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