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THE THEORY OF IONIZATION OF
GASES BY COLLISION
THE THEORY OF
IONIZATION OF GASES
BY COLLISION
• * * tº
. o- \
JOHN sº TownsPND M.A. F.R.S.
Wykeham PROFEssor of PHYSICs, Oxfor D,
FELLow of New Col.LEGE, Oxford,
ForMERLY FELLow of TRINITY College, CAMBRIDGE.
NEW YORK
D. WAN NOSTRAND COMPANY
23 Murray AND 27 WARREN STREETs
1910
PREFACE
AFTER studying the changes which take place in the
conductivity of gases through which ions are passing
under various conditions, I was led to propose the
theory of ionization by collision to explain the develop-
ment of currents in gases. The accuracy of the theory
has been established by a large number of experiments
specially arranged to measure conductivities which could
be compared with the values obtained from theoretical
considerations. The researches have brought to light
the fact that ionization by collision takes place when
comparatively small potential differences are established
between electrodes in a gas at a suitable pressure, so
that considerable multiplication of the ions may be
obtained by this process with forty or fifty volts, and
consequently with comparatively Small velocities of the
IOIlS.
This is in marked contrast with the previously known
cases of ionization produced by the motion of ions
through a gas, where the velocity is always very high,
Sometimes approaching that of light. Such, for instance,
is the case for the negative particles emitted by radio-
active substances, or for the particles composing the
cathode rays and the Lenard rays when the ions move
wiſh velocities that are acquired under the action of
Some thousands of volts.
The collision theory of the genesis of ions might have
been framed from a consideration of some of Stoletow's
298785
vi PREFACE
experiments on ultra-violet light which were made in
1888, as considerable increases in the current of electri-
city through a gas were obtained with Small voltages;
but this method of explaining these experiments does
not seem to have been suggested until (Nature, August 9,
1900) the theory had been substantially verified by the
results of experiments on the conductivity produced by
Röntgen rays. The explanation previously given of
Stoletow’s experiments was founded on a theory of
electrical surface layers, but the fatal objection to it is
that it fails to explain the fact that the current in the
gas increases in geometric proportion as the distance
between the plates increases in arithmetic proportion
when the electric force remains constant. The agree-
ment on the other hand between the experiments and
the numbers calculated on the collision theory is so
accurate that the phenomena are now attributed to the
effects produced by collisions.
It is obvious from the large effects that may be
obtained by the multiplication of ions with comparatively
small electric forces that the process of ionization by
collision is of fundamental importance in the develop-
ment of large currents, and affords an explanation of
many phenomena in connection with the discharge of
electricity through gases. In particular, I may mention
the application of these principles to account for spark-
ing in gases. The exact value of the Sparking potential,
agreeing with the experimental determinations within a
few per cent., may be calculated for a uniform field for
different pressures of the gas, and the leading features of
the more complicated phenomena obtained with large
currents may also be accounted for.
The various points in connection with the theory which
PREFACE vii
have been fairly well established by experimental methods
have hitherto only appeared in different original papers,
and those who are interested in the subject have found
it difficult to get a general outline of the principles which
are involved. The text-books that have been written on
the conductivity of gases do not contain a good description
of the methods by which the fundamental principles are
established, and the attempts that have been made to
explain well-known phenomena by the aid of the theory
of collisions involve so many arbitrary assumptions, and
are so inaccurate, that the results obtained are in many
cases of no value.
It is in the hope of removing some of these difficulties
that I have undertaken to write this account of the theory
of ionization by collision and to discuss some of the
phenomena in which the effects produced by collisions
appear to play a predominant part.
In arranging this work for publication I have derived
great advantage from suggestions that have been made
to me by Mr. C. E. Haselfoot; my best thanks are due
to him for the assistance he has thus rendered, and also
for having undertaken to correct the proofs.
CONTENTS
CELAPTER I
IONIZATION BY NEGATIVE IONS
SECTION
1.
10.
11.
12.
Variation of current with electric force.
Variation of current with distance between the elec.
trodes when the force is constant e o
Determination of a from experiments with ultra-violet
light. e & e G e e & º -
Determination of a from experiments with Röntgen
rays . º tº te tº º e º º e
Conductivity produced by Röntgen rays in a gas
between cylindrical electrodes . & º º &
Negative ions generated by various methods have the
same ionizing power . º e e º e e
Comparison of the masses of negative ions in liquids
and gases; corpuscular state º º
Representation of the values of a by a single curve
Agreement between the experimental results and the
..., & # (X
equation p".f (:) e
º Q. X
Properties of the curves =f (#)
ſº p
Maximum value of a ; mean free path of negative
ions . e - e e • e e º te
Comparison of the values of a with the expression
_NV
X
Ne
PAGE
1S
21
22
24
X CoNTENTS
SECTION
13. Molecular dimensions deduced from the mean free
paths of negative ions . º o * e
14. Application of the theory to Stoletow's experiments
15. Determination of the pressure corresponding to the
maximum value of a for a given force
16. Comparison of the velocities of ions and molecules
CEIAPTER II
IONIZATION By PosLTIVE IONS
17. Conductivity between parallel plates when positive and
negative ions generate others by collisions
18. Agreement between experiments and theory e
19. Curves representing 3/p as a function of X/p; com-
parison of effects produced by positive and negative
ions
CEIAPTER III
SPARKING POTENTIAL IN A UNIFORM ELECTRIC FIELD
20. Description of phenomena accompanying discharges;
Sparking potential . º ë e e
21. Potential required to maintain a discharge .
22. Properties of pointed and cylindrical electrodes
23. Condition for sparking in a uniform electric field.
Tables giving the sparking potentials determined
theoretically and experimentally
24. Effects of initial ionization .
CEIAPTER IV
PAGE
29
30
34
36
38
42
51
53
54
TEIEORY OF ELECTRIC DISCHARGES IN FIELDS OF FORCE WHICH
25.
26.
ARE NOT UNIFORM.
Description of phenomena to be investigated
Condition for the maintenance of a current by effects
of collisions in any field of force
63
65
CoNTENTS
xi
SECTION
27. Currents accompanied by a positive charge in the gas .
28. Increase of force at cathode accompanied by decrease
of potential required to maintain a current
between the electrodes for pressures above the
critical pressure * & e * & «s ©
29. Simple experiments to illustrate the effect of con-
centrating the force near the cathode e º º
30. Comparison of the effects of concentrating the force
near the cathode and near the anode ; explanation
of sparking potentials for positive and negative
points * e • • ſº g & wº
31. Phenomena at pressures below the critical pressure
32. Processes of ionization that may account for effects
obtained at low pressures e e &
33. Cathode fall of potential; ionization in the space near
the cathode when the cathode fall of potential is
established e © e e º &
34. Sparking potential at atmospheric pressure for very
short distances between the electrodes ©
35. Remarks on processes of ionization which account for
various phenomena . g e e § * e
36. Examination of some other theories of the sparking
- potential
PAGE
67
68
69
2
.
4.
7
5
ſ
7
80
81
82
THE THEORY OF IONIZATION OF
GASES BY COLLISION
CHAPTER I
IONIZATION BY NEGATIVE IONS
1. Wariation of current with electric force.
THE process of ionization by collision between ions
and molecules of a gas may be examined by investigating
D
i
Electric Force.
Tigure 1.
the currents between parallel plate electrodes when ultra-
violet light falls on the negative electrode or when the
gas is ionized by Röntgen rays. If the gas is at a high
pressure, the current increases with the electric force and
} I.G. B
2 THE THEORY OF IONIZATION OF GASEs
attains a maximum value, which is not exceeded unless
very large forces are used. It is possible, however, by
reducing the pressure of the gas, to make the ions
travel with sufficient velocity to generate others by
collisions with molecules, even when the potential
differences employed are small, and thus with a few
hundred volts to obtain large increases in the current."
The curve, figure 1, showing the connection between
the current and electric force in a gas at low pressure,
may be taken as illustrating this effect.
In the first stage, A B, the current between the plates
increases with the electro-motive force. The rate of
increase diminishes as the force increases, and the current
tends to attain a maximum value.
In the Second stage, B C, the current remains prac-
tically constant and shows only small variations for
large changes in the force. If the ions are produced by
the action of Röntgen rays or Becquerel rays, the con-
stant value is attained when the force is sufficiently great
to collect all the positive and negative ions on the
electrodes, but before this value is reached an appreciable
number of the ions is lost by recombination. Again, the
ions may be produced by the action of ultra-violet light
On the negative electrode. In this case, if the force is
too small, Some of the ions do not reach the positive
electrode, but diffuse through the gas to the negative
electrode.
In the third stage, C D, when the force is still further
increased, there is a large increase in the conductivity.
This can be explained on the hypothesis that new ions
are generated by collisions, at first practically by negative
* Nature, Wol. lxii., August 9, 1900.
IonizATION By NEGATIVE Ions 3
ions alone, but as the force increases and the sparking
potential is approached, the positive ions also acquire
the property of producing others to an appreciable
extent.
2. Wariation of current with distance between the electrodes
when the force is constant.
In the earlier experiments which were made to test
the theory the initial ionization was produced by the
action of Röntgen rays. The simplest conditions, how-
ever, are realized when the initial ionization consists of
negative ions set free from a metal surface by a beam of
ultra-violet light.
When the light falls on a metal plate a number no of
negative ions are set free which can be made to travel
various distances through a gas under any required force
to a parallel plate positively charged. If no new ions are
produced by collisions the number reaching the positive
plate will be n0, and the current will be independent of
the distance between the plates. If, however, the ions
produce others by collisions with the molecules of the
gas between the plates, the number reaching the positive
plate will increase and will depend on the distance
between the plates. In fact, if each negative ion set free
from the metal plate produces a new negative ions in
going through a centimetre of the gas, and if the new
ions produced in the gas have exactly the same property
of generating others by collisions, then the number that
arrive at the positive plate will be ne” where l is the
distance between the plates. For let n be number of ions
produced in a layer of thickness, a, measured from the
negative electrode, the number n including the original
B 2
4 THE THEORY OF IonizATION OF GASEs
no ions. In passing through a path of length da:
these ions produce mada, new ions by collisions, so
that dm-mada. This equation on integration gives.
logn=aw-Hconstant, or n=nge” since no is the value
of n corresponding to a:-0.
The quantity a depends only on the electric force and
pressure of the gas, and if these are constant the charges
n1, m2, etc., acquired by the positive plate for different
distances li, lo, etc., between the plates will be
le
all Ol.
m1=170e “; m2=n0e "; etc.
Hence for equal increments of the distance the ratios of
the successive charges will be the same, viz. –
a(lº - li).
*="*=etc. =e
711 ?? 2
The conditions specified above are easily realized in
practice, and experiments show that this simple expo-
mential law" for the increase of the current with the
distance l is accurately true for small distances between
the plates; for larger distances the conductivity rises
more rapidly owing, as will be explained later, to the
effect of positive ions; but for simplicity the currents
between plates separated by short distances will be
considered first.
1 These results showed definitely that it was necessary to abandon
the older theories of surface layers by which these phenomena were
formerly explained. Such theories provided, it is true, a possible
explanation of the increase of current with increase of force, as it
is conceivable that the greater the force the greater the number of
ions drawn from the plate. But it is clear that if the force at the
surface is kept constant an increase in the distance between the
plates should not affect the current, and this is quite contrary to the
experimental results.
Ionization By NEGATIVE IONS 5
3. Determination of a from experiments with ultra-violet
light.
The principal parts of the apparatus which was
used to find the conductivities are shown in figure 2.
The currents take place in the gas between the plates
A and B. The upper plate from which the ions are set
free was of zinc and was fixed to a micrometer Screw for
== |
+
P A ; : I A E
. . ;
§
ſº
.
t
| 1
º
ii;
"If
º
*
§
Figure 2.
adjusting the distance between the plates. Insulation
was provided by the ebonite pillars E. The lower plate
B was of quartz silvered on the upper side so as to have
a conducting surface in contact with the gas. A series
of fine parallel lines were ruled on the quartz forming a
transparent grating over a Small area at the centre of
the plate. The light from a spark gap S in a leyden
jar circuit passed through a quartz lens and the grating,
6 THE THEORY OF IONIZATION OF GASEs
and falling on the upper plate set free negative ions
from the zinc surface. The apparatus was covered with
an air-tight glass cover, so that the pressure of the gas
could be reduced, and suitable means were provided for
turning the micrometer screw and making connection
with the plates. The plate B was raised to any required
potential by connecting it to a number of small accumu-
lators. In a set of experiments in which it was necessary
to maintain a uniform force, the number of cells used
was proportional to the distance between the plates.
When the currents passing between the plates are small
the conductivities may be measured by a sensitive
quadrant electrometer connected to the upper plate,
but with large currents it is necessary to use an induc-
tion balance in order to get accurate determinations."
The currents must in all cases be comparatively small
so that the uniformity of field may not be disturbed by
the electrostatic force produced by the separation of the
ions in the gas.
The following examples may be given of experi-
ments with air at 2.5 millimetres pressure. With a
force of 350 volts per centimetre, the currents for
distances of 1, 3, and 5 millimetres between the plates
were proportional to 1, 2:06, and 4.22 respectively. For
the same distances the currents were proportional to
1, 4:24, and 183 when the force was 525 volts per
centimetre. The values of a for the different forces
may be obtained from the currents. Thus for 350 volts
per centimetre the value of a is given by the equation
.2a 2:06_4.2%
- T - 2:06:
! See paper, Philosophical Magazine, November, 1903.
Ion IZATION By NEGATIVE IONS 7
4. Determination of a from experiments with Röntgen rays.
When the initial ionization is not produced at an
electrode, but consists of positive and negative ions
generated in the gas between the electrodes, as, for
instance, by Röntgen rays or by Becquerel rays, the effect
produced by collisions of the negative ions may be
calculated as follows:–Let each negative ion produce a
others by collisions whilst passing through a centimetre
of the gas, and let no be the total number produced by
the rays between the plates when at a distance l apart.
