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Observations of resonance cones near the lower hybrid frequency
Volume 70A, number 3
PHYSICS LETTERS
5 March 1979
OBSERVATIONS OF RESONANCE CONES NEAR ThE LOWER HYBRID FREQUENCY K. LUCKS, M. KRAMER and H. SCHLUTER Institut für Experimentaiphysik II, Ruhr-Universitat, Bochum, Germany
Received 17 May 1978 Revised manuscript received 16 October 1978
The existence of resonance cones in the lower hybrid domain has been verified experimentally by an rf double probe technique. The observations are in good agreement with cold plasma theory. Measurements at small cone angles demonstrate the influence of ion terms, limiting the existence of cones to frequencies above the lower hybrid frequency.
In 1971 Fisher and Gould [1] and subsequently q exp(+ iwt) many other authors [2,3] have presented experimental evidence for the existence of resonance cones. Their experiments were concerned with a domain of param2 ~‘ eters =where the+ion motion can be neglected (w inw~(w? w~j)/(w~ + ~&~~e)). This study
cludes the lower hybrid regime proper (w ~ where the influence of ion motion is discernible; when ion terms are taken into account, the existence of resonance cones is excluded from the regime w < w~. The experimental arrangement used is similar to
that of Fisher and Gould: one coaxial probe serves as transmitting antenna, the other rotating relative to the first, as receiving antenna. The method employed lends itself to local measurements which are of particular interest for diagnostic purposes. Both amplitude and phase informations are used. In a simple model the transmitting probe can be considered as oscillating point charge. In the quasi. static approximation one must solve Poisson’s equation V~ K V~=—qexp(+i~t)~(r),
where the charge distribution of the point charge is given by a s-function and the dielectric tensor K in the anisotropic case (B 0 = Boe~* 0) has the nonzero cornponents ~ = ~ = K1, ~ = —K~~ = KH and K55 = Ku. For a cold plasma the solution of Poisson’s equation can be performed directly, yielding
1’2
Ø(p,z)
4~0KIC11’ [p2/K1+ z2/K11] 1 2 in cylindrical coordinates.case), The potential becomescone singular (in the collisionless if the resonance condition
tan2O
=
K /K
=
C
—
1
I
II
+ w2./(c~— w2) + w2 /(w2 — w2)
2 2 2 2 (2) t0pi~ — Wpe/W is fulfilled. The cone angle °cdefines the direction =
—
l~1
1
—
along which the potential goes to infinity, which is the case if K 1 and K11 have a different sign. The experiment has been performed in two alternative, stationary plasmas, a hot cathode discharge and a microwave discharge produced by a Lisitano coil [4] (f 2.45 GHz). The plasma column (diameter 3.6 cm) was confined by a magnetic field of 50—450 r. The hydrogen plasmas used typically had densities of and 3 (measured by means of a cavity) electron 109_lOlltemperatures cm— of a few eV (determined with
electric probes) at gas pressures of 10_3_102 Torr. The two coaxial probes revolve around the common axis perpendicular to the axis of the plasma column. The tips of the probes (i.e. the linear conductor is exposed a few mm) served as transmitting and receiving antenna, respectively. The transmission probe was fed by an rf signal (f 1—110 MHz) and the received signal was compared with a reference signal in amplitude 205
Volume 70A, number 3
PHYSICS LETTERS
5 March 1979
and phase by a network analyzer. A typical measurement of amplitude and phase versus transmission angle is shown in fig. 1. The curves
-
AMPLITUDE
show the resonance behaviour theoretically expected. In fig. 2a amplitude curves for different magnetic fields are plotted. As expected the cone angle O~decreases with increasing field strength. The comparison between observed and computed cone angle is presented in fig. 2b.
