PHYSICAL REVIEW A VOLUME 31, NUMBER 4 APRIL 1985
Probability of electrical breakdown: Evidence for a transition between the Townsend
and streamer breakdown mechanisms
R. V. Hodges, R. N. Varney, and J. F. Riley
Lockheed Palo Alto Research Laboratory, Organization 93-50, Building 204, 3251 Hanover Street,
Palo Alto, California 94304
(Received 12 September 1984)
Spark-breakdown delay times were measured for N2, H2, Ar, SF6, and CC12F2 in a uniform field
gap provided with a small current ( —10 ' A) of free electrons by uv illumination of the cathode.
Laue plots of the delay times yielded straight lines with slope iP, where i is the photocurrent and P
is the breakdown probability. The dependence of the breakdown probability on voltage for N2, H&,
and Ar was in good agreement with predictions of the Townsend breakdown mechanism. In SF6
and CC12F2, a transition was observed with increasing pressure from a dependence that agreed with
the Townsend theory to a more gradual rise with voltage, characteristic of a streamer mechanism.
This transition was ascribed to a decrease in the secondary-ionization coefficient with increasing
pressure in SF6 and CC12F2, which resulted in an average electron-avalanche size at the static breakdown
voltage that approached the critical value for streamer formation. A unified breakdownprobability
theory, for which the Townsend and streamer mechanisms are limiting cases, was
developed to account for the data over the full pressure range. The implications of these results for
measurement of the static breakdown voltage and the secondary-ionization coefficient are discussed.
I. INTRODUCTION
Under normal conditions. a gas is a good electrical insulator.
Electrical breakdown is the sudden transition of a
gas to a conducting state. The ability of a gas to serve as
either an insulator or a conductor has great technological
significance. Controlled electrical breakdown is the basis
of the operation of gas-filled switches. Interest in the
physics of breakdown has increased because of the importance
of high-power switches in fusion reactors, lasers,
directed-energy weapons, and electromagnetic pulse generators.
' Gases are used as insulation to prevent breakdown
in high-voltage circuits and transmission lines. The
probability of breakdown is an important consideration in
the design of high-voltage systems. A high probability of
breakdown is required for the reliable performance of
switches, whereas a very low probability of breakdown is
necessary for insulation.
Electrical breakdown occurs when two conditions are
satisfied. First, the applied voltage must equal or exceed
a minimum value (the static breakdown voltage V, ).
Second, a free electron must be present in the gas. A free
electron, accelerated by the electric field, produces more
free electrons and positive ions in ionizing collisions with
gas molecules. This process is described by the primaryionization
coefficient a, the average number of ionizing
collisions experienced by one electron moving 1 cm along
the direction of the electric field. An electron, departing
from the cathode, creates an electron avalanche that
grows exponentially to an average size n =exp(ad), where
d is the electrode spacing. New avalanches are begun
when positive ions, photons, or excited neutrals created in
previous avalanches impinge on the cathode and eject
electrons. This process is described by the secondaryionization
coefficient cu/a, the average number of secondary
electrons released from the cathode per positive
ion produced in the gas. Although the secondaryionization
coefficient is defined relative to the number of
positive ions, its value includes the contributions of all
secondary agents.
The development of current in a series of electron
avalanches is the basis of the breakdown mechanism proposed
in its initial form by Townsend. The criterion
for breakdown in the Townsend theory is that each electron
avalanche produce on the average at least one secondary
electron at the cathode, so that the discharge is
self-sustaining. This criterion is expressed mathematically
as
p=(co/a)n =1. (1)
Since a and co are both functions of V/(Xd), the Townsend
criterion defines a breakdown voltage VT, for a given
gas number density X, and electrode spacing d.
The Townsend theory successfully accounted for the
dependence of the breakdown voltage on gas density and
electrode spacing. Other experimental evidence, however,
was obtained which appeared to be inconsistent with the
Townsend mechanism. The time between application
of voltage and electrical breakdown was measured. In
spark gaps at atmospheric pressure with electrode spacing
—1 cm, very short delay times ( & 10 s) were obtained.
Since positive ion transit times across the gap are
—10 s, this time was too short to have involved a series
of successive avalanches produced by ions impinging on
the cathode. (Secondary action by photons was not considered.
) Photographs of spark discharges revealed the existence
of narrow, luminous "streamers" originating at the
anode or in midgap, rather than the broad development
from the cathode, expected for a Townsend discharge.
The streamer theory was developed to account for these
and other observations. In the streamer theory, breakdown
occurs when one avalanche attains a critical size n, .
At the critical size, space charge creates regions of high
31 2610 1985 The American Physical Society
31 PROBABILITY OF ELECTRICAL BREAKDOWN: EVIDENCE. . . 2611
field around the head of the avalanche where free electrons
can multiply efficiently. It was proposed that radiation
from the avalanche produces free electrons by photoionization
in these regions, propagating the avalanche to
form the observed streamers. The production of secondary
electrons at the cathode is not involved. The
streamer breakdown criterion is
n =n„or (1/n, )n =1 . (2)
It is apparent from Eqs. (1) and (2) that the form of the
breakdown criteria of the two theories is the same. They
differ in the mechanism of current growth in the prebreakdown
stage of spark development: secondary electrons
from the cathode (co/a) in the Townsend model and
a space-charge ( n, ) induced process in the gas in the
streamer model. The streamer theory has generated controversy
because of the difficulty in quantitatively describing
the ionization process. ' ' ' " The ability of an
avalanche to produce photons energetic enough to ionize
in a pure gas has been questioned. ' Other ionization processes,
associative ionization of excited atoms' and runaway
electrons, ' have been advanced to account for
streamer propagation.
