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An exact theory of ionization waves (striations)
Volume 40A, number 3
PHYSICS LETTERS
17 July 1972
AN EXACT THEORY OF IONIZATION WAVES (STRIATIONS) K. ROHLENA, T. R 0 ~ I ~ K A and L. P E I ~ R E K
Institute of Physics, CzechoslovakAcademy of Sciences, Prague, Czechoslovak& Received 11 April 1972 A theory of the moving striations at low-currents is presented based on a direct solution of the Boltzmann equation and on the ion and metastable continuity equations. Four out of the six obtained wave-varietiesare known from experiment. We calculated the dispersion of the slow and fast ionization waves (striations) in neon using an earlier published model of a metastable-guided (slow) and an ion-guided (fast) ionization wave [ 1], but applying the direct solution of the electron Boltzmann equation to include fully the space-resonances found in [2, 3]. The solution was accomplished for a low discharge current where the electron-electron interaction is negligible in comparison with the electron-atom elastic and inelastic interactions. Quasineutrality was supposed to be fulfilled for the wavelengths in question, and the mean free paths assumed to be much shorter than the wavelength, so that the influence of the macroscopic momentum of ions included in the low-pressure theory of Swain and Brown [4] and causing electroacoustic effects could be completely neglected. The one-dimensional equations for small deviations were used in the following form: On + bi ~z (nE) = + Va** + n O--tPoo~ V2a** -- ri--~ '
(1)
Onm ~)2nm 0t - D m 0z 2
(2)
V ° a - ua** - 2V2a°*
nm
Tmw
n m is the metastable density, n the ion density, Voa the metastable production rate, Va** the stepwise ionization rate, V2a~. the ionizing dual metastable collision rate, Vo.* the direct ionization rate, rmw and riw the metastable and ion wall life times, D m the metastable diffusion and b i the ion mobility. All the collision rates v mean the number of the collisional processes in question per time and volume unit and have the dimension sec-1 cm-3. The Boltzmann equation for the electrons takes the form:
1
a
E
~__~(Ufl)_ng[_ ~ ~(QdU2fo) ...i
+ (U + Ua) Qa (U + Ua) fo (U + Ua) - UQafo l = 0, ..I (3)
alo
alo
O--z- - E ~
+ng adfl = 0 ;
(4)
fo and f l are the symmetric and axial parts of the electron distribution, U the voltequivalent of the electron kinetic energy, ng the neutral gas density, m andM the electron and ion masses, Qd the momentum transfer cross-section, Qa the excitation cross section, Ua the inelastic threshold energy. The dispersion was calculated in the usual way for the real wavenumber k and complex frequency 60 + i~, (7 is the time increment). As it has been shown in [2], the solution of Boltzmann equation for a disturbance which is periodic in space shows sharp resonances for selected wavelengths, which can be expressed by an analytical approximation [3] as
x = (Wa/qL3 (1 +p);
(s)
q = 1, 2;p = (ngQdUa/E) 2 2m/3M '~ 1 is the (small) ratio of the elastic and inelastic energy losses. The space resonances modify strongly the dispersion curves of the ionization waves. Fig. 1 is an example of the dispersion and increment curves of the slow ionization wave, computed exactly from eqs. (1)-(3) for one numerical set of parameters. The ionization rates (as well as other coefficients for the electron gas) were computed directly from the electron energy distribution and collision cross-sections. At the two resonant wavenumbers sharp maxima appear on 239
Volume 40A, number 3
PHYSICS LETTERS
slow short wave p. It has not been observed. Another increment maximum, with an amplified wave, lies between the two resonances, and is evidently of the "hydrodynamic" origin [ 1]. It should correspond to the rare but observed fast wave variety r. A similar "hydrodynamic" maximum of the slow metastableguided wave (n') seen in fig. 1 gives a damped wave for the chosen numerical case. No wave of this type is known from experiments. So the four wave varieties (three connected with the electron gas resonances and one of hydrodynamic origin) of the six possible according to the theory (see table 1) are known experimentally.
r
E--.2
"d........