In the layer of thickness da; at a distance a from the
positive electrode a number * are generated by the
rays, and in passing through the distance a this number
is increased to *…* So that the total number of
negative ions n reaching the positive electrode is
| 'noe *da. n(*–1)
77, E ,-I-= —I-
The ratio . may be obtained from the curve giving
0
the connection between the current and the electrić
force, no being the constant current when no new ions
are produced by collisions, and n the current corre-
sponding to a force X. The value of a for the force X
al
is therefore given by the equation -: +'. The
accuracy of this formula has been tested experimentally,
and it has been found that for a given force the value of
+. varies with the distance l, but that the values of a
0
8 THE THEORY OF Ionization OF GASES
obtained for the different distances are the same within
the limits of experimental error.
The curves, figures 3 and 4, may be taken as
examples. In each figure the curve corresponding to a
distance of two centimetres between the plates rises
more rapidly than that corresponding to one centimetre,
S2O
230
24-O
aOO
i
12O
ôo 160 &40 320 460 560 640 720.
Wolts per centimetre.
Current-electric force curves for air at 1:1 mm.
pressure, with distances of 2, 1 and 5 cms.
between the plates.
Figure 3.
and in the curve corresponding to 5 centimetre the rate
of increase with the force is much less than in either of
the other two cases. It will be noticed that the curves are
1 The sets of curves at different pressures may be found in the
paper Philosophical Magazine, February, 1901.
IONIZATION By NEGATIVE IONS 9
practically parallel to the axis of a for the low forces, so
that no, the number of ions produced by the rays, is
proportional to the currents for forces of 10 or 15 volts
per centimetre. The following numbers may be taken
as illustrations:
In the experiments at a pressure of 1:1 millimetre, and
(60
ſé0
120
100
80
i
20
80 I60 240 320 400
Wolts per centimetre.
Current-electric force curves for air at 385 mm.
pressure, with distances of 2, 1 and 5 cms.
between the plates.
Figure 4.
a force of 160 volts per centimetre the value of a deduced
from the curve, figure 3, corresponding to a distance of
2 centimetres between the plates is 2:02, and from the
second curve the value 1'98 was obtained. With the
same force and a pressure of 385 millimetre a increases,
* * * *
10 THE THEORY OF IONIZATION OF GASEs
the values obtained from the curves, figure 4, being 28,
29, and 29. Retaining the same force and making further
reductions in the pressure, a is found to diminish; its
mean value deduced from similar experiments at 171
millimetre pressure being 2:15.
When the pressure is constant, a increases with the
force and approaches a maximum value, which is
attained when the force is so great that a new pair of
ions are formed at each collision. For the smaller forces
a has a smaller value since the velocity acquired by an
ion along a free path is only large enough to produce
others by collision in the case of the longer free
paths. -
When the force is constant and the pressure reduced,
as in the experiments at 160 volts per centimetre which
have been quoted, the value of a increases, attains a
maximum, and finally diminishes again. This also
agrees with the theory, for at high pressures the free
paths are very short and the ions do not acquire a large
velocity and do not ionize the molecules with which they
collide. As the pressure is reduced the free paths are
increased, so that along the longer paths a high velocity
is attained. The proportion of collisions which result
in new ions being formed thus increases, but the total
number of collisions per centimetre diminishes. When
the pressure is reduced beyond a certain point the
values of a begin to diminish, as the number of molecules
with which an ion collides may become very small.
The results of the experiments may thus be seen to be
in general agreement with the theory, but the variations
of a obtained by altering the force and pressure will be
examined more fully when the curves connecting a, X,
and p are explained.
IONIZATION By NEGATIVE IONS 11
5. Conductivity produced by Röntgen rays in a gas between
cylindrical electrodes.
There is one point in connection with the current-
electric force curves obtained with Röntgen rays which
remains to be decided by experiment. In this case both
positive and negative ions are produced initially in the
gas by the rays, so that it is not evident from the
experiments with parallel plates that the results are due
to negative ions. Obviously the effects might be due to
positive ions if it happened that they had the property
of generating others by collisions while the negative
ions were inactive. It is, however, easy to show by using
electrodes of different shapes that the effects must be
attributed to negative ions." If the gas is contained in
a spherical conductor which acts as one of the electrodes,
the other electrode being a small sphere at the centre,
or if a cylinder and a co-axial wire of Small radius are
used as electrodes, then for small differences of potential
the current through the gas is the same in both directions
and corresponds to the total number of ions generated
by the rays; but when the difference of potential is
large the currents are no longer equal. It has been
found that when the outer electrode is negative and the
electromotive force is increased a large increase in the
current is obtained. In this case all the negative ions
produced by the rays traverse the field of strong electric
force in the neighbourhood of the Small inner electrode,
and acquire sufficient velocity to generate others by
collisions with molecules of the gas. On the other
hand, when the outer electrode is positive and similar
1 See Nature, Vol. lxii., August 9, 1900.
12 THE THEORY OF IONIZATION OF GASEs
increases in electromotive force are made, the cor-
responding increases of current are very Small. In this
case the negative ions travel outwards, and only a few
of those produced by the rays pass through the field of
strong electric force, so that the number of new ions pro-
duced by collisions is comparatively small. Experiments
800
5 O
50 (OO 130 a.oO aso 300 J3o 4oo
Electromotive Force.
Figure 5.
on the currents between concentric cylinders have been
made by Mr. Kirkby, which show in a striking manner
the effect of changing the direction of the electric force.
The curves, figure 5, for air at a pressure of 3-58 milli-
metres give the currents between a cylinder of 4:15
centimetres diameter and a concentric wire of 206
millimetres diameter when the air is ionized by Röntgen
rays."
1 P. J. Kirkby, Philosophical Magazine, February, 1902.
Ion IZATION By NEGATIVE IONS 13
When the cylinder is negative new ions are produced
as the force increases, so that when the potential differ-
ence between the wire and cylinder is 360 volts the
current is ten times as great as that which is due to
the ions that are produced by the direct action of the
rays.
| When the cylinder is positive there is no appreciable
change in the current until the potential of the cylinder
is more than 300 volts above the wire, and with 360 volts
the current increases only by one-fifth of its initial value.
These results show that in the previous experiments
with parallel plate electrodes the increase of conductivity
must have been due to negative ions.
|
6. Negative ions generated by various methods have the
| same ionizing power.
Without considering the absolute values of a that
have been found, it is possible to deduce some interest-
ng conclusions as to the properties of negative ions.
Since a is independent of the distance between the
)lates l, the new ions must generate others by collisions
) exactly the same extent as the original ions by which
ey are themselves produced. From this it follows that
le negative ions generated by collisions in any gas must
each case be the same as those produced by the
xternal agencies,
Now when ultra-violet light is used, the ions set free
from the zinc plate are the same for different gases;
hence the negative ions produced by collisions from the
molecules of different gases and vapours must be all
the same, being identical with those set free from the
egative electrode. The gases for which the values of
a have been determined by experiments in which the
14 THE THEORY OF Ionization of GASEs
initial ionization was produced by ultra-violet light, are
air, hydrogen, carbon dioxide, water vapour, hydrochlorić
acid," nitrogen,” argon,” and helium.” A high degree
of accuracy can be obtained by this method, and the
experiments are simpler than those in which Röntgen
rays are used to generate the initial ionization. |
Again experiments with Röntgen rays have been made
with air," hydrogen, and carbon dioxide, and it is º
that the values of a are the same over large ranges o
pressure as those obtained for these gases with º
violet light. Thus under all variations of pressure an
electric force the negative ions produced in gases, either:
by Röntgen rays or by collisions, follow the same changes
in ionizing power as the negative ions set free from a
metal plate by ultra-violet light. It may be concluded,
therefore, that they are all identical, and this is Sup-
ported by independent experiments, which show that
the charges on negative ions produced by various methods
are all equal to a fived atomic charge.”
It is interesting to remark that the positive ions
although they have the same charge" as the negative
* See papers by the author, Philosophical Magazine, June, 190
and April, 1903.
* H. E. Hurst, Philosophical Magazine, April, 1906.
* E. W. B. Gill and F. B. Pidduck, Philosophical Magazine
August, 1908.
* See papers by the author, Philosophical Magazine, February
1901, and by the author and Mr. Kirkby, Philosophical Magazine,
June, 1901.
* See papers on Diffusion of Ions, Philosophical Transactions,
Wol. cxciii., 1899, and Vol. cxcv., 1900, also papers in Wols. lxxx...,
lxxxi., lxxxii. of the Proceedings of the Royal Society.
* Recent experiments show that in some cases positive ions have
twice the atomic charge, but as the charges on positive ions
generated by collisions have not yet been specially investigated,
IONIZATION By NEGATIVE Ions 15
have very different physical properties in gases at low
preSSures. It will be seen from the results obtained,
below that the positive ions must acquire a very much
larger kinetic energy than the negative ions before new
ions can be produced by their collisions with molecules.
This difference between the positive and negative ions
shows that the condition that new ions should be gene-
rated by a collision is not determined by the kinetic
energy of the colliding particle. Thus with equal kinetic
energies the ionizing power of the negative ions is much
greater than that of the positive, and this can only be
Wattributed to their possessing a larger velocity, and,
consequently, also a much smaller mass. Hence, since
hydrogen is one of the gases which have been examined,
it follows that the mass of the negative ion in any gas is
less than that of the positive ion in hydrogen.
A particular instance of this result was previously
established, in the case of negative ions set free from
a metal by ultra-violet light. The ratio of the mass
to the charge on the ion has been found, by Sir J. J.
Thomson, by means of experiments on the effect of a
magnetic field on the motion of the ion," and it was
deduced that the mass of an ion set free by ultra-violet
light was of the order 1/1000 of the mass of a hydrogen
atom.
7. Comparison of the masses of negative ions in liquids and
gases ; corpuscular state.
In this connection it is interesting to observe that
it will be assumed for simplicity that they have single atomic
charges. See papers by the author and Mr. C. E. Haselfoot,
Proceedings of the Royal Society, A. Wols. lxxx., lxxxi., 1908, and
Wol. lxxxii., 1909.
1 J. J. Thomson, Philosophical Magazine, December, 1899.
16 THE THEORY OF IonizATION OF GASEs
****
the ions in liquids are quite different from the ions in
gases, although they have the same charges. Thus, in
a solution of hydrochloric acid, the chlorine atom (which
is the constituent of the molecule with the larger mass)
has a negative charge and the hydrogen atom a positive
charge. In the gaseous state, on the other hand, hydro-
chloric acid conducts at low pressures just as other
gases, and the negative ion is associated with a mass
which is small compared with that of the positive ion.
At first sight the conclusions to which the experiments
on the effects of collisions lead as to the masses of the
ions appear to be inconsistent with experiments on the
velocities of the ions and the rates of diffusion, which]
show that at high pressures there is no great difference
between the apparent masses of positive and negative
ions. In these cases the ions move in the gas as if
comparatively large masses were connected with the
atomic charges, a fact deduced from determinations of
the rates of diffusion, which show, for example, that the
rate of diffusion of ions in hydrogen is Small as compared
with the rate of diffusion of carbonic acid into hydrogen.
The result thus obtained may be explained on the Sup-
position that each ion is accompanied by a group of
molecules which are attracted by the strong electric field
in the neighbourhood of the ion. The number of mole-
cules of the gas which are so affected would be determined
principally by the charge on the ion, so that there would
not be any great difference between the positive and
negative ions as they have the same charges. The
groups of molecules would remain attached to the ions
when they are not moving with large velocities, but when
large forces are acting and the ions move with large
velocities in the gas at low pressure, the effect of the
Ion IZATION By NEGATIVE IONS 17
charge in attracting a molecule would only last for a
short interval, so that for large values of X/p the ions
Would move with their own proper masses. The
tl Insition from one state to the other takes place
gradually as the force rises, and it is probable in the
Case of the negative ions that the corpuscular state
revails even when the force is not large enough to cause
hew ions to be produced by collisions. One result which
points to this conclusion is that when X is small there is
no very rapid diminution of a with the force as might
be expected if the mass of the negative ion tended to
increase.
Other experiments show also that the mass of the
negative ion is much less than that of the positive ion
when the forces acting are too small to produce ions by
collision. Thus experiments on diffusion" of ions in dry
air at low pressure show that when X|p='09, the mass of
the negative ion is much less than that of the positive
ion, whereas even for X=20 and p-1, the value of
a is inappreciable and only attains the value '12 when
X=60.
8. Representation of the values of a by a single curve.
In order to examine the values of a for different
forces and pressures, it is convenient to present the
experimental results in a special way by means of a curve
for each gas. The method leads to great simplification
as the value of a corresponding to any force and pressure
may be obtained immediately from the curve. If a be
determined for a force X and pressure p, and the points
whose co-ordinates are alp and X/p be marked on a
diagram, it will be found that they all lie on the same curve,
1 Proceedings of the Royal Society, 1908, Vol. lxxxi., p. 470.
I.G. C
18 THE THEORY OF IONIZATION OF GASEs
or if the experiments be examined it will be noticed that
a' corresponds to a force X' and a pressure p', then at a
pressure 2p', the value of a will 2a' when the force i f
*X', 2 being any multiplier.
In the later experiments the pressures were chosen so
as to show this by inspection." Thus, with air at à
pressure of 1 millimetre and a force of 350 volts per
centimetre, a-5:25; at 2 millimetres pressure and
with a force of 700 volts per centimetre, a-10'5.
Another example may be taken from the experiments
with hydrogen : i
p=8 mm., X=1050 volts per cnn., az-14-8 i
p=4 mm., X= 525 volts per cm., a- 7.4
p=2 mm., X= 262 volts per cm., a- 3.7 |
The results of all the experiments for any one gas can
therefore be recorded by a table o es of alp and X/p
or by means of a curve” representing on of these ratios as
a function of the other :=f (...) -
}
9. Agreement between the experimºntal results and
ion *— })
the equation pT.f ( te
|
It is easy to see that the #! requires that this
relation should exist between the variables a, X, and p.
