 [alpha] is a maximum, is
proportional to X; this follows at once from the fact that [alpha] 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 i0[epsilon]^{[alpha]l} or
i0[epsilon]^{XlF(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 i0[epsilon]^{VlF(X/p)},
and the greatest value it can have is i0[epsilon]^{V/60}. Thus the
ratio between the current between the plates when there is ionization
and when there is none cannot be greater than [epsilon]^{V/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, [alpha] 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 [alpha] with the strength of the current.

Townsend has shown that we can deduce from the values of [alpha] 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, [alpha] = 1/[lambda] and is independent
of the strength of the electric field. Hence we see that the value of
[alpha], 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 [alpha] 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 = i0[epsilon]^nx and i = i0[epsilon]^nl,
when l is the distance between the plates, we see that

nu / mv = [epsilon]^(nx) / ([epsilon]^(nl) - [epsilon]^(nx)),

n v [epsilon]^(nx)
-- = -- . -------------------------------.
m u [epsilon]^(ne) - [epsilon]^(nx)

Since v/u is a very small quantity we see that n will be less than m
except when [epsilon]^nl - [epsilon]^nx 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 = i0[epsilon]^[alpha]l 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

qe
i = i0[epsilon]^{[alpha]l} + -------(e^{[alpha]l} - 1).
[alpha]

With regard to the minimum energy which must be possessed by a
corpuscle to enable it to produce ions by collision, Townsend (loc.
cit.) came to the conclusion that to ionize air the corpuscle must
possess an amount of energy equal to that acquired by the fall of its
charge through a potential difference of about 2 volts. This is also
the value arrived at by H. A. Wilson by entirely different
considerations. Stark, however, gives 17 volts as the minimum for
ionization. The energy depends upon the nature of the gas; recent
experiments by Dawes and Gill and Pedduck (_Phil. Mag._, Aug. 1908)
have shown that it is smaller for helium than for air, hydrogen, or
carbonic acid gas.

If there is no external source of ionization and no emission of corpuscles from the cathode, then it is evident that even if some corpuscles happened to be present in the gas when the electric field were applied, we could not get a permanent current by the aid of collisions made by these corpuscles. For under the electric field, the corpuscles would be driven from the cathode to the anode, and in a short time all the corpuscles originally present in the gas and those produced by them would be driven from the gas against the anode, and if there was no source from which fresh corpuscles could be introduced into the gas the current would cease. The current, however, could be maintained indefinitely if the positive ions in their journey back to the cathode also produced ions by collisions, for then we should have a kind of regenerative process by which the supply of corpuscles could be continually renewed. To maintain the current it is not necessary that the ionization resulting from the positive ions should be anything like as great as that from the negative, as the investigation given below shows a very small amount of ionization by the positive ions will suffice to maintain the current. The existence of ionization by collision with positive ions has been proved by Townsend. Another method by which the current could be and is maintained is by the anode emitting corpuscles under the impact of the positive ions driven against it by the electric field. J. J. Thomson has shown by direct experiment that positively electrified particles when they strike against a metal plate cause the metal to emit corpuscles (J. J. Thomson, _Proc. Camb. Phil. Soc._ 13, p. 212; Austin, _Phys. Rev._ 22, p. 312). If we assume that the number of corpuscles emitted by the plate in one second is proportional to the energy in the positive ions which strike the plate in that second, we can readily find an expression for the difference of potential which will maintain without any external ionization a current of electricity through the gas. As this investigation brings into prominence many of the most important features of the electric discharge, we shall consider it in some detail.

Let us suppose that the electrodes are parallel plates of metal at
right angles to the axis of x, and that at the cathode x = 0 and at the
anode x = d, d being thus the distance between the plates. Let us also
suppose that the current of electricity flowing between the plates is
so small that the electrification between the plates due to the
accumulation of ions is not sufficient to disturb appreciably the
electric field, which we regard as uniform between the plates, the
electric force being equal to V/d, where V is the potential difference
between the plates. The number of positive ions produced per second in
a layer of gas between the planes x and x+dx is [alpha]nu.dx. Here n is
the number of corpuscles per unit volume, [alpha] the coefficient of
ionization (for strong electric field [alpha] = 1/[lambda]', where
[lambda]' is the mean free path of a corpuscle), and u the velocity of
a corpuscle parallel to x. We have seen that nu = i0[epsilon]^[alpha]x,
where i0 is the number of corpuscles emitted per second by unit area of
the cathode. Thus the number of positive ions produced in the layer is
[alpha]i0[epsilon]^[alpha]x dx. If these went straight to the cathode
without a collision, each of them would have received an amount of
kinetic energy Vex/d when they struck the cathode, and the energy of
the group of ions would be Vex/d.[alpha]i0[epsilon]^dx dx. The positive
ions will, however, collide with the molecules of the gas through which
they are passing, and this will diminish the energy they possess when
they reach the cathode.

