- Email: [email protected]
Ionization cross sections for electrons (0.6–20 keV) in noble and diatomic gases
Schram,
I% I,.
De Heer, Van
Physica
F. J.
der Wiel,
Kistemaker,
31
94-112 M. J.
J.
1965
IONIZATION (0.6-20
CROSS
keV)
SECTIONS
IN NOBLE
by B. I,. SCHRAM,
F. J.
DE
J.
AND HEER,
FOR
ELECTRONS
DIATOMIC M. J.
GASES
VAN DER
WIEL
and
KISTEMAKER
Synopsis Absolute attention
has been paid to the elimination
measured for fast electrons (energy Xe, H s, 1~s. Na and 0s. Detailed of disturbing effects caused by secondary
electrons
and to the influence
effects
range
ionization
0.6-20
a McLeod
keV)
gauge.
cross
incident
sections on He,
have Ne,
been
Ar,
of diffusion
Kr,
In all cases the experimental (r =
4na$R ___
on the pressure cross sections
Mi” In
measurements
with
can be represented
by
CiEel.
EL?1
The relation
between
son of our values
Mia and the optical
with
those
obtained
oscillator from
strength
different
is discussed
experiments
and a compari-
and
from
theory
is made.
I. Introduction. Determination of absolute excitation and ionization cross sections for electrons in gases is important for several reasons. The most obvious ones are applications of these data in plasma physics, astrophysics, upper atmosphere physics and radiation chemistry. Moreover it is possible to deduce information related to optical oscillator strengths for atoms and molecules from these total collision cross sections as was pointed out for instance by Miller and Platzmanl)s) and Fanoa). Measurements of absolute ionization cross sections for high energy electrons are very scarce, the only ones are those of Smith4) (up to 4.5 keV) and Sommermeyer and Dresels) (l-50 keV), which differ in magnitude up to 60%. Therefore, it is useful to remeasure these cross sections and to obtain the results for more gases. The most probable cause of differences between the results cited is the insufficient attention, that has been paid to disturbing secondary effects. We have made a critical analysis of all these effects and have succeeded in suppressing them. In this work absolute ionization cross sections of fast electrons were determined for the rare gases He, Ne, Ar, Kr and Xe and the diatomic gases Hz, Ds, Ns and 02. In the literature two different methods for measuring ionization cross sections are mentioned: firstly the condenser technique, where the current -
94 -
IONIZATION
of positive
CROSS SECTIONS
ions - relative
FOR ELECTRONS
to the primary
electron
IN GASES
current
95
- is determined,
and secondly the counting technique, where the individual ionization events are recorded either in a Wilson chamber-e) or with G. M. counters7) in a coincidence circuit. In this investigation the first method was chosen. II.
Apparatus
our apparatus
and
experimental
procedure.
1 COLLIMAtOR ___ELECTRONGUN
Fig. 1. Schematic
A. General.
A survey
of
is given in fig. 1.
view of the apparatus
COLL.CHAMBER
COLD TRAP
l_MAGNETCOlLS
used for measuring
ionization
cross sections.
The apparatus consists of a stainless steel cylindrical vessel, divided in two parts: the vacuum chamber and the collision chamber, which contain the electron source and the electrode system respectively. Both chambers are connected via the collimator and a bypass. An electron beam, generated by the electron source, is shot into the collision chamber and the current is measured in a Faraday cage. Positive ions, produced along a well defined pathlength, are collected on one of the condenser plates of the electrode system. Target gases are introduced into the collision chamber via a needle valve and their pressure is determined with a McLeod gauge. An axial magnetic field is produced by two magnet coils around both chambers. B. Vacuum. The vacuum chamber is connected to a 650 Z/soil diffusion pump. This pump has a specially designed water cooled chevron baffles), which needs no extra freon or liquid air cooling. During the measurements the bypass is closed and, when using the cold trap, the final pressure in the collision chamber is 5 x 10-S torr. Differential pumping through the collimator (conductance 0.3 Z/sfor air) enables us to operate the electron source at a pressure of 10-s torr, while maintaining a much higher target gas pressure, 10-4 - 10-a tori-, in the collision chamber. The absolute pressure of the gases is measured with a McLeod gauge,
96
B.
L. SCHRAM,
F. J. DE HEER,
M. J. V. D. WIEL
AND
J. KISTEMrlKER
constructed in the laboratory; its capillary has a diameter of 0.7 mm, constant within lx,, over the whole length, and the bulb volume amounts to 600 ems. Recent experimentsg) 10) showed, that a McLeod often gives incorrect results, the error being dependent on gas, temperature and dimensions of the instrument. This effect is a consequence of the steady stream of mercury vapour from the mercury reservoir to the cold trap (A in fig. 2) between the McLeod and the collision chamber. The target gas has to diffuse against this mercury stream, which gives rise to a difference between the pressures in the glass bulb and in the collision chamber. To eliminate this effect, we followed the suggestion made by Ishii and Na kay amaa). Their idea was to cool the walls of the McLeod just above the
Fig. 2. McLeod
gauge, equipped
with the normal cold trap A and an extra cold trap B. TABLE
I
reservoir (see B in fig. 2), so that the mercury vapour pressure is lowered sufficiently. With this method we found higher pressure readings for all gases; the corrections are listed in table I. As these values are dependent on dimensions of the McLeod and room
IONIZATION
CROSS
SECTIONS
temperature, they are representative be used as universal corrections.
FOR ELECTRONS
IN GASES
only for our instrument
97 and cannot
C. Magnetic field. An axial magnetic field - maximum 400 gauss was necessary for the focusing of the electron beam and, as will be discussed elsewhere (section IV A), also provided a valuable check on the suppression of secondary effects. External to the magnetic coils around the collision and vacuum chamber a cylindrical iron shield is fastened, in order to prevent asymmetric disturbances of the axial field, caused by the iron frame on which the whole apparatus is mounted and by magnetic stray fields of other apparatus in the neighbourhood. D. Electron source. The electron source is a Philips 6-AW-59 electron gun from a television tube. The normal operating voltage is 17 kV, but, by applying the axial magnetic field, we could use it over the whole energy range of 0.6-20 keV, obtaining an electron current variable up to 5 x lo-4 A. The diameter of the electron beam was limited to 3 mm by a collimator, located between vacuum and collision chamber, which also serves as a pumping resistance (see B). A John Fluke High Voltage Supply (model 408 A/J) was used for the energy range of 0.6-6 keV and a Hursant High Voltage Supply (model 30 B) for the range of 3-20 keV. Both were calibrated by a compensation method. E. Electrode system. The electrode system consists of a number of gold plated stainless steel plates, separated by glass spacers. Figs. 3 and 4 show the electrode system together with the potential configuration along the axis. The positive ions, produced in the symmetric condenser region between plates 6 and 7, can be directed to either of these electrodes by means of a small transverse electric field. With the aid of guard plates (4. 5, 8 and 9) we attained homogeneity of the ion collecting field, as was checked with the conducting sheet methodrr). Two sets of flat plates (a and b) are fastened on both sides of condenser and guard plates (fig. 4A) ; they are insulated from electrodes 6 and 7 and their function will be described later. To avoid surface charges on the insulating spacers at the back side of the electrode system the half diaphragms 2, 3, 12 and 13 are supplied with cylindrical shields as is depicted in fig. 4B. The electron beam current is detected at the collector, which consists of 2 parts: electrode 15, a hollow cone, into which points the half cone 14.
B.
98
L. SCHRAM,
Fig. 3. Upper part
F. J. DE HEER,
M. J. \:. 1). WIET. ANI)
: Cross section of the electrode system parallel to the electron beam;
-:zoov; the following potentials mere applied: collimator 4, 6 and 8: +80V; 5, 7and 9: OV; 10: +12OV; 3: +4OV; 13: + 100 V; Lower
part:
J. KIS’L-EMllKER
Potential
distribution
14: -I-4oov;
1: +1oov; 11: +4OV;
15: +zoov.
along the axis of the clectrodc
Fig. 4 A
: Cross section of the electrode
system
perpendicular
2: $12OV; 12: -/ 15OV;
Fig. 4 13: Perspective
system.
view of electrotlcs
2 and 3 or 12 and 13.
to the electron
beam.
The beam current is measured with a General Radio tube volt meter, the ion current with a Philips GM 6020 tube volt meter. III. Evaluation of the cross sections from the experinzental results. The gross ionization cross sections c (including multiple ionization processes) are calculated from the formula: CT=
Ii.
I+ I_ ln,
I_@
3.535 x 10’”
T 273
where I+ is the current of positive ions, measured on electrode 7 (see also section IV C); I_ the electron beam current, measured in the Faraday cage (see section IV B) ;
IONIZATION
CROSS
_._~~..
Z n p
SECTIONS
-~~
FOR
ELECTRONS
~~___
the length from which ions are extracted the number of gas molecules per ems; the pressure of the gas in torr
IN GASES
_____~
; (see section
99 ~~~__ - ~
IV E) ;
and
T
the temperature
of the gas in “K, for which was taken room temperature.
