1. Interpreting the behavior of a quarter-wave transmission line resonator in a
magnetized plasma
G. S. Gogna, S. K. Karkari, and M. M. Turner
Citation: Physics of Plasmas (1994-present) 21, 123510 (2014); doi: 10.1063/1.4904037
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2. Interpreting the behavior of a quarter-wave transmission line resonator
in a magnetized plasma
G. S. Gogna,1,a)
S. K. Karkari,2,b)
and M. M. Turner1
1
NCPST, School of Physical Sciences, Dublin City University, Dublin 9, Ireland
2
Institute for Plasma Research, Bhat, Gandhinagar, Gujarat, India
(Received 1 July 2014; accepted 29 November 2014; published online 16 December 2014)
The quarter wave resonator immersed in a strongly magnetized plasma displays two possible
resonances occurring either below or above its resonance frequency in vacuum, fo. This fact was
demonstrated in our recent articles [G. S. Gogna and S. K. Karkari, Appl. Phys. Lett. 96, 151503
(2010); S. K. Karkari, G. S. Gogna, D. Boilson, M. M. Turner, and A. Simonin, Contrib. Plasma
Phys. 50(9), 903 (2010)], where the experiments were carried out over a limited range of magnetic
fields at a constant electron density, ne. In this paper, we present the observation of dual resonances
occurring over the frequency scan and find that ne calculated by considering the lower resonance
frequency is 25%–30% smaller than that calculated using the upper resonance frequency with
respect to fo. At a given magnetic field strength, the resonances tend to shift away from fo as the
background density is increased. The lower resonance tends to saturate when its value approaches
electron cyclotron frequency, fce. Interpretation of these resonance conditions are revisited by
examining the behavior of the resonance frequency response as a function of ne. A qualitative
discussion is presented which highlights the practical application of the hairpin resonator for
interpreting ne in a strongly magnetized plasma. VC 2014 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4904037]
I. INTRODUCTION
Microwave based diagnostics1–4
are at the heart of
many professional hi-tech systems across the world where
they are extensively used in wide range of research applica-
tions including radio-communication, medical diagnostic,
surface engineering, and evaluation of material properties.
In plasma physics, applications of microwave diagnostics
are well known from the study of ionospheric plasma. The
plasma behaves as a dielectric medium since it has a natural
tendency to react as an absorber, transmitter, or reflector of
electromagnetic (EM) waves. The dielectric property of
plasma is a function of electron density, ne.2,5–8
Therefore
when electromagnetic waves travel through the plasma, they
are either reflected or absorbed after encountering respective
cut-offs and resonance inside the plasma.32–35
The microwave
techniques can therefore enable precise measurement of elec-
tron density. A commonly used plasma diagnostic for deter-
mining ne in laboratory plasmas is based on microwave
interferometer. However, the technique is limited to the mea-
surement of line-averaged density in the plasma.
The concept of measuring local electron density in
weakly magnetized plasma using a quarter wave resonator
was introduced by Stenzel in 1976.9
In recent years, the tech-
nique has attracted noticeable interest for diagnosing indus-
trial plasmas.10–19
These resonance probes, also popularly
known as hairpin probes due to their characteristic shape,
have been successfully applied in many industrial plasma
systems including deposition plasmas and dual radio-
frequency operated confined Capacitively Coupled Plasma
(CCP) discharge.10,12
The basic principle behind hairpin
probe is based on creating a standing wave that corresponds
to the plasma permittivity in the near field region around the
resonator. Beside the hairpin probes,9,20
several variations of
microwave resonator probes have been developed. The im-
portant ones are the plasma oscillation,21–24
plasma absorp-
tion,25
plasma transmission,26
plasma cut-off,27
multipole
resonance,28
LC resonance,29
and curling probes.30
These
probes differ in terms of the measurement technique but their
operating principles are based on dielectric response of the
plasma towards the incident EM wave. These probing techni-
ques have an advantage over the microwave interferome-
ter,2–4
because they are capable of providing a local
measurement of electron density.
