12
-
-
21
.
3
-
12
d
::
*
.
.
.
.
1.:
:
E
0.
L
.
:
2011,
::.
.
..
.
..
.
'
..
..
OP
.
.
.
14
.
1
23
*
I OF I
ORNL P
1539
.
+
.
1
ri
.
171
.
homenagement
a
1,0.
1 l
•
}
E
.
*
.
.
T
...! .
w
i
A
stres
s
on the that
t
h ese lines in the intervenir en cuenta
10
4
7
•
.
-
1
·
1
•
.
i
O
-
.
-
-
-
-
.
1
14 .
.
1
.
I'
.
.
-
1
A.
:
.
36
.
.
N
S
?
*
7.
....
MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS -1963
'.
..
.
BRN VA-1539
SEP 20 1965
MOVING STRIATIONS PRODUCED BY A SIMPLE THERMAL INSTABILITY*
Igor Alexeff
Oak Ridge National Laboratory
ABSTRACT
A simple thermal instability is investigated as a possible cause
of moving striations in discharge tubes. The mathematical treatment
predicts which gases should and which should not exhibit the insta-
bility. The frequency of oscillation of a cylindrical discharge, due
to the instability, also is predicted. Not discussed is the velocity
with which the instability appears to propagate longitudinally down
the discharge. The theoretical predictions are compared with the
experimental striation data of several authors, and reasonable
agreement is found.
-LEGAL NOTICE -

I This report
sa prepared as an accoud of hororubent sponsored work. Neither the Vatted
de Manus wa nurudy or prestation, and or lapu do wa rupect to demor
may, con Mon, a wobes at the marimidos com bond in de report, or the doma
ad my wa, mount, where a proces declarda due mu y box bedring
petredeky omal rights; or
8. And y Wa Madina se rupect to be made of her deg med tre
200 od ng balormatics, apperstes, method, or grocese diacioned in this report
Mwadudu won, partan xtera mea
leden helt m
maryna contro de Conducta, at maphored me conductor, bu o de
soch saployee or contractor of the Commission, or employee of such contractor properes,
domain, or provides second to, uy tornada perb We progna contract
vol du Coulalahon, a HI toplog via ma contractue,

RELEASED FOR ANNOUNCEMENT

IN NUCLEAR SCIENCE ABSTRACTS
*Research sponsored by the U.S. Atomic Energy Commission under
contract with the Union Carbide Corporation.
ta

This paper was rubmitted for publication
in the open Uiterature at least months
print to the issuanco date of this Micro-
an. Since the U.S.A.B.C. has no evi.
der.ce that it baw boon published, the pa-
per is being distributed in Microcard
form as a proprint.
1

I.
INTRODUCTION
Moving striations are bright concentrations of light that move
at high speed through discharge tubes. Typical moving striations
*
are shown in the image-converter photograph of Fig. 1. Although moving
striations bare been observed for many years and have been theoretically
discussed by numerous authors, a simple, complete theory apparently
Fring
here
does not exist. The most accepted theory is that of Pekerek,“ who
describes a striation as a progressive ionization ox breakdown front.
On this basis he obtains a dispersion relation which qualitatively
predicts the striation phase and group velocities.
In this paper we do not discuss the velocity of moving striations.
Instead, mathematical expressions are obtained that predict which
VA
geses should exhibit a thermal instability and what the frequency of
the instability should be.
The theoretical predictions concerning
the instability are compared with the experimental observations of
striations of several authors, and fair agreement is found.
II.
THEORETICAL DISCUSSION
The basic, new assumption in this investigation is that moving
striations are primarily & radial, not a longitudinal phenomenon.
This assumption is based on the striation structure, as shown in