So (I is the gross ionization cross section, expressed in ems/molecule. We verified whether the ion current was proportional both to the electron current and the gas pressure. 5
I,/l_rlO-‘A I
Fig. 5. Measured
ion
current
relative target
to the
electron
current
as a function
of the
gas pressure.
In fig. 5 we plotted I+/I_ as a function of target gas pressure for a representative measurement in He. At about 2 x 10-s torr a deviation from linear relationship sets in; our measurements, therefore, were carried out in the pressure region between 2 x 10-J and 1.5 x 10-s torr. The same choice was made for all other gases, which behaved analogously. Every cross section was evaluated from the slope of a graph like fig. 5. IV. General discussion on disturbing effects. A. Ionization by secondary electrons. Secondary electrons, which may have ionization cross sections much higher than the primary ones, are released from the target gas molecules in the ionization process as well as from the electrode material by result of impacting particles. The behaviour of these electrons is complicated by the presence of both a magnetic and an electric field.
100
_
.~~
B. L. SCHRAM,
F. J. DE HEER,
M. J. V. D. WIEL
AND
J. KISTEMAKER
In these combined fields they move along trochoidalis) tracks, which can be considered as a superposition of a helical motion parallel to the magnetic field and a drifting motion perpendicular to both fields. In our apparatus, where the normal ion collecting field amounts to 30 V/cm and the magnetic field varies from 150-400 gauss (resulting in a drift velocity of i lo7 cm/s) the following rough classification for these electrons can be made, dependent on their energy correlated to the motion Parallel to the magnetic field, viz.: a. less than 0.1 eV b. between c.
0.1 and 5 eV
more than 5 eV.
(The boundary values are very rough approximations). Of course, only those electrons matter, which have a total energy higher than the ionization potential. For electrons of group a the trochoidal drifting motion will be the most important, whereas those of group c, having a velocity parallel to the magnetic field much higher than the drift velocity, travel along helices, which nearly coincide with the magnetic field lines. The motion of electrons of group b is a combination of both possibilities. Evidently, the only way to prevent secondary electrons from entering the condenser region from the outside, is the application of an electric field parallel to the magnetic field, so that electrons are reflected from this region by a potential hill. This leads to an important rule: The condenser region should have the lowest potential of the whole system (see fig. 3). The only electrodes, hit by the primary beam, are the beam defining collimator (which is knife-edged) and the collector. The other diaphragms, notably 10 and 11, with openings large compared to the beam diameter, can be hit merely by scattered fast electrons. Secondary electrons, raised by these impacts, are suppressed by the suitable potential configuration. Then we have to consider the secondary electrons released from the gas molecules. Any slow electron created in the condenser region between electrodes 6 and 7 is moving in a trochoid, whose radius is given by Y = m,EleH”, where m, is the electrons mass, E the strength of the electric field, e the electron charge and H the magnetic field strength. Now the radius of the trochoidal movement is so small (in our experiment of the order of 2 x IO-2 mm) that the electrons will certainly not reach the positive electrode of the condenser (6). The electrons of category a, where the drifting motion perpendicular to both fields is predominant, will drift from between the electrode system, following more or less closely the aequipotential surface, in which they were initially born. It is necessary to catch these electrons lest they may float around on our apparatus and cause uncontrollable effects. Therefore, two
IONIZATION
CROSS SECTIONS
sets of extra plates are attached
FOR ELECTRONS
IN GASES
to the side of the electrode
101 system
(see
fig. 4A). In the 1 mm spacing between electrodes a and b, which have the same potentials as 6 respectively 7, the electric field strength is increased and this causes an increase of the radius of the trochoidal electron movement, so that the electrons land on the positive plate a. The electrodes 6 and 7 are insulated from a and b so the collection of slow electrons does not disturb the measurement of the positive ions. The electrons of group c, with a predominant velocity parallel to the magnetic field, will travel along the magnetic field lines and will be accelerated to the more positive parts of the electrode system (e.g. the Faraday cage). The electrons of group b will be partly caught at the positive side plate a and partly at positive electrodes next to the condenser region. All effects, caused by secondary electrons from outside the condenser region, are dependent on the magnetic field strength as their paths are influenced by this field. Because our cross sections were found to be independent on this field, we may claim a sufficient suppression of these effects. Evidently, ionization by fast secondary electrons produced in the condenser region itself, cannot be avoided; however, as its contribution is proportional to the square of the gas pressure, it can be neglected as long as the cross section is independent of the gas pressure (see also appendix). B. Effects influencing the electron beam current. Theelectron beam collector consists of two conical parts, on both of which together the beam current is measured. Fig. 6 shows the measuring circuit and the applied voltages, taken from highly insulated batteries.
Fig. 6. Schematic
view of the measurement
of the electron
Three effects might give a false determination
beam current.
of the beam current:
1. Escape of secondary electrons. The electric field, due to the potential difference of 200 V across both collector parts, proves to be strong enough to trap all secondary electrons somewhere in the collector. See fig. 7. 2.
Ionization in the collector. Again due to the 200 V potential difference, both positive ions and electrons reach one of the two collector parts; there is no net result in the total current. 3. Collections of slow electrons. It was discussed before (section IVA) that a part of the electrons released
102
B. L. SCHRAM,
F. J. DE HEER,
M. J. V. D. WIEL
AND
J. KISTEMAKER
from the gas molecules will reach the collector. This effect, inevitable because of our choice of potential configuration, is proportional both to the cross section and gas pressure, so it can be kept insignificant by limiting
Electron
energy
Ar Pressure
+-0
~~7~~~ ~~_ ~~_-. 50
100
5keV
6.W’ton
r 150
200 --~-
Fig. 7. Measured
collector
current
as a function
electrodes
of
250 V15-Vtd
the
inV
applied
voltage
between
14 and 15.
the pressure range. By giving electrodes 12 and 13 a negative potential and watching the change in beam current, this effect can be verified. It was always less than 1y/,, except in the case of Kr and Xe, where a correction was applied at low electron energies. C. Effects influencing of the positive ion current processes :
the positive current. The measurement on electrode 7 might be disturbed by three
1. Escape of secondary electrons. All electrons, ejected by impact of ions, electrons or photons are returned to the same plate under the influence of the magnetic field. 2.
Impact of fast scattered electrons. Primary electrons, scattered over an angle cc, can hit the ion collecting electrode when sin C(> (H%Pe/32mE,~)~ where d is the distance between the condenser plates; H the magnetic field strength; e the carge of an electron; and E,l the energy of the deflected primary electron. Assuming that the tracks of these fast electrons are not influenced by the small ion collecting field, we can reverse this field and collect only fast scattered electrons. A negative current i- is measured, which provides a correction on I+. Some values of this correction, relative to the I+, are listed in table II. The contribution of negative ions to the correction can be neglected because of the very small probability of their formation.
IONIZATION
CROSS
SECTIONS
FOR
TABLE Corrections
on ion current
I400
I,
3 keV 0.1%
400
1%
1,
-
4% 0.6%
160 400
,,
1% _
6% OS:/,
160 >>
I
0.8%
0.5% -
IN GASES
103
II
due to scattered
T
1 keV 160 gauss
ELECTRONS
fast electrons 10 keV
20 keV
2% 0.2%
4% 0.7% 6%
1y, 7%
-
2%
3.
Collection of ions, formed outside the condenser region. Due to the potential configuration in our system (a potential well for ions at the condenser region - see fig. 3), it is possible for ions from outside this region to reach the condenser plates. In the graph of the saturation current (fig. 8) the bump at about 10 V can be ascribed to these ions.
H=LOOGauss Electronenergy
-100
-50
0
50
5keV
100 --VsinV
Fig. 8. Graph showing the current measured at electrode 7 by varying the potential of electrode 6.
