BRIT. J. APPL. PHYS. (J. PHYS. D),
1968, SER. 2,
VOL.
1.
PRINTED IN GREAT BRITAIN
Ion mobility and liquid motion in liquefied argon
T. H. DEYt and T. J. LEWIS$
Materials Laboratory, Queen Mary College, London
MS. received 3 r d January 1968
Abstract. Current-voltage characteristics and ion mobilities have been determined for
liquefied argon of reasonable purity at fields up to 10 kv cm-1 using a triode cell with
a radioactive emitter. The current was space-charge limited up to about 1 kv cm-1,
but at higher fields there was only a partial saturation, indicating the influence of
current-induced liquid motion. Evidence that the grid electrode of the triode cell
produced a strong blocking action to ion neutralization was also found.
The current transients obtained by switching the grid potential provided values of the
mobility of Ar2+ and 02-ions in the liquid and also offered evidence of free electrons
and electron attachment to form the 0 2 - ion. The mobility of both ions was influenced
by self-induced liquid motion. Although the lowest measured mobility was about
3.6 x 10-4 cm2 v-1 s-1, extrapolation to zero current suggested a true ion mobility,
uninfluenced by liquid motion, of about 2 x 10-4 cm2 v-1 s-1. Differences between
values reported earlier, which are significantly larger than this value, are probably due
to liquid motion. The velocity of both ions at higher values of cell current varied
directly with the square root of the drift field, the combined effects of local polarization about the ion and induced liquid motion probably being responsible.
1. Introduction
There have been several reports of the mobility of positive and negative ions in liquefied
argon, and in common with most dielectric liquids the values lie in the range
10-3-10-4 cm2 v-1 s-1. A second negative charge carrier has also been detected, which is
probably a quasi-free electron with an apparent mobility which can exceed 102 cm2 v-1 s-1
(Davidson and Larsh 1948, 1950, Hutchinson 1948, Marshall 1954, Malkin and Schultz
1951, Williams 1957, Swan 1964, Schnyders et al. 1965). In these measurements primary
ionizing radiation is used to release electrons and positive ions. These may then be lost
again by recombination or, in the case of electrons, may be captured by impurities of
which oxygen and nitrogen are the most important. Swan (1963) found that oxygen was
about one hundred times more efficient than nitrogen in electron trapping. In so-called
'pure' argon (99.995 % purity) oxygen impurity is comparable with nitrogen and so clearly
is the most likely cause of electron trapping, producing 02- ions with mobilities in the range
already quoted. The capture cross section varies approximately inversely with the drift
field (Swan 1963). Because of this the determination of free-electron mobilities at fields
less than 10 kv cm-1 over drift distances greater than 1 mm becomes difficult unless the
oxygen level is exceedingly low. Schnyders et al. (1965), using a drift distance of 1 cm and
an oxygen impurity of 1 part in lo9, found an electron mobility of 420 cm2 v-1 s-1 in a field
of only 200 v cm-1.
Although the trapping process has been studied, there has been less interest in the
~ atmospheric
behaviour of the negative ( O r ) ion so formed. At a temperature of 8 7 " and
pressure, a mobility of 7.8 x 10-4 cm2 v-1 s-1 has been reported (Davis et al. 1962b). The
positive (Arz+) ion has about the same mobility as the negative one (Williams 1957, Davis
et al. 1962a, Henson 1964). Henson found values ranging from 6 to 9-75 x 10-4 cm2 v-1 s-1
for drift fields up to 4.3 kv cm-1, but the mobility changed in steps at critical values of the
field which depended on the operating conditions of the ion source.
t Now at the Admiralty Surface Weapons Establishment, Portsdown, Cosham, Hants.
$ Now at the School of Engineering Science, University College of North Wales, Bangor, Caerns.
1019
1020
T. H. Dey and T. J . Lewis
In the present investigation the velocity of both positive and negative ions in normal
(99.995 % pure) argon with an oxygen concentration of 5 p.p.m. and in purified argon with
an oxygen concentration of 1 p.p.m. or less have been measured in drift fields up to
10 kv cm-1, the critical range for electron trapping. Direct evidence of electron trapping
has been obtained simultaneously with the determination of ion mobility. The measurements have demonstrated that significant liquid motion is induced, which has a marked
influence on the values of ion mobilities deduced.