,,', ,'
-2
/
s
/
I
h
s'
p
7
- 1,
~,r%/ / j /
UQ I
i
Fig. 1. Amplification and dispersion curves of the slow waves in neon. 1/R = 1.27 mA/em, poR = 3 cm • tort, nm/ng = 3.16 × 10 -6 (degree of excitation), rm/rmw = 0.29 (ratio of the total and wall metastable life-times). A flat hydrodynamic maximum is seen between the both resonant ones: s' and pvariety of the ionization waves. the time-increment curve accompanied by very steeply falling parts on the dispersion curves. The resonant wavenumbers correspond to those observed for the s' and p varieties of the slow wave, with the correct values of Nov~k's potentials [5, 6] over the wavelength and with an instability around the resonant points. Fig. 2 shows the dispersion and increment curves computed in the same way for the fast wave. Again, the dispersion and increment curves are strongly influenced by the resonances. The first resonance corresponds to the well-known fast wave s, with the same Nov£k's potential as has the slow s' wave. The second resonance has a negative time-increment in this case and represents a fast counterpart " p " to the 1
ss s/ s
-1
7~// ss s
J
17 July 1972
[ I r
"\
p \ ,q\. VI U a , Ek
•
Fig. 2. Amplification and dispersion curves of the fast waves in neon. 1/R = 0.109 mA/cm, poR = 1 cm • torr, nm/ng = 3.16 × 10- 6 (degreeof excitation), r i = riw (ion life-time is given by the wall diffusion). The hydrodynamic maximum can be interpreted as the r-wave. The second resonance produces an unobserved ~p'~-wave. 240
Table 1 Time- and space-selected wave varieties for neon at low discharge current. Space-selection
First Hydrodynamic Second resonance maximum resonance q=l q=2 Eh =I~9V Eh= 13V
Eh= 9.5 V
Time scale selection Slow (metastable -guided) w a v e
s' variety "r"-variety
p -variety
observed not observed
observed
Fast (ion-guided) wave
s variety r -variety observed observed
"p'" variety not observed
In addition to the explanation of the constant potential law over the wavelength, also other observed properties of the ionization waves in the low current neon discharge are obtained by the theory, e.g., the high ratio of the absolute value of the group velocity to the phase velocity. The time-scale separation should not take place at high current densities [1], where the life time of metastable atoms is shorter than the life time of ions, and the space-resonance selection should disappear at high currents as the electron gas is maxwellized. This is in agreement with observations at high currents [7] and corresponding ionization wave theories [8].
References [ 1] L. PekArek, K. Magek and K. Rohlena, Czech. J. Phys. B 20 (1970) 879. [2] T. Rfi$i~ka, Thesis, Inst. of Physics of Czech. Acad. Sci. Prague (1970).
Volume 40A, number 3
PHYSICS LETTERS
[3] T. RS[i~ka and K. Rohlena, 10th ICPIG, Oxford (1971) 287. [4] D.W. Swain and S.C. Brown, Phys. Fluids 14 (1971) 1383. [5] M. Nov~ik,Czech. J. Phys. B 10 (1960) 954.
17 July 1972
[6] V. Krej~, K. Ma~ek, L. L~ska and V. Pe'fina, Beitr. Plasmaphys. 7 (1967) 413. [7] K. Wojaczek, Beitr. Plasmaphys. 6 (1966) 319 and 11 (1971) 335. [8] K. Wojaczek, Beitr. Plasmaphys. 1 (1961) 30.
241
PHYSICS LETTERS
17 July 1972
AN EXACT THEORY OF IONIZATION WAVES (STRIATIONS) K. ROHLENA, T. R 0 ~ I ~ K A and L. P E I ~ R E K
Institute of Physics, CzechoslovakAcademy of Sciences, Prague, Czechoslovak& Received 11 April 1972 A theory of the moving striations at low-currents is presented based on a direct solution of the Boltzmann equation and on the ion and metastable continuity equations. Four out of the six obtained wave-varietiesare known from experiment. We calculated the dispersion of the slow and fast ionization waves (striations) in neon using an earlier published model of a metastable-guided (slow) and an ion-guided (fast) ionization wave [ 1], but applying the direct solution of the electron Boltzmann equation to include fully the space-resonances found in [2, 3]. The solution was accomplished for a low discharge current where the electron-electron interaction is negligible in comparison with the electron-atom elastic and inelastic interactions. Quasineutrality was supposed to be fulfilled for the wavelengths in question, and the mean free paths assumed to be much shorter than the wavelength, so that the influence of the macroscopic momentum of ions included in the low-pressure theory of Swain and Brown [4] and causing electroacoustic effects could be completely neglected. The one-dimensional equations for small deviations were used in the following form: On + bi ~z (nE) = + Va** + n O--tPoo~ V2a** -- ri--~ '
(1)
Onm ~)2nm 0t - D m 0z 2
(2)
V ° a - ua** - 2V2a°*
nm
Tmw
n m is the metastable density, n the ion density, Voa the metastable production rate, Va** the stepwise ionization rate, V2a~. the ionizing dual metastable collision rate, Vo.* the direct ionization rate, rmw and riw the metastable and ion wall life times, D m the metastable diffusion and b i the ion mobility. All the collision rates v mean the number of the collisional processes in question per time and volume unit and have the dimension sec-1 cm-3. The Boltzmann equation for the electrons takes the form:
1
a
E
~__~(Ufl)_ng[_ ~ ~(QdU2fo) ...i
+ (U + Ua) Qa (U + Ua) fo (U + Ua) - UQafo l = 0, ..I (3)
alo
alo
O--z- - E ~
+ng adfl = 0 ;
(4)
fo and f l are the symmetric and axial parts of the electron distribution, U the voltequivalent of the electron kinetic energy, ng the neutral gas density, m andM the electron and ion masses, Qd the momentum transfer cross-section, Qa the excitation cross section, Ua the inelastic threshold energy. The dispersion was calculated in the usual way for the real wavenumber k and complex frequency 60 + i~, (7 is the time increment). As it has been shown in [2], the solution of Boltzmann equation for a disturbance which is periodic in space shows sharp resonances for selected wavelengths, which can be expressed by an analytical approximation [3] as
x = (Wa/qL3 (1 +p);
(s)
q = 1, 2;p = (ngQdUa/E) 2 2m/3M '~ 1 is the (small) ratio of the elastic and inelastic energy losses. The space resonances modify strongly the dispersion curves of the ionization waves. Fig. 1 is an example of the dispersion and increment curves of the slow ionization wave, computed exactly from eqs. (1)-(3) for one numerical set of parameters. The ionization rates (as well as other coefficients for the electron gas) were computed directly from the electron energy distribution and collision cross-sections. At the two resonant wavenumbers sharp maxima appear on 239
Volume 40A, number 3
PHYSICS LETTERS
slow short wave p. It has not been observed. Another increment maximum, with an amplified wave, lies between the two resonances, and is evidently of the "hydrodynamic" origin [ 1]. It should correspond to the rare but observed fast wave variety r. A similar "hydrodynamic" maximum of the slow metastableguided wave (n') seen in fig. 1 gives a damped wave for the chosen numerical case. No wave of this type is known from experiments. So the four wave varieties (three connected with the electron gas resonances and one of hydrodynamic origin) of the six possible according to the theory (see table 1) are known experimentally.