At higher field strengths between the condenser electrodes these ions are more strongly deflected and reach one of the guard electrodes. Always the saturation value I+ = i+ + i- was taken to represent the true ion current. D. Effects caused by X rays. Only few measurements exist on the absolute intensity of the X radiation emitted under electron impact. Met c hni krs) has determined the intensity of the Ka and the continuous radiation for Cu. According to his results the number of emitted X ray quanta of the continuous radiation per sterad and per incident electron (of 30 keV) is 5 x IO- 5. This number will be for 20 keV electrons on gold: 6 x 10-5, as the intensity is proportional to the atomic number of the target material and to the square of the applied voltage. Now if we take the con-
104
B. L. SCHRAM,
F. J. DE HEER,
M. J. V. D. WIEL
AND
J. KISTEMAKER
tribution of the characteristic radiation to be the tenfold of the continuous radiation and note that the opening of our collector is 1O-2 sterad, we estimate the number of X ray quanta emitted from the collector per incident electron and entering our ionization chamber to be about 6 x 10-s. We can distinguish two effects of the X radiation: the direct ionization of the gases or ionization by photoelectrons released from the electrodematerial. As the ionization cross sections of X rays are smaller than for electrons and as the efficiency for producing photoelectrons from gold will be less than loo-A14)) 1 ‘t IS . c1ear that these effects contribute to such a small amount in our measurements that they may be discarded. Another disturbing effect, namely the release of secondary electrons from the ion collecting electrode, which would be observed as a positive current, is ruled out by the magnetic field (see section IVC 1). With good reason we can say that disturbing effects due to X rays can be fully neglected. E. Correction for increase in path length of electrons. It can be shown that the relative increase of the path lengthis) for an electron entering a magnetic field under an angle CLwith the field lines, is 1/ 1 + tg%. The beam diameter was measured at the end of the collision chamber by making an image on a piece of paper. From this test we concluded that the maximum value of C( is 0.08 rad., so the correction averaged over all angles (assuming an equal probability for all values of C( between 0 and 0.08 rad.) is of the order of 1 X,,. The relative path length increase because of the cycloidal motion, which is due to the electric field between the condenser plates, is of negligible magnitude (1 in 105) for fast electrons. So clearly for neither of these effects corrections are required in this work. V. Results The gross ionization cross sections were measured for He, Ne, Ar, Kr, Xe, Hz, 112, N.L and 02. The results for these gases are listed in table III. The cross sections at six energies (0.6, 1, 2, 4, 8 and 16 keV) were evaluated from graphs like fig. 5, which procedure was done at least three times; the values listed in the table are an average of these determinations. The data at the remaining energies were obtained from one measurement at constant pressure and normalized to the absolute ones. The reproducibility over a period of two months was better than 3%, which covers all random errors. In addition to this accidental error, the values are subjected to a possible systematic error which might be 6%, consisting of the following contributions: 57; in the absolute pressure determinations, due to unknown capillary effects in the McLeod, 2% in the current measurements (l’:& in the
IONIZATION
CROSS
SECTIONS
FOR
TABLE Gross ionization
I
Gas
He
\Efd 0.6 keV 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0
__-
0.159 0.144 0.132 0.121 0.111
cross sections
I
ELECTRONS
III
for electrons
I
I
in units of lo-‘6
I
I
cm2
I
Ne
Ar
0.430
1.20
0.293
0.296
1.15
0.393 0.362
1.09
0.260
0.261 0.234
1.02 0.926
0.33 1
0.986 0.914
105
IN GASES
1.52
2.32 2.17
0.234
_~
I
~_
1.30 1.17
0.213
0.844
1.06 0.977
0.844
2.01
0.212 0.196
0.195
0.781
0.906
0.0963
0.309 0.270
1.39 1.30
0.723
1.13
1.77
0.168
0.168
0.79 1
0.0840
0.244
0.648
1.01
0.149
0.147
0.0766
0.221
0.601
0.916
1.56 1.43
0.680 0.607
0.132
0.547
0.634
0.0694 0.0627
0.201
0.542
0.841
0.133 0.121
0.499
0.186
0.769
0.577 0.531
0.158
0.657
0.108 0.09 15
0.459
0.0528 0.0450
0.495 0.420
0.38 1
0.449
0.136 0.121
0.362 0.320
0.570 0.502
0.0775
0.328
0.386 0.341
0.284
0.451
0.258
1.30 1.20
0.120
1.02 0.884
0.0913 0.078 1
0.778
0.0681
0.0683
0.290
0.701 0.637
0.0610
0.0615
0.262
0.409
0.055 1
0.0554
0.238
0.109
0.706
3.5 4.0
0.0358
4.5
0.0323
0.108 0.0992
5.0
0.0295
0.0914
0.240
0.381
0.587
0.0504
0.0504
0.218
0.257
5.5
0.0273 0.0257
0.0844
0.351
0.545
0.0465
0.0463
0.327
0.508
0.0434
0.0432
0.203 0.187
0.236
0.079 1
0.219 0.207
0.0224
0.0697
0.183
0.289
0.448
0.0381
0.0380
0.166
0.193
8.0 9.0
0.0201
0.0626
0.162
0.260
0.404
0.0339
0.0340
0.149
0.175
0.0181
0.0572
0.149
0.236
0.366
0.0307
0.0309
0.135
0.159
10.0 12.0
0.0168
0.0526 0.0456
0.136
0.216 0.187
0.337 0.289
0.0280 0.0239
0.0281
0.146
0.165
0.256
0.0211
0.0240 0.0211
0.123 0.106
6.0 7.0
14.0
0.0399
0.0144
0.120 0.104
0.0401
16.0
0.0129 0.0114
0.23 1
0.0188
0.0104
0.0329
0.0936 0.0848
0.148
18.0
0.136
0.212
0.0171
20.0
0.0095
0.0302
0.0777
0.124
0.197
0.0158
0.0360
!
0.305 0.279
0.220
0.0933
0.125 0.110
0.0186
0.0839
0.0995
0.0170
0.0770
0.0913
0.0159
0.0703
0.0842
absolute calibration of each tube voltmeter) and 1o/o in the electron energy. The purity of the gases was checked with a mass-spectrometer: in all cases the contribution of foreign ions (at 70 eV) was less than 2x,, so the error introduced by impurities can be neglected. A comparison for helium, neon and argon with the work of S m i t ha) shows that his values are consistently higher than ours by lo-20%. Although for argon part of the difference may be explained by the McLeod deviation, there is a distinct discrepancy for He and Ne. Supposedly Smith’s values at high electron energies are enlarged due to ionization by secondary electrons, for instance from the entrance slit and slit Ss in his apparatus (see appendix). The scattering of fast electrons (see section IVC 2) may explain the fact that at energies above 3 keV Smith’s values come closer to ours. Our measurements for krypton and xenon cannot be compared with other ones as the only absolute measurements published till now are confined to the region below 100 eV16). For the molecular gases a comparison can be made with the work of
106
H.
L. SCHRAM,
Tat e and Smit
hi7).
F. J. DE HEER,
Although
same general trend as before, than our values.
M. J. V. D. WIEL
AND
J. KISTEMAKER
their data end at 750 eV we find here the their cross sections
being higher by 1O-2Oo/o
The results of Sommermeyer and Dresel5) come out lower than ours. However, their measuring technique does not seem to be very reliable ; e.g. no correct saturation current is obtained. The difference found between hydrogen and deuterium is too small to attribute any significance to it. VI. Comparison with theory. Theoretical calculations of ionization cross sections by charged particles have been made on the basis of the Born approximation. Starting from the equations originally derived by Bet he I*), it was shown by Miller and Platzmanr) 2), that the ionization cross section of high energy electrons can be represented by 4naiR G= ~ Ed
IM$ciEel
(1)
where G is the total cross section in cmz/atom or molecule, a0 the first Bohr radius of hydrogen, R the Rydberg energy, E,l the electron energy and ci a constant. In the case of excitation of a discrete level the constant M; is equal to the square of the dipole matrix element divided by a;, which is equivalent to fJ(E%/R), where fn represents the optical oscillator strength and En is the excitation energy. When treating the ionization of atoms this quotient has to be integrated over the continuum; this leads to
s
df 1qJ; zzz m.dE
R E
dE
In the case of molecules a complication arises due to the so-called superexcited statesrg) (states with excitation energy exceeding the ionization potential) which will decay either by pre-ionization or by dissociation. Denoting by q(E) the efficiency of ionization when the molecule has acquired an energy E, we have now:
In formula (1) a correction has to be made for relativistic effects at electron energies exceeding 1 keV. If we define E:, by E:, = &rn,$ (mo = rest mass of the electron), formula ( 1) becomes 20) :
IONIZATION
CROSS
where Eel, the accelerating
SECTIONS
voltage,
FOR ELECTRONS
107
IN GASES
is related to E:, by: 1
1
1 + (-%z/~oc~)~ I In our energy range (2) reduces to.
(approximation at 20 keV better than 274,). In order to see whether our cross sections satisfy the theoretical relation a plot of aEd,/4naiR versus InEAr was made, similar to the graphs presented by Millers). In fig. 9 we have plotted our results for helium, together with those of several other authorsa) 21) 22). The energy dependence of our cross sections is in excellent agreement with the theory, as could be expected
I
( , 5.10*
, , , ,
Fig. 9. Ionization
o
(
lo&
5.103
__
measurements
This work;
,’
I
I
2.10’
lo3
A
in helium gas compared authors : Liska; Smith; x
with measurements q
2.104 E,l,in eV
of other
Harrison.
because the Born-Bethe approximation is valid when the energy of the incoming particle is much higher than the energy transferred to the atom or molecule in the ionization process. Fig. 9 clearly shows, that the results of Smit ha), often quoted as standards, do not give a straight line. The relative values of Liskass) certainly have a wrong energy dependence, and those of H anle and Rie desa), which
108
B.
L.
SCHRAM,
F.
J.
DE
HEER,
M.
J.
V.
D.
WIEL
AND
J.
seem to have a correct energy dependence, cannot be compared as they have been presented only in graphical form.
KISTEMAKER
accurately,
Graphs like fig. 9 were made for each gas and in all cases a straight was obtained. The values of A+?: and ci, calculated from the slopes A48 and the intercepts
\‘alws (TX
/-
~~
Hc Ne Ar Iir
Xe
Sate: ‘l‘lrc possible citimatcd
to be
ci in C\‘_~_
!
