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Experimental control of Stewart's theoretical model of large amplitude moving striations
Physica 92C (1977) 290-292 © North-Holland Publishing Company
EXPERIMENTAL CONTROL OF STEWART'S THEORETICAL MODEL OF LARGE AMPLITUDE MOVING STRIATIONS G. VAN DEN BERGE and M. VANMARCKE
Laboratorium voor Natuurkunde, Ri/ksuniversiteit, Rozier 44, B-9000 Gent, Belgium Received 2 March 1977
The longitudinal variation of the electron concentration in large amplitude moving striations, computed theoretically by Stewart, has been tested experimentally. The measurements are carried out by means of a sampling probe technique in the glow discharge of neon (1 = 105 mA, 2R = 5.6 cm, P0 = 0.79 torr) and of argon (I = 75 mA, 2R = 5.7 cm, PO = 0.46 torr). It is found that the measured dependence of the concentration is not consistent with the theory.
bution o f the electrons. This assumption is certainly not valid in the case of self-excited moving striations where even double-peaked distributions may be present [2]. In the present paper an attempt is made for an improved experimental control of Stewart's equation, by measuring the electron energy distributions during a period of the moving striations, and by comparing the production rate of ion pairs, calculated following eq. (1) with data based on the measured energy distributions.
1. Introduction The longitudinal variation of the electron concentration in large amplitude moving striations, computed theoretically by Stewart [1 ], is given by the equation On + ~ 2 - 4 0 ~ 2
~t
~T/
la+(nOE z On D a n - - - o f \ ~ + Ez-~t )=P ,
(1)
where
n
= n(r, z, t) = the electron concentration, r, z, t = the radial and the axial distance (measured in the direction anode to cathode) and the time, R = the radius o f the discharge tube, Da = the ambipolar diffusion coefficient, /a+ = .the mobility of the positive ions, of = the phase velocity of the moving striations, Ez = the axial electric field, and = the production rate of ion pairs (electrons P + ions) per unit of volume by direct ionization.
2. Method By the use o f the sampling probe technique described in [2], time-resolved measurements o f the electron energy distribution F(e) and the axial electric field E z at different phases of a moving striation, have been made. Let P1 represent the production rate given by Stewart's eq. (1). It is calculated as follows. The electron concentration n and the mean electron energy (e) at the time t during a moving striation period are computed from the energy distribution F(e) measured at the same time
In order to control eq. (1) experimentally Stewart calculates the production rate P of ion pairs by graphical differentiation o f the experimental n versus t and E z versus t curves. In this investigation the n versus t relationship is based on Langmuir probe measurements, whereby the author assumes a Maxwell energy distri-
n(t) = ~ F(e) ' de, 0 290
G. Van Den Berge and M. Vanmarcke/Control o f Stewart's striation m o d e l
291
se the electron ionization efficiency at the gas pressure
(e(t))= ~ eF(e), de/ S F(e)" de. o o The derivatives an~at and aEz/at are obtained by graphical differentiation of the experimental n and Ez versus t curves. The ambipolar diffusion coefficient in the positive column of a glow discharge is given by
P0" In the case where no electron energies larger than 2e i are to be expected, as is the case in our experiment, the ionization efficiency s e can be approximated by the relation S e = CPO ( e -- ei),
e
D a = bt+
e
where #+ is the ion mobility, k the Boltzmann constant and - e the charge of the electron. In this expression the electron temperature Te is obtained from
where c is a known gas constant [3]. By taking into account eqs. (3)-(5), the production rate P2 is given by (6) e
\ml
= ]kT ;
ei
the ion mobility/% is obtained from the reduced mobility ta+, 0 (at 0°C and 760 torr):
Eq. (6) has been used to compute P2 directly from the measured energy distribution F(e).
760 /% =P+,0 PO
3. Results
where P0 is the reduced gas pressure in torr. On the other hand the production rate of ion pairs can be derived directly from the electron energy distribution F(e). Its numerical value P2 is given by the basic equation
,"2 --
wi(e)" fc(e) • F ( e ) . de,
(2)
ei
where e i = the ionization energy of the gas, Wi = the probability of direct ionization, and fc = the collision frequency per electron of energy e. It is known that Wi(e) = k e • Se(e),
(3)
f (e)
(4)
=
m being the electron mass, ~ the mean free path, and
(5)
The above measuring and calculation methods have been applied to the study of moving striations in discharges in neon ( I = 105 mA, 2R = 5.6 cm, p 0 = 0.79 torr) and in argon (1 = 75 mA, 2R = 5.7 cm, P0 = 0.46 torr). Details about the measurement of the electron energy distribution F(e) and of the electric field Ez are given for the neon and argon striations, respectively, in .[2] and [4]. The reduced mobilities /.t+, 0 o f N e + and Ar + are 4.2 and 1.6 cm2/V • s, respectively [5] ; the gas constants c are 5.6 and 71 ion pairs/torr • V . m [3]. The calculated ionization functions P1 and P2, and the light intensity measured in arbitrary units, have been plotted in terms of the time t in figs. 1 and 2. The striation period is 0.635 ms in neon and 1.00 ms in argon. The amplitude of the function P1 computed according to the Stewart equation is observed to follow closely the light intensity; this empirical result is in complete agreement with the result obtained by Stewart [1 ]. On the other hand the relation between the function/2, obtained directly from the measured distribution function, and the light intensity, appears to be far less simple. Moreover, the peak values o f P 1 and P2 differ by a factor 8 and 13,
292
G. Van Den Berge and M. Vanmarcke/Control of Stewart's striation model I
~3
\
.¢_
.3
L3 2
o
1
I
I
o
I
I
O.4
0.2
I
i
0
0.6
I 0.2
0
o'6
0./,
08
1.0
t (ms)
o.s
~.o
t~ms)
o-s
to
tc~
---2 o~
2
| _
o
0.2
0.4
0.6
tlmsj
~
A
0 o
i
7
0.2
o.4
;
* o.s
'm .a
~2 0
~ = 0
-
~ ~ 0.2
_~
_
~ 0./.