In passing through a centimette in a gas an ion
traverses free paths of various lengths between the
collisions. The chance of produéing a new ion by
collision will depend on the velocity at impact, and this
* Many examples may be found in theºesearches publish ki
#. and December, 1
* This result was first obtained from the experiments h
Röntgen rays. See Philosophical Magazine, February, 1901.
the Philosophical Magazine, November, 190
IONIZATION By NEGATIVE IONS 19
is determined by the force X and the length of the path
which is terminated by the collision. The lengths of the
free paths are inversely proportional to the pressure, so
that if the pressure is increased from p to 2p all the
free paths will be reduced to 1/2 of their original value.
If the force X remained unaltered, the velocities on
Jºº.
*9.
L-T
2.
~ Aiſ—
L-T
e
22
2.
º
/.
z
/
2.
Ž
foo aoo 300 4oo 500 GOC 720 &oo soo fooo
X--p.
(X in volts per centimetre, p in millimetres of mercury.)
Figure 6.
collision would be reduced, but if the force is increased
to 2X the velocities will be restored to their original
values and the number of ions arising from a given
º: of collisions will be the same as before. Since
tal number of collisions per centimetre is increased
in the same ratio as the pressure, the value of a will
be increased to 2a, when X and p become 2X and
C 2
20 THE THEORY OF Ion IZATION OF GASEs
2p respectively. Hence the three variables must be
connected by an equation of the form *=f (...). The
p p
experiments confirm this result very accurately for all
the gases, and so afford an important verification of the
theory.
3 J’.
: |2: Jºž
f 21 lºſº
21.2%
2.
y > 2.
/1 -º - -
2O 4-O ©o 3O WOO /20 /4 o Woo /8o &OO &2O
X--p.
(X in volts per centimetre, p in millimetres of mercury.)
Figure 7.
The curves representing the values of alp corre-
sponding to Some of the larger values of X/p are given in
Figs, 6 and 7. -
The values of alp for the smaller values of X/p are
given in Fig. 8 for some of the gases.
IonizATION By NEGATIVE IONS 21
10. Properties of the curves *=f ().
p p
The properties of the curves are in general agree-
ment with the indications of the theory. For simplicity
the pressure may be taken as unity which has been fixed
as the pressure due to 1 millimetre of mercury. The
21 LT
2. ><
92 || Lºſ
Z || Dr.
21 iſ T
/
Z ||
A 22
º
Ží a'OO 3OO 4CAO Tjø
X+p.
(X in volts per centimetre, p in millimetres of mercury.)
Figure 8.
co-ordinates of any point then represent the electric
force X and the number a of molecules of the gas at a
millimetre pressure that are ionized by a single ion in
moving through one centimetre under this force. For
small forces the quantity a practically vanishes; the
ions in this case do not acquire sufficient velocity along
any of their free paths to generate others by collisions.
22 THE THEORY OF IONIZATION OF GASEs
As the force increases a acquires a small value and
Subsequently increases still further, the ions during these
stages acquiring sufficient velocity along some of their
free paths to produce others by collisions. With the
Smaller forces such velocity is acquired only along the
longer free paths, but with large forces the number of
paths along which an ion gains this velocity becomes
more numerous. When a becomes large its rate
of increase with the force diminishes, and finally
a approaches a maximum value which should be
attained when a new pair of ions is produced at each
collision.
11. Maximum value of a ; mean free path of negative ions.
The maximum value cannot be obtained experi-
mentally by these methods, as sparking would ensue.
when large forces are used, unless the amount of gas
between the electrodes is very small, and in this case
the calculations would be inaccurate, as the distance
between the plates would not be large compared with
the free paths. The maximum values of a for a pressure
of 1 millimetre may nevertheless be deduced from the
curves by the following method.
Let it be assumed that one pair of ions is formed when
the velocity w at collision exceeds a certain value. Also
t it be supposed that the velocity of a negative ion is so
imuch reduced by collision with a molecule that it prac-
tically starts from rest along its new path. An ion will
then acquire the requisite velocity under the force X
when it travels freely along a distance y such that
mni”
2
eXy->
IonizATION By NEGATIVE Tons 23
or when Xy exceeds a certain potential W. The value of
a corresponding to X represents, therefore, the number
of free paths per centimetre which exceed the value
gy–W/X. It is easy to express a as a function of X
and the mean free path. Let N be the total number of
collisions that an ion makes with molecules in travelling
through a distance of 1 centimetre in a gas at 1 milli-
metre pressure, so that * is the mean free path. Let m
be the number of paths per centimetre which exceed the
distance y. In going through the element dy of these paths
the number of collisions will be proportional to ndy,
hence – drº-knay where k is a constant. On integrating,
this equation gives n=ce-" , the constant c being
equal to N, which represents the number of paths which
exceed the value y=0. The constant k may be found
in terms of the mean free path, for since —ydn is the
sum of the paths whose lengths are between y and
y-H dy, the sum of all the paths in 1 centimetre is
—ſpan=| NkyeT"dy=1,
which gives k=N.
Hence the number of paths per centimetre which exceed
the distance y is N.-Ny. which is therefore the value of
a when v=.
The connection, therefore, between a and X when
- tº _NY g C. NVP)
p=1 is a-Neº X (or in general t-Neº- 3) where N,
- ſp f
the maximum value of a, is the number of collisions
that an ion makes in travelling a distance of 1
24 THE THEORY OF IonizATION OF GASES
centimetre in the gas at 1 millimetre pressure, and
the constant W represents the potential difference
between the ends of a path along which an ion
acquires sufficient velocity to generate others by
collisions.
12. Comparison of the values of a with the
g _NV
expression Neº X.
The following tables give values of a for the gases
at 1 millimetre pressure, the forces X being expressed
in volts per centimetre, and also the values of the
- NW
quantity Neº X, the constants N and V having been
chosen so that the formula should agree with the
experimental results for the larger forces.
The values of a corresponding to any other pressure
may be obtained by taking the numbers to indicate X|p
and alp instead of X and a respectively.
TABLE I.
AIR.—N=14:6; W=25'0.
X. 1000 800 || 700 | 600 || 500 | 400 || 300 | 200 100
a found -
experimentally. 10.5 9-3 || 8-7 || 7-9 || 7-0 || 5'82 || 4:4 || 2:6 || 72
NV
Ne X 10.1 | 9-3 || S-6 || 7-9 || 7-0
5
8
5
4’3 2'34 '38
IonizATION By NEGATIVE IONS
25
TABLE II.
NITROGEN.—N=12:4; W=27-6.
X. 600 500 400 300 200 100
a found
experimentally. || 7-0 || 6-2 5' 2 3'95 || 2:3 •42
_NV
Ne X 7:0 6.25 5°3 3.95 2° 24 '41
TABLE III.
HYDROGEN.—N=5-0; W =26.
X. 400 300 200 100 50 30
a found
experimentally. || 3:7 3°3 2' 62 | 1.36 • 36 ‘OS
_NV .
Ne X 3-64 || 3:24 2.62 1'35 • 37 '066
TABLE IV.
CARBON DIOxIDE.—N=20; W=23-3.
X. 1200 | 1000 | 800 || 700 60() 500 | 400 300 200 | 100
a found
experimentally. 13-7 | 12:6||11–0|| 10:2| 9-1 || 7-8 || 6’4|4-8 2-8 '82
_NV -
Ne X 13-6 | 12°5 || 11-2 || 10-3 || 9-2 || 7-9 || 6’2 || 4-24 2-0 20
26 THE THEORY OF IONIZATION OF GASEs
TABLE W.
HYDROCHLORIC ACID.—N=22:2; W=16.5.
|X. 1500 || 1000 | SOO || 700 | 600 500 | 400 || 300 |200|| 100
a found
experimentally. 17.5 15.4 ||14-0||1310||11.9 10-5 | 89 || 6-8 || 4-1 || 1:21
_NV
Neº X 17°5 15°4 || 14-0 || 13-2 || 12-0 |10-65| 8-9 || 6′5 || 3-5 || 57
TABLE WI.
WATER WAPOUR.—N=12.9 ; W=22'4.
X. 1000 || 900 | 800 || 700 | 600 500 | 400 || 300 |200|| 100
a found
experimentally. 9-7 | 9.4 9-0 || 8:5 |7-95 || 7-2 || 6′35 | 5-2 || 3:6 | 1.31
NV
Ne TX 9-7 || 9-35 | S-9 || 8'54 || 7-96 || 7-2 || 6-25 || 4-9 || 3-0 || '71
TABLE VII.
ARGON.—N=13:6; W=17-3.
X. 600 500 | 400 300 | 200 100 50
a found
experimentally. 9-2 || 8.5 || 7-5 | 6’2 || 4:4 || 2:0 '58
NV
Ne TTX 9.2 8' 5 7.5 6'2 4'2 1°3 • 12
IONIZATION By NEGATIVE IONS 27
TABLE VIII.
HELIUM.—N=2'4; W=14-5.
X. 60 50 40 30 20 10
a found
experimentally. 1:35 | 1:20 || 1:00 77 “40 • 12
_NV
Ne TX 1'34 || 1:20 | 1.01 •75 '42 •074
The above tables include the results of the most
accurate experiments, and are probably correct to within
2 or 3 per cent. The values of a/p for the large values of
X/p given in some of the papers are not very reliable, so
they have not been included. The difficulty arises in
these cases from the action of the positive ions. When
the positive ions are not producing any appreciable effect
the value of a can always be determined accurately by
taking the ratio of the currents at two different distances
between the plates, even when the free path of the ion
is a considerable fraction of the Smaller distance, as it is
only the change of distance which is involved in the
calculation. With large forces and small pressures,
however, it is necessary to take into consideration the
action of the positive ions. In this case the expression
for the ratios of the currents at different distances
involves the absolute distance, and for accurate deter-
minations it is desirable to use distances between the
plates which would be large compared with the free
paths. But there is a limit to the extent to which such
experiments can be carried, for with large forces and
small pressures sparking takes place when the distance
28 THE THEORY OF IonizATION OF GASEs
exceeds a certain critical value. On this account the
determinations of a/p for values of X/p larger than those
which are given in the above tables have not been
obtained with a high degree of accuracy.
The tables show that in all cases the theory is in
accordance with the experimental results over a wide
range when the forces are large. For the Smaller forces
the values of a are larger than those given by the
expression Ne—º. There are many causes which may
contribute to this difference, but it is impossible to give
any reliable explanation of it as so little is known as to
what exactly takes place when an ion collides with a
molecule.
It has been assumed that one effect of a collision is to
reduce the velocity of an ion to a relatively small value,
an assumption which seems reasonable when the velocities
are so large that the ions tend to produce changes in
the molecules by impact. In these cases a large
proportion of the kinetic energy of the ion would be
absorbed by the molecule, so that after a collision the
ion would begin to move under the electric force with
a small velocity. But when the velocities on impact
are much below that which is required to ionize a
molecule, there is no reason for supposing that a large
proportion of the kinetic energy of the ion is lost on
collision; it may thus acquire the critical velocity more
frequently than is shown by the calculations for the
smaller forces. It is also probable that there are other
circumstances besides the velocity of the negative ion
which determine what takes place on collision, and if so,
that ions may be produced on some few occasions when
the negative ion collides with a velocity less than that
IONIZATION By NEGATIVE Ions 29
corresponding to the potential fall W. A theory worked
Out on these lines would make the expression for a in
terms of X very complicated, and would not add much
to the information which is obtained by comparing the
experiments with the simple formula involving one
exponential term.
13. Molecular dimensions deduced from the mean free
paths of negative ions.
The mean free paths of the negative ions in the
different gases at 1 millimetre pressure are the recipro-
cals of the numbers N. As might be expected, the mean
free path is longer in hydrogen than in other gases, and
in general the sectional area of the molecule which is
proportional to N is larger in the heavier gases. The
most notable exception occurs with helium. Although
the density of this gas is twice that of hydrogen, a
negative ion makes twice as many (5–3–2'4) collisions in
travelling a given distance in hydrogen as it does in
helium at the same pressure.
It is interesting to compare the mean free paths of
the ions as obtained by this method with the mean free
paths of molecules of gases obtained from the determina-
tions of the viscosity. If a collision occurs between two
molecules when their centres come within a distance
20 from each other, then a negative ion, being of Small
dimensions, would on this view collide with the molecule
when it came within a distance a of the centre. It will,
however, be seen that the ion makes fewer collisions
than would occur under these circumstances, or, in other
words, that the ion must come within a distance R from
30 THE THEORY OF Ionization of GASEs
the centre of the molecule in order that ionization may
result from the collision, the distance R being less than
0. For if m is the number of molecules in a cubic
centimetre of a gas at 760 millimetres pressure and
15° C., then TE*m=NX760, and the quantity troºm may
be deduced from the coefficients of viscosity. The
values of RX 10° and a × 10° in centimetres for the
different gases are given in the following table, the
number m being taken as 3 × 10” which is the value
deduced from the recent investigations by Professor
Perrin.” The values of a have been deduced for the
Same value of m from the expression for the viscosity
given by Professor Jeans.”
TABLE IX.
Air N, H, CO, HCl H.O | A He
N 14.6 12°4 || 5-0 |20-0 || 22°2 | 12:9 13-6 || 2.4
R × 108 1-08 || 1:00 '63 || 1:26 || 1:33 | 1.02 | 1.04 || 44
or X 108 1:64 | 1.67 || 1:18 || 2:00 | 1.93 | 1.92 | 1.60 | 1.04
14, Application of the theory to Stoletow's experiments.
The experiments which have been described were
arranged expressly for the purpose of obtaining results
from which the values of a could be easily obtained,
as no reliable investigations had previously been made
* Jean Perrin, Comptes Rendus, Octobre 5, 1908.
* Jeans, “Dynamical Theory of Gases,” p. 250. The values of
10° a given in this treatise on p. 251 correspond to mE4×10°.