1
In fig. 3a comparison between experimental data and theoretical predictions — calculated from eq. (2) — is shown for a wide range of parameters. The influence of ion terms is only important for small angles; their influence is best demonstrated for the asymptotic case —~ 00. The measured values related to this case are
+90
obtained in the following manner using a variable wave frequency: As long as the peak of the potential
PHASE
-90
-60
—30
+30
0 ~9’/dog
+60
measured increases with decreasing frequency, the po0max = 0~> 0; as soon as tential has its maximum at the observed behaviour is reversed the case = 00 is surpassed, the peak of the potential staying fixed at °max= 0. The maximum of the curve thus obtained
Fig. 1. Amplitude and phase of transmitted signal versus transmission angle;f 110 MHz, B 9 cm3, p = 1.6 X iO~Tory. The theoretical curves are 0 = 174 (dashed) ~ n~= 2.8 X i0 fitted to the experimental traces by using the effective collision frequency vce 2.4 X w in formula ~
(~)
C
A
~12~
~—i6
~\
2C
///~+
N N
~
N~28
~---
AMPLITUDEJC2db I
-90
I
I
-60
-32
I
0
32
~ I
+30
÷60
+90
o
10
20
30
-~/deg
Fig. 2. (a) Transmission curves for different magnetic fields; (b) Resonance cone angle versus magnetic field, f = 110 MHz, ~e = 4.7 X i0~cm3, p 6.4 X 10~Torr. 206
Volume 70A, number 3
PHYSICS LETTERS
~ff
_
25°
10’
—i—---—
5 March 1979
L)~~LI
~1
10
~ ~
•
0
2
-
/
.
0
-
I
I
Fig. 3. Diagram of resonance cone angle (hydrogen); comparison of measured values (points) and theoretical predictions (solid curves); Wpe,i: electron, ion plasma frequency; We ~: electron, ion cyclotron frequency; ~0: geometric mean of we and wj.
signals the frequency attached to = 00 i.e. to the condition w = w~.Fig. 4 demonstrates this behaviour of the potential by depicting measured values near 0c = 00 normalized to the case U = 900 as a function of w. The theoretical curve is fitted to the experimental data by evaluating eq. (1) with the proper collision frequency. The inclusion of ion terms is vital for analysing the observed behaviour at small cone —
—
0
___________~___~~_
8
12
16
20
Fig. 4. Potential amplitude, normalized, versus frequency; 9 cm3, p = 6.4 x l0~ Torr; the 0 = 116 r,curve ~e is1.6fitted X i0to the measured points by using theoretical the effective collision frequencies vce = 2.0 X i08 s~1~ = 4.7 X 10~s_i.
B
fig. 3. The comparison of density values thus obtained with those obtained from cavity and Langmuir probe measurements yield good agreement, confirming the usefulness of cone measurements as diagnostic tool pointed out by Fisher and Gould. The range of appli. cability is not only limited by experimental accuracy, but also due to ion motion as illustrated above: the applied frequencies have to surpass wth ( w~).
angles.
The measurements can be well explained by cold
This work has been supported by the Sonder-
plasma theory. In some cases, however, fine structures
forschungsbereich Plasmaphysik Bochum/JUlich and
of the cone patterns were observed. Fisher and Gould [1] explained this behaviour by interference phenomena due to thermal effects. Burrel [5] presented detailed theoretical and experimental work concerning these effects for the case of negligible ion motion. Using his formula for the angular spacing of interference maxima, the pattern observed in this work can be reconciled reasonably well with theory. For small cone angles, where ion terms might complicate matters, interference maxima are observed to disappear. The cone measurements may be applied to determine the electron density, using the values given in
by the Deutsche Forschungsgemeinschaft (Schwerpunktprogram “Fusionsorientierte Plasmaphysik”). We are indebted to Mr. Schmidt for his assistance. References [1J R.K. Fisher and R.W. Gould, Phys. Fluids 14(1971)857. [2] R.J. Briggs and R.R. Parker, Phys. Rev. Lett. 29 (1972) [3] P. Bellan and M. Porkolab, Phys. Rev. Lett. 34 (1975) 124. [4] G.G. Lisitano et al., Appl. Phys. Lett. 16 (1970) 122. [51K.H. Burrel, Phys. Fluids 18 (1975) 1716.
