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Reflection of cylindrical ion acoustic waves
Volume 89A, number 2
PHYSICS LETTERS
26 April 1982
REFLECTION OF CYLINDRICAL ION ACOUSTIC WAVES L. SCHOTT Department of Physics, University of Saskatchewan, Saskatoon, Canada S7N OWO Received 22 December 1981
The amplitude of a cylindrical ion acoustic wave reflected by a plane grid is measured as a function of the angle of incidence. The reflected wavefronts deviate slightly from the cylindrical shape because of the ion drift to the grid.
During the last decade the reflection of ion acoustic waves (IAW) has been studied by many researchers. A review of the work on this subject has been published by Ishihara et al. [ 1]. The experiments of lAW reflection fall roughly into two categories, (i) reflection of spontaneously excited waves from the solid walls of a discharge chamber leading to standing wave patterns (see e.g. refs. [2,4]), and (ii) reflection of waves by a grid excited by applying a voltage signal to a probe [5] or to a similar second grid which is parallel to the first ([6,7]). Attempts to reflect wave packets from solid walls have failed so far. The density perturbation in front of a grid extends normally only a few Debye lengths into the plasma, so that for lAW reflection from grids fluid dynamic damping [8] is unimportant in contrast to reflection from solid walls where the greater particle losses lead to a more extensive perturbation of the plasma. In this letter we report on measurements of the reflected amplitude o f cylindrical lAW from a plane grid. The experiments are carried out in an argon plasma at a pressure of 5 × 10 -5 Torr. The plasma density is n = 3.6 × 108 cm -3, the electron temperature T e = 1.2 eV, and the ion temperature Ti, inferred from Landau damping measurements, 0.13 eV. The plasma is produced in a magnetic multipole device described elsewhere [9]. The geometry of the experimental setup is shown in fig. 1. The reflector, a stainless steel grid (30 cm × 30 cm, 8.66 lines/cm, transparency 69.7%), can be rotated through an angle 13( - 8 5 ° ~
,
:
~
/R-
.,/"
'~
o?h X ''-'~
I refl,ctor
virtual exciter
Fig. 1. Experimental setup. mm diameter and 50 cm length mounted to the frame supporting the reflector at a distance o f 3 cm from the common axis of rotation. The lAW signals are picked up with a longitudinally movable cylindrical probe, whose current collecting tip made of nichrome wire of 0.3 mm diameter and 11 mm length is bent up (out of the plane of the paper in fig. 1) to be parallel with the plane of the exciter. The probe is kept at a potential slightly positive with respect to the plasma and samples the electron component of the lAW. The waves are launched by applying negative square pulses of 2.5 /as width and 1 V amplitude through a 1 #F capacitor to the floating exciter. With this mode of excitation the region of overlap of the reflected and the incident wave signals close to reflector is smaller than with multiple cycle toneburst excitation. From the time of flight diagram for near normal incidence shown in fig. 2 (/3 ~ 2.5 ° to allow the probe to pass the exciter) it is seen that the lAW is reflected at a distance r of approximately 2 mm from the reflector. Measurements o f n close to the reflector show that the point of the maximum curvature Id2n/
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
Volume 89A, number 2 40
PHYSICS LETTERS
--
o
o
o
o o
30--
o o o
t(~s)
o
o
--
°
o o
o
o
o
°
~'-reflected
wave
o
2(: Q
o
o o o o
o
x
x
o x
IG
x
x
x
x
x
x x
~lncident
Xxx I I
I
I 2
x
x
wave
x
x x xI x
I
x
x x
x x x
r (cm)
XlX 3
I
I 4
I
Fig. 2. Time of flight t versus position r of incident and reflected wave near normal incidence 03 = 2.5°). dr 21 occurs at r ~ 7 mm. Consequently, the plane of reflection is inside the quasineutral layer. From the slopes of the curves in fig. 2 we rind the phase velocity of the incident wave to be 2.2 × 105 cm/s and that of the reflected wave 1.8 × 105 cm/s, from which we ob-
26 April 1982
tain an ion drift velocity of 2 × 104 cm/s orientated toward the reflector. Extrapolation of the curve of the reflected wave to its intercept with the negative r axis gives the location of the "virtual image" of the exciter (see also fig. 1) at r = - 2 . 6 cm, i.e. the exciter and its virtual image are symmetric with respect to the plane of reflection (at r ~ 2 mm) as ~xpected. Fig. 3 shows measured probe signals for the radius of the reflected wavefront R --- 4.1 cm and/3 = 2.5 °, 30 ° and 60 °. The directly coupled signal (D) arrives at the probe immediately after the applied signal (A), while signal (I) is the lAW which travels directly from the exciter to the probe. Note, the phase of the reflected wave (R) is clearly inverted in contrast to the inphase reflection of plane waves observed by other workers [6,7]. The reflected amplitude A r as a function of the angle of incidence 0, which varies with the position on the cylindrical wavefronts and is a function of the angle/3, is plotted in fig. 4. A r is obtained by dividing the raw data for the reflected signals by the amplitude of the incident signal measured at a distance of 2 cm from the exciter. This takes account of the angular
I
2.5*
A 30*
60*
5 jJS
Fig. 3. Experimental traces of direct coupled signal (D), IAW signal travelling directly from exciter to the probe (I), and reflected lAW signal (R) for/3 = 2.5° , 30° , and 60°. 83
V o l u m e 89A, n u m b e r 2
PHYSICS LETTERS
26 April 1982
1.5-1,4 -- X
X
X
1.3--
Ar
1.2 --
X X
(orb. units) 1.0--
°X
X
(o)
0.8-
X
o
X o Xo
o.~ ~(o) 0.4 [--
0
0
(o)
0.2
o
I
I
I0
I
I
20
I
I
30
(degrees)
I
i
I
40
I
50
Fig. 4. Reflected amplitude, A r , as a f u n c t i o n o f angle o f incidence, 0.