IONIZATION By NEGATIVE Ions 31
from which these values could be deduced. Several
experimental investigations, however, have been made
by other physicists of the conductivity produced by
external agencies in gases at low pressures. Among
the earliest were those by Stoletow," who showed
that there are large variations in the currents
between parallel plates due to changes in the force
and pressure when ultra-violet light acts on the negative
electrode.
Most of the effects which were observed may obviously
be attributed to the development of large currents by
the collisions of the ions with the molecules of the gas,
and a detailed examination of all these experiments is
unnecessary. One result, however, obtained by Stoletow,
may be mentioned, as it requires some consideration to
See how it can be explained by the theory.
It was found that when the distance and potential fall
between the electrodes is constant, the current increases
as the pressure of the gas is diminished, that at a certain
pressure P the current attains a maximum value, and
that when the pressure is reduced below the value P the
current diminishes. The pressure P, at which the
current attains the maximum value, is proportional to
the force X and independent of the distance between the
plates.
The following table is given by Stoletow for air,”E being
the electromotive force between the two plates, the unit
being the electromotive force of a Clark cell, l the distance
between the plates in millimetres, and P the pressure,
* Stoletow, Comptes Rendus, t. Cvii., p. 91, 1888; Journal de
Physique, Série ii., t. ix., 1890.
* Stoletow, Journal de Physique, loc. cit,
32 THE THEORY OF IonizATION of GASEs
in millimetres of mercury, for which the current is
a maximum.
TABLE X.
Pºv 1 a
E ! P |E × 10
165 • 25 25°3 383
165 •47 13° 5 384
65 •47 5' 3 383
100 '83 4-7 389
65 •83 3.0 383
60 '83 2.8 386
65 1.91 1-3 382
65 3.71 •67 382
40 3' 60 ‘43 387
The agreement between the figures in the last column
shows that P is proportional to the force E/l.
When E is expressed in volts and l in centimetres, the
value of the pressure for which the current is a maximum
is P= 3.
372
distances between the plates were small, so that there
can be no appreciable effect due to ionization by the action
of positive ions. There may be a small variation of the
number of ions set free from the plate due to the varia-
tion in the pressure, but the principal variations must
be due to the ionization produced by the motion of the
negative ions. The pressure for which the current is a
maximum should therefore be practically the same as
that at which a becomes a maximum. This pressure can
be deduced by differentiating the formula, a=|f(;).
In these experiments the potentials and the
IONIZATION By NEGATIVE Ions 33
with respect to p and equating ; to zero. The
following condition for a maximum value of a is thus
obtained:
f(;) _X |(})=0.
p/ p p
It is obvious that P is proportional to X since the
equation involves only the ratio X/p. In order to find
the value of X/P which satisfies this equation it is not
necessary to know the form of the function f as the
required ratio may be found for each gas from the
curves (figures 6 and 7). The a. and y co-ordinates of any
point on one of the curves being X/p and alp respectively,
it follows from the above condition that y=% al
the point corresponding to the maximum value of a.
}=} shows that the tangent to the
30 00
curve at the point (a,J) must pass through the
origin, so that a. and y are obtained by drawing a
tangent to the curve from the origin. In the curve
for air the point of contact of the tangent is near
the point where Xp has the value 870, and this
value is in good agreement with the result obtained
by Stoletow. It is not easy, however, to judge the
exact point at which the tangent touches the curve,
so that the following method is perhaps more
satisfactory.
The equation
* See paper by the author, Philosophical Magazine, February,
1901; also J. J. Thomson, “Conduction of Electricity through
Gases,” 1903 edition, pp. 233, 342, where the same result is given
in a different notation.
I.G. D
34 THE THEORY OF IoWTZATION OF GASES
15. Determination of the pressure corresponding to the
maximum value of a for a given force.
Since the values of alp determined experimentally
agree very closely with those given by the formula
NW
:=N* for the larger values of X/p, the pressure
for which a is a maximum when X is constant
may be obtained by equating to Zero the differential
NVp
coefficient of peºx with respect to p. The equation
in X|p thus obtained is 1—º–0.
X
Thus for any gas the pressure P, which gives the
greatest conductivity with a force X, is P=v, and
the maximum value of a is NPs "= X.
We
The factor by which the original number of ions is
multiplied owing to the effects of collisions depends on
al or #=% where v is the potential difference
between the plates. Thus in the case in which no ions
are set free from the negative electrode the number that
arrive at the positive electrode has a maximum value
noe", independent of the distance l between the plates
when a given potential difference v is established between
them. In order that this equation may hold accurately
it is necessary that the negative ions should have the
mean velocity corresponding to the force X along the
whole distance l. This condition will not be established
until the ions have travelled a short distance from the
Ion.TZATION By NEGATIVE IONS 35
negative electrode; beyond this it holds, so that the
factor by which the current is multiplied by increasing
the distance from li to lo is given accurately by
X(lº—ll)
the term e Ve º and this result is confirmed by
experiment. The potential difference between the
plates when the first distance li is used should
not be less than W, or l1 should be greater than
W/X, and for the larger distances the potential lx.
should not approach the sparking potentials, other-
wise the effects produced by positive ions would be
appreciable.
Table XI. gives the values of W and NW for the different
gases.
TABLE XI.
AIR N2 H2 CO2 HC1. H2O A. He
V 25 27-6 || 26 23-3 || 16.5 22°4 || 17-3 || 14-5
NW 365 || 342 || 130 || 466 || 366 289 || 235 | 34-8
The greatest conductivity is obtained with a force X
X e
NW, X being
For air this theory
when the pressure in millimetres P=
expressed in volts per centimetre.
gives P=; which is in good agreement with Stoletow's
- X
result P=#3.
The maximum value of a is 2. Or al=#, v, the
& eW eV
potential difference between the plates, being expressed
in volts, and l in centimetres.
D 2
36 THE THEORY OF IonizATION of GASEs
It thus appears that with a given potential v the best
conductivity is obtained with those gases for which V is
Small, as with helium, hydrochloric acid, and argon.
The second of these gases would give results in accord-
ance with the formulae over large ranges of potentials,
since positive ions do not produce appreciable effects in
this gas until fairly large potentials are attained.
16. Comparison of the velocities of ions and molecules.
It is interesting to see how the velocity acquired by
an ion moving freely under an electric force compares
with the velocity of agitation when its motion is in
equilibrium with the surrounding molecules of a gas
at normal temperature, 15° C. This point has been
examined by Professor Schuster in one of his Bakerian"
lectures, and the following is essentially the method
adopted.
If v is the velocity acquired by an ion of mass M in
travelling freely along a path over which the potential
º g 1Ma-º.
difference is W, then #Mv T300'
expressed in electrostatic units, and V in volts. The
velocity of agitation w is given by the equation
#Mmu°–10° where m is the number of molecules in a
gas at normal pressure and temperature. Hence
the charge e being
4)2
#27V, 0.9.
me being 1:23x10".
Hence in order to generate ions by collision it is
necessary that the velocity of the negative ion should be
1 Proceedings of the Royal Society, Vol. xlvii., 1890.
IONIZATION By NEGATIVE Ions 37
much greater than the velocity of agitation that it would
have in a gas at 15° C.
The velocity acquired by a negative ion under the
action of the potential difference W may be also compared
with the velocity of the 3 particles emitted by radio-
active bodies, or with that of the cathode rays. In the
former case some of the particles travel with a velocity
of 2.85 × 10" centimetres per second, and in the latter
case the velocity is much smaller, and depends on the
circumstances under which they are generated. The ratio
of the charge e to the mass p of the particle has been
found by Simon" to be 1.86 X 107 for cathode particles
travelling with an average speed of 7 × 10° centimetres
per second, e being expressed in electro-magnetic units.
The potential required to impart a velocity of 7 × 10°
centimetres per second to a particle for which the ratio
has this value is 13,000 volts, which is large compared
with the Voltages required to generate ions by collisions.
Taking the latter to be of the order of 20 volts and the
ratio to have the above value for negative ions
generated by collisions in gases, the velocity they require
to generate others by collisions is of the order of
2.7 × 10° centimetres per second.
* Simon, Wied. Annal., p. 589, 1899.
CHAPTER II
IONIZATION BY POSITIVE IONS
17. Conductivity between parallel plates when positive
and negative ions generate others by collisions.
The effects produced by the motion of positive ions
may be deduced from determinations of the currents
that pass between parallel plates while ultra-violet light
is falling on the negative electrode, if the forces and the
distances between the plates are sufficiently large.
It has already been shown that when X/p and l are
Small, and l is varied whilst X and p are kept constant,
the number of negative ions reaching the positive
electrode is nº”, where a is a constant depending
on X and p. For large values of X/p and l the number
of negative ions reaching the positive electrode is
greater than nº". showing that some other form of
ionization has come into play. This stage is attained
even when the potential between the plates is much less
than that required to produce a spark. It will be seen
from the following investigations that all the features of
the new process of ionization can be explained on the
supposition that it arises from the action of the positive
ions.
Further, it is obvious that if both positive and negative
ions produce others in sufficient numbers, a current
would be obtained which would last after the supply
IONIZATION By PosLTIVE Ions 39
from the negative electrode was cut off, and a continuous
discharge would ensue. The investigations show how
the potential required to produce a continuous discharge
may be found on the assumption that the whole ioniza-
tion is produced by collisions of positive and negative
ions in a gas, and, as will be seen, there is a very accurate
agreement. Over a large range of pressures between the
potentials thus calculated and the sparking potentials
determined experimentally.
In making an investigation of the currents which
would be produced between parallel plates when both
positive and negative ions generate others by collisions,
the results of the experiments may be anticipated, and
it may be assumed that the positive and negative ions
produced by the impact of a positive ion with a neutral
molecule are identical with the positive and negative ions
produced by the impact of a negative ion. In applying
the theory, therefore, it is necessary only to consider one
kind of positive ion and one kind of negative ion in each
gas, but the positive ions, unlike the negative ions, are
most probably different in different gases. -
If a number no of negative ions start from the negative
plate and move through a distance l from the positive
plate they will generate others in the gas, and a number
nº” will reach the positive electrode. Thus n(*–1)
positive ions are produced in the gas and move in
the opposite direction. When these also ionize the gas
the total number n of negative ions that reach the
positive electrode exceeds nº".
Of the number n—no of each kind generated in the
gas, let r be produced in the layer of gas between the
negative plate and a parallel plane at a distance a, and
40 THE THEORY OF Tonization of GASEs
let r" be produced in the layer between this plane and
the positive electrode.
Then m=n0+r-Hr'.
Let a be the number of ions produced by a negative
ion in moving through one centimetre of the gas.
Let 3 be the corresponding number for a positive ion.
The number of ions dr generated between the two
planes at distances a, and ac-H da; is given by the
expression dria(no-Hr)da;+3r'da, since the number of
negative ions travelling through the length da; is no-Hr,
and these produce a (no-Hr)da, and the number of positive
ions passing through the same element of length da; is
r", and they produce 3r'da. Substituting n—no–r for r"
the equation for r becomes
#=(a-b) (nºr)+én,
which on integration gives
no-Hr- —ºº-ºº:
The constant c is determined by the condition r=0
when ac-0.
Hence the value of r is given by the equation
Ion.TZATION By PosſTIVE Ions 41
and the value of m in terms of the other constants is
obtained by substituting for r its value n—no when ac-l,
l being the distance between the plates. Thus
a—3)l
70 - ??, (a –8). 3) e
Sºy (a–3)l
a—3e
This gives the number of negative ions n arriving at
the positive electrode when no negative ions start from
the negative electrode, and both positive and negative
ions generate others by collisions with molecules of the
gaS.
It will be noticed that when the distance between the
plates l has a certain value S given by the equation
(a — 8)S_ — 1 ... / Q.
a — 3e =0, Of S=== log (...)
the denominator of the fraction in the expression for n
becomes Zero, so that n becomes infinite. This shows
that a current would continue to flow indefinitely after
the supply of the negative ions no from the surface of
the negative electrode ceases. The importance of this
conclusion in connection with sparking potentials will be
considered later."
For distances between the plates shorter than S the
denominator of the fraction expressing n is positive, and
the current becomes Zero after the light ceases to act on
the electrode. The values of n are then finite, but greater
than the number of ions nº." which would reach the
positive electrode if the negative ions alone produced
others by collisions.
* See section 23.
42 THE THEORY OF IONIZATION OF GASEs
18. Agreement between experiments and theory.
To illustrate the theory the experiments" may be
quoted in which the currents between two parallel plates
in air at one millimetre pressure were determined, the
electric force being 350 volts per centimetre in each
case. The distance between the plates l is given in
centimetres in the first line of Table XII. ; the second
line gives the charge q acquired by one of the plates
while the light was acting on the negative electrode for
a fixed time; the third and fourth lines give the values
a—3)l
al and (a–3). 3)
a—Be (a—3)l
and 3–'0141.
Of
respectively, where a=5’25
The values of al agree only with the experiments at
the shorter distances, but the numbers in the fourth
line show that the positive ions (although 3 is small
compared with a) produce an appreciable effect when
the plates are about 6 millimetres apart and have a very
large effect for larger distances.
TABLE XII.
l 2 || 4 || 6 || 8 || 10 || 1:1
Q 2-86 8-3 || 24-2 || 81 | 373 || 2:250
al 2.86 8.2 23°4 | 66°5 190 322
_a\,(a- 8)!
*::= 2.87 | 83 24.8 80 || 380 2150
a — 8e
i See papers in Electrician, April 3, 1903, and Philosophical
Magazine, November, 1903, on “The Genesis of Ions by the
Motion of Positive Ions in a Gas.”
IonizATION By PosLTIVE TONs 43
The currents are given in arbitrary units, and the
values of a and 8 may be found from the ratios of the
currents at three distances. The agreement between the
expression for n involving a and 3 and the currents over
Several distances shows that the experiments are
accurately explained by the theory.