0.108
1: 0.005
1.87 4.50
1~ 0.01 : 0.04
0.0319
7.51
1 0.03 1 0.08
0.037 0.281
HZ 1)s
0.721 0.720
1_ 0.003 : 0.008
sz
3.85 4.75
_ 0.02
\yitcrnatic
C$
l_y~mmi
0.489
11.75
02
by means of a least square analysis n/r: In ci, are listed in table IV.
of ‘1113 and
.IE~im
~rrcr
-~ 0.005 0.0006
I
0.049
,
0.00 1 ~,~ 0.001
I
0.286 0.070 0.053
scctiolrs
0.002
~1 0.001
0.035
‘~ 0.03 of th? cross
line
0.004 0.001 0.001
irom
which
.1/<2 ;LII~ ci arc calculated,
is
6”‘,.
The values in table IV can be compared with results obtained by other investigators and from other experiments. Miller’s”) work, listed in the second column is based mainly on that of Smitha); Hart and Platzman24) have given the best value from a consideration of several experiments.
\‘alut?s
lir Xt:
H? 112
7.51 11.75 0.721
’
j
I 0.711
N\1?
0.720 3.85
3.6
02
4.75
3.5
0.71
of dli2
’
0.478
~
I
The values in the fourth column were calculated by Fanos) from the results of McClu re7), who measured the ionization cross sections of electrons with very high energy (0.2-l .5 MeV).
IONIZATION
CROSS
SECTIONS
A failure in these measurements
FOR ELECTRONS
was pointed
IN GASES
out by Fano:
109
the addition
of hydrogen to the noble gases has the consequence that excitations of the noble gases will be converted into ionization of hydrogen. As is expected, the greatest influence is to be found for helium and neon; for argon the influence is small and for hydrogen this effect is absent. Comparison with our data proves these statements. In the theory outlined above it was shown that i@ is related to the optical oscillator strength. A comparison can be made with values of 44: calculated from photoionization data for helium zs), neon 26) and argon 27). The values for helium and neon, which were obtained by numerical integration of the photoionization cross sections listed in tables in ref. 2.5 and 26, are in good agreement with our values obtained from the electronionization cross sections. The value for argon was obtained by using the photoionization data read from a graph in Samson’s article and so will be less accurate. Still for argon the difference is remarkable, but as Samson states his data might be too low (leading to a total oscillator strength of 5.91 for the M-shell electrons) and realizing that our value may be enhanced by contributions of multiple ionizations the real difference is probably smaller than calculated here. From the original treatment by Bethe it can be seen that at sufficient high velocity the cross sections for electrons and protons of the same velocity should be equal. H ooperss) has measured the gross ionization cross sections for protons in several gases and his results are listed in the fourth column of table V. Most cases show good agreement except hydrogen and oxygen for which no explanation can be given. Theoretical calculations on photoionization cross sections are limited to a few cases. Detailed calculations for helium have been published recently by Stewart and Webbza). These authors calculated the photoionization cross sections with three different wave functions namely Coulomb, Hartree and Hartree Fock functions and, moreover, they have taken three different methods by using successively transition elements of the dipole length, the dipole velocity and the dipole acceleration form. Taking the average of their results as they did themselves for comparing with experimental data a value of h4: = 0.53 was calculated. In view of the large spread in their data this number is in good agreement with our experimental value. For neon and argon calculations of the photoionization from the outer atomic subshells were performed by Coop e r so). His results give for neon Mz = 1.7 1 for ionization from the 2p shell and for argon Mi = 2.58 for ionization from the 3p shell. Clearly Cooper’s values are lower than our experimental values which however, include also inner shell ionizations. Besides this method of integrating the theoretically calculated photoionization cross sections, a comparison with theoretical values can be made, based on the application of the sum rules for oscillator strengthssr). The
110
n.
L. SCHRAM,
F.
J.
DE
HEER,
M. J.
V. I).
WIEL
AND
J.
KISTEMAKER
sum rule that we need is:
Here ~‘0 is the ground state wave function and ri is the position vector of the i-th electron. Especially for helium very accurate theoretical results are available, so that this method should give a good test on our experiment. For the righthand side of equation (3) Pe kerisaa) has calculated a value of 0.7525. Many calculations have been made on the oscillator strengths of the discrete transitions of which we have selected those of Schiff and Pekerissa) and those of Dalgarno and Stcwartaa). A summation of their results yields $ _A_ ,t- L’ E,jR and an estimation So for
= 0.251,
of the higher levels results in a contribution
of about 0.01.
00
I.1’.
a value of 0.49 is calculated, in excellent agreement with our experimental value in table V. Again, however, we should mention that this experimental value of 0.489 can be influenced by a necessary correction for multiple ionization which might bc of the order of 2% in the case of heliumas). Acknowledgements. The authors wish to express their thanks to Professor Dr. R. L. Platzman for suggesting this problem and for many helpful discussions. They thank Mr. E. de Haas for constructing the apparatus and Ir. R. A. Timmer for his assistance in the early stage of this work. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for Pure Scientific Research).
APPENDIX
In section IV ;Z we arrived at the conclusion that a check on the absence of secondary electrons is found by the independency of the cross sections
IONIZATION
on the strength
CROSS
SECTIONS
of the magnetic
FOR
ELECTRONS
field. However,
IN
GASES
111
care should be taken in the
application of this criterion. In general one can say that when the secondary electrons are not suppressed, the number of them present in or near the beam of primary electrons, depends on the ratio of the drifting velocity - E/B - to their velocity parallel to the magnetic field. The greater part of the secondary electrons will have energies below 50 eV, corresponding to velocities up to 4 x 108 cm/s. In our experiment the drift velocities ranged from 2.1 x 107 cm/s (at 150 gauss) to 8 x 106 cm/s (at 400 gauss). We noticed that when we did not suppress completely the secondary electrons, e.g. originating from the collector, the cross sections increased with higher magnetic field strength. Apparently with low magnetic field some of these electrons are deflected which would else have reached the condenser region at higher magnetic fields. At very high magnetic fields a saturation value will be obtained for G, namely when all secondary electrons are kept so close to the beam that they traverse the whole condenser region in axial direction. Clearly the criterion mentioned above can be used only at magnetic field strengths below the value where saturation sets in. In the experiments of Smith4) this condition is not fulfilled. There the drift velocity is 2.7 x 106 cm/s (at 250 gauss). Because of the smaller dimensions of Smith’s apparatus and the very unfavourable ratio of velocities the mentioned saturation is reached at much lower magnetic fields than in our case. When, as we supposed, the secondary electrons originated at the entrance slit they will all arrive at the electron collector. Probably the collection of electrons on the box T (see ref. 4) at magnetic fields below 200 gauss should be explained by the deflection of slow electrons. So we see that in the work of Smith the presence of secondary electrons is very probable despite the fact that his cross sections are independent of the magnetic field. Received
30-7-64
REFERENCES
1) 2) 3) 4)
Miller, Miller, Fano, Smith,
W. F. and Platzman,
R. L., Proc. roy. Sot. 70 (1957) 299.
W. F., Thesis, Purdue University U., Phys. Rev. 95 (1954) 1198.
1956.
P. T., Phys. Rev. 36 (1930) 1293.
K. and Dresel, H., Z. Phys. 141 (1955) 307. 5) Sommermeyer, E. J. and Terroux, F. R., Proc. roy Sot. A126 (1930) 289. 6) Williams, G. W., Phys. Rev. 90 (1953) 796. 7) McClure, G., Advances in Vacuum Science and Technology, Vol. I, p. 335, Pergamon 8) Zinsmeister, Press, London ( 1958). K. and Ishii, H., Vacuum Symposium Transactions 8-I (1961) 519. 9) Nakayama, 10) Rol, P. K. and De Vries, A. E., To be published. W. J. and Soroka, W. W., Analog Methods, 11) Karplus, Sciences, McGraw Hill, New York (1959) p. 408.
McGraw
Hill Series in Engineering
IONIZATION
112 1-Y
Pierce,
13)
Metchnik,
J.
CROSS
Ii., Theory
SECTIONS
and design
V., Proc.
ray.
Sot.
of electron
83 (1963)
CValker,
\I;. C., Wainfan,
N. and Weissler,
15)
Asundi,
Ii. Ii., Proc.
Sot.
Asundi, ‘rate,
18)
Uethe,
19)
Platzman,
20)
Mott,
21)
University Harrison,
22)
Liska,
23) 24)
Hanlc, Hart,
25)
York (1961), \:ol. I, p. 103. Baker, 1). J,, Bedo, D. E. and Tomboulian,
27) 28)
1~. L., J.
5 (1930)
l’hys.
N. F. and Massey,
J.
Contr.
Rev. 39 (1932)
Phys.
15 (1963)
al (1960) 853. H. S. \V., The Theory of Atomic Univ.