Despite their popularity in industrial plasma applica-
tions, resonant probe utility has remained essentially under-
explored in magnetized plasma system. Recently, the hairpin
probe was applied to measure ne in a strongly magnetized
plasma, existing near the extraction grid of the negative ion
source.14
Consecutively, the basic resonance characteristics
were investigated with a laboratory plasma set-up.13
In con-
trast to weakly magnetized plasma, the results indicate the
possibility of resonance frequency occurring below the
probe’s resonance frequency in vacuum. This behavior is
similar to that observed for a dielectric medium introduced
between the resonator pins. However, ne obtained from the
lower branch of resonance frequency was inconclusive due
to a restricted operating parameter range in those experimen-
tal conditions.13
Another key aspect concerning the application of hairpin
probes in magnetized plasma is the presence of non-uniform
a)
Author to whom correspondence should be addressed. Electronic mail:
gurusharansingh.gogna@gmail.com
b)
skarkari@ipr.res.in
1070-664X/2014/21(12)/123510/8/$30.00 VC 2014 AIP Publishing LLC21, 123510-1
PHYSICS OF PLASMAS 21, 123510 (2014)
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3. electric field distribution around the hairpin, particularly
when the magnetic field introduces anisotropy in the plasma
dielectric. Recent experiments were performed to understand
the validity of cold plasma approximation resulting in different
permittivity components in magnetized plasma.1,36–38
As
reported in our earlier paper,14
ne obtained by considering the
perpendicular component of plasma dielectric using the upper
resonance frequency fairly matched with the positive ion den-
sity measured by the conventional Langmuir probe. However,
the performance of dual resonances was not discussed in detail.
This paper presents the systematic study of resonance
characteristics of hairpin probe in the presence of ambient
magnetic field over a broad range of electron cyclotron fre-
quency, fce (0–3.0 GHz) and electron plasma frequency, fpe
(0–1.0 GHz). In contrast to previous results,13
we observe
the existence of dual resonance condition satisfying the same
ne. The behavior of these resonances is discussed in terms of
the analytical model for the probe’s resonances, by consider-
ing the perpendicular component of plasma permittivity for
deriving ne.
This paper is organized as follows: In Sec. II, we present
a brief description about the anisotropic plasma dielectric
model applicable to the hairpin. The experimental set-up for
investigating the performance of the hairpin in magnetized
plasma is presented in Sec. III. This is followed by results
and discussion in Secs. IV and V, respectively. The summary
and conclusion are presented in Sec. VI.
II. HAIRPIN PROBE THEORY FOR APPLICATION IN
MAGNETIZED PLASMA
The hairpin probe is a quarter-wavelength resonator
which is comprised of a two-wire parallel transmission line
that includes one short-circuited end while the other end is
kept open. The length of each arm is equivalent to a quarter
wavelength while the effective path-length is half a wave-
length from tip to tip. A given dielectric medium will support
a quarter-wave that will correspond to a unique value of driv-
ing frequency. During this condition of resonance, an intense
time varying electric field is sustained in the region around
the open ends of the hairpin. As the plasma dielectric
changes, the resonance frequency of the hairpin also changes
in response to the quarter-wave which is accommodated
along the length of the hairpin.
The resonance condition of the quarter wavelength
probe is given by
fr ¼
c
2 2l þ wð Þ
ffiffiffiffiffi
jp
p ¼
fo
ffiffiffiffiffi
jp
p ; (1)
where l is the length of the hairpin, w is the separation
between the wires, and jp is the relative dielectric constant
of the plasma.
The dielectric constant of plasma in the presence of a
magnetic field is a complex dielectric tensor given by31
p ¼ ojp ¼ o
j? Àjj 0
jj j? 0
0 0 jjj
0
B
@
1
C
A; (2)
where the plasma permittivity components are
jjj ¼ 1 À
f2
pe
f2
; (3)
j? ¼ 1 À
f2
pe
f2 À f2
ce
; (4)
j ¼
fce
f
f2
pe
f2 À f2
ce
: (5)
The plasma dielectric components are shown in Fig. 1.
The components, jjj and j? are real whereas j is imagi-
nary. The jjj component is same as permittivity in a non-
magnetized plasma, hence there is no effect of external
magnetic field, whereas the component of plasma permittiv-
ity transverse to the magnetic field, j? is dependent on fce.
The imaginary (out-phase) component, j introduces a
phase-shift of 90
with respect to the incident wave and this
is the component which is mainly responsible for the absorp-
tion of the wave in the plasma.
In the far-field region, the contribution of the imaginary
quantity in the effective permittivity is known to excite dif-
ferent wave modes, such as right hand circularly polarized
(RCP) and left hand circularly polarized (LCP) modes propa-
gating parallel to the magnetic field. Other important modes
are ordinary (O-mode) and extraordinary (X-mode) modes.
These modes are observed when the propagation vector ~k is
perpendicular to the magnetic field. Many different wave-
modes can be excited in the far field region of the resonator
when acting as a radiating antenna. However, their effect is
less significant while considering the near-field region which
is the active region around the hairpin.
The near field region around the hairpin is dominated by
the radiating electric ~E and magnetic fields ~H, which are
characterized by mutually orthogonal in-phase oscillations in
~E and ~H. Therefore, j? component is responsible for affect-
ing the quarter wave-length within the resonant structure.
When j? is considered as the effective permittivity in
the formula of quarter wavelength resonance given in Eq.