The above example of currents developed in air,
in which p=1, X=350, a-5:25, and 3–'0141, may
be compared with the results of an experiment in
which p=2, X=700, since according to the theory
a and 8 should in this case have twice the above
values, and the current corresponding to any distance
should be the same as that at double the distance in the
previous experiment.
Table XIII. gives the values of the currents q in this
C8,S6. -
TABLE XIII.
l •1 •2 •3 | "4 •5
Q 2-9 8-3 23'S S0 374
(a–8)e (a. *** 8)!
*====== 2.87 8'3 24'6 SO 3S0
a — 8e (a — 8)!
The third line gives the calculated values when
a=10-5 and 8–'0282. The currents at 1, 2, 3, 4,
and 5 millimetres are the same as those in the
previous experiment when the distances were double
these values.
For comparison with other gases the following examples
may be taken from among the experiments which have
been made with hydrogen.
44 THE THEORY OF Ionization of GASEs
TABLE XIV.
HYDROGEN.—p=2 mms. ; X=262 volts per Crm. ;
a=37; 6='041. -
! 4. •6 •8 1-0
Q 4'6 10-0 22.7 65
— Q)e(a-3)!
e-bº. 4-55 9-9 22.7 67
0. –3e (a - 8)
TABLE XW.
HYDROGEN.—p=4 mms. ; X=525 volts per cm.;
a=7'4; 8–'082.
l •2 •3 '4 •5
Q 4°6 9-9 22-3 66
(a *g 6)." 8)!
4'55 9-9 22.7 67
a — 8e (a – 8)!
Several experiments of this kind have been made with
a number of gases, and they all show that when X and p
are multiplied by a factor the values of both a and 3
are multiplied by the same factor. This agrees with
the theory which requires that the connection between 3
and the variables X and p should be of the same form
as the relation “ =f(...) So that := (;). The values
p p O ſ)
of 8 may therefore be obtained from a curve whose
ordinates are 8/p and X|p.
A point to be noticed is that with a given potential
IONIZATION By POSITIVE Ions 45
and a given quantity of gas between the plates the
current is independent of the distance l. This follows
immediately from elementary considerations, for when
the number of molecules between the plates is constant
an ion makes a fixed number of collisions, and the fall
of potential along each path depends only on the
potential difference between the plates. Under these
circumstances the total number of ions generated by
positive and negative ions must be independent of the
distance between the plates.
19. Curves representing 3/p as a function of X/p; com-
parison of effects produced by positive and negative ions.
The values of 3/p in terms of X/p for the different
gases are shown by the curves figure 9. The numbers
obtained for 8 in argon and nitrogen are practically the
same, so that a single curve is given for the values of
Blp for these two gases."
It will be observed that for the same force and
pressure 3 is much less than a in any gas. It can be
The values of 8/p for helium are probably not as accurate as
those for other gases. They have been deduced from the values of
a and a few determinations of the Sparking potential obtained for
pure helium (E. W. B. Gill and F. B. Pidduck, Philosophical
Magazine, August, 1908). When the Sparking potential W is
known for a given pressure p and distance S between the plates,
then s—º and the corresponding value of a can be found from
the curve, giving aſp in terms of Xlp. The value of 8 may then
be deduced from the equation 2–6,(*-8) *=0. The presence
of impurities in helium gives rise to comparatively large errors in
the values of a and 3, and it is intended to make further experiments
in which particular care will be taken to maintain the helium free
from impurities.
46 THE THEORY OF Ionization of GASES
shown from the results that have been obtained that {
positive ions are much less efficient than negative ions
in producing new ions by collision even when they have
the same kinetic energy. For, assuming that the
positive ions are large compared with negative ions and
of the same order of magnitude as the molecules of the
too zoo 3oo 4oo 3oo Goo ſoo boo aoo tooo too 1200 13oo 17oo 13.
Y-–p.
(X in volts per centimetre, p in millimetres of mercury.)
Figure 9.
gas from which they are derived, each positive ion would
make four times as many collisions with molecules as a
negative ion in travelling the same distance in the gas.
The free paths of a negative ion in a gas at pressure p
will therefore be the same as the free paths of a positive
ion in a gas at pressure p/4, and if the same electric force
Ion IZATION By PosLTIVE IONS 47
X be acting in each case the energy acquired between
the collisions will also be the same. The relative
ionizing powers may therefore be found by comparing
the value of a in a gas at a certain pressure with the
value of 3 in the gas at a quarter of the pressure when a
constant force X is acting. Take as an example the
case of hydrogen. With the force X=50 volts per
centimetre and pressure 1 millimetre a="35. The
curves giving 3 show that ;-000 when ; =200, so
that 3=-022 when X=50 and p-25. Hence when
positive and negative ions collide an equal number of
times with molecules and acquire equal kinetic energies
along their free paths the negative ion produces sixteen
times as many ions as the positive ion. The ratio of the
activities of the positive and negative ions as estimated
in this way varies with the force X. Under a force of
75 volts per centimetre a=9 when p-1, and 8–'065
when p-'25, so that the effect of the negative ion is
fourteen times that of the positive ion.
The rate of increase of the ionizing power of positive
ions is therefore greater than that of negative ions, and
for very large forces 8/p would theoretically be four times
as great as the maximum value of aſp. But it is
impossible to test the accuracy of the theory on this
point, since the smaller values of 3 do not supply
sufficient data for the purpose, and the larger values
cannot be obtained by the experimental method that has
been used, as Sparking takes place before 8/p rises above
a small fraction. The above figures nevertheless show
that for Small values of the kinetic energy the negative
ions are much more efficient than positive ions in pro-
ducing new ions by collision.
48 THE THEORY OF IONIZATION OF GASEs
If this difference between the positive and negative
ions is due to the relatively large mass of the former, it
is to be expected that the difference would be more
marked in the heavier gases, assuming that the mass of
the positive ion is of the same order as that of a molecule
of the gas. It is interesting, therefore, to make similar
calculations for other gases and to compare them with
the results obtained for hydrogen. For although these
comparisons give no accurate information as to the ratio
of the masses of the positive ions in different gases, the
calculations afford some reason for believing that the
positive ions in other gases are greater than those in
hydrogen.
A matter of importance that arises in making com-
parisons of the relative activity of positive and negative
ions in different gases is to decide what force to select in
each case, as the relative activity varies with the force.
The principle which is here adopted is to take in each
case such a force as will give a value of a in the gas at a
millimetre pressure that is a definite fraction of N, the
maximum value of a. Thus in the examples that have
already been given for hydrogen at
50 volts per centimetre a="35=-07 N,
75 , , 9 3 a= '9–18 N,
N being equal to five for hydrogen.
Taking the forces that give a="16 N in each case,
the following numbers may be obtained from the experi-
mental results. For air N=14-6, and the force which
corresponds to a-14-6X 16=2:38 is 190 volts per
centimetre when p-1. The value of 3 when X=190
and p-25 is obtained from the point on the curve
corresponding to }=10. This gives #–164. SO
Ionization By PosLTIVE Ions 49
that £8–'041. These values of a and 8 are in the ratio
of 57: 1. Again, for argon N=13-6, and
a=2°17 when X=105 and p-1,
6='025 when X=105 and p-25,
giving the ratio a: 8=86:1.
For carbon dioxide N=20 and
a=8°2 when X=215 and p-1,
8='003 when X=215 and p-25
giving the ratio a 3=1070 : 1.
It thus appears that the greater the molecular weight
of the gas the greater the difference between positive and
negative ions as regards their efficiency in producing
new ions by collisions.
In the case of pure helium the range of forces for
which observations of a and 3 have been made is very
Small, and it would not be possible to deduce numbers
from them similar to those found for other gases at the
point at which a-'16 N. The values of a and 3 that
have been found for an impure specimen of the gas
containing about 98 per cent. of helium may be used.
They give the following results:
a="5, X=25, p=1,
(3=-0095, X=25, p="25,
which gives a-52×8.
This result shows that the positive ions in helium are
less efficient than those in hydrogen, but it is remarkable
that the ratio of a to 3 should be nearly the same as
for air.
When comparisons of the relative efficiency of positive
and negative ions in pure helium and in air are made,
using smaller forces which correspond to values of a
I.G. E
50 THE THEORY OF IonizATION OF GASEs
equal to 1XN in each case, a similar result is obtained
a being 708 for both gases.
Again in nitrogen the range of values of 8 does not
admit of comparisons being made at the point a-'16N,
but for a-‘1 × N the corresponding value of 8 gives
a=263.
The relative ionizing powers of negative and positive
ions in different gases thus obtained are—for hydrogen
15:1, air 57:1, argon 86 : 1, carbon dioxide 1070: 1, and
helium 52: 1. The first four numbers may be said to
support the assumption that the positive ions are of the
Same dimensions as the molecules, since the positive
ions differ most from the negative for the gases with
large molecular weights.
CHAPTER III
SPARRING POTENTIAL IN A UNIFORM ET, ECTRIC FIELD
20. Description of phenomena accompanying discharges ;
sparking potential.
The various phenomena relating to the discharge
of electricity through gases at low pressures have been
investigated experimentally by many physicists, and the
principal facts in connection with them have been well
established. With regard to the Sparking potential,
it will be seen that the researches which have been
carried out on the effects of collisions, suggest an
obvious method" of obtaining a simple theory of the
sparking potentials for parallel plates in a gas at
different pressures and also a consistent explanation of
the large difference that exists between the sparking
potential and the potential required to maintain the
discharge at the higher pressures.
The principal features of the phenomena which are to
be considered may be briefly stated as follows:—When
a potential difference is established between two elec-
trodes in a gas, the gas behaves as an insulator unless
the potential exceeds a certain definite value, which is
called the sparking potential. The sparking potential
is very sharply defined, and it has been found that the
1 See papers by the author in Electrician, April 3, 1903, and
Philosophical Magazine, November, 1903, and March, 1905.
E 2
52 THE THEORY OF IonizATION of GASEs
gas insulates well for a potential which is a few volts
less than that required to initiate a discharge. When
the applied potential is equal to or greater than the
Sparking potential, the insulation breaks down, and a
comparatively large current, accompanied by a glow,
passes through the gas. For a fixed distance between
the electrodes the sparking potential changes with the
pressure of the gas. At high pressures the potential is
large; it diminishes with the pressure and reaches a
minimum for a certain critical value of the pressure.
i
Pressure.
Figure 10.
When further reductions are made in the pressure the
sparking potential rises rapidly, and attains very high
values for the lower pressures.
The variations in the Sparking potential may be illus-
trated by the curve, P Q R, figure 10, O A being the
critical pressure and Q A the minimum sparking
potential.
When parallel plates are used as electrodes the spark-
ing potential depends only on the quantity of gas between
the electrodes, which is measured by the product of the
pressure p and distance S between the electrodes. This
SPARKING PotRNTIAL IN A UNIFORM ELECTRIC FIELD 53
is known as Paschen's' law, and it has been found by
Carr” to hold for large ranges of low pressures in the
neighbourhood of the critical pressure. -
All the experimental results at different distances and
pressures may therefore be represented by a single curve,
whose ordinates are the sparking potential and the pro-
duct of the pressure and distance between the plates. In
particular, it will be noticed that the critical pressure
corresponding to the minimum sparking potential is
inversely proportional to the distance between the plates.
21. Potential required to maintain a discharge.
When a spark discharge takes place between parallel
plate electrodes the gas immediately becomes a con-
ductor, and allows a comparatively large current to pass
through. When the pressure is greater than the critical
pressure the potential required to maintain the current
is very much less than the sparking potential, and the
potential difference between the electrodes diminishes as
the current increases. When the pressure is less than
the critical pressure a small current is produced when the
sparking potential is applied, and in order to increase
the current it is necessary to increase the potential
difference between the electrodes.”
The potential required to maintain a current may be
represented by the dotted line P' Q R'. At a point B,
where the pressure is greater than the critical pressure,
the gas insulates until the potential is equal to R. B. If
this potential be applied by a battery of cells through
a resistance, immediately the spark occurs a large current
* Paschen, Wied. Ann., xxxvii., 1889.
* W. R. Carr, Philosophical Transactions A, Wol. cci., pp. 403—
433, 1903.
* J. A. Brown, Philosophical Magazine, September, 1906.
54 THE THEORY OF IonizATION OF GASEs
passes through the gas, and the potential difference at the
electrodes falls to R' B.
The following example of an experiment" made with
air at 4:31 millimetres pressure between parallel plates
at 8 millimetres apart illustrates the phenomena. The
gas insulated when a potential difference of 601 volts
was established between the electrodes, but a spark took
place when the potential was raised to 603 volts, and a
current of ‘0052 ampère was established, which was
maintained by a potential fall of 350 volts between the
electrodes, the remainder of the 603 volts being taken
up in sending the current through a large external
resistance. In this case the difference between the
sparking potential and the potential required to maintain
the current was 253 volts.
22. Properties of pointed and cylindrical electrodes.
When electrodes of different shapes are used, as
when a discharge takes place from a point to a plane, the
sparking potential depends on the direction of the force.
For pressures above the critical pressure the smaller
potential is obtained when the point is negatively
charged, and for pressures below a certain value the
Smaller potential corresponds to the opposite direction
of the electric force.
Thus the variation of potential with the pressure may
be represented by a curve similar to P Q R when the
point is positive, and when the force is reversed the
corresponding curve has a relative position similar to
P' Q R'. The two curves intersect near the point
corresponding to the minimum sparking potential, which
varies according to the shape of the electrodes.
| Philosophical Magazine, December, 1904,
SPARKING PotRNTIAL IN A UNIFORM ELECTRIC FIELD 55
Exactly similar phenomena are observed when the
discharge takes place through a gas in the Space between
two coaxial cylindrical electrodes: for pressures above
the critical pressure it is easier to produce a Spark when
the inner cylinder is negative, and for lower pressures
the smaller sparking potential is obtained when the
inner cylinder is positively charged.