_-: ~
. 0.6
_ t (ms)
Fig. 1. Light intensity and calculated ionization functions PI and P2 as a function of time in a moving striation of a neon plasma.
in the neon and argon striations, respectively. The P2 values.are somewhat less precise than the P1 values, owing to the integration difficulties at the tail of the energy distribution. However, this is not relevant to the mentioned large numerical discrepancy. The discrepancy between P1 and P2 is probably due to the fact that Stewart excludes in the ionization term P1 ion pair production by stepwise ionization of the gas. As reported in [2], ion pair production by collisions between excited atoms and electrons, and by mutual collisions between excited atoms is to be expected, together with direct ionization of the gas.
Acknowledgements This work has been supported by a grant from the
0==
~...,.
0
0.2
:
A 0.4
.
.L 0.6
Fig. 2. Light intensity and calculated ionization functions P1 and P2 as a function of time in a moving striation of an argon plasma.
Nationaal Fonds voor Wetenschappelijk Onderzoek, Brussels. The authors are indebted to Professor P. Mortier for his encouragement and criticisms.
References [1] A.B. Stewart, J. Appl. Phys. 27 (1956) 911. [2] G. Van Den Berge, Physica 83C (1976) 227. [3 ] A. yon Engel, Ionized Gases (Oxford Univ. Press, London, 1965) p. 63. [4] G. Van Den Berge, Proefschrift, Rijksuniversiteit Gent (1972). [5] H.S.W. Massey, E. H. S. Burhop and H. B. Gilbody, Electronic and Ionic Impact Phenomena, Vol, 3 (Oxford Univ. Press, London, 1971) p. 1995.
EXPERIMENTAL CONTROL OF STEWART'S THEORETICAL MODEL OF LARGE AMPLITUDE MOVING STRIATIONS G. VAN DEN BERGE and M. VANMARCKE
Laboratorium voor Natuurkunde, Ri/ksuniversiteit, Rozier 44, B-9000 Gent, Belgium Received 2 March 1977
The longitudinal variation of the electron concentration in large amplitude moving striations, computed theoretically by Stewart, has been tested experimentally. The measurements are carried out by means of a sampling probe technique in the glow discharge of neon (1 = 105 mA, 2R = 5.6 cm, P0 = 0.79 torr) and of argon (I = 75 mA, 2R = 5.7 cm, PO = 0.46 torr). It is found that the measured dependence of the concentration is not consistent with the theory.
bution o f the electrons. This assumption is certainly not valid in the case of self-excited moving striations where even double-peaked distributions may be present [2]. In the present paper an attempt is made for an improved experimental control of Stewart's equation, by measuring the electron energy distributions during a period of the moving striations, and by comparing the production rate of ion pairs, calculated following eq. (1) with data based on the measured energy distributions.
1. Introduction The longitudinal variation of the electron concentration in large amplitude moving striations, computed theoretically by Stewart [1 ], is given by the equation On + ~ 2 - 4 0 ~ 2
~t
~T/
la+(nOE z On D a n - - - o f \ ~ + Ez-~t )=P ,
(1)
where
n
= n(r, z, t) = the electron concentration, r, z, t = the radial and the axial distance (measured in the direction anode to cathode) and the time, R = the radius o f the discharge tube, Da = the ambipolar diffusion coefficient, /a+ = .the mobility of the positive ions, of = the phase velocity of the moving striations, Ez = the axial electric field, and = the production rate of ion pairs (electrons P + ions) per unit of volume by direct ionization.