r
E--.2
"d........
,,', ,'
-2
/
s
/
I
h
s'
p
7
- 1,
~,r%/ / j /
UQ I
i
Fig. 1. Amplification and dispersion curves of the slow waves in neon. 1/R = 1.27 mA/em, poR = 3 cm • tort, nm/ng = 3.16 × 10 -6 (degree of excitation), rm/rmw = 0.29 (ratio of the total and wall metastable life-times). A flat hydrodynamic maximum is seen between the both resonant ones: s' and pvariety of the ionization waves. the time-increment curve accompanied by very steeply falling parts on the dispersion curves. The resonant wavenumbers correspond to those observed for the s' and p varieties of the slow wave, with the correct values of Nov~k's potentials [5, 6] over the wavelength and with an instability around the resonant points. Fig. 2 shows the dispersion and increment curves computed in the same way for the fast wave. Again, the dispersion and increment curves are strongly influenced by the resonances. The first resonance corresponds to the well-known fast wave s, with the same Nov£k's potential as has the slow s' wave. The second resonance has a negative time-increment in this case and represents a fast counterpart " p " to the 1
ss s/ s
-1
7~// ss s
J
17 July 1972
[ I r
"\
p \ ,q\. VI U a , Ek
•
Fig. 2. Amplification and dispersion curves of the fast waves in neon. 1/R = 0.109 mA/cm, poR = 1 cm • torr, nm/ng = 3.16 × 10- 6 (degreeof excitation), r i = riw (ion life-time is given by the wall diffusion). The hydrodynamic maximum can be interpreted as the r-wave. The second resonance produces an unobserved ~p'~-wave. 240
Table 1 Time- and space-selected wave varieties for neon at low discharge current. Space-selection
First Hydrodynamic Second resonance maximum resonance q=l q=2 Eh =I~9V Eh= 13V
Eh= 9.5 V
Time scale selection Slow (metastable -guided) w a v e
s' variety "r"-variety
p -variety
observed not observed
observed
Fast (ion-guided) wave
s variety r -variety observed observed
"p'" variety not observed
In addition to the explanation of the constant potential law over the wavelength, also other observed properties of the ionization waves in the low current neon discharge are obtained by the theory, e.g., the high ratio of the absolute value of the group velocity to the phase velocity. The time-scale separation should not take place at high current densities [1], where the life time of metastable atoms is shorter than the life time of ions, and the space-resonance selection should disappear at high currents as the electron gas is maxwellized. This is in agreement with observations at high currents [7] and corresponding ionization wave theories [8].
References [ 1] L. PekArek, K. Magek and K. Rohlena, Czech. J. Phys. B 20 (1970) 879. [2] T. Rfi$i~ka, Thesis, Inst. of Physics of Czech. Acad. Sci. Prague (1970).
Volume 40A, number 3
PHYSICS LETTERS
[3] T. RS[i~ka and K. Rohlena, 10th ICPIG, Oxford (1971) 287. [4] D.W. Swain and S.C. Brown, Phys. Fluids 14 (1971) 1383. [5] M. Nov~ik,Czech. J. Phys. B 10 (1960) 954.
17 July 1972
[6] V. Krej~, K. Ma~ek, L. L~ska and V. Pe'fina, Beitr. Plasmaphys. 7 (1967) 413. [7] K. Wojaczek, Beitr. Plasmaphys. 6 (1966) 319 and 11 (1971) 335. [8] K. Wojaczek, Beitr. Plasmaphys. 1 (1961) 30.
241