23. Condition for sparking in a uniform electric field.
The sparking potential between parallel plate
electrodes can be deduced immediately from the formula
__ (a-b),("-8)
?? E700
O. - 3e (a — 8) l
which represents the number of ions n reaching the
positive electrode when no negative ions are started by
ultra-violet light from the negative electrode. The
quantity n increases as the distance l between the plates
increases, the force X being maintained at a constant
, already obtained in section 18,
value, and when 6.0-8 )l becomes equal to a the
denominator of the above fraction vanishes, and m becomes
infinite. A current will therefore continue to flow for an
indefinite time after the light ceases to act when the
distance between the plates attains the value S given by
the equation
a -6."-P8–0.
Thus the sparking potential S X is known for the
particular distance S and pressure p of the gas.
In order to test the theory experimentally, the sparking
distances S corresponding to given values of X and p
were found by means of the formula S= # log (#)
56 THE THEORY OF Ionization of GASES
a and 8 having been found in terms of X and p by
the methods that have been described.
The plates were then set at the exact distances apart S,
and the potential W required to produce a spark was
determined. The results of the experiments' made with
air, hydrogen, nitrogen, carbon dioxide, argon, and helium
are given in Tables XVI., XVII., XVIII., XIX., XX., and
XXI., and they show the close agreement that exists
between the theoretical sparking potential SX and the
sparking potential W determined experimentally. The
product of the pressure p and distance S is given
in the last column of each table to illustrate Paschen’s
law.
TABLE XVI.
AIR.—Sparking Potentials.
X. p S SX W pS
1050 8 •765 803 803 6-12
1400 8 "431 601 603 3:45
1050 6 • 572 601 604 3'43
700 4 •871 610 615 3°48
1050 4 '454 477 480 1-82
525 2 '91 481 488 1.82
700 2 • 575 403 407 1-15
350 1 1.13 395 398 1°13
437 1 •832 364 365 '83
350 '66 •965 338 340 •64
437 '66 •766 335 336 • 505
1 The sparking potentials for air and hydrogen are taken from
the papers by the author, Philosophical Magazine, November, 1903,
and by the author and Mr. Hurst, Philosophical Magazine,
December, 1904; those for carbon dioxide and nitrogen are taken
from the paper by Mr. Hurst, Philosophical Magazine, April, 1906;
and those for argon and helium from the paper by Messrs. E. W. B.
Gill and F. B. Pidduck, Philosophical Magazine, August, 1908.
:
i
©
i
:
SPARKING PoTENTIAL IN A UNIFORM ELECTRIC FIELD 57
HYDROGEN.—Sparking Potentials.
TABLE XVII.
X p S SX W pS
1050 20 "643 675 675 12.9
875 16 •710 621 619 11'4
1050 16 '463 485 490 7'4
700 12 •791 555 561 9°5
1050 12 •353 370 389 4-23
25 8 •927 486 487 7-42
700 8 • 57 . 399 395 4'56
050 8 •306 322 322 2:45
350 4 1:14 399 395 4'56
525 4 •613 322 323 2:45
700 4 '405 283 282 1-62
350 2 •810 283 287 1-62
525 2 '501 269 273 1:00
350 1 •806 282 289 •81
TABLE XVIII.
NITROGEN.—Sparking Potentials.
X p S SX V pS
700 4 •72 502 507 2.88
4 •6 460 2-4
2 •85 382 1-70
525 2 •665 349 344 1°33
2 •65 339 1:30
2 ‘58 332 1°16
1 1-0 310 1°00
1. •868 300 •868
1 •601 298 '601
525 1 •58 304 300 *58
1 •5 310 •5
1 *4 330 '4
58 THE THEORY OF Ionization of GASEs
TABLE XIX.
CARBON DIOxIDE.-Sparking Potentials.
X p S SX W pS
1400 2 •369 516 517 •738
700 1 •736 516 509 •736
875 1 • 571 500 495 • 571
1050 1 •465 488 491 *465
525 ‘5 •929 488 485 •464
700 ‘5 •706 494 497 • 353
875 '5 •613 537 530 *306
525 • 25 1'06 558 564 • 265
TABLE XX.
ARGON.—Sparking Potentials.
X p S SX V pS
505 10 1.087 549 549 10.87
610 4. •471 287 276 1-88
410 4 •947 388 380 3.79
618 2 •378 234 233 •76
401 2 •618 248 245 1:24
615 1 •399 245 248 •399
517 1 •482 249 244 •482
401 1 •580 233 235 •580
309 1 •770 248 237 •770
402 '66 '599 241 248 •395
301 '66 '802 242 238 •528
SPARKING PotRNTIAL IN A UNIFORM ELECTRIC FIELD 59
TABLE XXI.
HELIUM (before purification with liquid air).”—Sparking Potentials,
X p S SX V pS
404 13 •674 272 * 8.76
305 8 •845 258 262 6-76
385 1() •650 250 252 6'50
305 6 •781 238 236 4-69
304 4 •760 231 234 3-04
506 6 •462 234 232 2.77
772 6 •326 254 246 1.96
304 2 •863 262 262 1-73
402 2 •694 279 274 1 - 39
The sparking potentials in volts are also given in
the following curves, figure 11, the abscissae being the
product of the pressure and the distance between the
plates.
The accuracy of the theory is thus established over a
range of pressures extending from the critical pressure
to a pressure twelve times as great for any distance
between the plates. The theory has not been tested for
pressures below the critical pressure, as the values of
a and 8 were not determined for the lower pressures for
the reasons already explained.
It may readily be seen from the theory that the
potential SX is a function of pS only, and con-
sequently that Paschen's law is satisfied. For if v
denotes the theoretical sparking potential SX and m
the product pS which is proportional to the quantity
of gas between the electrodes, then a=f(;) =pf(...)
1 The sparking potentials for pure helium are much lower;
the minimum does not exceed 200 volts,
60 THE THEORY OF IonizATION OF GASEs
and 3-pºſº). Substituting these expressions for
a and 3 in the equation for S, viz.,
(a–3) S=log a-log 3,
a relation
miſſ.)-(+)=log{ſ()/.(...)
Sparking
Potential
in volts.
3OO
2.
Jº
> __T
5 OO 2' - L-Tº
Jºſal—
\ 9. Aſ _|_1~
* $º 2-T
4-OO Z A–Z Pi—
42 - Härike wort) -
N fore
^ º He(**H | Hel H
200 -
2 4. <> 3 /O
p XS.
p measured in millimetres of mercury.
S } % ,, centimetres.
Figure 11.
is obtained, which shows that the sparking potential v is
a function of m only. -
24. Effects of initial ionization.
The determinations of the sparking potentials given
in the above tables were made by increasing the potential
applied to the electrodes and observing the point at
SPARKING Pot'ENTIAL IN A UNIFORM ELECTRIC FIELD 61
which sparking took place when ultra-violet light of
Small intensity was acting on the negative electrodes.
The potentials thus obtained were very definite, and
showed no irregularities, being independent of the length
of time during which the gas is subjected to the electric
force. They were in most cases about two volts below the
potential required to spark when no light is acting on
the negative electrode.
The lowering of the sparking potential due to the
action of ultra-violet light, which was discovered by
Hertz," is more noticeable when the light is strong and
the pressure high. According to this theory, it is due to
(a_g),(*-8)'
the large current proportional to n=no Eº
a—e." £8)l
which ensues when X and no are large, before the
denominator of the fraction vanishes. The polarization
due to the separation of the positive and negative ions dis-
turbs the uniformity of the field between the plates in such
a way as to facilitate the discharge, as will be explained
in section 27. The charge in the gas which causes the
polarization is proportional to the intensity of the light,
and inversely proportional to the velocity of the ions.
This latter quantity diminishes as the pressure of the
gas between the plates increases, since X/p for sparking
diminishes as the quantity pS rises. It is therefore
necessary to use ultra-violet light of very small intensity
in order to determine with accuracy the sparking
potential which corresponds to the theoretical value
derived from the equation a- eſa-B) S_
When no light is falling on the electrodes the sparking
O.
1 Hertz, Wied Ann., xxxi., 1887.
62 THE THEORY OF Ionization of GASEs
is due to the multiplication of the few ions which under
all conditions are present in the gas. Theoretically any
small number of ions would be sufficient to initiate a
discharge when the plates are at the sparking distance,
S corresponding to the electric force, and that a few
ions are being continually generated in a gas even when
it is contained in a closed vessel has been shown by
Geitel' and C. T. R. Wilson.” But the number of such
ions is very Small, and being generated throughout the
volume of the gas, they are not on the average as
efficient for starting a spark as ions coming from the
negative electrode, each of which traverses the whole dis-
tance between the electrodes. This accounts for the small
difference of two or three volts between the ordinary
sparking potential and that obtained when ultra-violet
light of Small intensity falls on the negative electrode.
Geitel, Physikalische Zeitschr., ii., 1900.
* C. T. R. Wilson, Cam. Phil. Soc., xi., 1900.
CHAPTER IV
THEORY OF ELECTRIC DISCEIARGES IN FIELDS OF FORCE
WHICH ARE NOT UNIFORM
25. Description of phenomena to be investigated.
The theory may also be applied to discharges
between electrodes in gases in which the electric force
varies from point to point along the path of the current ;
but in these cases the investigations cannot be carried
out so completely as those which deal with the sparking
potential in a uniform field between two parallel plates.
Thus the potential required to produce a point discharge
cannot be expressed in a form that admits of being
compared with experimental results; but it can be
shown from the theory that the potential is smaller
when the point is negative than when the point is
positive throughout the range of forces and pressures
for which a and 8 have been determined. Another case
which is of interest is the potential required to maintain
a current between two parallel plates. When the
current is small the uniformity of the field is not
appreciably disturbed by the charge in the gas, so that
the potential required to maintain the current is the
same as the sparking potential. But when the current
increases the charge in the gas increases the force near
the negative electrode, and under these circumstances it
can be shown that the potential between the electrodes
which would maintain the current is less than the
64 THE THEORY OF Ionization of GASES
sparking potential. The difference between these two
potentials arises from the same effects as produce the
difference between the sparking potentials for negative
and positive points.
It is necessary to point out that there are limits to the
ranges of values of the current, electric force, and
pressure for which the collision theory, as it is here
discussed in a simple form, can be expected to give a
complete explanation of the experimental results. When
the current is large the gas becomes heated, and the
number of molecules in the path of the discharge is
diminished ; also a certain proportion of the positive and
negative ions recombine, and this introduces a further
complication. When the pressures are very low, and
large forces are required to produce a discharge, radia-
tions, such as cathode rays and Röntgen rays, are
emitted, which contribute to the formation of ions and
thus facilitate the discharge. A comparison of the
sparking potentials for cylinders and parallel plates
shows that radiation effects are probably of importance
when the pressure is somewhat below the critical
pressure. The investigations will therefore be confined
at first to pressures above the critical pressure and to
the smaller currents, for which it is highly improbable
that there are any other processes of ionization taking
place that add appreciably to the number of ions
generated by collisions.
In order to examine the experimental results, and to
See how far they may be explained by the theory, it is
necessary to find the condition that a given field of force
should suffice to maintain a current when the Supply of
ions is kept up by the effects of collisions. An expression
may be found for this condition in the general case by
THEORY OF ELECTRIC DISCHARGEs 65
a method similar to that which was used to determine
the condition for sparking in a uniform field, but it is
interesting to investigate the problem on Somewhat
different principles."
26. Condition for the maintenance of a current by
effects of collisions in any field of force.
Let the current pass between two parallel plate
electrodes at a distance l apart, and let a be the
distance of any point in the gas from the negative
electrode. Let u be the velocity of the negative ions,
a the number of molecules ionized by a negative ion in
travelling through a centimetre of the gas, m1 the number
of negative ions per cubic centimetre, v, 8, and n2 similar
quantities for positive ions. These quantities depend
on the force X, which is supposed to vary from point to
point. If e is the charge on an ion, them e(niu-Hºnav)
represents the current in the gas, so that miu-Hmov=c
is constant along the path of the discharge.
When the current is steady there are no variations
with respect to the time at any point in the gas, So
that
dini
d
"dº TT dº (n11)+amſu-Hſºn, v=0,
dna_
l
and II –– i. (mov)+aniu-H/3m39–0.
Hence i. (n11)=aniu-H 3n2w-(a-6)niu-HBc,
So that
(a-6)da. (a-6)da, —ſ(a-6)da:
nº-º' +' 8ce f da,
* See papers by the author, Philosophical Magazine, March,
1905, and June, 1906, -
I.G. F
66 THE THEORY OF IONIZATION OF GASEs
Let Z(x) denote the quantity Ja—ºr, then niw
may be expressed in the form
niu-AZ(a)--c/ (a|. [Z(x)]T 'bar,
and similarly
11.2%) =BZ@-częſ TZºl tºº 'ad.
Since all the ions are produced in the gas, the following
conditions hold at the electrodes:—
At the negative electrode we(), ni–0, and nºv=c.
At the positive electrode wal, nº=0, and nºw-c.
The first two conditions give A=0 and B-c, and the
Second two give -
1=Z(l)|[Z@IT'x 8da, and
l, –1
1=| [Z(a)] × ada,
which are not independent, since
(C —1 —1
ſe-ozo"dº-Zaï'-1,
So that the equation
1–ſ.ſe-ºn,
represents the condition which must be satisfied by the
values of a and 8 along the path from the negative
electrode wi- 0 to the positive electrode ac-l in order
that a continuous current should be maintained. r
When the current is very small, so that the field of
force between the plates is uniform, a and 3 are constant,
and the above condition reduces to
1= *(*-*–1) or a-6"Tº
THEORY OF ELECTRIC DISCHARGEs 67
which is the equation that determines the sparking
potential. Hence the sparking potential may be defined
as the potential which is required to maintain a very
Small current in the gas.