The calculation of breakdown probability in the framework
of either of these breakdown mechanisms requires
that the statistical nature of current development be considered.
Although for many purposes collisional ionization
in the gas and release of secondary electrons from the
cathode may be represented by the average values a and
co/a, they are in reality stochastic processes. Current
growth in an electrical discharge is subject to statistical
fluctuations. A current flowing in the gas does not invariably
lead to electrical breakdown even though the applied
voltage exceeds V, . Theoretical expressions for breakdown
probability have been formulated for both the
Townsend' ' and streamer' ' models.
Breakdown probability can be studied experimentally
by measurements of the spark delay time. The time between
application of voltage to a spark gap and collapse
of the gap resistance is the sum of the statistical and formative
delay times. The statistical delay time is the time
between application of the voltage and the appearance of a
free electron that initiates breakdown. The formative delay
time is the time required for the current to build into a
detectable discharge. Statistical delay times follow the
distribution first applied by von Laue, '
N, /No ——exp( iPt), —
where N, /No is the fraction of delay times greater than t,
i is the rate of appearance of free electrons, and P is the
probability that an electron initiates breakdown. The
average statistical delay time v equals 1/iP.
Many workers have reported measurements of statistical
delay times in discharge gaps, but few have been done
under conditions in which i was controlled so that P
could be evaluated. If i is maintained constant, as by
photoemission from the cathode, changes in the average
delay time as a function of other parameters can be attributed
to changes in P. We report here measurements of
spark delay times as a function of voltage, pressure, and
free electron current in N2, Hz, Ar, SF6, and CClzF2. The
II. APPARATUS
The discharge gap and associated electronics are diagrammed
in Fig. 1. The gap was contained in an aluminum
chamber evacuated to a base pressure of 0.3 Pa {2
mTorr) by a mechanical pump. The cathode was a 1.25cm-diam
steel or molybdenum cylinder. The cathode surface
was polished on a mechanical polishing wheel with
I-pm diamond paste, then cleaned with xylene and
ethanol. The anode was a hollow steel cylinder with a
20-pm mesh (cut from a particle sieve) spot-welded to the
end. This mesh size was sufficiently small to assure a
uniform field over the electrode spacing of 0.5 mm. The
accuracy of the spacing was estimated to be +10%%uo.
Light from a Hg-Xe arc lamp passed through two apertures,
a CaF window and the anode mesh, and struck the
cathode surface, releasing photoelectrons. With the
chamber evacuated and 20 V applied across the gap, pho-
nr
V
Hg-Xe
ARC LAMP
SCOPE
GENER ATOR
PEDESTAL
GENERATOR
OV
CaF
Wl NDO W
COUNTER
ANODE
HI GH
VO LTAGE
PS
SPAR
GAP
CATHODE
FIG. 1. Apparatus for measuring spark delay times.
results for N2, H2, and Ar were well described by the
Townsend theory. The data for SF6 and CC12F2 fit the
Townsend model at low pressure, the streamer model at
higher pressure, and neither model at intermediate pressures.
Good agreement was obtained over the entire pressure
range with a new equation that combined elements of
both the Townsend and streamer theories. The Townsend
and streamer breakdown probabilities are limiting cases of
this theory for co/a large or zero. The results provide evidence
for a transition between the two mechanisms and
suggest the following regimes of validity: Townsend
model for n «n„and streamer model for n )n, In. the
transition region both cathode secondary processes and
space-charge field distortion are required to account for
the data.
2612 R. V. HODGES, R. N. VARNEY, AND J. F. RILEY 31
tocurrents in the range 10 ' —10 ' A were measured
with a Cary vibrating-reed electrometer. The photoelectron
current was varied by placing wire screens in the
light path. The attenuation factors of the screens were
determined from measurements of the current with and
without the screens in place.
The cathode was connected to an adjustable power supply
through a quenching resistor (0.5—3.0 MQ). The
voltage on the anode was controlled by a custom-built
voltage pedestal generator. A schematic of the operation
of the pedestal generator and measurement of the spark
delay time is shown in Fig. 2. A voltage of —300 V was
applied to the anode so that the gap voltage was less than
V, . When the pedestal was activated manually with a
button or automatically after a preset cycle time, the
anode voltage was switched to ground. At the same time
a logic gate opened, sending a 100-kHz square wave from
a pulse generator to a counter. The pedestal remained on
for a maximum of 300 ms. If breakdown occurred in less
than 300 ms, the voltage pedestal was switched off and
the gate to the counter was closed. The spark delay time
was the ratio of the number of counts and the pulse generator
frequency.
The following procedure was used to acquire the data.
Gas was admitted to the discharge chamber to the desired
pressure (read on a Wallace-Tiernan gauge). Research
grade gases were used without further purification. The
pedestal generator was set to trigger automatically every 5
sec and the cathode was gradually made more negative
until breakdowns began to occur. Several hundred breakdowns
were used to "condition" the cathode. Electrode
damage was minimal since the only charge passed during
the discharge was that stored on the gap capacitance.