Fig. 1.
The spacing of the striations is generally greater than
several radii of the discharge tube.
Therefore, the main plasma loss
must be radially to the wall, since the plasma is not confined in any
manner. Consequently, a radial oscillation process is assumed, in
spite of the apparent longitudinal motion of the striations.
If striations are caused by radiai cscillations, then an
investigation of a small section of the 43charge tube should be
sufficient to describe the phenomenon. Consider a section of the
discharge tube so short in length that within it the plasma conditions
can be considered uniform. For the sake of simplicity, we replace
functions that depend on radius by suitable average values.
The rate .
.
of heat input per unit volume to the electron gas is given by jºo
,
where oe is the electrical conductivity of the discharge. The rate
of heat loss from the electron gas 18 given by a Of T, where a 18 a.
constant, Op is the thermal conductivity of the electron gas, and T
is the electron temperature.
Let B be the ratio of heat input to heat loss,
For o and Ome one substitutes the theoretical values found in the
literature, 3
Here e is the electron charge, n is the electron density, L is the
electron's mean free path in the gas, m is the electron mass, v is
the electron average velocity, and k is Boltzmann's constant. For L
we substitute (NO), where N is the gas density, and O is the electron-
gas molecule scattering cross section.
The quantity B then is expressed
88
(2)
Within the discharge, two quantities are conserved: first the current
density ), because the discharge tube is fed by a high-impedance source;
and second, the neutral gas density N, because the degree of ionization
18 low, and because striation velocities can exceed sonic speeds giving
the gas no time to pove.
In the above equation for B the quantities in the parenthesis are
constants. The scattering cross section O is a direct function of
electron velocity, and hence electron temperature. However, for temp- .
erature increases of short duration compared to ion loss processes,
the electron density n in the discharge plasma in proportional to the
lategral with respect to time to the ion generation rate (T),
tist
26/0 (T) dt.
(3)
.
Thus if the electron temperature T increases suddenly, o responds
immediately while a responds more slowly. Conversely, if I decreases
O again responds immediately, while n responds only at a rate :
corresponding to ion loss processes.
We are now ready to discuss the conditions for thermal instability
in the discharge tube. Assume that o increases more rapidly than T/.
Equation (2) then reveals that the discharge is unstable. A slight
perturbation that increases the local temperature T results in more
heat input relative to heat 1088, and the temperature risee rapidly
10 a "runaway" fashion. Conversely, a slight decrease in the local
temperature results in a catastrophic drop in temperature. However,
18 o increases less rapidly than T/C, any temperature perturbation
corrects itsell and the discharge 18 stable.
.
Buable.
Knowing the basic cause of the thermal instability, we can infer
the rest of the oscillation cycle.
The electron temperature T first
rises in a runaway fashion until it reaches some limiting value,
perhaps caused by n eventualiy becoming large enough to bring B equal
to unity. However, even when T ceases to rise the integral form of n
given by Eq. (3) means that a still rises. Therefore Eq. (2) shows
that ß must finally become less than unity. The thermal instability
now reverses. The lower T becomes the less the heat input relative
to the heat loss, and a catastrophic cooling occurs. The result is a
cold, dense plasma. This cold, dense plasma persists until the elec-
tron density a drops sufficiently for B again to become greater than
unity. The cycle then repeats.
Thus, the cycle of the thermal instability is a sudden rise and
fall of the electron temperature, coupled with a sudden rise and slow
Fig. 2 hori
decay of the electron density ag is illustrated in Fig. 2.
This
theoretical cycle corresponds closely to the striation cycle
.
experimentally observed. 4,5
The frequency of oscillation of the thermal instability is
determined by the slowest process present in the cycle. This appears
to be the rate of loss of the ions. For pressures less than a few
torr, the main ion loss is by radial ambi-polar diffusion. The basic
differential equation is
:
.
:
Here k is Boltzmann's constant, T is the electror temperature, m, is
the lon mass, a is the lon density, v. is the lon-molecule collision
rate, and t refers to time. For a cylindrical tube of radius ro, the
time t required for the lon density to drop x factors of e 18 given by
t = X 5.78
(5)
kr.
.
The frequency of oscillation is the reciprocal of the above quantity.
In comparing this equation with experiments for which x was not
measured, a reasonable approximation is to set x equal to unity.
In summary, the discharge tube exhibits a strong thermal
instability if operating in a regime where the scattering cross
section for electrons on gas atoms increases more rapidly with electron