(1), we obtain a bi-quadratic equation in terms of fr that
depends on independent variables fce, fpe, and fo as
f4
r À f2
r ð f2
o þ f2
ce þ f2
peÞ þ f2
ce f2
o ¼ 0: (6)
FIG. 1. Hairpin resonator and sketch of plasma permittivity components in a
magnetized plasma.
123510-2 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)
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4. Out of the four real roots of fr, two are found to be non-
physical, as fr cannot be negative. The remaining two posi-
tive roots are given by fr1 and fr2 as follows:
fr1;r2 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðF=2Þ6
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðF=2Þ2
À f2
o f2
ce
qr
; (7)
where F ¼ f2
o þ f2
ce þ f2
pe.
Further solving Eq. (1) in terms of fpe as a function of fr,
fce, and fo we obtain
ne 1010
cmÀ3ð Þ ¼ 1:23 Â 1 À
f2
ce
f2
r
!
f2
r À f2
o
À Á
: (8)
The above equation is simplified by normalizing the fre-
quency values by GHz. The factor of 1.23 is the net value of
the constant in the relation between ne and f2
pe; ne ¼ pme
e2 f2
pe.
Equation (8) suggests that for a given ambient B-field
(or fce) and ne, two possible values of fr exist namely, fr1 and
fr2. The upper frequency, fr1 fo stay remains above fce. On
the other hand, the second resonance fr2 stay below fo and fce.
Therefore theoretically there are two possible resonance fre-
quencies for a given value of ne.
III. EXPERIMENTAL SETUP
The basic construction of the hairpin probe is well
described in the literature.9,10,20
Briefly, the probe comprises
of a 50 X coaxial line terminated by a small loop antenna at
its closed end. The hairpin structure is attached close to the
loop antenna over a ceramic tube (diameter - 4.0 mm) holder
that shields the coaxial line and the loop from direct expo-
sure to the plasma as shown in Fig. 2. Therefore the hairpin
is closely coupled but insulated from the loop antenna.
The wires are made of platinum-rhodium alloy (diameter -
0.125 mm) with a typical length between 25.0 mm and
30.0 mm and width between 3.0 mm and 4.0 mm. The basic
set-up for the hairpin probe requires a variable frequency
microwave oscillator, a directional coupler in conjunction
with a Schottky diode for measuring the reflected output
voltage in the oscilloscope. When the applied frequency is
tuned to the resonance frequency, a marked drop in the
reflected power is observed. Hence the resonance frequency
is determined.
In order to verify the role of external magnetic field on
the characteristic resonances, we refer to the same experi-
mental set-up as discussed in previous paper.13
Here, we
present a simple overview of the experimental plan. The
inductively coupled plasma39
(see Fig. 3) of argon gas was
produced at 13.56 MHz. The main experiment was carried
out in the diffusion chamber. The hairpin is placed between
the pole pieces of the two permanent magnets and the mag-
netic field at the center was varied by adjusting the relative
position of the poles. Fig. 4(a) shows the hairpin when
placed normal to the B-field lines while Fig. 4(b) shows the
hairpin when placed along the B-field lines (this arrangement
is generally called a L-shaped hairpin probe). In both situa-
tions, the hairpin is located in the uniform B-field region.
The magnetic field strength is measured at all positions
between the magnetic poles using a standard Hall probe. The
B-field strengths that are measured normal to the magnetic
axis are shown in the Fig. 5. The data are shown for the con-
dition when the magnetic poles are kept very close to each
other with gap of 6.0 cm. It is clear that the B-field is uniform
nearer to the hairpin (length - 26 mm and width - 3.5 mm).
Since the electric field of the hairpin is concentrated mainly
at its open-end, the magnetized electrons are mostly affected
FIG. 2. Hairpin probe setup, where X is linear ramp voltage amplitude proportional to frequency and Y is reflected signal amplitude corresponds to each
frequency.
FIG. 3. Applied radio frequency (13.56 MHz) inductive source, NS–north
and south poles of the magnets, PS–power supply and MU–matching unit.
123510-3 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)
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5. in this region. The experiments are carried out in the range
between 100 W and 500 W where the rf power coupling was
stable with minimal reflected power. The pressure is varied
in a range between 1.0 Â 10À3
mbar and 8.0 Â 10À3
mbar.
IV. EXPERIMENTAL RESULTS
Figures 6–11 show the characteristic resonance signals
for specific cases as a function of scanning frequency. In all
the waveforms, the vacuum resonance data are subtracted
from those obtained in the presence of ambient plasma. This
eliminates the common mode noise in the signal, which
arises due to spurious reflections that are uncharacteristic of
the probe’s resonance. Therefore, the resultant data show an
inverted peak corresponding to vacuum frequency fo. The
positive peaks indicated by fr1 and fr2 are the resonances
observed in the plasma. Therefore in accordance with Eq.