After conditioning, the chamber was evacuated and refilled.
Spark delay times were measured and recorded
manually beginning at the lowest voltage at which breakdowns
were observed. Fifteen to forty measurements were
made at each voltage. The voltage was increased in steps
up to 250 V overvoltage, then decreased back to the
minimum. Finally, measurements were repeated at one of
the higher overvoltages to check that the photocurrent
had not changed significantly. Delay times were occasionally
measured with the light blocked and found to
be much greater (usually &300 ms) than those with illumination.
This confirmed that sources of electrons other
than the photocurrent could be neglected.
III. RESULTS
Plots of log(X, I%0) versus t yielded straight lines
which were fit by the method of least squares. The slope,
given by the Laue equation [Eq. (3)], is iP The. average
delay time r is the reciprocal of the slope, 1/(iP). Figure
3 shows Laue plots for three voltages in N2 at 21.3 kPa
(160 Torr). We assume that the photocurrent i is independent
of voltage and th'at the differences in slope are due to
different values of the breakdown probability P.
The dependence of the average delay time ~ on the illumination
intensity for N2, SF6, and CC12F2 was measured
at several different voltages. Results for SF6 at two
voltages are shown in Fig. 4. The average delay time was
inversely proportional to the photocurrent, as predicted by
the Laue equation [Eq. (3)] for statistical delay times.
This verified that the formative delay time was negligible
compared to the statistical delay time for these small pho-
tocurrents.
The dependence of the average delay times on voltage
for several pressures of N2, H2, Ar, SF6, and CC12F2 is
shown in Figs. 5—14. The right ordinate applies to the
experimental points. Each point is the reciprocal of an
average delay time and represents the slope of a plot like
those in Fig. 3. For each of the curves in Figs. 5—14, as
the applied voltage increases, the delay time approaches
an asymptotic value which is determined by i (the rate of
appearance of electrons) since the value of P is approaching
one. The asymptotic value establishes the position of
1.0
PULSE
GENERATOR
COUNTER
GATE
COUNTER
I NPUT
0-
ANODE
VOLTAGE
-300-
BREAKDOMfN
0.1
Y
CATHODE
VOLTAGE
1.0 2. 0 3.0 0.0
DELAY TIME (ms)
5.0 6.0
T IME
FIG. 2. Electrode voltages and timing circuit signals.
FIG. 3. Laue plots of spark delay times in N2 at 21.3 kPa
(160 Torr).
31 PROBABILITY OF ELECTRICAL BREAKDOWN: EVIDENCE. . . 2613
1600
I
1700
l
1800
I
250 —-
200
150
1.0—
0
I-
CQ
Cl
O
CL
z 0.1—
O
I2
LU
CK',
Cl
0
—10
2
12) —10
100
0.01 I I I
1600 1700
APPLI ED YO LTAG E (V)
l
1800
50
FIG. 6. Experimental average spark delay times and calculated
initiation probabilities for N2 at 53.3 kPa (400 Torr). Parameters:
d =0.45 mm, co/a =2.9& 10
0
0 0.2 0.4 0.6 0.8 1.0
the scale of the left ordinate, which gives the breakdown
probability. The major sources of experimental uncertainty
are statistical variations and variations in the photocurrent.
From the scatter of the experimental points
about a smooth curve, we estimate the uncertainty in ~ to
be +30%. Longer conditioning periods usually resulted
LICHT ATTENUATION FACTOR
FIG. 4. Reciprocal of average spark delay time in 26.7 kPa
(200 Torr) SF6, electrode spacing 0.45 mm, as a function of
cathode illumination intensity.
800
1
900
1
'I 000
o oo
in reduced scatter.
The curves are calculated breakdown probabilities obtained
from the equations discussed below. Literature
values for a and g (the electron attachment coefficient)
for Nz, Hi, Ar, SF6, and CC12Fz were taken from Refs.
19, 20, and 21, 19 and 22, and 23, respectively. The adjustable
parameters in the calculations were cola and n,
Although the nominal electrode spacing was 0.5 mm, significantly
better fits were obtained for d =0.45 mm,
which is within the experimental uncertainty.
800 90OI
t
1000 0 3—10
1.0—
~ 0.1-
O
Cl
hC
LU
CQ
21.3 kP8
PER IMENT
UAT ION (12)
—103
l
g
h
I-
K
0.1—
O
CL
K
0Cl
hC
Lll
& O. O1-
CQ
40.0 kpa
o EXPER IMENT
EQUATION (12)
2 h—10
—10
0.01 l
800
1 i
900
APPLIED VOLT ACE (V)
1
10OO
0.001 l I I I
800 900
APPL. I ED VOLTAGE (V)
I
1000
FICi. 5. Experimental average spark delay times and calculated
initiation probabilities for N2 at 21.3 kPa (160 Torr). Parameters:
d =0.45 mm, co/a =1.2&& 10
FIG. 7. Experimental average spark delay times and calculated
initiation probabilities for Hq at 40.0 kPa (300 Torr). Parameters:
d =0.45 mm, ~/a=1. 2)&10
2614
goo 1O0o
yARNEY, AND J. . R&LEYR. y HODGES, R. N.