2. Method By the use o f the sampling probe technique described in [2], time-resolved measurements o f the electron energy distribution F(e) and the axial electric field E z at different phases of a moving striation, have been made. Let P1 represent the production rate given by Stewart's eq. (1). It is calculated as follows. The electron concentration n and the mean electron energy (e) at the time t during a moving striation period are computed from the energy distribution F(e) measured at the same time
In order to control eq. (1) experimentally Stewart calculates the production rate P of ion pairs by graphical differentiation o f the experimental n versus t and E z versus t curves. In this investigation the n versus t relationship is based on Langmuir probe measurements, whereby the author assumes a Maxwell energy distri-
n(t) = ~ F(e) ' de, 0 290
G. Van Den Berge and M. Vanmarcke/Control o f Stewart's striation m o d e l
291
se the electron ionization efficiency at the gas pressure
(e(t))= ~ eF(e), de/ S F(e)" de. o o The derivatives an~at and aEz/at are obtained by graphical differentiation of the experimental n and Ez versus t curves. The ambipolar diffusion coefficient in the positive column of a glow discharge is given by
P0" In the case where no electron energies larger than 2e i are to be expected, as is the case in our experiment, the ionization efficiency s e can be approximated by the relation S e = CPO ( e -- ei),
e
D a = bt+
e
where #+ is the ion mobility, k the Boltzmann constant and - e the charge of the electron. In this expression the electron temperature Te is obtained from
where c is a known gas constant [3]. By taking into account eqs. (3)-(5), the production rate P2 is given by (6) e
\ml
ei
the ion mobility/% is obtained from the reduced mobility ta+, 0 (at 0°C and 760 torr):
Eq. (6) has been used to compute P2 directly from the measured energy distribution F(e).
760 /% =P+,0 PO
3. Results
where P0 is the reduced gas pressure in torr. On the other hand the production rate of ion pairs can be derived directly from the electron energy distribution F(e). Its numerical value P2 is given by the basic equation
,"2 --
wi(e)" fc(e) • F ( e ) . de,
(2)
ei
where e i = the ionization energy of the gas, Wi = the probability of direct ionization, and fc = the collision frequency per electron of energy e. It is known that Wi(e) = k e • Se(e),
(3)
f (e)
(4)
=
m being the electron mass, ~ the mean free path, and
(5)
The above measuring and calculation methods have been applied to the study of moving striations in discharges in neon ( I = 105 mA, 2R = 5.6 cm, p 0 = 0.79 torr) and in argon (1 = 75 mA, 2R = 5.7 cm, P0 = 0.46 torr). Details about the measurement of the electron energy distribution F(e) and of the electric field Ez are given for the neon and argon striations, respectively, in .[2] and [4]. The reduced mobilities /.t+, 0 o f N e + and Ar + are 4.2 and 1.6 cm2/V • s, respectively [5] ; the gas constants c are 5.6 and 71 ion pairs/torr • V . m [3]. The calculated ionization functions P1 and P2, and the light intensity measured in arbitrary units, have been plotted in terms of the time t in figs. 1 and 2. The striation period is 0.635 ms in neon and 1.00 ms in argon. The amplitude of the function P1 computed according to the Stewart equation is observed to follow closely the light intensity; this empirical result is in complete agreement with the result obtained by Stewart [1 ]. On the other hand the relation between the function/2, obtained directly from the measured distribution function, and the light intensity, appears to be far less simple. Moreover, the peak values o f P 1 and P2 differ by a factor 8 and 13,
292
G. Van Den Berge and M. Vanmarcke/Control of Stewart's striation model I
~3
\
.¢_
.3
L3 2
o
1
I
I
o
I
I
O.4
0.2
I
i
0
0.6
I 0.2
0
o'6
0./,
08
1.0
t (ms)
o.s
~.o
t~ms)
o-s
to
tc~
---2 o~
2
| _
o
0.2
0.4
0.6
tlmsj
~
A
0 o
i
7
0.2
o.4
;
* o.s
'm .a
~2 0
~ = 0
-
~ ~ 0.2
_~
_
~ 0./.
_-: ~
. 0.6
_ t (ms)
Fig. 1. Light intensity and calculated ionization functions PI and P2 as a function of time in a moving striation of a neon plasma.
in the neon and argon striations, respectively. The P2 values.are somewhat less precise than the P1 values, owing to the integration difficulties at the tail of the energy distribution. However, this is not relevant to the mentioned large numerical discrepancy. The discrepancy between P1 and P2 is probably due to the fact that Stewart excludes in the ionization term P1 ion pair production by stepwise ionization of the gas. As reported in [2], ion pair production by collisions between excited atoms and electrons, and by mutual collisions between excited atoms is to be expected, together with direct ionization of the gas.
Acknowledgements This work has been supported by a grant from the
0==
~...,.
0
0.2
:
A 0.4
.
.L 0.6
Fig. 2. Light intensity and calculated ionization functions P1 and P2 as a function of time in a moving striation of an argon plasma.
Nationaal Fonds voor Wetenschappelijk Onderzoek, Brussels. The authors are indebted to Professor P. Mortier for his encouragement and criticisms.
References [1] A.B. Stewart, J. Appl. Phys. 27 (1956) 911. [2] G. Van Den Berge, Physica 83C (1976) 227. [3 ] A. yon Engel, Ionized Gases (Oxford Univ. Press, London, 1965) p. 63. [4] G. Van Den Berge, Proefschrift, Rijksuniversiteit Gent (1972). [5] H.S.W. Massey, E. H. S. Burhop and H. B. Gilbody, Electronic and Ionic Impact Phenomena, Vol, 3 (Oxford Univ. Press, London, 1971) p. 1995.