Press,
Washington
D. H., Phys.
1123. Hoopcr,
J. LV., Harmer,
Schiff,
33) 34)
I)algarno,
35)
Stanton,
D. H., Phys.
D. S., Martin,
Dalgarrlo, Pekeris,
T. G., Proc.
Rev. If8
(1962)
C. L., Phys.
A. and Stewart, H. A. and Monahan,
Rev.
133
1). \V., and
2nd ed. p. 367, Oxford
Phys.
Academic
(1964)
124
D. S., Phys.
Mc Daniel,
Sac. 82 (963)
Rev. 134
A. L., Proc. Phys. J. E., Phys.
A70 (1964) Sot.
(1957)
532.
802.
A 640. A78
(1960)
Rev. 119 (1960)
(1961)
Press
Inc.
New
1471.
A 1525.
681.
A. and Lynn, N., Proc. Phys. Sot. C. I_, l’hys. Rev. 115 (1959) 1216. 1s. and Pekeris,
1366.
41.
Rev.
J. A. R., J. Opt. Sot. Am. 54 (1954) 1198. J. IV., McDaniel, E. W., Martin, D. 1%‘. and Harmer,
31)
p. 27.
D. C. 1956).
Samson, Hooprr,
J. \I:., I’hys.
1949
169.
I>. L. and Tomboulian,
Cooper,
z(i (1955)
Collisions
Ederer,
30)
York,
270.
LV. and Riede, D., Z. I’hps. 133 (1952) 537. II. J. and I’latzman, Ii. L., Mechanisms in Radiobiology,
29)
New
325.
Rev. 46 (1934)
(1962) 2000. Stewart, A. L., and Webb,
32)
Nostrand,
Radium
Press 1949. H., Thesis (Catholic W., Phys.
Van
G. L., J. appl.
M. V., J. Electron
1’. T., Phys.
H. A., Arm. Phys.
beams,
IN GASES
372.
17)
26)
1~. Ii. and Kurepa,
81 (1963)
16)
J. T. and Smith,
ELECTRONS
515.
14)
Phys.
FOR
49.
711.
E.
W.,
Rev. 121 Phys.
(1961)
Rev.
185
I% I,.
De Heer, Van
Physica
F. J.
der Wiel,
Kistemaker,
31
94-112 M. J.
J.
1965
IONIZATION (0.6-20
CROSS
keV)
SECTIONS
IN NOBLE
by B. I,. SCHRAM,
F. J.
DE
J.
AND HEER,
FOR
ELECTRONS
DIATOMIC M. J.
GASES
VAN DER
WIEL
and
KISTEMAKER
Synopsis Absolute attention
has been paid to the elimination
measured for fast electrons (energy Xe, H s, 1~s. Na and 0s. Detailed of disturbing effects caused by secondary
electrons
and to the influence
effects
range
ionization
0.6-20
a McLeod
keV)
gauge.
cross
incident
sections on He,
have Ne,
been
Ar,
of diffusion
Kr,
In all cases the experimental (r =
4na$R ___
on the pressure cross sections
Mi” In
measurements
with
can be represented
by
CiEel.
EL?1
The relation
between
son of our values
Mia and the optical
with
those
obtained
oscillator from
strength
different
is discussed
experiments
and a compari-
and
from
theory
is made.
I. Introduction. Determination of absolute excitation and ionization cross sections for electrons in gases is important for several reasons. The most obvious ones are applications of these data in plasma physics, astrophysics, upper atmosphere physics and radiation chemistry. Moreover it is possible to deduce information related to optical oscillator strengths for atoms and molecules from these total collision cross sections as was pointed out for instance by Miller and Platzmanl)s) and Fanoa). Measurements of absolute ionization cross sections for high energy electrons are very scarce, the only ones are those of Smith4) (up to 4.5 keV) and Sommermeyer and Dresels) (l-50 keV), which differ in magnitude up to 60%. Therefore, it is useful to remeasure these cross sections and to obtain the results for more gases. The most probable cause of differences between the results cited is the insufficient attention, that has been paid to disturbing secondary effects. We have made a critical analysis of all these effects and have succeeded in suppressing them. In this work absolute ionization cross sections of fast electrons were determined for the rare gases He, Ne, Ar, Kr and Xe and the diatomic gases Hz, Ds, Ns and 02. In the literature two different methods for measuring ionization cross sections are mentioned: firstly the condenser technique, where the current -
94 -
IONIZATION
of positive
CROSS SECTIONS
ions - relative
FOR ELECTRONS
to the primary
electron
IN GASES
current
95
- is determined,
and secondly the counting technique, where the individual ionization events are recorded either in a Wilson chamber-e) or with G. M. counters7) in a coincidence circuit. In this investigation the first method was chosen. II.
Apparatus
our apparatus
and
experimental
procedure.
1 COLLIMAtOR ___ELECTRONGUN
Fig. 1. Schematic
A. General.
A survey
of
is given in fig. 1.
view of the apparatus
COLL.CHAMBER
COLD TRAP
l_MAGNETCOlLS
used for measuring
ionization
cross sections.
The apparatus consists of a stainless steel cylindrical vessel, divided in two parts: the vacuum chamber and the collision chamber, which contain the electron source and the electrode system respectively. Both chambers are connected via the collimator and a bypass. An electron beam, generated by the electron source, is shot into the collision chamber and the current is measured in a Faraday cage. Positive ions, produced along a well defined pathlength, are collected on one of the condenser plates of the electrode system. Target gases are introduced into the collision chamber via a needle valve and their pressure is determined with a McLeod gauge. An axial magnetic field is produced by two magnet coils around both chambers. B. Vacuum. The vacuum chamber is connected to a 650 Z/soil diffusion pump. This pump has a specially designed water cooled chevron baffles), which needs no extra freon or liquid air cooling. During the measurements the bypass is closed and, when using the cold trap, the final pressure in the collision chamber is 5 x 10-S torr. Differential pumping through the collimator (conductance 0.3 Z/sfor air) enables us to operate the electron source at a pressure of 10-s torr, while maintaining a much higher target gas pressure, 10-4 - 10-a tori-, in the collision chamber. The absolute pressure of the gases is measured with a McLeod gauge,
96
B.
L. SCHRAM,
F. J. DE HEER,
M. J. V. D. WIEL
AND
J. KISTEMrlKER
constructed in the laboratory; its capillary has a diameter of 0.7 mm, constant within lx,, over the whole length, and the bulb volume amounts to 600 ems. Recent experimentsg) 10) showed, that a McLeod often gives incorrect results, the error being dependent on gas, temperature and dimensions of the instrument. This effect is a consequence of the steady stream of mercury vapour from the mercury reservoir to the cold trap (A in fig. 2) between the McLeod and the collision chamber. The target gas has to diffuse against this mercury stream, which gives rise to a difference between the pressures in the glass bulb and in the collision chamber. To eliminate this effect, we followed the suggestion made by Ishii and Na kay amaa). Their idea was to cool the walls of the McLeod just above the
Fig. 2. McLeod
gauge, equipped
with the normal cold trap A and an extra cold trap B. TABLE
I
reservoir (see B in fig. 2), so that the mercury vapour pressure is lowered sufficiently. With this method we found higher pressure readings for all gases; the corrections are listed in table I. As these values are dependent on dimensions of the McLeod and room
IONIZATION
CROSS
SECTIONS
temperature, they are representative be used as universal corrections.
FOR ELECTRONS
IN GASES
only for our instrument
97 and cannot
C. Magnetic field. An axial magnetic field - maximum 400 gauss was necessary for the focusing of the electron beam and, as will be discussed elsewhere (section IV A), also provided a valuable check on the suppression of secondary effects. External to the magnetic coils around the collision and vacuum chamber a cylindrical iron shield is fastened, in order to prevent asymmetric disturbances of the axial field, caused by the iron frame on which the whole apparatus is mounted and by magnetic stray fields of other apparatus in the neighbourhood. D. Electron source. The electron source is a Philips 6-AW-59 electron gun from a television tube. The normal operating voltage is 17 kV, but, by applying the axial magnetic field, we could use it over the whole energy range of 0.6-20 keV, obtaining an electron current variable up to 5 x lo-4 A. The diameter of the electron beam was limited to 3 mm by a collimator, located between vacuum and collision chamber, which also serves as a pumping resistance (see B). A John Fluke High Voltage Supply (model 408 A/J) was used for the energy range of 0.6-6 keV and a Hursant High Voltage Supply (model 30 B) for the range of 3-20 keV. Both were calibrated by a compensation method. E. Electrode system. The electrode system consists of a number of gold plated stainless steel plates, separated by glass spacers. Figs. 3 and 4 show the electrode system together with the potential configuration along the axis. The positive ions, produced in the symmetric condenser region between plates 6 and 7, can be directed to either of these electrodes by means of a small transverse electric field. With the aid of guard plates (4. 5, 8 and 9) we attained homogeneity of the ion collecting field, as was checked with the conducting sheet methodrr). Two sets of flat plates (a and b) are fastened on both sides of condenser and guard plates (fig. 4A) ; they are insulated from electrodes 6 and 7 and their function will be described later. To avoid surface charges on the insulating spacers at the back side of the electrode system the half diaphragms 2, 3, 12 and 13 are supplied with cylindrical shields as is depicted in fig. 4B. The electron beam current is detected at the collector, which consists of 2 parts: electrode 15, a hollow cone, into which points the half cone 14.