2. Experimental procedure
The method adopted (figure 1) was similar to that of Davis et al. (1962a). The triode
test cell contained a platinum emitter electrode 0.7 cm in diameter coated with a 10-3 Ci
deposit of americium 241 in oxide form covered with a thin gold film. The a particles
emitted into the argon liquid from this source had an attenuated energy between 4.1 and
4.9 MeV. Ions produced by the 01 particles in the emitter region were drawn across the
0.4 mm emitter space and through a grid electrode to enter the drift space between the grid
and a copper collector electrode. The grid, consisting of nickel-covered copper, was
5 x 10-3 in. thick and had a circular pattern of holes at a spacing of 200 per linear inch
covering a diameter of 1 cm and giving 50 % geometrical transmission. The collector had a
guard ring so that a uniform drift field existed between grid and collector. The drift
distance d between grid and collector was set to 0.5 cm so that with a drift field Ed of
10 kv cm-1 and an assumed ion mobility of 10-3 cm2 v-1 s-1, space-charge-limited conditions were avoided with the chosen radioactive source.
f i e l d E,
f i e l d Ed
Figure 1. The triode system. E, americium 241 a emitter deposited on platinum; G, grid;
C, collector; F, collector guard. EG = 0.4 mm, GC= 5 mm.
By appropriate choice of potentials, ions of either polarity could be introduced into the
drift space. The resultant current transients were measured by means of an electrometer
amplifier and oscilloscope connected across the collector resistor R (figure 1). In order to
generate satisfactory signals the resistor R had to have a value greater than 107 Q, The
unavoidable effective shunt capacitance was 1 PF and the input time constant was thus
not less than about 10-5 s. Ion transit times could be measured easily but the unavoidably
large time constant prevented measurement of electron transit times in the microsecond
range at fields greater than 1 kv cm-1. Normally the grid was maintained at a constant
high potential with respect to the collector and, to release carriers, the emitter potential
was switched relative to the grid via a delayed pulse from the display oscilloscope.
The various electrodes were rigidly mounted on PTFE supports and the whole cell was
suspended within a cylindrical glass envelope at the end of a tube which also formed the
screen for the collector current lead. The tube and connections were brought out through
a demountable metal flange sealed to the glass envelope. A copper heat sink was sealed to
the lower end of the glass envelope and was immersed in liquid nitrogen to a point such
that argon gas could be condensed into the cell. The temperature gradient across the
Ion mobility and liquid motion in liquefied argon
1021
electrode system was small and in the reverse of that for normal boiling action so that thermal
currents were minimized.
The cell was evacuated to a pressure of 0.1 mtorr and was repeatedly flushed with pure
argon before admitting the gas sample which during experiments was maintained at a pressure of about 5 torr above atmospheric pressure. The vapour pressure of argon gas is a
very sensitive indicator of the liquid temperature and was adjusted to hold the temperature
at 8 7 " ~ .
The argon gas was normally of 99.995 % purity with the following probable impurities
(p.p.m.): Nz, 1-10; 0 2 , 0 - 5 ; CO2,O-5; H 2 , l ; H20,O-6. A purified argon was also used, in
which oxygen was removed to less than 1 p.p.m. by passing the normal gas through a
solution of chromous chloride and then through silica gel and phosphorous pentoxide
columns. A Hersch cell, according to the design of Dewey (1961), was used to measure
the oxygen content of the argon gas stream and had a lower detection limit of 1 p.p.m.
3. Results
3.1. Static characteristics
The static collector-current-drift-field characteristics, showing the degree of control
exercised by the grid, are given in figure 2. These characteristics are little modified whether
positive or negative ions are injected or whether normal or purified argon is used. In early
measurements there was an inexplicable degree of scatter and lack of reproducibility but
subseauent exDeriments showed that this was due to the slow establishment of stable
conditions in (he neighbourhood of the grid
-- ~
0225
-0
.
*
1
1
0
2
4
Drift f i e l d Ed
6
8
IO
(kv cm-1)
Figure 2 . Static collector-current-drift-field
characteristics. Results are applicable to
positive or negative ions in normal or purified
argon.
0
5
€e ( k v cm-1)
Figure 3. Static collector-current-emitterfield characteristics. The characteristic for
the diode system obtained by removing the
grid is also shown.
The current follows a space-charge-limited law for low fields but ultimately becomes
quasi-saturated, the degree of saturation decreasing as the level of charge injection rises.