& )00 1200
gp
31
700
)
1.0—
800
I
900
i
1.0— O
V eQ~~ ~
10
—lo3
cO 0. 1—
0
0-
X
0
W
EQ0 p1~~
EXPER [MENT
EqUAT~ON (12) 2—10
IV. DISCUSSION
wn robabilityA. Theoretical breakdown p
10I
900800
APPLI ED VOLTAGE (V)
k delay times and cacalculat-G.. 8. p.xperimental g
l Pa (760 Torr). Paramal
average spar e
ame-
F
abilities for Ar at 101 aed initiation probabi sties a
=0.45 mm, cu/o;=3. 2)& 1ters: d =
1
K
O
o.oot —3
-
t
R tMENT
ATlON (& 3) '
m
AT~ON (&&) -10
AT|ON {&&)
FOR ~i~= 7.S x 10T
'to
12000 1000 g )00900
APPLf ED VOLTAGE (V)
al average spark delay times and calcu-p
bilities for SF6 at 9. a1 t d initiation proba i i
'
5 mm, co/o, =1.2&
[E=7.5X10 ' [Eq. (17)], n, =5&&1 q.
(17)].
la h The stochastic nature.anche grout . e
~ ~ ~
Th statistics of
o co n results in sta is i
h Th I' d'
bution v (n or o an
trons in a'n a nonattaching gas is
1.0
1600
I
1700 1 800
I
~l
~ ~ ~ ~~o~~ ~
~G~ ~ ~
1.0—
Q3
CQ
0
~0.1—
K
0O
hC
UJ
QC
tQ
800 900
I
1000
1
—104
—103
I-
0.1—
CQ
0lX
0
X
0
mp. 01
DD
1
l
f
l
!
!
!
!O
—102
kPa
—10R!MENT
AT I ON (13)
AT!ON (14)
AT )ON {17)
0.01—
10
1000
II
800 900
APPLt ED VOLTAGE (V)
calculat-1 average spar e a1 a g k delay times and calc
ed initiation probab'
' '
.o ailities for SF6 at 4.0 a
a=2.9d =0.45 mm, co/a=3. 2&(
=5&&10' [Eq. (17)].
ters:
=1)&10'[Eq. (14)], n, =~10 ' [Eq. (17)], n, = )&
0.001
1600
I
1700
APPLl ED VOLTAGE (V)
1800
vera e spark delay times and calcu-' g
initiation probabilities o
co/a=1. 6X 10 [Eq.
n, =5&&10' [Eq. (17)].
31 PROBABILITY OF ELECTRICAL BREAKDO%'N: EVIDENCE. . . 2615
800
1
900 1000 1900 2000 2100
1.0—
0-
I-
CQ
CQ
O
0.1—lY
Q
hC
LLI
o
CQ e
0.01 o
Q
0
7 kPa
ENT
ON (13)
~ ~ ~ EQ UAT l ON ('1 0)
EQ UAT I ON (17) —10
'l. 0—
l-
CQ
CQ
O
EL
Q. 0.1
K
O
A
CQ ooo
6
0
O
0 kPa
ON (13)
ON (fe)
ON (17)
—fp 2
I l l
900 1000
APPLl ED VOLT AG E {Y)
FIG. 12. Experimental average spark delay times and calculated
initiation probabilities for CClzF2 at 6.7 kPa (50 Torr).
Parameters: d =0.45 mm, co/a =3.7 X 10 [Eq (13)],
co/a=3. 2X10 [Eq. (17)], n, =1X10 [Eq. (14)], n, =5X10
[Eq. (17)].
I
800
n —1
1 1
u(n)= —1 ——
n n
1 n
——exp (4)
n n
where n =exp(ad). The number of positive ions in an
avalanche is only one less than the number of electrons.
In electronegative gases negative ions are formed by attachment
of electrons to neutral molecules. This process
l
1900 2000 2100
APPLl ED VOLTAGE (V)
FIG. 14. Experimental average spark delay times and calculated
initiation probabilities f'or CC12F2 at 34.0 kPa (255 Torr).
Parameters: d =0.45 mm, co/a=6. 7X 10 [Eq. (13)], co/a
=1.8X10 [Eq. (17)], n, =1 X107 [Eq. (14)], n, =5X10 [Eq.
(17)].
is described by the attachment coefficient q, the number
of attaching collisions per electron per centimeter travel in
the direction of the electric field. The number of electrons
and positive ions cannot be taken to be equal, but
differ by the number of negative ions. In a fraction g/a
of all avalanches, current development is aborted by attachment
of all the electrons. The average number of
electrons and positive ions in unaborted avalanches is
given by
exp[(a —ri )dJ
n~= 1 —g/a
1000
1
& 500
I
a ~ ~
1600
exp[(a —g)d J
np=
(1—g/a)
l-
CQ
4C
CQ
CK
0
Cl
hC
LU
0.01—
20. 0 kPa
—10
2
Sx10
PERIMENT
UAT! ON (13)
u AT&ON (1&)
UATlON (17)-10
p'=(a)/a)(1 v]/a)n~ =1 . — (7)
The probability distribution for electrons and positive ions
is approximately
g/a for n*=0
u'(n*) = (8)
(1—g/a) exp( n /n ) for —n & 0,
The asterisks denote parameters in an electron-attaching
gas. The Townsend breakdown criterion in attaching
gases is
I, s l
1 000 1500
APPLl ED VOLTAGE (V)
1600
FIG. 13. Experimental average spark delay times and calculated
initiation probabilities for CC12F2 at 20.0 kPa (150 Torr).