anisotropy of the incident amplitude caused by ion drift and by the presence of the probe holder (when the probe is between exciter and reflector). The circles correspond to R = 7.6 cm, i.e. to a reflected wavefront enclosing the exciter. The exciter obstructs the reflected wave in an angular region corresponding to the circles in brackets. The crosses represent data measured on a wavefront of radius R = 4.1 cm. Clearly, the reflectance decreases with increasing 0. The absolute value of the reflectance for normal incidence (/3 = 0 °) was determined by extrapolating the measured data to the sheath edge and found to be 14%. As a consequence of the drift the angle of reflection is slightly smaller than the angle of incidence. For the same reason the wave fronts deviate from the exact cylindrical shape. If the reflected and incident wavefronts were exactly cylindrical, the phase of a signal emitted by the exciter at a fixed time should arrive at a surface of radius R centered at the virtual image of the exciter at the same time t independent of/3. In the presence of an ion drift velocity u toward the reflector the incident wave travels faster (downstream) than the reflected wave (upstream). The downstream path travelled by the incident wave increases while the upstream path of the reflected wave decreases when 1/31becomes larger. Consequently, the phase of the reflected wave on a cylindrical surface appears advanced for/3 :/: 0 compared to that at/3 = 0. A 84
simple analysis shows that for u ~ Vph(Vph = phase velocity of the IAW for u = 0) the corresponding difference in time of arrival At = t(/~) -- t(0 °) on a cylindrical surface centred at the virtual image is
At.= (u/o2h)(r cos/3 -
r0),
(1)
where r 0 = r(/3 = 0). We found qualitative agreement between measured and predicted values at At (compare table 1). In conclusion, we have presented the first experimental evidence on the dependence of IAW reflection on the angle of incidence and have shown that the wavefronts deviate from the exact cylindrical shape as a consequence of ion drift. Table 1 Theoretical and experimental values of difference in time o f arrival At o f t h e reflected wave o n a cylindrical surface of radius R = 7.6 cm for various angles I~ I. The experimental data are averages o f m e a s u r e m e n t s o n b o t h sides o f the normal of t h e reflector. *At too small to be measurable. I#l (degrees)
Attheor.(t~s)
Atexp.(US)
0 10 30 50 60 70
0 -0.03 -0.23 -0.66 0.97 -1.37
* * * -0.7±0.2 -1.3±0.2 -1.9±0.2
Volume 89A, number 2
PHYSICS LETTERS
This work has been supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
R eferen ces [1] O. Ishihara, I. Alexeff, H.J. Doucet and W.D. Jones, Phys. Fluids 21 (1978) 2211. [2] I. Alexeff and R.V. Neidigh, Phys. Rev. Lett. 7 (1961) 223.
26 April 1982
[3] F.W. Crawford, Phys. Rev. Lett. 6 (1961) 663. [4] I. Alexeff, W. Halchin, W.D. Jones and J.F. Potts, Oak Ridge National Laboratory ORNL-4010 (1966). [5] S. Aksornkitti, H.C.S. Hsuan, K.E. Lonngren and I. Alexeff, Phys. Fluids 11 (1968) 1838. [6] T. Nakamura, T. Nomura and T. Itoh, Proc. Int. Conf. Plasma Phys. (Nagoya, Japan) Vol. 1 (1980) p. 119. [7] E.M. Barkhudarov, A. Sh. Dzagnidze, V.F. Lygin and D.D. Tskhakaya, Fiz. Plazmy 6 (1980) 1041 [Soy. J. Plasma Phys. 6 (1980) 571 ]. [8] L. Schott, Phys. Fluids 18 (1975) 645. [9] L. Schott, Rev. Sci. Instrum. (1978) 491.