B.
98
L. SCHRAM,
Fig. 3. Upper part
F. J. DE HEER,
M. J. \:. 1). WIET. ANI)
: Cross section of the electrode system parallel to the electron beam;
-:zoov; the following potentials mere applied: collimator 4, 6 and 8: +80V; 5, 7and 9: OV; 10: +12OV; 3: +4OV; 13: + 100 V; Lower
part:
J. KIS’L-EMllKER
Potential
distribution
14: -I-4oov;
1: +1oov; 11: +4OV;
15: +zoov.
along the axis of the clectrodc
Fig. 4 A
: Cross section of the electrode
system
perpendicular
2: $12OV; 12: -/ 15OV;
Fig. 4 13: Perspective
system.
view of electrotlcs
2 and 3 or 12 and 13.
to the electron
beam.
The beam current is measured with a General Radio tube volt meter, the ion current with a Philips GM 6020 tube volt meter. III. Evaluation of the cross sections from the experinzental results. The gross ionization cross sections c (including multiple ionization processes) are calculated from the formula: CT=
Ii.
I+ I_ ln,
I_@
3.535 x 10’”
T 273
where I+ is the current of positive ions, measured on electrode 7 (see also section IV C); I_ the electron beam current, measured in the Faraday cage (see section IV B) ;
IONIZATION
CROSS
_._~~..
Z n p
SECTIONS
-~~
FOR
ELECTRONS
~~___
the length from which ions are extracted the number of gas molecules per ems; the pressure of the gas in torr
IN GASES
_____~
; (see section
99 ~~~__ - ~
IV E) ;
and
T
the temperature
of the gas in “K, for which was taken room temperature.
So (I is the gross ionization cross section, expressed in ems/molecule. We verified whether the ion current was proportional both to the electron current and the gas pressure. 5
I,/l_rlO-‘A I
Fig. 5. Measured
ion
current
relative target
to the
electron
current
as a function
of the
gas pressure.
In fig. 5 we plotted I+/I_ as a function of target gas pressure for a representative measurement in He. At about 2 x 10-s torr a deviation from linear relationship sets in; our measurements, therefore, were carried out in the pressure region between 2 x 10-J and 1.5 x 10-s torr. The same choice was made for all other gases, which behaved analogously. Every cross section was evaluated from the slope of a graph like fig. 5. IV. General discussion on disturbing effects. A. Ionization by secondary electrons. Secondary electrons, which may have ionization cross sections much higher than the primary ones, are released from the target gas molecules in the ionization process as well as from the electrode material by result of impacting particles. The behaviour of these electrons is complicated by the presence of both a magnetic and an electric field.
100
_
.~~
B. L. SCHRAM,
F. J. DE HEER,
M. J. V. D. WIEL
AND
J. KISTEMAKER
In these combined fields they move along trochoidalis) tracks, which can be considered as a superposition of a helical motion parallel to the magnetic field and a drifting motion perpendicular to both fields. In our apparatus, where the normal ion collecting field amounts to 30 V/cm and the magnetic field varies from 150-400 gauss (resulting in a drift velocity of i lo7 cm/s) the following rough classification for these electrons can be made, dependent on their energy correlated to the motion Parallel to the magnetic field, viz.: a. less than 0.1 eV b. between c.
0.1 and 5 eV
more than 5 eV.
(The boundary values are very rough approximations). Of course, only those electrons matter, which have a total energy higher than the ionization potential. For electrons of group a the trochoidal drifting motion will be the most important, whereas those of group c, having a velocity parallel to the magnetic field much higher than the drift velocity, travel along helices, which nearly coincide with the magnetic field lines. The motion of electrons of group b is a combination of both possibilities. Evidently, the only way to prevent secondary electrons from entering the condenser region from the outside, is the application of an electric field parallel to the magnetic field, so that electrons are reflected from this region by a potential hill. This leads to an important rule: The condenser region should have the lowest potential of the whole system (see fig. 3). The only electrodes, hit by the primary beam, are the beam defining collimator (which is knife-edged) and the collector. The other diaphragms, notably 10 and 11, with openings large compared to the beam diameter, can be hit merely by scattered fast electrons. Secondary electrons, raised by these impacts, are suppressed by the suitable potential configuration. Then we have to consider the secondary electrons released from the gas molecules. Any slow electron created in the condenser region between electrodes 6 and 7 is moving in a trochoid, whose radius is given by Y = m,EleH”, where m, is the electrons mass, E the strength of the electric field, e the electron charge and H the magnetic field strength. Now the radius of the trochoidal movement is so small (in our experiment of the order of 2 x IO-2 mm) that the electrons will certainly not reach the positive electrode of the condenser (6). The electrons of category a, where the drifting motion perpendicular to both fields is predominant, will drift from between the electrode system, following more or less closely the aequipotential surface, in which they were initially born. It is necessary to catch these electrons lest they may float around on our apparatus and cause uncontrollable effects. Therefore, two
IONIZATION
CROSS SECTIONS
sets of extra plates are attached
FOR ELECTRONS
IN GASES
to the side of the electrode
101 system
(see
fig. 4A). In the 1 mm spacing between electrodes a and b, which have the same potentials as 6 respectively 7, the electric field strength is increased and this causes an increase of the radius of the trochoidal electron movement, so that the electrons land on the positive plate a. The electrodes 6 and 7 are insulated from a and b so the collection of slow electrons does not disturb the measurement of the positive ions. The electrons of group c, with a predominant velocity parallel to the magnetic field, will travel along the magnetic field lines and will be accelerated to the more positive parts of the electrode system (e.g. the Faraday cage). The electrons of group b will be partly caught at the positive side plate a and partly at positive electrodes next to the condenser region. All effects, caused by secondary electrons from outside the condenser region, are dependent on the magnetic field strength as their paths are influenced by this field. Because our cross sections were found to be independent on this field, we may claim a sufficient suppression of these effects. Evidently, ionization by fast secondary electrons produced in the condenser region itself, cannot be avoided; however, as its contribution is proportional to the square of the gas pressure, it can be neglected as long as the cross section is independent of the gas pressure (see also appendix). B. Effects influencing the electron beam current. Theelectron beam collector consists of two conical parts, on both of which together the beam current is measured. Fig. 6 shows the measuring circuit and the applied voltages, taken from highly insulated batteries.
Fig. 6. Schematic
view of the measurement
of the electron
Three effects might give a false determination
beam current.
of the beam current:
1. Escape of secondary electrons. The electric field, due to the potential difference of 200 V across both collector parts, proves to be strong enough to trap all secondary electrons somewhere in the collector. See fig. 7. 2.
Ionization in the collector. Again due to the 200 V potential difference, both positive ions and electrons reach one of the two collector parts; there is no net result in the total current. 3. Collections of slow electrons. It was discussed before (section IVA) that a part of the electrons released
102
B. L. SCHRAM,
F. J. DE HEER,
M. J. V. D. WIEL
AND
J. KISTEMAKER
from the gas molecules will reach the collector. This effect, inevitable because of our choice of potential configuration, is proportional both to the cross section and gas pressure, so it can be kept insignificant by limiting
Electron
energy
Ar Pressure
+-0
~~7~~~ ~~_ ~~_-. 50
100
5keV
6.W’ton
r 150
200 --~-
Fig. 7. Measured
collector
current
as a function
electrodes
of
250 V15-Vtd
the
inV
applied
voltage
between
14 and 15.
the pressure range. By giving electrodes 12 and 13 a negative potential and watching the change in beam current, this effect can be verified. It was always less than 1y/,, except in the case of Kr and Xe, where a correction was applied at low electron energies. C. Effects influencing of the positive ion current processes :
the positive current. The measurement on electrode 7 might be disturbed by three
1. Escape of secondary electrons. All electrons, ejected by impact of ions, electrons or photons are returned to the same plate under the influence of the magnetic field. 2.
Impact of fast scattered electrons. Primary electrons, scattered over an angle cc, can hit the ion collecting electrode when sin C(> (H%Pe/32mE,~)~ where d is the distance between the condenser plates; H the magnetic field strength; e the carge of an electron; and E,l the energy of the deflected primary electron. Assuming that the tracks of these fast electrons are not influenced by the small ion collecting field, we can reverse this field and collect only fast scattered electrons. A negative current i- is measured, which provides a correction on I+. Some values of this correction, relative to the I+, are listed in table II. The contribution of negative ions to the correction can be neglected because of the very small probability of their formation.
IONIZATION
CROSS
SECTIONS
FOR
TABLE Corrections
on ion current
I400
I,
3 keV 0.1%
400
1%
1,
-
4% 0.6%
160 400
,,
1% _
6% OS:/,
160 >>
I
0.8%
0.5% -
IN GASES
103
II
due to scattered
T
1 keV 160 gauss
ELECTRONS
fast electrons 10 keV
20 keV
2% 0.2%
4% 0.7% 6%
1y, 7%
-
2%
3.