Parameters: d =0.45 mm, co/a =3.4 X 10 [Eq. (13)],
co /a =5 X 10 [Eq. (17)], n, =2.8 X 10 [Eq. (14)], n, =5 X 10
[Eq. (17)].
with n =n,* or n~.
Townsend mechanism. Wijsman'" and Legler' considered
the stochastic nature of collisional ionization and
secondary electron ejection from the cathode to calculate
the breakdown probability. The probability w (nz, m) that
an avalanche that produces n~ positive ions results in the
release of m secondary electrons at the cathode is given by
the binomial distribution
2616 R. V. HODGES, R. N. VARNEY, AND J. F. RILEY 3I
good, but at the lower pressures Eq. (14) predicted a much
more gradual slope than was measured and gave significant
breakdown probabilities only with unreasonably
small critical avalanche sizes n, & 10 . This value is much
below the critical size of 10 typical of streamer calcula-
tions.
unified breakdown-probability theory. Neither the
Townsend nor the streamer mechanisms alone can account
for the behavior of the spark delay time in SF6 and
CClzF2 over the full range of pressures investigated. The
assumptions underlying either one of these mechanisms
are not valid over the entire range. The streamer mechanism
neglects secondary electrons produced at the
cathode. The Townsend mechanism neglects field distortion
by space charge. A more realistic description of the
breakdown process is a buildup of space charge in a sequence
of avalanches until the field distortion precipitates
collapse of the gap voltage and establishment of a glow or
arc discharge. We define electrical breakdown as the occurrence
of a sequence of aualanches in which the total
number ofpositive ions produced achieves a critical value
The drift velocity of ions is a factor of several hundred
less than that of electrons. If the secondary electrons at
the cathode are produced by photons [this is probably the
case for SF6 (Refs. 32 and 33)], the time between
avalanche generations is approximately equal to the electron
transit time. The positive ions from many generations
of avalanches will accumulate in the gap.
Legler" has derived the expression for the probability
distribution V(n) of the total size of a finite sequence of
avalanches as
w(nz, m)= ~ (co/a) (1—cv/a) ~ (9)
Therefore, the probability u(m) that the first avalanche
produces m avalanches in the second generation is
00 m
u(m)= g w(nz, m) u(nz)=
(p+ 1)m+1
(10)
In Wijsman's' and I.egler's" derivation, electrical breakdown
is defined as the occurrence ofan infinite succession
of electron aualanches If. 1 —P is the probability that an
avalanche series begun by one electron eventually terminates,
the probability of termination of a series that has
m avalanches in the second generation is (1—P) . There-
fore,
1 P= g—u(m)(1 P)—m=0
The solution of this equation yields the probability of
breakdown,
0 for p&1
P=' 1 —1/p for p&1, (12)
for nonattaching gases. This analysis was extended by Tagashira
to include electron attachment,
0 for p*&1
(1—71/a)(1 —1/p" ) for p,
* ~ 1 . (13)
Later Kondo and co-workers considered the effect of de-
tachment.
Electrical breakdown probabilities calculated from Eq.
(12) or (13) are plotted with the experimental points in
Figs. 5—14. There was good agreement between the experimental
points and calculated curves for N2, H2, and
Ar at all pressures and for SF6 and CC12F2 at lower pressures.
As the pressure of CClzF2 and SF6 increased, a
more gradual slope in the voltage dependence of the average
spark delay time was observed, which was not predicted
by Eq. (13).
These results are consistent with literature reports.
Similar experiments showed agreement with Eq. (12) for
N2 (Refs. 27 and 28), Ne (Ref. 29), and H2 —rare-gas mixtures.
However, for SF6 at 54 kPa (405 Torr) with electrode
spacing 1.52 mm, Crowe ' reported a more gradual
slope resembling our higher-pressure results.
Streamer mechanism. Electrical breakdown in streamer
theory is defined as the occurrence of one avalanche that
achieves the critical size. The probability of electrical
breakdown is the probability that a free electrori produces
an avalanche larger than the critical size n, . The probability
can be calculated from the integral of the
avalanche-size distribution, which for electron-attaching
gases is'
1 I&(2p' nln)
V(n) =—, exp[ —(1+p)n ln],
n p'~'n/n
(15)
where Ii is the first-order Bessel function. The integral
of the distribution over all n has one of two results depending
on the sign chosen for a square root,
1 for p & 1
o 1/p for p)1.V ndn='
These results correlate with the probability for an infinite
sequence [Eq. (12)]. For p & 1, all avalanche sequences are
finite, whereas for p & 1, the fraction 1 —1/p is infinite.
The probability of electrical breakdown is the probability
of an infinite avalanche sequence plus the probability
of a finite sequence that achieves the critical size
(1—il/a) I V(n~ )dn~ for p*& 1
C
(17)
(1—g/a) 1 —1/p'+ I V(nz )dn~ for p* ~ 1 .