In the space-charge-limited range a square law is followed up to about 1 kv cm-1 and, if the
usual space-charge equation is applied for this range, the effective ion mobility has a value
of about 1.3 x
cm2 v-l s-l. As will become clear, this value is almost certainly
73
1022
T. H. Dey and T. J. Lewis
influenced by liquid motion and is in agreement with values deduced from ion transit times
under conditions in which the ion current is sufficient to induce liquid motion (see figure lo).
In the saturation range the current varies with emitter field E e in the manner shown in
figure 3, and there is a marked tendency towards current limitation when Ee exceeds about
2.5 kv cm-1. When the grid was removed, however, the resultant diode system did not
produce any such tendency and the current rose continuously with E e as shown. The severe
collector current limitation induced by the grid may be due to more grid current being
drawn as Ee increases or, more probably, a space charge develops in the emitter space
about the grid wires which then limits the passage of ions through the grid. The establishment of this condition would be a time-dependent process. It was found that the collector
current increased with time after application of the emitter field (figure 4(u)), the initial value
established depending also on the magnitude and polarity of E e prior to the test. When the
grid and collector were connected together, so forming a single collector for current from
the emitter, the current decreased as shown in figure 4(b). It is important to note that the
current in both configurations tended to the same limit, and this strongly suggests that the
grid current was ultimately blocked off completely, probably by a space charge of ions
collecting about the grid wires which would offer a relatively small area for neutralization.
Figure 4. Variation of current after switching on an emitter field of 425 v cm-I: ( U ) collector
current, E d = 1.6 kv cm-l; (b) collector+grid current, E d = & The differences between positive
and negative ions are not significant.
This effect is reminiscent of the blocking action at the anode of a diode system under breakdown conditions (Swan and Lewis 1961). In the diode configuration, with the grid connected to the collector and a blocking action at the grid, most of the current will enter the
collector. The drift field E d will be very low, if not zero, and it is most likely therefore that
liquid motion carries ions to the collector from the emitter space. With normal potentials
on the grid and collector, liquid motion will have an even greater influence on the measurements of ion mobility, as will be seen below.
During a mobility measurement the emitter field E e is switched on and off repetitively for
2.5 s periods. Operating in this mode it was found that the collector current grew with time
in a manner similar to that shown in figure 4(a).
The current-stress characteristics (figure 2) are not unlike those obtained by Secker and
Lewis (1965) and others for induced conduction in hexane. In the quasi-saturation region
at stresses above about 4 kv cm-1 the current varies as Ed1i4. Although this sort of
behaviour has been attributed to a field-sensitive supply of carriers, it will seem more
probable in the present case that increased liquid motion draws additional ions through
into the drift field from the space-charge region around the grid wires.
Ion mobility and liquid motion in liquefied argon
1023
3 -2. Transient characteristics
The ideal transient collector current for a single ion species caused by switching the
emitter field on and off is illustrated in figure 5(u). There is a linear change between zero
and maximum current I , over times Tn and Tf corresponding to the transit of the ion front
across the drift space, and also delays tn and tf due to ion motion through the emitter space
into the drift region. Actual records are shown in figure 6(u) for injection below the spacecharge limit. Transit times can be estimated with an error of 10%. The trapped
electron ( 0 2 - ) transient in normal argon is similar to that of the positive (Ar+) ion because
of the high probability of electron trapping in the emitter space. The ideal shape of figure
5(a) is not obtained because diffusion causes the times to be less well-defined.
---0.2 s
(’)
Figure 5 . Ideal collector current transients
caused by switching the emitter field: (a)
Arz- or 0 2 - ions only in normal argon;
(b) free electrons plus ions in purified argon.
Tn and Tf are ion transit times when Ee
switched on and off, tn and t p are delay times
due to ion transit across emitter-grid space.
F.E.T. is free-electron transient.
Time-
Positive ions
Trapped electrons
Figure 6. Examples of actual ion transients
in normal argon for a drift field of 1 kv cm-1:
(U) low value of Ee without space charge;
(b) high value of E, with space-charge
distortion. (Reduced current amplification.)
At high levels of injection space charges disturb the drift field (figure 6(b)) and therefore
distort the transient, particularly at switch-on. The switch-off transient is less affected and
significantly faster and more consistent than the corresponding transients at low levels of
injection (figure 6(u)).