27. Currents accompanied by a positive charge in the gas.
In order to investigate the effect of the increase
of current on the potential, it is necessary to consider
how the force between the plates is affected by the
electricity in the gas. The positive ions, being of larger
mass, travel more slowly than the negative ions, so that
there is a greater number of positive than of negative ions
in the gas at any time. This excess of positive electricity
causes an appreciable disturbance in the uniformity of
the field as the current rises, and since all the positive
ions must pass through the gas at the negative electrode,
the positive charge is greatest in the neighbourhood of
that electrode. The effect of this charge is to increase
the force near the negative electrode and to diminish it
in other parts of the field. The ionizing power of the
positive ions is much increased in the field of strong
electric force, and when that field is near the negative
electrode the supply of ions is kept up when the potential
difference between the electrodes is less than the sparking
potential. *.
Experiments on currents through rarefied gases in
discharge tubes show that the force is greater near the
cathode than at other points of the discharge. The
phenomena which attend these discharges are compli-
cated, and have given rise to numerous speculations, but
the principal variation in the electric force was first
explained by Professor Schuster" in the manner given
* Bakerian Lecture, Proc, Roy, Soc., xlvii., 1890, p. 541.
F 2
68 THE THEORY OF Ion.TZATION OF GASEs
above. He assumed that the positive and negative ions
in the gas have equal charges, the same as the atomic
charges on ions in liquid electrolytes, a hypothesis which
subsequent experiments have confirmed, and attributed
the positive charge in the gas to the difference in mass
between the positive and negative ions.
28. Increase of force at cathode accompanied by decrease
of potential required to maintain a current between
the electrodes for pressures above the critical pressure.
The decrease in the potential difference between the
electrodes that accompanies the increase of the electric
force at the cathode may also be determined theoretically
by finding the potential required to maintain a current
in a field made up of two parts in each of which the
force is constant, the value of the constant being
greater in the part near the negative electrode.
Let a and 3 have the values ai and 81, corresponding to
a force X1, through a space of thickness a on the side of
the cathode, and the values as and 32, corresponding to a
force X2, in the rest of the field of thickness b.
e-a—xie-b—º
Negative Positive
electrode || al{3, a,3, electrode
(CEO 2=l.
Xi i Xa
In this case the condition for the maintenance of the
current between the plates becomes
a—e." Tº", 6–2.."T",” a
a1–31 (33—a, * - L. &
In order to apply this formula to a definite case, let the
gas between the electrodes be hydrogen at 10 millimetres
THEORY OF ELECTRIC DISCHARGEs 69
pressure, and let a force of 80 volts per millimetre act
in a layer extending 2 millimetres from the cathode,
and a force of 50 volts per millimetre in the rest of the
field. The values of ai, 31, a2, 39, as obtained from the
curves giving aſp and 3/p in terms of X/p, are as follows:
a1=10:0, B1=-081, ag=3-6, 32–'021,
so that the distance b may be found from the above
equation, since a is given equal to 2. The value of b
thus determined is 8 millimetres, and hence the total fall
of potential between the electrodes is 80×2+50×8
=560 volts, which is about 20 volts less than the
sparking potential in hydrogen at 10 millimetres pressure
between plates a centimetre apart.
Larger differences between the potentials are obtained
when further increases are made in the force near the
cathode. Thus, if X1=200 volts per millimetre in the
first millimetre of gas near the cathode and 40 volts per
millimetre in the rest of the field, a similar calculation
gives b-3:25, so that the fall of potential between the
electrodes is 330 volts, the corresponding sparking
potential between the electrodes, 4.25 millimetres apart,
being 385 volts. It is thus obvious, from the theoretical
results that have been obtained, that the potential
required to maintain the current in the gas diminishes
as the current increases, owing to the increase of the
force near the negative electrode.
29. Simple experiments to illustrate the effect of
concentrating the force near the cathode.
These results are in accordance with the experi-
mental investigations" for pressures above the critical
* Philosophical Magazine, June, 1906.
70 THE THEORY OF Ionization of GASEs
pressure, as may be seen from the following experiments
with air. Instead of depending on the charge on the
positive ions to intensify the field near the negative
electrode, a gauze may be used near the negative electrode
and maintained at any required potential.
Two electrodes, A and B, were set up, as shown in the
accompanying diagram, figure 12, in an air-tight glass
tube, and a grating of fine wire G was placed near the
negative electrode, and could be charged to any potential
G.
i
!
I
|
|
|
|
|
|
f
|
|
|
|
| . - - -
Figure 12.
ſ
W' intermediate between those of A and B by means of
a separate connection. The sparking potential between A
and B was then found to depend on the potential of the
grating. The following results" were obtained, in which
W' represents the potential difference between the
grating and the negative electrode and W the sparking
potential between A and B:— -
W. 100 150 170 210 240
V 700 680 640 600 550
1 Philosophical Magazine, June, 1906.
THEORY OF ELECTRIC DISCHARGEs 71
Thus the sparking potential between A and B was
lowered by 150 volts by concentrating the force near the
cathode.
The essential points on which this theory of the
diminution of the potential required to maintain a
current below the sparking potential is based may be
stated briefly as follows:—
The supply of ions by which the current is maintained
depends chiefly on the negative ions, and of these the
most efficient are those generated near the negative
electrode, since they travel nearly the whole length of
the discharge. When the force is increased near that
electrode and the total fall of potential between the
electrodes is diminished, two effects are produced: the
number of ions generated by one of the negative ions in
travelling between the electrodes is diminished, and the
number of ions generated by a positive ion near the nega-
tive electrode is increased, thus increasing the number of
the most active negative ions. Now the rate of increase
of 3 with the force X bears to 3 a much greater
ratio than the corresponding rate of increase of a bears
to a |o: #~. #| and on this depends the main-
tenance of the current in spite of the large diminution
which takes place in the potential difference between the
electrodes. When this potential is reduced the average
value of a along the path of the discharge is diminished,
and the effect of the negative ions, which depends on an
exponential term (such as Noe”), is greatly reduced;
but, owing to the large increase in the value of 3, a
compensating effect of the same order is introduced (in
72 THE THEORY OF IonizATION OF GASES
the quantity No) by the increased activity of the positive
ions at the negative electrode.
30. Comparison of the effects of concentrating the
force near the cathode and near the anode ;
explanation of sparking potentials for positive
and negative points.
It is interesting to examine the equation
2–6."-", e-gº”
a1–61 |32-az
more generally and to contrast the effects of increasing
the force in the parts of the field near the cathode and
near the anode.
For a field of force chosen arbitrarily the sum of the two
fractions in the above equation has a value Y which is
different from unity. When the values of a and 8 are
small, the forces do not suffice to maintain a current,
and each of the fractions becomes equal to unity, so that
Y=2. As the values of a and 3 increase the fractions
diminish, and their sum becomes equal to unity when
the forces are sufficient to maintain a discharge. Hence,
when the value of Y lies between 1 and 2 for a system
of values of a and 8, the forces between the electrodes
will not suffice to maintain a current.
The effect of reversing a field may be found by inter-
changing aſ and 81, and as and 39, and if this be done in
the two numerical examples given above, it will be found
that Y is greater than unity in each case, so that when
the forces are reversed they are not sufficient to maintain
a current.
A very simple case arises when one of the forces Xa,
acting through the distance b, is so Small that the
=1
THEORY OF ELECTRIC DISCHARGES 73
corresponding value of 82 may be neglected. If X2 be
on the side of the negative electrode, the condition for the
maintenance of a current is
81–a1e-*T*, * 1.
31–a1 Cl2
which shows that a-s, "Tº *=0, or that the
potential fall in the distance a on the positive side must
be the sparking potential Va for parallel plates at the
distance a apart. Hence the potential difference between
the electrodes is in this case
Wa-HX2b.
If the forces are reversed so that X2 is on the side of the
positive electrode, the condition for the maintenance of a
current reduces to
(a1–81)a
a1–31e —agb
a1–61
=1,
which shows that a1 must exceed a.(61-6 Da. and con-
sequently the potential fall along the distance a is less
than the sparking potential Wa, so that the potential
difference between the electrodes must be less than
X2b-HVa.
The most familiar examples of phenomena which may
be explained on this principle are point discharges.
When the point is negative, the strong field is near the
negative electrode, so that the potential required to
produce a discharge is less than when the point is
positive. The same result is obtained when a gas is
contained between two concentric cylinders: the lower
74 THE THEORY OF Ionization of GASEs
sparking potential corresponds to the case in which the
inner cylinder is negative.
The theory thus gives a satisfactory explanation of the
variation of the potential with the current for parallel plate
electrodes and also of the difference between the sparking
potentials for point discharges when the direction of the
force is reversed.
31. Phenomena at pressures below the critical pressure,
The preceding results can only be considered to
hold for pressures above the critical pressure, as the
experiments by which the values of a and 3 were
determined show that for these pressures the ionization
of the gas is completely accounted for by the action of
the positive and negative ions in generating others by
collisions with molecules. The curves which have been
obtained for the values of a and 3 show that alp does
not vary much with the force for large values of X/p, so
that a diminishes nearly in direct proportion to the
pressure when the force is large and the pressure is below
the critical pressure. The question then arises whether
the quantity 3 increases sufficiently rapidly to account
for the sparking potentials obtained experimentally for
the low pressures. It is easy to calculate roughly
the values of 8 which are necessary, and they do not
appear to be greater than the numbers that might be
expected to correspond to points on continuations of the
curves for 3/p, but the question cannot be decided in this
way, as it is difficult to apply the theory of collisions
accurately when the free paths of the ions become con-
siderable fractions of the distance between the plates.
Some evidence of the processes which influence the
sparking at the low pressures may, however, be obtained
THEORY OF ELECTRIC DISCHARGEs 75
by examining either the variations of the potential
difference between the plates for different currents or
the sparking potentials between conductors arranged so
as to produce a variation of the force along the path of
the discharge.
If the supply of ions were maintained by the effects
of collisions at the lower pressures, the sparking potential
for a gas between parallel plates should continue to be
greater than the potential difference which maintains a
current, but experiments show that at the lower pressures
the potential increases with the current." This might be
due to a rise of temperature of the gas and a correspond-
ing reduction in the number of molecules between the
plates, in which case the potential maintaining the current
would tend to increase in the same way as the sparking
potential increases when the pressure diminishes. This
suggestion has not yet been tested experimentally, so
that it is uncertain whether it affords a complete explana-
tion of the effects obtained.
These phenomena are most probably related to the
point discharges for the lower pressures, and a theory
which would explain why sparking is produced more
easily from a positively charged point in a gas at a low
pressure than from the same point negatively charged
would apply also in some degree to the potential required
to maintain a current.
32. Processes of ionization that may account for effects
obtained at low pressures.
In these cases, as well as in the discharge between
cylinders, the lower sparking potential is obtained when
the greater force in the gas is in the neighbourhood of
* J. A. Brown, Philosophical Magazine, September, 1906.
76 THE THEORY OF IonizATION of GASEs
the positive electrode. It would appear from this that
some new process of ionization is called into play which
acts in addition to the effects of collisions. The force in
the neighbourhood of the positive electrode affects prin-
cipally the negative ions, all of which pass through that
region. The potentials which are obtained experimentally
can therefore be explained on the hypothesis that some
form of radiation is emitted from the anode when the
negative ions impinge on it. Some such effect might
be expected, as it is known that Röntgen rays are pro-
duced by the same cause when the pressure is very low.
The pressures under consideration are, however, large
compared with any pressure in a vacuum tube at which
Röntgen rays can be detected outside the tube by
ordinary methods, but it is possible that non-penetrating
rays of a similar kind may be produced, capable of
ionizing the gas inside the tube, at these comparatively
high pressures. The experimental results may thus
be explained in a general way, but it is impossible
at this stage to form a complete theory of the sparking
potentials at pressures below the critical pressure, as
they have not yet been examined very extensively
with a view to obtaining evidence as to the relative
importance of the processes of ionization that take place.
It may be pointed out, in connection with the possible
existence of radiations of a non-penetrating character
inside a discharge tube, that E. Wiedemann" has found
that, when a current is established in a gas, easily absorbed
rays (termed Entladungstrahlen) are emitted by the
luminous portions of the discharge, and Professor Sir
J. J. Thomson has shown that these rays have the
* Wied. Ann., lx., p. 269.
THEORY OF ELECTRIC DISCHARGEs 77
property of ionizing the gas." A radiation of this kind
would obviously affect the potential required to maintain
a current, but if the intensity of the radiation is pro-
portional to the square of the number of ions per cubic
centimetre of the discharge, as Professor H. A. Wilson's”
investigations on the ionization in the positive column
seem to indicate, the rays would be inappreciable with
Small currents, and would not affect sparking potentials.
33. Cathode fall of potential; ionization in the space
near the cathode when the cathode fall of potential
is established.
In gases above the critical pressure the currents
when large are accompanied by the phenomena known
as the cathode fall of potential. The polarization in the
gas as measured by this fall does not vary with the
current, and the potential difference between the
electrodes also remains approximately constant. At
this stage the effects which take place in the gas are
SO complicated, involving recombination and various
changes in the velocities of the ions, that no theory has
been proposed which explains the distribution of force in
the path of the discharge and accounts for the remarkable
fact that certain potentials are almost independent of the
current. It is, however, easy to show in a general way
that the ionization arising from collisions in these cases
may be sufficient to maintain the current.
The condition which must be satisfied by the values
of a and 3 along the path of the discharge between
electrodes at a distancel apart may be written in the form
* J. J. Thomson, Proceedings Cambridge Philosophical Society,
Wol. x., Pt. ii., p. 74.
* Philosophical Magazine, July, 1903,
78 THE THEORY OF IONIZATION OF GASES
- 1— | b ...ſº -0)dºdg- | l, ...ſº —a)dad30,
O b
and if the values of a and 3 are very small in the column
of gas of length l—b in contact with the positive electrode,
the quantity on the right of the above equation becomes
very small, and the integral
[...ſº-º:da:
becomes nearly equal to unity, so that the potential
fall along the distance b (from ac-0 to a;=b) at the
negative electrode is nearly the same as the potential
that would maintain the same current between electrodes
at a distance b apart, the pressure of the gas being
unchanged.