C
For electron-attaching gases the probability distribution
V(n) was multiplied by the factor (1—q/a), p was replaced
by p*, and n was taken to be the average number
of positive ions n &. Note that this equation is not a simple
sum of Eqs. (13) and (14), but includes finite sequences
of avalanches with total size greater than n, .
Unfortunately, an analytical solution of the indefinite
Breakdown probabilities calculated from this equation for
SF6 and CC12F2 with n*=nz are p1otted in Figs. 9—14.
At the highest pressures agreement with experiment was
P*=t u'(n*)dn*={1—g/a)exp( n, /n *) . (14)—n
PROBABILITY OF ELECTRICAL BREAKDOWN: EVIDENCE. . .
1.0
0.1
CQ
CQ
CL
0.001'
r
r
/
/
r
r
r
r
f
r
I
r
1600
~ ~ ~ ~ ~ a~ ~ ~
~ ~ ~
.~
e
e
FINITE
SEQUENCES
.......I NFINITE
SEQUENCES
TOTAL
VT.
1700 1800
APPLIED VOLTAGE (Y)
FIG. 15. Contributions of finite and infinite avalanche sequences
to the calculated breakdown probability in SF6 at 26.7
kPa (see Fig. 11).
integral of V(n) is not available. For n, /L'~ /n large the
integral can be approximated (see the Appendix). Calculated
curves from Eq. (.17) are plotted with the experimental
data for SF6 and CC12F2 in Figs. 9—14. The agreement
is good over the entire pressure range. The curves
were fit by trial and error. The values given for the parameters
are reasonable, but are not necessarily unique.
The distinction between finite and infinite avalanche sequences
has mathematical, but not physical significance,
since, of course, all real avalanche sequences are finite
ones, which either die out or build up current to the point
of voltage collapse. The mathematical distinction is useful
to show the relationship between Eq. (17) and the
Wijsman-Legler theory, which includes only infinite sequences.
Figure 15 shows separately the contributions of
finite and infinite sequences of avalanches to the total
theoretical breakdown probability from Eq. (17) for SF6 at
27.6 kPa (see Fig. 11). At voltages below Vr, p" & 1 and
only finite sequences contribute to the breakdown probability.
Above Vz- the probability of an infinite sequence is
nonzero and the fraction of finite avalanches decreases.
The Townsend and streamer breakdown probabilities
are limiting cases of the unified theory. For co/u=O (no
cathode secondary effect), p=O and V(n)=U(n), so that
Eq. (17) becomes the streamer probability, Eq. (14). For
m/a large, breakdown occurs by the Townsend mechanism
at a voltage at which n is still much smaller than n, .
Many avalanches are required to accumulate the critical
space charge, and finite avalanche series make a negligible
contribution to the breakdown probability, i.e.,
V n dn-0.n
B. Transition between the Townsend
and streamer mechanisms
There has been much discussion in the literature regarding
the conditions under which the Townsend and
streamer mechanisms apply. ' Townsend's work was carried
out at low voltages that required the use of low gas
densities and small electrode spacings. With the development
of high-voltage techniques, the range of experimental
data was extended to larger Nd. The difficulties in interpreting
these results by the Townsend mechanism led
to the proposal of the streamer theory. The first proponents
of the streamer theory suggested a transition between
the two mechanisms in air at Xd =6.6X 10' cm
(200 Torr cm), with the Townsend mechanism being valid
below this value and the streamer mechanism applying at
larger values. More recently measurements of formative
delay times as a function of Xd and voltage have suggested
a different boundary. Discontinuities in delay time
versus voltage curves were observed and attributed to a
transition between the two mechanisms with overvoltage,
rather than Xd, being the critical parameter. The Townsend
mechanism was considered valid for low overvoltage
(&20%), the streamer mechanism at high overvoltage
(& 20%)
The present results represent another type of experiment
that provides evidence for a transition between the
two mechanisms. The breakdown probabilities in Figs.
9—11 and 12—14, respectively, show a transition at low
overvoltages from a voltage dependence consistent with
the Townsend model [Eq. (13)] to a. dependence which fits
the streamer equation [Eq. (14)j. The unified theory [Eq.
(17)] describes the transition. The dominant character of
Eq. (17) depends on the relative values of the average
avalanche size n and the critical space charge n, . For
n &&n, many avalanche generations are required to accumulate.
sufficient space charge and the Townsend equation
is an acceptable approximation. As n approaches n, the
probability that one avalanche is large enough to cause
breakdown becomes s'ignificant and the effect of space
charge must be included. This suggests the regimes of validity
of the two models: Townsend theory for n «n,
and streamer theory for n &n, This d.efinition of the
transition is equivalent to the assignment of the models to
different ranges of overvoltage, since n is a strong function
of voltage. The overvoltage at which the transition
occurs depends on the relative values of co/a and I/n, If.co/a is so small that the Townsend criterion is not satisfied
until n approaches n„ the influence of space charge
will be important at 0% overvoltage.
Raether has discussed this transition in similar terms.
Methane and ether ordinarily have a very low m/u, so
that the streamer mechanism is valid at the static breakdown
voltage. However, if a copper cathode is coated
with CuI to increase ~j'a, successions of Townsend
avalanches can be observed in these gases.