3 . 3 . Electron transients
When purified argon is used free electrons are detectable. These electrons have a finite
probability of reaching the collector and cause the collector current transient to exhibit
rapid ‘step’ changes corresponding to the electron transit time as shown in figures 5(b)
and 7.
If a simple collision model for electron trapping by oxygen molecules is used (Swan
1963), the number of electrons in the drift space during the switch-off transient decreases
with time according to the factor exp (- vet/l), where ve is the electron drift velocity and I is
the mean free distance before an electron becomes attached to an 0 2 molecule. Assuming
that the drift velocity of 0 2 - ions is very much less than De, so that they are effectively
stationary during the electron transit, the free and trapped (Oz-) electron currents, If and
It respectively, are then of the form given in the Appendix. I f and It depend on the parameter d/l, where d is the drift distance, in the manner shown in figure 8. In the range
lO%d/l%O-I, with a=O*ld, the transient is sensitive to the value of d/l and it is possible to
estimate values of 1 from actual transient records such as figure 7 by comparison with the
theoretical transients. Assuming that oxygen is the major trapping impurity, 1 can then be
T. H. Dey and T. J . Lewis
1024
0.4 sb
4
T
V
Z
.
.
+
-
Time’
-
Ed=I kv cm-1, E,= 425
V cm-1,
‘
I,= 8 pA
02-075 p p m
0.1 5
J+-----+%$z.
-
Ed=I O kv
cm-I, €,=425
V cm-1,
&=32 pA
O2 5 p.p.m.
0.2 s
-
*
I
:;y-
Ed=lOkV c m - l , E e = 1 ~ 8 k vcm-I,I,=IIO
0,-2,5
p.p.m.
pA
Figure 7. Combined free and trapped
electron transients in pure argon. Estimates
of oxygen concentration obtained from the
‘off’ transients using figure 8 are shown.
dive
Time
Figure 8. Theoretical transient electron currents If+ I t at switch-off for various values of
d/l. The transient If would appear to be
instantaneous on the time scale chosen for It.
related to the attachment coefficient discussed by Swan (1964) and the corresponding oxygen
concentrations determined as indicated in figure 7.
The free-electron transient did not appear for the first few fillings of the cell even when
pure argon was used, but it became more prominent subsequently and reached a maximum
after 20 fillings. Unfortunately it was never large enough to permit accurate direct determination of the free-electron transit times.
3.4. Zon mobilities
Ton movement induces liquid motion which makes the ‘off’ transient faster and more
consistent than the ‘on’ transient. Residual motion at the moment of switching on the
emitter is not controlled, whereas the steady current prior to the ‘off’ transient is conducive
to a well-defined and steady liquid motion. It is notable that other workers (Henson
1964, Davis et al. 1962a, b) quote ion transit times determined only from ‘off’ transients,
The ion velocities vi, measured from ‘off’ transients, are shown in figure 9 as functions of
drift field Ed. The velocity of negative ions is less in purified than in normal argon, whereas positive ions have the same velocity in both. In purified argon a component of the
collector current Imwill be due to electrons which traverse the drift space without being
attached to oxygen molecules. These free electrons, having higher mobilities and interacting less with the medium, will produce less liquid motion than the corresponding ions.
More direct evidence of liquid motion is afforded by figure 10. At constant Ed, ci increases
with Zm in a manner similar to that reported for ions in hexane (Secker and Lewis 1965).
By using an idealized model which takes account of the cell geometry and liquid properties,
it is possible to show (Kopylov 1964) that at constant stress the liquid velocity ~ ‘ 1should
depend linearly on the logarithm of the current. According to this model, if 1.i is written
as pEd+vl, where p is the true mobility of the ion in the liquid, then we expect that for
constant Ed
vi = C1+ Cz In Im
where C1 and CZdepend on p, Ed and the cell properties. The results of figure 10 roughly
confirm this expectation, but unfortunately it does not appear to be possible to deduce p.
Extrapolation in figure 10 to zero Im, however, indicates that p is likely to be less than
2 x 10-4 cm2 v-1 s-1, which is appreciably less than the apparent values for significant
values of Im.
-
(0)
0.81
m
I
&.\
\\\
-Y
‘ ,.;
-
-
.5
2-
‘
\ \‘
.
‘‘3,.
3
.-
*::,
-.q
a‘.
U
-
*e*
0
0
>
-s
)
‘-3.
e‘.::...