85
PHYSICS LETTERS
26 April 1982
REFLECTION OF CYLINDRICAL ION ACOUSTIC WAVES L. SCHOTT Department of Physics, University of Saskatchewan, Saskatoon, Canada S7N OWO Received 22 December 1981
The amplitude of a cylindrical ion acoustic wave reflected by a plane grid is measured as a function of the angle of incidence. The reflected wavefronts deviate slightly from the cylindrical shape because of the ion drift to the grid.
During the last decade the reflection of ion acoustic waves (IAW) has been studied by many researchers. A review of the work on this subject has been published by Ishihara et al. [ 1]. The experiments of lAW reflection fall roughly into two categories, (i) reflection of spontaneously excited waves from the solid walls of a discharge chamber leading to standing wave patterns (see e.g. refs. [2,4]), and (ii) reflection of waves by a grid excited by applying a voltage signal to a probe [5] or to a similar second grid which is parallel to the first ([6,7]). Attempts to reflect wave packets from solid walls have failed so far. The density perturbation in front of a grid extends normally only a few Debye lengths into the plasma, so that for lAW reflection from grids fluid dynamic damping [8] is unimportant in contrast to reflection from solid walls where the greater particle losses lead to a more extensive perturbation of the plasma. In this letter we report on measurements of the reflected amplitude o f cylindrical lAW from a plane grid. The experiments are carried out in an argon plasma at a pressure of 5 × 10 -5 Torr. The plasma density is n = 3.6 × 108 cm -3, the electron temperature T e = 1.2 eV, and the ion temperature Ti, inferred from Landau damping measurements, 0.13 eV. The plasma is produced in a magnetic multipole device described elsewhere [9]. The geometry of the experimental setup is shown in fig. 1. The reflector, a stainless steel grid (30 cm × 30 cm, 8.66 lines/cm, transparency 69.7%), can be rotated through an angle 13( - 8 5 ° ~
,
:
~
/R-
.,/"
'~
o?h X ''-'~
I refl,ctor
virtual exciter
Fig. 1. Experimental setup. mm diameter and 50 cm length mounted to the frame supporting the reflector at a distance o f 3 cm from the common axis of rotation. The lAW signals are picked up with a longitudinally movable cylindrical probe, whose current collecting tip made of nichrome wire of 0.3 mm diameter and 11 mm length is bent up (out of the plane of the paper in fig. 1) to be parallel with the plane of the exciter. The probe is kept at a potential slightly positive with respect to the plasma and samples the electron component of the lAW. The waves are launched by applying negative square pulses of 2.5 /as width and 1 V amplitude through a 1 #F capacitor to the floating exciter. With this mode of excitation the region of overlap of the reflected and the incident wave signals close to reflector is smaller than with multiple cycle toneburst excitation. From the time of flight diagram for near normal incidence shown in fig. 2 (/3 ~ 2.5 ° to allow the probe to pass the exciter) it is seen that the lAW is reflected at a distance r of approximately 2 mm from the reflector. Measurements o f n close to the reflector show that the point of the maximum curvature Id2n/
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
Volume 89A, number 2 40
PHYSICS LETTERS
--
o
o
o
o o
30--
o o o
t(~s)
o
o
--
°
o o
o
o
o
°
~'-reflected
wave
o
2(: Q
o
o o o o
o
x
x
o x
IG
x
x
x
x
x
x x
~lncident
Xxx I I
I
I 2
x
x
wave
x
x x xI x
I
x
x x
x x x
r (cm)
XlX 3
I
I 4
I
Fig. 2. Time of flight t versus position r of incident and reflected wave near normal incidence 03 = 2.5°). dr 21 occurs at r ~ 7 mm. Consequently, the plane of reflection is inside the quasineutral layer. From the slopes of the curves in fig. 2 we rind the phase velocity of the incident wave to be 2.2 × 105 cm/s and that of the reflected wave 1.8 × 105 cm/s, from which we ob-
26 April 1982
tain an ion drift velocity of 2 × 104 cm/s orientated toward the reflector. Extrapolation of the curve of the reflected wave to its intercept with the negative r axis gives the location of the "virtual image" of the exciter (see also fig. 1) at r = - 2 . 6 cm, i.e. the exciter and its virtual image are symmetric with respect to the plane of reflection (at r ~ 2 mm) as ~xpected. Fig. 3 shows measured probe signals for the radius of the reflected wavefront R --- 4.1 cm and/3 = 2.5 °, 30 ° and 60 °. The directly coupled signal (D) arrives at the probe immediately after the applied signal (A), while signal (I) is the lAW which travels directly from the exciter to the probe. Note, the phase of the reflected wave (R) is clearly inverted in contrast to the inphase reflection of plane waves observed by other workers [6,7]. The reflected amplitude A r as a function of the angle of incidence 0, which varies with the position on the cylindrical wavefronts and is a function of the angle/3, is plotted in fig. 4. A r is obtained by dividing the raw data for the reflected signals by the amplitude of the incident signal measured at a distance of 2 cm from the exciter. This takes account of the angular
I
2.5*
A 30*
60*
5 jJS
Fig. 3. Experimental traces of direct coupled signal (D), IAW signal travelling directly from exciter to the probe (I), and reflected lAW signal (R) for/3 = 2.5° , 30° , and 60°. 83
V o l u m e 89A, n u m b e r 2
PHYSICS LETTERS
26 April 1982
1.5-1,4 -- X
X
X
1.3--
Ar
1.2 --
X X
(orb. units) 1.0--
°X
X
(o)
0.8-
X
o
X o Xo
o.~ ~(o) 0.4 [--
0
0
(o)
0.2
o
I
I
I0
I
I
20
I
I
30
(degrees)
I
i
I
40
I
50
Fig. 4. Reflected amplitude, A r , as a f u n c t i o n o f angle o f incidence, 0.