The application of this principle to SF6 is illustrated in
Fig. 16. Our experimental breakdown voltage measurements
and those of Bhalla and Craggs are used to construct
a Paschen curve giving the value of E/N at breakdown
as a function of %d. Calculated curves are also
2618 R. V. HODGES, R. N, VARNEY, AND J. F. RILEY 31
20
P4
U
E 10
8THIS
PAPER
REF. 38
n*=5x10P
—&*=3.0x 10
P
(EIN) I.
LU
l
0.1
11
1.0
Nd (10 cm )
I I
10
FIG. 16. E/N at the static breakdown voltage as a function
of Nd for SF6.
plotted that show the values of E/N and Nd that satisfy
the condition n z
——const. The solid curve represents
n z
——n, =5 X 10, which is the value for the critical
avalanche size that gave reasonable fits to the data in
Figs. 9—11. The dashed curve, which passes through our
experimental breakdown voltage at 4 kPa (Nd =4.5 X 10'
cm ), is shown for comparison and represents
n~ =3.4&10". These curves are members of the family
of curves defined by the equation knz ——1, where k is a
constant. Comparison with the breakdown criteria, Eq.
(1) or (2), shows that these curves give the Paschen curves
expected if co/a or 1/n, were constant. The experimental
points for SF6 do not lie on a single curve, but fall on
curves of smaller k with increasing Nd. The value of
co/a obtained from fits of the probability curves in Figs.
9—11 decreases from 2.9X10 at 4 kPa (30 Torr) to
1.6X 10 at 26.7 kPa (200 Torr). A similar trend occurs
in CC12F2. A decrease in co/a with increasing pressure is
characteristic of a photon secondary effect, which is reduced
by increased collisional quenching of photonemitting
excited states and absorption of photons in the
gas. ' ' The decrease may also be due to increased electron
attachment near the cathode surface.
The transition from Townsend to streamer behavior
with increasing pressure in SF6 and CC12F2 is a consequence
of the decrease in m/a, which causes the Paschen
curve to approach the line n~ =n, (Fig. 16). In SF6 at 4
kPa (Nd =4.5 X 10' cm ) with co/a=2. 9X 10
breakdown occurs by the Townsend mechanism at
n ~ &&n, . The breakdown probability is well described by
the Townsend model [Eq. (13)] and V, =VT (Fig. 9).
Transition to a streamer mechanism will occur at an overvoltage
at which n z approaches n, . At 9.3 kPa
(Nd=1.05X10' cm ), co/a=7. 5X10 and satisfaction
of the Townsend criterion requires an average
avalanche size n.&
—10 . The probability that only a few
avalanche generations produce space-charge field distortion
is significant. Both secondary action at the cathode
and space-charge field distortion are required to account
for the dependence of the, breakdown probability on voltage
(Fig. 10). A significant breakdown probability
exists below the voltage at which the Townsend criterion
is satisfied, i.e., V, & VT. At 26.7 kPa (Nd =3.0
X 10' cm ) a value for the secondary-ionization coefficient,
co/a=1. 6X 10,gives a reasonable fit to the data
(Fig. 11). An average avalanche size approximately equal
to n, would be required to satisfy the Townsend
criterion. Space charge dominates breakdown at the static
breakdown voltage and the streamer breakdown probability
fits the experimental data. Chalmers and co-workers
investigated spark development in SF6 at Xd =10' cm
by time-resolved photography and found evidence for
space-charge field distortion in early stages of current
growth at low overvoltage.
These considerations also apply to measurements of
co/n, obtained from the Townsend breakdown criterion
[Eq. (1)] or from the curvature in plots of prebreakdown
currents. Several authors ' ' ' have reported values
for SF6 and CC12F2 in the range 10 —10 . It is likely
that space-charge field distortion affected these measurements.
The Townsend criterion cannot be used if
V, & VT. blarney and co-workers have shown that upcurving
in plots of prebreakdown current can be caused by
distortion of the field by space charge.
C. Breakdown probability and the critical nature
of the static breakdown voltage
The breakdown criteria [Eqs. (1) and (2)] are ordinarily
interpreted to define a critical voltage V„below which
breakdown will not occur. However, because of the stochastic
nature of current growth, the breakdown probability
is not a step function from 0 to 1 at this voltage, but
increases over a range of voltages. The probability function
I' must be considered in the interpretation of measurements
of V, .
The outstanding characteristic of our experimental data
is the markedly more gradual rise of logP versus voltage
seen particularly in Figs. 11 and 14 for higher pressures in
SF6 and CC12F2. We have attributed this difference to the
onset of space-charge effects (streamer breakdown). It is
notable that, according to the Townsend theory, there is a
well-defined voltage threshold for breakdown, shown in
Eq. (12), wherein the breakdown probability is zero for
p & 1. Above the threshold the probability rises extremely
rapidly, so that the critical nature of the threshold is only
mildly affected by statistical factors. There is no such
clearly marked threshold shown in the streamer case, Eq.
(14), for which the breakdown probability varies as
exp( n, /n ), wh—ich declines progressively more steeply as
n decreases, but still never becomes precisely zero.