It is possible to show the connection between the
cathode fall of potential and the minimum sparking
potential by using the above investigation in connection
with the following experimental results.
The sparking potential has a minimum value corre-
sponding to a certain amount of gas between the plates,
which may be measured by the product ph, where p
is the pressure and b the critical distance between the
plates for the pressure p. In this case the sparking
potential does not differ much from the potential required
to maintain a discharge in the gas. Thus in a set of
experiments with hydrogen at a pressure of 2.55 milli-
metres, the following potentials were obtained between
electrodes 44 centimetres apart : 273 volts for a
very small current and 272, 255, and 250 volts for
currents of 2x10, "2×10, and 4.6×107" ampères
respectively. Within an error of about 4 per cent. the
THEORY OF ELECTRIC DISCHARGEs 79
value of the potential may be taken as 260 volts for any
of the currents.
If, therefore, the plates are at a large distance apart
and the fall of potential along a layer of thickness b near
the negative electrode has attained the value of 260
volts, b being the distance between electrodes for which
the pressure in the gas is the critical pressure, the
current will be maintained by the ions generated in
that layer, even when the force in the rest of the dis-
charge is not sufficient to cause ions to be generated by
collision. The experiment quoted shows that the
potential fall along the distance b need not vary much
with the current.
Thus the cathode fall at a given pressure p should
be independent of the current and of the distance
between the electrodes, should be equal to the minimum
sparking potential, and should extend over a distance b,
where p X b is a measure of the amount of gas corre-
sponding to the minimum sparking potential, and b should
consequently vary inversely as p. These conclusions
are in general agreement with experiment.
The cathode fall of potential as determined experimen-
tally is independent of the length of the discharge and
the intensity of the current, and has been shown by
Strutt' to be practically the same as the minimum
sparking potential. Experiments also show that the
thickness b1 of the layer of gas across which the fall of
potential takes place increases as the pressure diminishes,
though it cannot be found very accurately as the force is
very small at a short distance from the cathode. The
product pbi, as given by most observers, is less than pb,
1 Hon. R. J. Strutt, Philosophical Transactions, Wol. cxciii.,
p. 377, 1900.
80 THE THEORY OF IonizATION OF GASEs
but since the force is very small along the distance
b–b1, the experiments show that for a layer of thick-
ness b the potential fall is practically the minimum
sparking potential, and the product pb may be taken
as constant.
34. Sparking potential at atmospheric pressure for very
short distances between the electrodes.
Some experiments made by Earhart' dealing with
very short Spark gaps in air at atmospheric pressure
Seem to be at variance with the accumulated evidence,
which shows that there is a minimum spark potential
below which it is impossible to obtain a discharge
through a gas. This minimum is about 340 volts for
air, but in Some cases, when the electrodes were very close
together, it was found that a current passed between
them when the potential difference was as low as 30 volts.
It was at first supposed that the current passed through
the air between the electrodes, but recent experiments
made by Almy” show that the electrodes are liable to be
drawn together and to come into contact, owing to the large
electrostatic force between them when they are a short
distance apart and a potential difference is established
between them. It was found that when the apparatus
was constructed so that the electrodes do not become
displaced a potential of 330 volts is not sufficient to
produce a discharge through air at atmospheric pressure,
while 360 volts is sufficient to do so. This result is in
accordance with the theory that has been given for the
sparking potential, and establishes the interesting fact
* Earhart, Philosophical Magazine, 1901, p. 147.
* J. E. Almy, Philosophical Magazine, September, 1908,
THEORY OF ELECTRIC DISCHARGES 81
that the large forces developed at the surfaces of con-
ductors separated by a fraction of the wave length of
sodium light, and differing in potential by 300 volts, are
not sufficient to make electricity pass from the conductor
to the surrounding gas.
35. Remarks on processes of ionization which account for
various phenomena.
In the preceding account of the development of
currents in gases it has been shown that the conduc-
tivities may be explained on the theory that all the
ionization is to be attributed to ions produced in the gas
by positive and negative ions. Some of the phenomena
that have been discussed might also be fairly well
accounted for by supposing that the negative ions ionize
the molecules of the gas, and that the positive ions have
only the power of setting free negative ions when they
impinge on the negative electrode. There are good
reasons, however, for preferring the former method of
explaining the phenomena that occur at pressures above
the critical pressure, and of showing the connection
between them. In the first place, if the positive ions
acted by setting free negative ions from the cathode,
their effect would presumably depend on the metal of the
electrode, and the sparking potentials would show varia-
tions depending on the metal of which the electrode
was made. This point has been examined recently by
Carr (loc. cit.), and the results of several careful experi-
ments have shown that the Sparking potential is
independent of the metal. This conclusion is in agree-
ment with the observations of most other physicists, but
some have found that with aluminium, and possibly also
with magnesium, the Sparking potentials are somewhat
lower than with other metals.
I.G. - G.
82 THE THEORY OF Ionization of GASEs
Again, if the case of point discharges be considered it
will be seen that when the point is positive the supply of
negative ions cannot be kept up by the action of positive
ions at the negative electrode. For this electrode might
be large and so distant from the point, that the force at
its surface would be too small to give the positive ions a
velocity of impact sufficient for the production of negative
ions. In this case it is obvious that the effect of the
positive ions must be attributed to their action on the
molecules of the gas, and the difference between the
Sparking potentials for positive and negative points
follows immediately from this theory, as has been
explained above. Another fact to be noticed is that
a positive ion makes only one collision with the electrode,
whereas it makes a large number of collisions with
molecules of the gas, and there is no reason for supposing
that a negative ion is more easily set free from a metal
than from the gas molecules. There is thus definite
evidence to show that molecules of the gas are ionized by
the positive ions, and for pressures above the critical
pressure a consistent explanation may be obtained of a
large number of phenomena on the theory that this is the
predominating effect produced by the positive ions.
On the other hand, at pressures below the critical
pressure the negative ions forming the cathode rays
make up an integral part of the current, so that for
these low pressures the ions coming from the negative
electrode should be taken into consideration, and when
they introduce appreciable effects the sparking potentials
might be expected to depend on the metal of the electrode.
36. Examination of some other theories of the sparking
potential.
Several expressions for the sparking potential in
terms of the pressure have been given by physicists,
THEORY OF ELECTRIC DISCHARGES 83
some of which are empirical and do not aim at giving
an explanation of the fundamental principles on which
the potentials depend, and others, depending on various
assumptions which attribute certain properties to posi-
tive and negative ions, appear to explain more or less
accurately some of the phenomena which are observed.
Professor Sir J. J. Thomson has recently given two
theories of the latter kind, and it is interesting to
examine how far they furnish results which account for
the various potentials and to what extent the assumptions
made in the process of the investigations agree with
those properties of ions which are found to accord with
simpler phenomena.
The first" of these investigations differs essentially
from that given in section 20 above, and deals with the
problem of finding the potential required to maintain a
current flowing between electrodes when the ordinary
cathode fall of potential is established. No distinction
is drawn between the sparking potential and the poten-
tial required to maintain a current. It is obvious that
the expression which is thus found cannot represent the
sparking potential, and the apparent agreement between
the formula and the sparking potentials obtained experi-
mentally results only from attributing convenient values
to undetermined constants. Clearly the formula for the
potential that is found in this way should represent the
potential required to maintain a comparatively large
current, if the assumptions on which the theory is based
are correct.
The current on this theory is kept up by the ionization
of the molecules of the gas by negative ions, and a
J. J. Thomson, “Conduction of Electricity through Gases,”
1903, p. 38.
84 THE THEORY OF Ionization of GASEs
supply of the latter is supposed to be generated at the
cathode by the impact against it of positive ions. The
expression for the sparking potential W as found on this
hypothesis is
gº-ºº: 1. (1+kWoe) c(1+w) 1 (y-Hw) d
V=V+5. log. kVoe tº- \ 3e T3a TX’
where Wo is the cathode fall of potential, 3, w, and k
are constants, y the ratio of the number of collisions in
which corpuscles remain attached to the molecule to the
total number of collisions, A the mean free path of a
corpuscle, c the thickness of the Crookes layer, and e the
charge on an ion.
It is difficult to understand the reason for many of the
assumptions that are made in obtaining this formula.
For instance, it is supposed that the positive ions all
traverse the Crookes layer, impinging on the cathode with
a kinetic energy eVo, a supposition which involves the
assumption that they make no collisions with the molecules
of the gas, although the length c is much longer than the
mean free path of the positive ions. This view is said to
be supported by experiments which show that solid
obstacles placed in the cathode dark space cast a shadow
on the electrode. But it must be remembered that the
solid obstacle which is placed near the cathode would
presumably take up a potential less than the cathode
fall of potential. The potential difference between the
cathode and the obstaclein question would therefore be less
than the minimum sparking potential, and no current
could be maintained between the cathode and the obstacle.
This should hold for any theory of the discharge, and
the fact that the gas is not luminous in the space between
the cathode and the obstacle supports equally all theories
that account for the minimum sparking potential.
THEORY OF ELECTRIC DISCHARGEs 85
With regard to the factor y, the experimental determina-
tions of currents between parallel plates show that the
supposed effect which it is intended to represent either
does not exist or else is so Small as to be inappreciable.
For let it be supposed that a certain number m of the
negative ions in going through the gas become attached
to molecules and lose the property of ionizing the gas ;
then if the conductivity be produced by ultra-violet light,
as described in section 2 above, the equations for the
determination of the quantity of electricity q arriving at
the positive electrode are #=an-n and #=m,
where n is the number of corpuscles which cross a plane
at a distance a from the negative electrode, anda, the
number of new ions generated in the distance da;, and ynda:
the number of corpuscles that become attached to
molecules in the distance da;. When ac-l, the distance
between the electrodes, (m+n)e is the total charge q
arriving at the positive electrode, e being the atomic
charge.
The above equations give
n=nº"T" and m=nº; (. 6–0–1)
when no corpuscles start from the negative electrode.
Hence the quantity q for the distancel between the plates
is given by the equation.
C. (a—y)l_ 710)
—E
a — y a—y'
Now it has been found experimentally that the values of
#=(m+n)=m,
wº al
q are given accurately by the formula q=qoe even
for the higher pressures and Small electric forces where
86 THE THEORY OF IONIZATION OF GASEs
it might be expected that y would be large compared
with a. The experimental evidence therefore shows
that y=0. If there is any adhesion of the negative
corpuscles to molecules in the circumstances under
consideration, the connection must be of a very loose and
transitory character."
In the 1906 edition of the treatise on “The Conduc-
tion of Electricity through Gases,” Professor Sir J. J.
Thomson gives another investigation of the sparking
potential.
The sparking potential for parallel plate electrodes is
obtained from the condition required to maintain a very
small current which does not disturb the uniformity of
the electric field, the lowering of the potential for large
currents is attributed to the increase of force at the
cathode, the effect of the positive ions in ionizing mole-
cules of the gas is also taken into consideration, and for
the larger distances between the plates this is regarded
as the predominating effect of the positive ions. It will
be noticed that it was on these principles, as has already
been explained,” that the theory which gives results in
accurate agreement with experimental determinations was
originally” worked out. t
The general expression for the potential is given by
the equation*
a—ye
a—y—B)d
(*-*_i.v. | ... (-,-6) _.(4–7–82, 1 }
a—y d ( a-y–3 (a—y–8)?" (a-y—8)”
* See section 7 above.
* Sections 20, 21, and 22.
* J. S. Townsend, Electrician, April, 1903; Philosophical Magazine,
November, 1903, and March, 1905.
* “The Conduction of Electricity through Gases,” 1906 edition,
pp. 494 and 495.
THEORY OF ELECTRIC DISCHARGEs 87
in obtaining which it is supposed that ions are generated
by the collisions of positive and negative ions with
molecules of the gas, and also that negative ions are set
free from the negative electrode when the positive ions
collide with it. For large distances between the electrodes
greater than the distance corresponding to the minimum
sparking potential the condition for sparking becomes
y=aT", y representing the effect of the positive
ions per centimetre in ionizing the gas. This equation is
nearly the same as the equation a=6." Tº )S (section 23),
since 3 is Small compared with a, and it has been seen
that this gives the correct expression for sparking.
With regard to the general expression for the con-
dition for sparking given above, it is difficult to use the
equation in order to make calculations of the potential, as
the numerical values of the effects involved are not
determined. From general principles, however, it
appears obvious that it could not be very accurate for
pressures or distances between the electrodes for which
the product p X d is less than the critical value, as it fails
to take into consideration processes of ionization which
would account for experimental results with electrodes of
various shapes and which would explain the variation of
the potential with the current between parallel plates.
As has already been explained in section 28, the
potential required to Spark or maintain a current in
a gas diminishes as the force becomes concentrated at
the cathode, when the ions are produced by the action
of positive and negative ions in the gas. In the general
formula quoted above the system of ionization considered
consists of the ions generated by positive and negative
ions from molecules of the gas and in addition ions set
88 THE THEORY OF IonizATION of GASEs
free from the negative electrode when positive ions
impinge on it. This additional process of ionization
would obviously be greater when the force is concen-
trated near the cathode than when the larger force
is at the anode. It is obvious, therefore, that when
these three processes of ionization are acting alone, the
potential required to give a spark or maintain a discharge
should diminish when the force at the cathode increases,
So that the results obtained from the general formula for
sparking potentials could not be correct at the lower
preSSures.
In such cases it is necessary to introduce Some impor-
tant factor in addition to the direct effects of collisions,
and a system which would account for the phenomena
has been suggested in section 32. It would be easy to
find mathematical expressions, based on various assump-
tions, to express the sparking potentials under these
conditions, but it is hardly desirable to make further
investigations of sparking potentials at low pressures
until some experimental evidence can be brought forward
in support of the principles that are involved.
OCT 2 ºf
BRADBURY, AGNEW, & Co. LD., PRINTERS, LoNDON AND TONBRIDGE
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