temperature then T/".
The frequency of oscillation v is given by
1*
here
III. COMPARISON WITH EXPERIMENT
A. Stability of Various Gases
The discussion presented above gives clear-cut predictions as to
which gases should and which should not show the thermal instability.
However, one must not be surprised if these predictions only separate
those gases which easily produce striations from those which produce
striations with difficulty. Our simplified treatment does not contain
-
WD
such complicating effects as the production of metastable atoms and
.
contamination of the gas. Another possible complication is that
.
. .
.
earlier workers oxy cave coufused moving striations with oscillatory
phenomena originating at the anode of the discharge. Finally, the
-
-
-
thermal instability may be only one of several striation mechanisms.
Predictions concerning the stability of various gasses are
presented in Table 1, along with comments concerning the observations
of moving striations. Diatomic gnses ere not discussed to avoid
difficulties caused by molecular dissociation. For determining o.
a standard reference of cross section data' suffices, except where
noted.
&
Table I here
Discussich of Comments from Table 1:
1.
In helium we bave never observed clear-cut moving striations,
although weals, strongly damped fluctuations were seen moving
away from the anode. On the other hand, Astou and Kikuchiº
have noted striations in helium.
In neon the curve of o
vs T rises in a stepwise fashion
in the energy range of a few electron volts.
Thus, one has
regions of stability and instability, depending on the average
electron temperature at which the discharge operates.
In mercury vapor Fouldsu bas observed moving striations,
although they were irregular.
On the other hand, Crawford
has not:: observed moving striations in mercury vapor 'under:
the same conditions.
4.
In cesiun vapor Robertson
has never observed moving
striations.
This observation agrees with the theoretical
prediction of stability as obtained from the latest croso
section data. 13
B.
Predicted and Observed Frequencies
Predicted frequencies of oscillation vn are compared with observed
frequencies v. in Table 2. The experimental data in Table 2 is intended
to be a sample of the more recent and more complete published material.
The ion-atom collision rate v. used in Eq. (6) 18 computed from the
ion-atom collision cross section, assuming that both the ions and
the gas atoms are at room temperature.
*
Table & bine
Discussion of Comments from Table 2:
5. The paper of Stewarts contains the most complete set of data
: for the requirements of Eq. (6).
6. The results of Sicha'' show a fluctuation of electron density
during the striation cycle of only 20%. This result is
smaller by a factor of about 50 than the results of Stewart,"
Paik, et al., ') and Sodomsky. Only if we assume that & .
density fluctuation of e occurs will the theoretical frequency
be in fair agreement with the experimental frequency.
7. 'At the highest pressure used by Donahue and It?ke,
the
main loss of plasma is possibly by recombination in the gas
rather than by wall 1088e8, Recombination in the inert gases
has been observed at pressures above a few torr.
In the
:
presence of increased plasma losses, the observed oscillation
frequency should be higher than that predicted in Eq. (6) as
18 indeed observed.
A quantitative measure of the agreement between theory and
experiment is obtained by computing an averape factor by which yo
differs from w. Since one does not know if the predicted or the
observed frequency is correct, the frequency difference is compared
to the average frequency to obtain the frequency ratio f,
f = (x - ) / 3/2 (7) + %)] .
Omitting the anomalous-appearing results of Sichat and the highest
pressure result of Donahue and Dieke, 15 we find the rms value for f
to be + 0.383 and the most probable value to be $ 0.258. Using the
most probable value for f we work backwards through Eq. (7) to find
1.7 > V/V> 1/1.7.
The above statistical discussion shows that the predicted and
observed frequencies shown in Table 2 agree fairly well. Remember
that, in computing the predicted frequencies from Eq. (6), we must
generally estimate some of the variables. In general, we do not
know the ion decay constant x, the electron temperature T, and the
100-gas atom collision rate v. (which depends on the rarely measured
gias terperature).
.
-10
-10
IV.
SUMMARY
This paper presents a simple model of a thermal instability which
may drive moving striations. The model predicts both which gases should
be unstable and what the frequency of the instability should be. Fair
agreement is found between the predictions of the model and the exper-
imental observations of moving striations by various authors.
No attempt is made in this paper to derive the apparent longitudinal
velocity of propagation of the instability. However, earlier experi-
mental work by usº now suggests that the instability generally
progresses down the discharge tube at a rate determined by ambi-polari
diffusion (see Eq. 4).
.
.
.
ACKNOWLEDGMENT