The relationship of the breakdown-probability equations
to the measurement of the breakdown voltage is illustrated
in the following calculation. A common procedure
for measuring V, is to apply a slowly rising voltage
ramp to a spark gap which is irradiated to provide initiating
electrons. The distribution function of the measured
breakdown voltage is given by
I' ( V) =1 —exp —f, dV'
~ &'
( V )&( V')
t
d V'/dt
where E(V) is the fraction of measurements that exceed
31 PROBABILITY OF ELECTRICAL BREAKDOWN: EVIDENCE. . . 2619
V, i is the rate of appearance of free electrons, and P is
the breakdown probability. In this equation the assumption
is made that the free electrons act independently.
This assumption w'ill be valid for i & 1/tf, where tf is the
formative delay time.
Distribution functions were calculated for N2 at 53.3
kPa (400 Torr) and SF6 at 26.7 kPa (200 Torr) at d =0.45
mm for a ramp rate of 1 V/s and for i =1 electron/s and
i =6.24)&10 electron/s (1 pA) (Fig. 17). For N2 with 1
electron/s a spread of several volts is predicted, but with 1
pA of free electrons the distribution rises very steeply and
is indistinguishable from a step function within experimental
voltage resolution. Under the latter conditions the
breakdown voltage measurement will yield a critical voltage
nearly identical with VT. Davies and Evans also
found that the statistical uncertainties in the Townsend
model are negligible in measurements of V„if a sufficient
number of free electrons are present.
A different situation occurs with SF6 at 26.7 kPa in
which space charge is significant. The streamer breakdown
probability does not define a critical voltage below
which the breakdown probability is zero. The more gradual
rise of the probability curve results in measured
breakdown voltages that depend strongly on experimental
conditions.
(1) Breakdown occurs by the Townsend mechanism if
the average avalanche size n is much smaller than the
critical size for streamer formation n, . As n approaches
n, the probability of rapid streamer breakdown becomes
significant.
(2) The overvoltage at which the transition to streamer
breakdown occurs depends on the relative values of n,
and the secondary-ionization coefficient co/a. If
co /a(1/n„breakdown at the static breakdown voltage
V, will occur by the rapid streamer process.
(3) When co/a&10, measurements of V, and co/a
should be interpreted with caution. The statistical uncertainty
in V, introduced by the streamer breakdown probability
must be assessed. Space-charge acceleration of
avalanche growth may affect measurements of co/a.
ACKNOWLEDGMENTS
The voltage pedestal generator was designed and built
by Dr. K. Bardin and Mr. K. Hing. We thank Dr. E.
Hansen for assistance in the mathematical development.
This research was supported by independent research and
development funds of Lockheed Missiles and Space Company,
Inc. (Palo Alto, CA).
V. CONCLUSIONS
O 1.0I-
I
U I i
z 0 8- iN I
2
U
~ 0.6O
i
i
SF—j
I- 6 —i =1
~ 0.4—
LQ ——i =6.
~ 0.2- I
cn i
/
I
&sso 1600 1650
T i 2i BREAKDOWN VOLTACE (V)
e /s
6
2x 10e/s
pA
YT (SF 6)
1700
FIG. 17.Breakdown
voltage distribution functions calculated
from Eq. (18) for Nz and SF6 for a voltage ramp rate of 1 V/s.
The breakdown mechanism near the static breakdown
voltage can be identified by the voltage dependence of the
breakdown probability. The Townsend mechanism
predicts a well-defined voltage threshold for breakdown
above which the probability rises rapidly from zero to
one. The probability of streamer breakdown does not
have a precise threshold and increases less steeply with
overvoltage. The breakdown probabilities measured in
N2, H2, and Ar agreed with the Townsend probability
equation. In SF6 and CC12F2 a transition was observed
between Townsend and streamer behavior with increasing
pressure. A unified probability theory, for which the
Townsend and streamer mechanisms are limiting cases,
accounted for the transition. The results of this paper do
not depend on the mechanism of streamer propagation,
but rather suggest the conditions under which space
charge affects the prebreakdown state of current growth.
APPENDIX
The approximation for the Bessel function
I)(z)-e'/(2m. z)'/ (A 1)
is accurate to within 10% for z ~2. Substitution of this
approximation yields the integral
V n dn
&
p
—3/4(~ / )1/2
X f n /
exp[ —(1 p / ) n—/n]dn
The variable transformation y =n ' gives the integral
f V(n)dn
(A2)
3/4(n /~)1/2f exp[( 1 pl/2)2y2/p]yzdy
C
(A3)
x =(n, /n)(1 —p' )
and the probability function
@(z)=(2/m' ) f e 'dt .
0
An analytical approximation for 4 is given by
@(x'/ )=1—(a~t+a2t +a3t )e
(A5)
(A6)
(A7)
with solution
f V(n)dn
p
—3/4(~/n )1/2I(e —x/~l/2) x 1/2[1 g)(x 1/2)]I
(A4)
where
2620 R. V. HODGES, R. N. VARNEY, AND J. F. RILEY 31
where
t =I /( I +px '
),
p =0.47047,
ai ——0.3480242,
a2 ———0.095 879 8,
a3 ——0.7478556 .
Inserting this approximation gives
f V(n)dn =p ~
(nln, )' e
C
X[m' —x (a &t+a2t +a3t )] . (A8)
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