207
PHYSICS LETTERS
5 March 1979
OBSERVATIONS OF RESONANCE CONES NEAR ThE LOWER HYBRID FREQUENCY K. LUCKS, M. KRAMER and H. SCHLUTER Institut für Experimentaiphysik II, Ruhr-Universitat, Bochum, Germany
Received 17 May 1978 Revised manuscript received 16 October 1978
The existence of resonance cones in the lower hybrid domain has been verified experimentally by an rf double probe technique. The observations are in good agreement with cold plasma theory. Measurements at small cone angles demonstrate the influence of ion terms, limiting the existence of cones to frequencies above the lower hybrid frequency.
In 1971 Fisher and Gould [1] and subsequently q exp(+ iwt) many other authors [2,3] have presented experimental evidence for the existence of resonance cones. Their experiments were concerned with a domain of param2 ~‘ eters =where the+ion motion can be neglected (w inw~(w? w~j)/(w~ + ~&~~e)). This study
cludes the lower hybrid regime proper (w ~ where the influence of ion motion is discernible; when ion terms are taken into account, the existence of resonance cones is excluded from the regime w < w~. The experimental arrangement used is similar to
that of Fisher and Gould: one coaxial probe serves as transmitting antenna, the other rotating relative to the first, as receiving antenna. The method employed lends itself to local measurements which are of particular interest for diagnostic purposes. Both amplitude and phase informations are used. In a simple model the transmitting probe can be considered as oscillating point charge. In the quasi. static approximation one must solve Poisson’s equation V~ K V~=—qexp(+i~t)~(r),
where the charge distribution of the point charge is given by a s-function and the dielectric tensor K in the anisotropic case (B 0 = Boe~* 0) has the nonzero cornponents ~ = ~ = K1, ~ = —K~~ = KH and K55 = Ku. For a cold plasma the solution of Poisson’s equation can be performed directly, yielding
1’2
Ø(p,z)
4~0KIC11’ [p2/K1+ z2/K11] 1 2 in cylindrical coordinates.case), The potential becomescone singular (in the collisionless if the resonance condition
tan2O
=
K /K
=
C
—
1
I
II
+ w2./(c~— w2) + w2 /(w2 — w2)
2 2 2 2 (2) t0pi~ — Wpe/W is fulfilled. The cone angle °cdefines the direction =
—
l~1
1
—
along which the potential goes to infinity, which is the case if K 1 and K11 have a different sign. The experiment has been performed in two alternative, stationary plasmas, a hot cathode discharge and a microwave discharge produced by a Lisitano coil [4] (f 2.45 GHz). The plasma column (diameter 3.6 cm) was confined by a magnetic field of 50—450 r. The hydrogen plasmas used typically had densities of and 3 (measured by means of a cavity) electron 109_lOlltemperatures cm— of a few eV (determined with
electric probes) at gas pressures of 10_3_102 Torr. The two coaxial probes revolve around the common axis perpendicular to the axis of the plasma column. The tips of the probes (i.e. the linear conductor is exposed a few mm) served as transmitting and receiving antenna, respectively. The transmission probe was fed by an rf signal (f 1—110 MHz) and the received signal was compared with a reference signal in amplitude 205
Volume 70A, number 3
PHYSICS LETTERS
5 March 1979
and phase by a network analyzer. A typical measurement of amplitude and phase versus transmission angle is shown in fig. 1. The curves
-
AMPLITUDE
show the resonance behaviour theoretically expected. In fig. 2a amplitude curves for different magnetic fields are plotted. As expected the cone angle O~decreases with increasing field strength. The comparison between observed and computed cone angle is presented in fig. 2b.