The diminution in the energy will increase in geometrical proportion
with the length of path travelled by the ion and will thus be
proportional to [epsilon]^-[beta]x, [beta] will be proportional to the
number of collisions and will thus be proportional to the pressure of
the gas. Thus the kinetic energy possessed by the ions when they reach
the cathode will be

[epsilon]^{-[beta]x} . V(ex/d) . [alpha]i0[epsilon]^{[alpha]x} dx,

and E, the total amount of energy in the positive ions which reach the
cathode in unit time, will be given by the equation
_
/d
E = | [epsilon]^{-[beta]x} . V(ex/d) . [alpha]i0[epsilon]^{[alpha]x} dx
_/0
_
Ve[alpha]i0 /d
= ----------- | [epsilon]^{-([beta]-[alpha])x}.x.dx
d _/0

Ve[alpha]i0 / 1 / 1 d \ \
= ----------- { ------------------ - [epsilon]^{-([beta]-[alpha])d} { ------------------ + ---------------- } } (1).
d \([beta]-[alpha])^2 \([beta]-[alpha])^2 ([beta]-[alpha])/ /

If the number of corpuscles emitted by the cathode in unit time is
proportional to this energy we have i0 = kE, where k is a constant;
hence by equation (1) we have

([beta]-[alpha])^2 d
V = ------------------ . --,
ke[alpha] I

where

I = 1 - [epsilon]^{-([beta]-[alpha])d} (1 + d([beta] - [alpha])).

Since both [beta] and [alpha] are proportional to the pressure, I and
([beta] - [alpha])^2d/[alpha] are both functions of pd, the product of
the pressure and the spark length, hence we see that V is expressed by
an equation of the form

1
V = -- [int](pd) (2),
ke

where [int](pd) denotes a function of pd, and neither p nor d enter
into the expression for V except in this product. Thus the potential
difference required to produce discharge is constant as long as the
product of the pressure and spark length remains constant; in other
words, the spark potential is constant as long as the mass of the gas
between the electrodes is constant. Thus, for example, if we halve the
pressure the same potential difference will produce a spark of twice
the length. This law, which was discovered by Paschen for fairly long
sparks (_Annalen_, 37, p. 79), and has been shown by Carr (_Phil.
Trans._, 1903) to hold for short ones, is one of the most important
properties of the electric discharge.

We see from the expression for V that when ([beta] - [alpha])d is very
large

V = ([beta] - [alpha])^2d/ke[alpha].

Thus V becomes infinite when d is infinite. Again when ([beta] -
[alpha])d is very small we find

V = 1/ke[alpha]d;

thus V is again infinite when d is nothing. There must therefore be
some value of d intermediate between zero and infinity for which V is
a minimum. This value is got by finding in the usual way the value of
d, which makes the expression for V given in equation (1) a minimum.
We find that d must satisfy the equation

/ \
1 = [epsilon]^{-([beta]-[alpha])d} {1 + ([beta] - [alpha])d + ([beta] - [alpha].d)^2}.
\ /

We find by a process of trial and error that ([beta]-[alpha])d = 1.8
is approximately a solution of this equation; hence the distance for
minimum potential is 1.8/([beta] - [alpha]). Since [beta] and [alpha]
are both proportional to the pressure, we see that the critical spark
length varies inversely as the pressure. If we substitute this value
in the expression for V we find that [=V], the minimum spark
potential, is given by

_ [beta] - [alpha] 2.2
V = ---------------- . ---.
[alpha] ke

Since [beta] and [alpha] are each proportional to the pressure, the
minimum potential is independent of the pressure of the gas. On this
view the minimum potential depends upon the metal of which the cathode
is made, since k measures the number of corpuscles emitted per unit
time by the cathode when struck by positive ions carrying unit energy,
and unless [beta] bears the same ratio to [alpha] for all gases the
minimum potential will also vary with the gas. The measurements which
have been made of the "cathode fall of potential," which as we shall
see is equal to the minimum potential required to produce a spark,
show that this quantity varies with the material of which the cathode
is made and also with the nature of the gas. Since a metal plate, when
bombarded by positive ions, emits corpuscles, the effect we have been
considering must play a part in the discharge; it is not, however, the
only effect which has to be considered, for as Townsend has shown,
positive ions when moving above a certain speed ionize the gas, and
cause it to emit corpuscles. It is thus necessary to take into account
the ionization of the positive ions.

Let m be the number of positive ions per unit volume, and w their
velocity, the number of collisions which occur in one second in one
cubic centimetre of the gas will be proportional to mwp, where p is
the pressure of the gas. Let the number of ions which result from
these collisions be [gamma]mw; [gamma] will be a function of p and of
the strength of the electric field. Let as before n be the number of
corpuscles per cubic centimetre, u their velocity, and [alpha]nu the
number of ions which result in one second from the collisions between
the corpuscles and the gas. The number of ions produced per second per
cubic centimetre is equal to [alpha]nu + [gamma]mw; hence when things
are in a steady state

d
--(nu) = [alpha]nu + [gamma]mw ,
dx

and

e(nu + mw) = i,

where e is the charge on the ion and i the current through the gas.
The solution of these equations when the field is uniform between the
plates, is

enu = C[epsilon]^{([alpha]-[gamma])x} - [gamma]i/([alpha] - [gamma]),

emw = -C[epsilon]^{([alpha]-[gamma])x} + [alpha]i/([alpha] - [gamma]),