Collection of ions, formed outside the condenser region. Due to the potential configuration in our system (a potential well for ions at the condenser region - see fig. 3), it is possible for ions from outside this region to reach the condenser plates. In the graph of the saturation current (fig. 8) the bump at about 10 V can be ascribed to these ions.
H=LOOGauss Electronenergy
-100
-50
0
50
5keV
100 --VsinV
Fig. 8. Graph showing the current measured at electrode 7 by varying the potential of electrode 6.
At higher field strengths between the condenser electrodes these ions are more strongly deflected and reach one of the guard electrodes. Always the saturation value I+ = i+ + i- was taken to represent the true ion current. D. Effects caused by X rays. Only few measurements exist on the absolute intensity of the X radiation emitted under electron impact. Met c hni krs) has determined the intensity of the Ka and the continuous radiation for Cu. According to his results the number of emitted X ray quanta of the continuous radiation per sterad and per incident electron (of 30 keV) is 5 x IO- 5. This number will be for 20 keV electrons on gold: 6 x 10-5, as the intensity is proportional to the atomic number of the target material and to the square of the applied voltage. Now if we take the con-
104
B. L. SCHRAM,
F. J. DE HEER,
M. J. V. D. WIEL
AND
J. KISTEMAKER
tribution of the characteristic radiation to be the tenfold of the continuous radiation and note that the opening of our collector is 1O-2 sterad, we estimate the number of X ray quanta emitted from the collector per incident electron and entering our ionization chamber to be about 6 x 10-s. We can distinguish two effects of the X radiation: the direct ionization of the gases or ionization by photoelectrons released from the electrodematerial. As the ionization cross sections of X rays are smaller than for electrons and as the efficiency for producing photoelectrons from gold will be less than loo-A14)) 1 ‘t IS . c1ear that these effects contribute to such a small amount in our measurements that they may be discarded. Another disturbing effect, namely the release of secondary electrons from the ion collecting electrode, which would be observed as a positive current, is ruled out by the magnetic field (see section IVC 1). With good reason we can say that disturbing effects due to X rays can be fully neglected. E. Correction for increase in path length of electrons. It can be shown that the relative increase of the path lengthis) for an electron entering a magnetic field under an angle CLwith the field lines, is 1/ 1 + tg%. The beam diameter was measured at the end of the collision chamber by making an image on a piece of paper. From this test we concluded that the maximum value of C( is 0.08 rad., so the correction averaged over all angles (assuming an equal probability for all values of C( between 0 and 0.08 rad.) is of the order of 1 X,,. The relative path length increase because of the cycloidal motion, which is due to the electric field between the condenser plates, is of negligible magnitude (1 in 105) for fast electrons. So clearly for neither of these effects corrections are required in this work. V. Results The gross ionization cross sections were measured for He, Ne, Ar, Kr, Xe, Hz, 112, N.L and 02. The results for these gases are listed in table III. The cross sections at six energies (0.6, 1, 2, 4, 8 and 16 keV) were evaluated from graphs like fig. 5, which procedure was done at least three times; the values listed in the table are an average of these determinations. The data at the remaining energies were obtained from one measurement at constant pressure and normalized to the absolute ones. The reproducibility over a period of two months was better than 3%, which covers all random errors. In addition to this accidental error, the values are subjected to a possible systematic error which might be 6%, consisting of the following contributions: 57; in the absolute pressure determinations, due to unknown capillary effects in the McLeod, 2% in the current measurements (l’:& in the
IONIZATION
CROSS
SECTIONS
FOR
TABLE Gross ionization
I
Gas
He
\Efd 0.6 keV 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0
__-
0.159 0.144 0.132 0.121 0.111
cross sections
I
ELECTRONS
III
for electrons
I
I
in units of lo-‘6
I
I
cm2
I
Ne
Ar
0.430
1.20
0.293
0.296
1.15
0.393 0.362
1.09
0.260
0.261 0.234
1.02 0.926
0.33 1
0.986 0.914
105
IN GASES
1.52
2.32 2.17
0.234
_~
I
~_
1.30 1.17
0.213
0.844
1.06 0.977
0.844
2.01
0.212 0.196
0.195
0.781
0.906
0.0963
0.309 0.270
1.39 1.30
0.723
1.13
1.77
0.168
0.168
0.79 1
0.0840
0.244
0.648
1.01
0.149
0.147
0.0766
0.221
0.601
0.916
1.56 1.43
0.680 0.607
0.132
0.547
0.634
0.0694 0.0627
0.201
0.542
0.841
0.133 0.121
0.499
0.186
0.769
0.577 0.531
0.158
0.657
0.108 0.09 15
0.459
0.0528 0.0450
0.495 0.420
0.38 1
0.449
0.136 0.121
0.362 0.320
0.570 0.502
0.0775
0.328
0.386 0.341
0.284
0.451
0.258
1.30 1.20
0.120
1.02 0.884
0.0913 0.078 1
0.778
0.0681
0.0683
0.290
0.701 0.637
0.0610
0.0615
0.262
0.409
0.055 1
0.0554
0.238
0.109
0.706
3.5 4.0
0.0358
4.5
0.0323
0.108 0.0992
5.0
0.0295
0.0914
0.240
0.381
0.587
0.0504
0.0504
0.218
0.257
5.5
0.0273 0.0257
0.0844
0.351
0.545
0.0465
0.0463
0.327
0.508
0.0434
0.0432
0.203 0.187
0.236
0.079 1
0.219 0.207
0.0224
0.0697
0.183
0.289
0.448
0.0381
0.0380
0.166
0.193
8.0 9.0
0.0201
0.0626
0.162
0.260
0.404
0.0339
0.0340
0.149
0.175
0.0181
0.0572
0.149
0.236
0.366
0.0307
0.0309
0.135
0.159
10.0 12.0
0.0168
0.0526 0.0456
0.136
0.216 0.187
0.337 0.289
0.0280 0.0239
0.0281
0.146
0.165
0.256
0.0211
0.0240 0.0211
0.123 0.106
6.0 7.0
14.0
0.0399
0.0144
0.120 0.104
0.0401
16.0
0.0129 0.0114
0.23 1
0.0188
0.0104
0.0329
0.0936 0.0848
0.148
18.0
0.136
0.212
0.0171
20.0
0.0095
0.0302
0.0777
0.124
0.197
0.0158
0.0360
!
0.305 0.279
0.220
0.0933
0.125 0.110
0.0186
0.0839
0.0995
0.0170
0.0770
0.0913
0.0159
0.0703
0.0842
absolute calibration of each tube voltmeter) and 1o/o in the electron energy. The purity of the gases was checked with a mass-spectrometer: in all cases the contribution of foreign ions (at 70 eV) was less than 2x,, so the error introduced by impurities can be neglected. A comparison for helium, neon and argon with the work of S m i t ha) shows that his values are consistently higher than ours by lo-20%. Although for argon part of the difference may be explained by the McLeod deviation, there is a distinct discrepancy for He and Ne. Supposedly Smith’s values at high electron energies are enlarged due to ionization by secondary electrons, for instance from the entrance slit and slit Ss in his apparatus (see appendix). The scattering of fast electrons (see section IVC 2) may explain the fact that at energies above 3 keV Smith’s values come closer to ours. Our measurements for krypton and xenon cannot be compared with other ones as the only absolute measurements published till now are confined to the region below 100 eV16). For the molecular gases a comparison can be made with the work of
106
H.
L. SCHRAM,
Tat e and Smit
hi7).
F. J. DE HEER,
Although
same general trend as before, than our values.
M. J. V. D. WIEL
AND
J. KISTEMAKER
their data end at 750 eV we find here the their cross sections
being higher by 1O-2Oo/o
The results of Sommermeyer and Dresel5) come out lower than ours. However, their measuring technique does not seem to be very reliable ; e.g. no correct saturation current is obtained. The difference found between hydrogen and deuterium is too small to attribute any significance to it. VI. Comparison with theory. Theoretical calculations of ionization cross sections by charged particles have been made on the basis of the Born approximation. Starting from the equations originally derived by Bet he I*), it was shown by Miller and Platzmanr) 2), that the ionization cross section of high energy electrons can be represented by 4naiR G= ~ Ed
IM$ciEel
(1)
where G is the total cross section in cmz/atom or molecule, a0 the first Bohr radius of hydrogen, R the Rydberg energy, E,l the electron energy and ci a constant. In the case of excitation of a discrete level the constant M; is equal to the square of the dipole matrix element divided by a;, which is equivalent to fJ(E%/R), where fn represents the optical oscillator strength and En is the excitation energy. When treating the ionization of atoms this quotient has to be integrated over the continuum; this leads to
s
df 1qJ; zzz m.dE
R E
dE
In the case of molecules a complication arises due to the so-called superexcited statesrg) (states with excitation energy exceeding the ionization potential) which will decay either by pre-ionization or by dissociation. Denoting by q(E) the efficiency of ionization when the molecule has acquired an energy E, we have now:
In formula (1) a correction has to be made for relativistic effects at electron energies exceeding 1 keV. If we define E:, by E:, = &rn,$ (mo = rest mass of the electron), formula ( 1) becomes 20) :
IONIZATION
CROSS
where Eel, the accelerating
SECTIONS
voltage,
FOR ELECTRONS
107
IN GASES
is related to E:, by: 1
1
1 + (-%z/~oc~)~ I In our energy range (2) reduces to.