1
In fig. 3a comparison between experimental data and theoretical predictions — calculated from eq. (2) — is shown for a wide range of parameters. The influence of ion terms is only important for small angles; their influence is best demonstrated for the asymptotic case —~ 00. The measured values related to this case are
+90
obtained in the following manner using a variable wave frequency: As long as the peak of the potential
PHASE
-90
-60
—30
+30
0 ~9’/dog
+60
measured increases with decreasing frequency, the po0max = 0~> 0; as soon as tential has its maximum at the observed behaviour is reversed the case = 00 is surpassed, the peak of the potential staying fixed at °max= 0. The maximum of the curve thus obtained
Fig. 1. Amplitude and phase of transmitted signal versus transmission angle;f 110 MHz, B 9 cm3, p = 1.6 X iO~Tory. The theoretical curves are 0 = 174 (dashed) ~ n~= 2.8 X i0 fitted to the experimental traces by using the effective collision frequency vce 2.4 X w in formula ~
(~)
C
A
~12~
~—i6
~\
2C
///~+
N N
~
N~28
~---
AMPLITUDEJC2db I
-90
I
I
-60
-32
I
0
32
~ I
+30
÷60
+90
o
10
20
30
-~/deg
Fig. 2. (a) Transmission curves for different magnetic fields; (b) Resonance cone angle versus magnetic field, f = 110 MHz, ~e = 4.7 X i0~cm3, p 6.4 X 10~Torr. 206
Volume 70A, number 3
PHYSICS LETTERS
~ff
_
25°
10’
—i—---—
5 March 1979
L)~~LI
~1
10
~ ~
•
0
2
-
/
.
0
-
I
I
Fig. 3. Diagram of resonance cone angle (hydrogen); comparison of measured values (points) and theoretical predictions (solid curves); Wpe,i: electron, ion plasma frequency; We ~: electron, ion cyclotron frequency; ~0: geometric mean of we and wj.
signals the frequency attached to = 00 i.e. to the condition w = w~.Fig. 4 demonstrates this behaviour of the potential by depicting measured values near 0c = 00 normalized to the case U = 900 as a function of w. The theoretical curve is fitted to the experimental data by evaluating eq. (1) with the proper collision frequency. The inclusion of ion terms is vital for analysing the observed behaviour at small cone —
—
0
___________~___~~_
8
12
16
20
Fig. 4. Potential amplitude, normalized, versus frequency; 9 cm3, p = 6.4 x l0~ Torr; the 0 = 116 r,curve ~e is1.6fitted X i0to the measured points by using theoretical the effective collision frequencies vce = 2.0 X i08 s~1~ = 4.7 X 10~s_i.
B
fig. 3. The comparison of density values thus obtained with those obtained from cavity and Langmuir probe measurements yield good agreement, confirming the usefulness of cone measurements as diagnostic tool pointed out by Fisher and Gould. The range of appli. cability is not only limited by experimental accuracy, but also due to ion motion as illustrated above: the applied frequencies have to surpass wth ( w~).
angles.
The measurements can be well explained by cold
This work has been supported by the Sonder-
plasma theory. In some cases, however, fine structures
forschungsbereich Plasmaphysik Bochum/JUlich and
of the cone patterns were observed. Fisher and Gould [1] explained this behaviour by interference phenomena due to thermal effects. Burrel [5] presented detailed theoretical and experimental work concerning these effects for the case of negligible ion motion. Using his formula for the angular spacing of interference maxima, the pattern observed in this work can be reconciled reasonably well with theory. For small cone angles, where ion terms might complicate matters, interference maxima are observed to disappear. The cone measurements may be applied to determine the electron density, using the values given in
by the Deutsche Forschungsgemeinschaft (Schwerpunktprogram “Fusionsorientierte Plasmaphysik”). We are indebted to Mr. Schmidt for his assistance. References [1J R.K. Fisher and R.W. Gould, Phys. Fluids 14(1971)857. [2] R.J. Briggs and R.R. Parker, Phys. Rev. Lett. 29 (1972) [3] P. Bellan and M. Porkolab, Phys. Rev. Lett. 34 (1975) 124. [4] G.G. Lisitano et al., Appl. Phys. Lett. 16 (1970) 122. [51K.H. Burrel, Phys. Fluids 18 (1975) 1716.
207