-
.....-’-.
4-
’. 4,
-
1
1
6-
8--
I
I
I I I I I
I
Liquid motion through the grid permits interaction between the emitter and drift spaces,
the delay times tn and tf due to ion transit in the emitter space reflecting this interaction and
tending to increase with Ed at constant Ee and vice versa.
When Im is constant V i varies as
(figure 9). At first sight this appears to be good
confirmation of the law proposed for gases (Wannier 1953) and applied to liquid argon by
considering it as a dense gas (Swan 1960). According to this law, in the situation where the
energy gain from the field is not small compared with the thermal energy, the ion velocity
where M is the ion mass and h is a mean free path for the ion.
is vi=(2/M2)1/4(ehEd)1/2,
If h is only a slowly varying function of Ed then V i varies as Ed1/2. There are, however,
I
0
I
I
I
I
20
40
60
80
I
100
&, ( P A )
Figure 10. Ion velocity as a function of current 1, for trapped electrons in normal argon (broken
curves) and positive ions in normal and pure argon (full curves).
1026
T. H. Dey and T. J. Lewis
several reasons why this agreement with experiment must be fortuitous. Firstly, using
reasonable values for the mass M (Arz+ or 0 2 - ) and experimentally determined values of t‘i
and E d , h turns out to be of the order of lO-l4 cm, a totally unacceptable value. Secondly,
the criterion given by Wannier for this equation to hold requires Ed$kT/2eh. For appropriate values of E d and liquid temperature T the resultant value of h makes Z’i much too
large. Thirdly, there is the evidence that liquid motion is making a significant contribution
which is not accounted for in the equation for vi.
Since both p and v i may be functions of E d it is again difficult to separate true ion motion
from liquid motion. However, as has been shown (Kopylov 1964), liquid flow should tend
to become independent of E d when Imis small, and since the characteristics of figure 9,
which extend to low currents, are strikingly similar, it would appear that the variation of ci
is probably due to the mobility p varying as Ed-112. Malkin and Schultz (1951) using a
single ‘a’ pulse method also found that the mobility of a negative carrier in argon varied
inversely as the square root of the field. At that time the carrier was considered to be a
free electron, but the suggestion has since been made that it was more likely to be a negative
impurity ion (Swan 1960).
From the present work it seems very likely that the differences in positive ion mobility
shown in the table are attributable to liquid motion. Unfortunately few measurements
have been made on the negative ion by other authors, but similar conditions will apply.
Even for electron mobilities, values will differ because of liquid motion since some electrons
in transit will become trapped (Swan 1963) and the ions so formed will then induce liquid
motion, thus influencing the apparent electron mobility.
Positive ion mobilities in liquefied argon
Reference
Mobility
(10-4 cm2 v-1 s-l)
Williams (1957)
Davis et al. (1962a)
Henson (1 964)
Present authors
23-32
6.10
6 0-9 * 75
3.6-15
The work of Henson (1964) on positive ion mobility is unique in that it shows step changes
in p at certain field strengths (or currents). Henson attributes these changes to the presence
of cluster ions which change size, and hence mobility, at critical fields. Such effects have
not been observed in the present work, which covers the same range, and it is plausible
that the phenomena observed by Henson were caused by changes in the pattern and magnitude of liquid motion in the cell rather than changes in true ion mobility.
4. Conclusions
The experiments have demonstrated that, even in an atomic liquid where short-range
Van der Waals forces might be expected to predominate, ion movement induces significant
liquid motion. This motion might be set up by direct momentum transfer in ion-atom
collisions, but there is also likely to be liquid polarization about each ion with a sufficient
binding energy at the temperatures of liquid argon to result in an atmosphere which moves
with the ion. Thus there could be considerably greater mass flow than that given by the ions
alone.
It has been suggested that the stable forms of the ions in question are Ar2+ and 0 2 - (Davis
et al. 1962a, b). Positive and negative ions appear to have similar mobilities and to induce
the same liquid motion which encourages the conclusion that the environments of the Arzand 0 2 - ions in argon liquid are similar.