The author acknowledges several very useful discussions with
Dr. W. D. Jones.
«ll.
Table 1
THE EXISTENCE OF THE
THERMAL INSTABILITY IN VARIOUS GASES
Average
Electron
Temperature
Stability
Prediction
Striation
Production
Gas
Comments
stable
difficult
3 eV
I ev
i ev
indefinite
easy
unstable
easy
J. ev
unstable
easy
1 eV
.
unstable..
easy
stable
difficult
1 ev
0.2 ev
ma
stable
not observed
-12-
Table ?
THE PREDICTED FREQUENCY OF THE THERMAL INSTABILITY
COMPARED WITH
THE OBSERVED FREQUENCY OF MOVING STRIATIONS
D
torr
Author
Ges
со
ev
CP8
cp8
Comments
Author_
Stewart5
Gas
A
o p
om torr
.856
e
eV
1.1
x
2.2
vp
cps
920
20
c ps Comments
1,000. 5
Paik,
A
0.8
1.3
1.0*
1.0+
2.3
2.3
17,500
10,800
31,000.-
3,800
et al.15
Sodomsky 16
Siche i7
NE
Ne
Ne
1.05
zl.O
21.0
0.5 1.04 1.9
2.9 1.04 0.2
1.1 1.0% 0.2
22,000
8,100
21,300
12,000
1,600.
16,500
.
6
6
Donahue,
A ;
6,800
et al. 18
c.75
0.75
0.75
2.1
4
12.0
1.04
?.0*
1.0*
1.04
1.0*
1.0*
3,250
A
1,190
5,000
1,333
1,500-
3,500
2,500
0.75
30
1.04
1.0*
477
-
7
Bletzinger,
Ne
1.25
1.0
1.0*
1.0*
15,000
3,200
'
et al. 19
-
*Value not published, assumed by us.
-13-
FIGURE CAPTIONS
· Fig. 1
Image-Converter Photograph of Moving Striations.
The striations move from the anode on the right
to the cathode on the left.
The gas 18 krypton
at about 30w pressure.
Fig. 2
Deduced Variation of Electron Temperature and
Ion Density During the Thermal Instability Cycle.
in a Fixed Element of Volume.
-14.
.
REFERENCES
1. Druyvesteyn, M. J., Physica 1, 273 and 1003 (1934).
Armstrong, E. B., K. G. Emeleus and T. R. Neill, Proc. Roy. Irish
Acad. 54A, 291 (1951).
Watanabe, S., and N. L. Oleson, Phys. Rev. 99, 1701 (1955).
Robertson, H. 8., Phys. Rev. 105, 368 (1957).
Nedospasov, A. V., J. Tech. Phys. (USSR) 3, 153. (1958).;::.
Chapnik, I. M., Soviet Phys. (JETP) 34, 1033 (1958).
Wojaczek, K., Beiträge aus der Plasmaphysik (E. Germany), 1., 1 (1962);
2, 50 (1962); 2, 56 (1962); 2, 122 (1962); 2, 134 (1962).
General Review:
Iruyvesteyn, M. J., and F. M. Penning, Revs. Modern Phys. 12, 87 (1940).
Francis, G., Handbuch der Physik XXII, 137 (1956).
-
*
2.
Pekarek, L., and V. Krejci, Czech. J. Phys. B 12, 450 (1962); B 13,
881 (1963); Proc. 5th Int. Conf., Ionization phenomena in Cases,
Vol. 1, H. Maecker, Ed. (North-Holland, Amsterdam, 1962), p. 573.
3. Joos, G., Theoretical Physics, 2nd Ec. (Hafner, New York, 1950),
pp. 445-447.
4. Pupp, W., Physik 2. 36, 61 (1935).
5. Stewart, A. B., J. Appl. Phys. 27, 911 (1956).
6. Cooper, A. W., J. Appl. Phys. 35, 2877 (1964).
7. McDaniel, E. W., Collision Phenomena in Ionized Gases (Wiley, New
York., 1964), pp. 116 and 117.
8. Aston, F. W., and T. Kikuchi, Proc. Roy. Soc. (London) A 98, 50
(1920-21).
9. Normand, C. E., Phys. Rev. 33, 1217 (1930).
20. Foulds, K.W.H., J. Electron. Control (GB) 2, 270 (1956-57).
21. Crawford, F. W., and H. R. Pagels, 'M. L. Report No...962, Microwave
Laboratory, Stanford University, October, 1962 (unpublished).
A
-
2
19
17
12. Pobertson, H. 8., and M. A. Hakeen, proc. 5th Int. Conf.:
Ionization Phenomena in Cases, Vol. 1, H. Maecker, Ed. (North-
Holland, Amsterdam, 1962), p. 550.
13. Garrett, W. R., and R. A. Mann, Phys. Rev. 130, 658 ! 1.963).
14. McDaniel, E. W., Collision Phenomena 1.n Ionized Gases (Wiley, New
York, 1964), p. 439.
15. Palk, 8. F., J. N. Shapiro, and K. D. Gilbert, J. Appl. Phys. 35
2573 (1964),
16. Sodomsky, K. F., J. Appl. Phys. 34, 1860 (1963).
Sicha, M., Czech. J. Phys. B 13, 499 (1963).
18. Donahue, F., and G. . Dleke, Phys. Rev. 81, 248 (1951).
19. Bletzinger, P., and A. Garscadden, J. Electron. Control (GB) 16,
269 (1964).
20.
Mulcahy, M. J., and J. J. Lennon, Proc. Phys. Soc. (London) 80,
626 (1962).
Jones, W. D., and... I.Alexeft, Bull. Am. Phys. Soc. 9, 469 (1964).
21.

4
.
.
.
1
FIG. 1.
.
,
...
:
PION DENSITY
C
1/
- ELECTRON TEMPERATURE
t
.
.
Y
3
TIME
DENSITY AND ELECTRON TEMPERATURE
. AS A FUNCTION OF TIME :::
i. .Fia. 2;. :
....
:
.::
.
.
€
.
C
.
END
-
+
=
DATE FILMED
9 / 13 / 66







:
L
')