anisotropy of the incident amplitude caused by ion drift and by the presence of the probe holder (when the probe is between exciter and reflector). The circles correspond to R = 7.6 cm, i.e. to a reflected wavefront enclosing the exciter. The exciter obstructs the reflected wave in an angular region corresponding to the circles in brackets. The crosses represent data measured on a wavefront of radius R = 4.1 cm. Clearly, the reflectance decreases with increasing 0. The absolute value of the reflectance for normal incidence (/3 = 0 °) was determined by extrapolating the measured data to the sheath edge and found to be 14%. As a consequence of the drift the angle of reflection is slightly smaller than the angle of incidence. For the same reason the wave fronts deviate from the exact cylindrical shape. If the reflected and incident wavefronts were exactly cylindrical, the phase of a signal emitted by the exciter at a fixed time should arrive at a surface of radius R centered at the virtual image of the exciter at the same time t independent of/3. In the presence of an ion drift velocity u toward the reflector the incident wave travels faster (downstream) than the reflected wave (upstream). The downstream path travelled by the incident wave increases while the upstream path of the reflected wave decreases when 1/31becomes larger. Consequently, the phase of the reflected wave on a cylindrical surface appears advanced for/3 :/: 0 compared to that at/3 = 0. A 84
simple analysis shows that for u ~ Vph(Vph = phase velocity of the IAW for u = 0) the corresponding difference in time of arrival At = t(/~) -- t(0 °) on a cylindrical surface centred at the virtual image is
At.= (u/o2h)(r cos/3 -
r0),
(1)
where r 0 = r(/3 = 0). We found qualitative agreement between measured and predicted values at At (compare table 1). In conclusion, we have presented the first experimental evidence on the dependence of IAW reflection on the angle of incidence and have shown that the wavefronts deviate from the exact cylindrical shape as a consequence of ion drift. Table 1 Theoretical and experimental values of difference in time o f arrival At o f t h e reflected wave o n a cylindrical surface of radius R = 7.6 cm for various angles I~ I. The experimental data are averages o f m e a s u r e m e n t s o n b o t h sides o f the normal of t h e reflector. *At too small to be measurable. I#l (degrees)
Attheor.(t~s)
Atexp.(US)
0 10 30 50 60 70
0 -0.03 -0.23 -0.66 0.97 -1.37
* * * -0.7±0.2 -1.3±0.2 -1.9±0.2
Volume 89A, number 2
PHYSICS LETTERS
This work has been supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
R eferen ces [1] O. Ishihara, I. Alexeff, H.J. Doucet and W.D. Jones, Phys. Fluids 21 (1978) 2211. [2] I. Alexeff and R.V. Neidigh, Phys. Rev. Lett. 7 (1961) 223.
26 April 1982
[3] F.W. Crawford, Phys. Rev. Lett. 6 (1961) 663. [4] I. Alexeff, W. Halchin, W.D. Jones and J.F. Potts, Oak Ridge National Laboratory ORNL-4010 (1966). [5] S. Aksornkitti, H.C.S. Hsuan, K.E. Lonngren and I. Alexeff, Phys. Fluids 11 (1968) 1838. [6] T. Nakamura, T. Nomura and T. Itoh, Proc. Int. Conf. Plasma Phys. (Nagoya, Japan) Vol. 1 (1980) p. 119. [7] E.M. Barkhudarov, A. Sh. Dzagnidze, V.F. Lygin and D.D. Tskhakaya, Fiz. Plazmy 6 (1980) 1041 [Soy. J. Plasma Phys. 6 (1980) 571 ]. [8] L. Schott, Phys. Fluids 18 (1975) 645. [9] L. Schott, Rev. Sci. Instrum. (1978) 491.
85