(approximation at 20 keV better than 274,). In order to see whether our cross sections satisfy the theoretical relation a plot of aEd,/4naiR versus InEAr was made, similar to the graphs presented by Millers). In fig. 9 we have plotted our results for helium, together with those of several other authorsa) 21) 22). The energy dependence of our cross sections is in excellent agreement with the theory, as could be expected
I
( , 5.10*
, , , ,
Fig. 9. Ionization
o
(
lo&
5.103
__
measurements
This work;
,’
I
I
2.10’
lo3
A
in helium gas compared authors : Liska; Smith; x
with measurements q
2.104 E,l,in eV
of other
Harrison.
because the Born-Bethe approximation is valid when the energy of the incoming particle is much higher than the energy transferred to the atom or molecule in the ionization process. Fig. 9 clearly shows, that the results of Smit ha), often quoted as standards, do not give a straight line. The relative values of Liskass) certainly have a wrong energy dependence, and those of H anle and Rie desa), which
108
B.
L.
SCHRAM,
F.
J.
DE
HEER,
M.
J.
V.
D.
WIEL
AND
J.
seem to have a correct energy dependence, cannot be compared as they have been presented only in graphical form.
KISTEMAKER
accurately,
Graphs like fig. 9 were made for each gas and in all cases a straight was obtained. The values of A+?: and ci, calculated from the slopes A48 and the intercepts
\‘alws (TX
/-
~~
Hc Ne Ar Iir
Xe
Sate: ‘l‘lrc possible citimatcd
to be
ci in C\‘_~_
!
0.108
1: 0.005
1.87 4.50
1~ 0.01 : 0.04
0.0319
7.51
1 0.03 1 0.08
0.037 0.281
HZ 1)s
0.721 0.720
1_ 0.003 : 0.008
sz
3.85 4.75
_ 0.02
\yitcrnatic
C$
l_y~mmi
0.489
11.75
02
by means of a least square analysis n/r: In ci, are listed in table IV.
of ‘1113 and
.IE~im
~rrcr
-~ 0.005 0.0006
I
0.049
,
0.00 1 ~,~ 0.001
I
0.286 0.070 0.053
scctiolrs
0.002
~1 0.001
0.035
‘~ 0.03 of th? cross
line
0.004 0.001 0.001
irom
which
.1/<2 ;LII~ ci arc calculated,
is
6”‘,.
The values in table IV can be compared with results obtained by other investigators and from other experiments. Miller’s”) work, listed in the second column is based mainly on that of Smitha); Hart and Platzman24) have given the best value from a consideration of several experiments.
\‘alut?s
lir Xt:
H? 112
7.51 11.75 0.721
’
j
I 0.711
N\1?
0.720 3.85
3.6
02
4.75
3.5
0.71
of dli2
’
0.478
~
I
The values in the fourth column were calculated by Fanos) from the results of McClu re7), who measured the ionization cross sections of electrons with very high energy (0.2-l .5 MeV).
IONIZATION
CROSS
SECTIONS
A failure in these measurements
FOR ELECTRONS
was pointed
IN GASES
out by Fano:
109
the addition
of hydrogen to the noble gases has the consequence that excitations of the noble gases will be converted into ionization of hydrogen. As is expected, the greatest influence is to be found for helium and neon; for argon the influence is small and for hydrogen this effect is absent. Comparison with our data proves these statements. In the theory outlined above it was shown that i@ is related to the optical oscillator strength. A comparison can be made with values of 44: calculated from photoionization data for helium zs), neon 26) and argon 27). The values for helium and neon, which were obtained by numerical integration of the photoionization cross sections listed in tables in ref. 2.5 and 26, are in good agreement with our values obtained from the electronionization cross sections. The value for argon was obtained by using the photoionization data read from a graph in Samson’s article and so will be less accurate. Still for argon the difference is remarkable, but as Samson states his data might be too low (leading to a total oscillator strength of 5.91 for the M-shell electrons) and realizing that our value may be enhanced by contributions of multiple ionizations the real difference is probably smaller than calculated here. From the original treatment by Bethe it can be seen that at sufficient high velocity the cross sections for electrons and protons of the same velocity should be equal. H ooperss) has measured the gross ionization cross sections for protons in several gases and his results are listed in the fourth column of table V. Most cases show good agreement except hydrogen and oxygen for which no explanation can be given. Theoretical calculations on photoionization cross sections are limited to a few cases. Detailed calculations for helium have been published recently by Stewart and Webbza). These authors calculated the photoionization cross sections with three different wave functions namely Coulomb, Hartree and Hartree Fock functions and, moreover, they have taken three different methods by using successively transition elements of the dipole length, the dipole velocity and the dipole acceleration form. Taking the average of their results as they did themselves for comparing with experimental data a value of h4: = 0.53 was calculated. In view of the large spread in their data this number is in good agreement with our experimental value. For neon and argon calculations of the photoionization from the outer atomic subshells were performed by Coop e r so). His results give for neon Mz = 1.7 1 for ionization from the 2p shell and for argon Mi = 2.58 for ionization from the 3p shell. Clearly Cooper’s values are lower than our experimental values which however, include also inner shell ionizations. Besides this method of integrating the theoretically calculated photoionization cross sections, a comparison with theoretical values can be made, based on the application of the sum rules for oscillator strengthssr). The
110
n.
L. SCHRAM,
F.
J.
DE
HEER,
M. J.
V. I).
WIEL
AND
J.
KISTEMAKER
sum rule that we need is:
Here ~‘0 is the ground state wave function and ri is the position vector of the i-th electron. Especially for helium very accurate theoretical results are available, so that this method should give a good test on our experiment. For the righthand side of equation (3) Pe kerisaa) has calculated a value of 0.7525. Many calculations have been made on the oscillator strengths of the discrete transitions of which we have selected those of Schiff and Pekerissa) and those of Dalgarno and Stcwartaa). A summation of their results yields $ _A_ ,t- L’ E,jR and an estimation So for
= 0.251,
of the higher levels results in a contribution
of about 0.01.
00
I.1’.
a value of 0.49 is calculated, in excellent agreement with our experimental value in table V. Again, however, we should mention that this experimental value of 0.489 can be influenced by a necessary correction for multiple ionization which might bc of the order of 2% in the case of heliumas). Acknowledgements. The authors wish to express their thanks to Professor Dr. R. L. Platzman for suggesting this problem and for many helpful discussions. They thank Mr. E. de Haas for constructing the apparatus and Ir. R. A. Timmer for his assistance in the early stage of this work. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (Foundation for Fundamental Research on Matter) and was made possible by financial support from the Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek (Netherlands Organization for Pure Scientific Research).
APPENDIX
In section IV ;Z we arrived at the conclusion that a check on the absence of secondary electrons is found by the independency of the cross sections
IONIZATION
on the strength
CROSS
SECTIONS
of the magnetic
FOR
ELECTRONS
field. However,
IN
GASES
111
care should be taken in the
application of this criterion. In general one can say that when the secondary electrons are not suppressed, the number of them present in or near the beam of primary electrons, depends on the ratio of the drifting velocity - E/B - to their velocity parallel to the magnetic field. The greater part of the secondary electrons will have energies below 50 eV, corresponding to velocities up to 4 x 108 cm/s. In our experiment the drift velocities ranged from 2.1 x 107 cm/s (at 150 gauss) to 8 x 106 cm/s (at 400 gauss). We noticed that when we did not suppress completely the secondary electrons, e.g. originating from the collector, the cross sections increased with higher magnetic field strength. Apparently with low magnetic field some of these electrons are deflected which would else have reached the condenser region at higher magnetic fields. At very high magnetic fields a saturation value will be obtained for G, namely when all secondary electrons are kept so close to the beam that they traverse the whole condenser region in axial direction. Clearly the criterion mentioned above can be used only at magnetic field strengths below the value where saturation sets in. In the experiments of Smith4) this condition is not fulfilled. There the drift velocity is 2.7 x 106 cm/s (at 250 gauss). Because of the smaller dimensions of Smith’s apparatus and the very unfavourable ratio of velocities the mentioned saturation is reached at much lower magnetic fields than in our case. When, as we supposed, the secondary electrons originated at the entrance slit they will all arrive at the electron collector. Probably the collection of electrons on the box T (see ref. 4) at magnetic fields below 200 gauss should be explained by the deflection of slow electrons. So we see that in the work of Smith the presence of secondary electrons is very probable despite the fact that his cross sections are independent of the magnetic field. Received
30-7-64
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Miller, Miller, Fano, Smith,
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IONIZATION
112 1-Y
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SECTIONS
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IN GASES
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ELECTRONS
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