Swan (1960) attempted to correlate the experimental value of positive ion mobility found
by Williams (see table) with various theoretical determinations. For example, using the
Langevin gas-kinetic theory or its modification which takes account, in some degree, of
cm2 v-l s-l respectively.
polarization scattering of the ion, the values were 60 and 32.4 x
Ion mobility and liquid motion in liquejied argon
1027
The value determined from viscosity data using Stokes's law was 15 x 10-4 cm2 v-1 s-1,
and that from the diffusion coefficient (deduced from self-diffusion data) using the Einstein
relationship was 28.6 x 10-4 cm2 v-1 s-1. These values are very much larger than the
latest experimental determinations. This is almost certainly because a gaseous collision
cross section was employed and polarization was not adequately considered. We have
already indicated that the high-field theory developed by Wannier (1953), even though
predicting correctly that the mobility should vary as Ed-'/', is not applicable on several
counts.
In the theory developed by Rice and Allnatt (1961) polarization is considered in detail,
and a model is developed in which polarization interaction contributes to the friction
experienced by a moving ion. Although dissipative collisions occur essentially as a result of
short-range forces between hard atomic cores, long-range polarization leads to electrostriction and to an increase in local density about each ion. Thus polarization indirectly
influences energy dissipation. The theory predicts an Arz+ ion mobility of
cm2 v-1 s-1 at a temperature of ~O'K, in good agreement with the experimental
5.93 x
value found by Davis et al. (1962a). In the case of the 0 2 - ion the theory produces
acceptable agreement with the experimental value found by Davis et al. (1962b)
(8 x 10-4 cm2 v-1 s-1 at 9 0 " ~and 5 atm pressure) provided the short-range interaction
potential is appropriately adjusted.
The present work has demonstrated, however, that appreciably lower values of ion
mobility may be obtained. In fact extrapolation to zero current and negligible liquid
velocity has indicated that the mobility of both Ar2+ and 0 2 - is probably about
2 x 10-4 cm2 v-1 s-1 (figure 10). There may be significance in the fact that this value appears
to be independent of Ed. Since the theory of Rice and Allnatt deals with a stationary liquid
through which the ions move, it would appear that the theory yields an overestimate of the
true mobility, and it may be rewarding to re-examine the form of interaction potential
employed.
At constant current away from the zero-current limit the mobility varies as E d - 1 / 2 (see
figure 9). In the theory of Rice and Allnatt there is no explicit field dependence. In a
stationary state, however, the energy gained by an ion from the field Ed must balance that
lost in the dissipative hard collisions. Allowing that the local density increase due to longrange polarization is a time-dependent relaxation process, it is reasonable to conclude that
the mobility will depend on Ea. The manner of the dependence requires further consideration, especially as the current will have induced liquid motion.
Acknowledgments
The authors wish to thank Professor M. W. Humphrey Davies for facilities provided at
Queen Mary College. One of the authors (T.H.D.) is indebted to the Science Research
Council for the award of a Research Studentship.
Appendix
If n is the rate of electron emission from the emitter and 0 is the transmission coefficient
at the grid, then in the steady state, allowing for trapping, electroiis reach a point distance
x from the grid in the collector space at a rate ne(x) given by
where a is the emitter-grid distance. The number of electrons in an interval dx is then
where t'e is the electron drift velocity.
1028
T. H. Dey and T. J. Lewis
At time t (<dice) after injection ceases the number of electrons remaining is
and the corresponding free-electron current is
( ):
If(t)=I~ exp -
~d-1jexp
( -7)- exp (-): )
where Im = en0.
When trapping occurs there is a composite transient due to both electrons and ions.
The trapped electron or ion transient is determined as follows. The total number of
trapped electrons remaining in the drift region at time t which were injected from the emitter
space is
nOvi-1 ( 1 -exp
I);-(
(d-Dit)
(AI)
where ci is the velocity of trapped electrons.
In the steady state electrons reach a point x at a rate ne(x) and the rate of production of
trapped electrons in an element dx at x is
( “1) exp ( -):
-~
dne(x) dx =n 02-1 exp dx
dx.
At time t after injection ceases the number of trapped electrons remaining which originated
in the element dx at x is
ntdx=nOui-lI-1 exp
(-;I (-:I
exp
( d - x - v i t ) dx
and the total number of trapped electrons generated in the drift region and remaining at
time t (< d / S ) is
Combining equations ( A l ) and (A2), it follows that the total number of trapped electrons
from both sources remaining is
and the trapped electron current transient is
The total transient current is then If+ I t , but Ifis significant only for times t < dive, which
are short compared with the time scale t < d/vi for It.
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Ion mobility and liquid motion in liquefied argon
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1029