(7), the observation clearly suggests that the magnetized
plasma dielectric supports two possible wave-modes. In one
case the phase-velocity is greater and in the other case it is
lower than the speed of light.
As evinced from the graphs presented in Figs. 6–11, the
quality of the resonance signal also displays noticeable fea-
tures with regard to the orientation of probe with respect to
the magnetic field. Inspecting Fig. 6, the resonance peaks are
well defined when the probe pins are introduced perpendicu-
lar to the magnetic field lines. In Fig. 7, the quality of the res-
onance signals is found to be strongly diminished as the
probe is pointed towards the direction of B-field. This fact is
also evident from Figs. 10 and 11. The broadness of the
FIG. 4. Front view of the plasma diffu-
sion chamber where SP is spherical
probe, HP is hairpin probe, and NS
represents north-south poles of two
permanent magnets.
FIG. 5. B-field strength measured normal to the magnetic axis, where d ¼ 0
shows the location of the center of the magnetic axis.
FIG. 6. Resonance signals for the case when ~k 90 ~B under conditions
fo ¼ 2.63 GHz, 200 W, 6.4 Â 10À3
mbar, and fce ¼ 2.80 GHz.
123510-4 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)
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6. resonance peak suggests the Ohmic dissipation of electro-
magnetic energy stored in the resonator. This can be primar-
ily due to strong interaction of plasma electrons in response
to the oscillating electric field between the pins which are or-
thogonal to the B-field.
Figures 8–11 plot the experimental data obtained under
various conditions. The chosen operating conditions fall
under two categories. Figures 8 and 10 fall under the condi-
tion when, fce fo; while Figs. 9 and 11, falls under fce fo.
In both cases, the resonance frequency fr1 shifts towards the
higher frequency as background density is increased. On the
other hand, initially fr2 responds by moving away from fo as
evinced in Figs. 9 and 11, however fr2 signal tends to saturate
near fce, if fce is chosen to be below fo, as found in Fig. 8.
From Eq. (7), one expects that the resonances fr1 and fr2
correspond to the same density. However, ne calculated from
fr2 gives one-quarter lower value of ne than those obtained
from fr1 (see Fig. 12). We found that there is no change in ne
when the probe is rotated about its axis, while the pins are
pointed in the direction perpendicular to the magnetic field
(see Fig. 13).
In order to investigate the role of j? and j given by
Eqs. (4) and (5), respectively, to the effective permittivity,
we plotted the possible values of fr as a function of fce based
FIG. 7. Resonance signals for the case when ~k k ~B under conditions
fo ¼ 2.68 GHz, 200 Watt, 6.4 Â 10À3
mbar, and fce ¼ 3.08 GHz.
FIG. 8. Resonance signals for the case when ~k 90 ~B and fo fce, where
fo ¼ 2.52 GHz and fce ¼ 2.30 GHz.
FIG. 9. Resonance signals for the case when ~k 90 ~B and fo fce where
fo ¼ 2.64 GHz and fce ¼ 2.80 GHz.
FIG. 10. Resonance signals for the case when ~k k ~B and fo fce where
fo ¼ 2.68 GHz and fce ¼ 1.00 GHz.
123510-5 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)
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7. on Eq. (7). Both axes are normalized with the vacuum reso-
nance frequency fo (see Fig. 14). The experimental data are
also plotted on the same graph, which clearly shows that the
response of fr1 with fce match very well with the analytical
model based on j? as opposed to jÂ. For the fr2, we have
shown only experimental data that falls on the predicted ana-
lytical curve.
The limitation in obtaining more data points for fr2 was
due to the fact that the resonance spectrum fell outside the
scanning frequency range available with the current micro-
wave source (2.0GHz–8.0 GHz). For example, fr2 ¼ 1.82GHz
if fce ¼ 2.28 GHz and fpe ¼ 0.73 GHz and hence fr2 will be out
of the detectable range of microwave source. On the other
hand fr2 tends to saturate about fce. In addition, it is difficult to
maintain the constant density to study the effect of the ambi-
ent B-field on fr2.
V. DISCUSSION
The resonance characteristics displayed in the Results
section needs a detailed discussion regarding the interpreta-
tion of hairpin resonances in magnetized plasma. To investi-
gate the behavior of these resonances, it is necessary to
understand the characteristics of j? as a function of fre-
quency. Figure 15 plots the j? by considering three cases;
(1) fpe fce, (2) fpe ¼ fce, and (3) fpe fce. The plots are scat-
tered across the opposing quadrant between the intersection
lines f ¼ fce and f ¼ fo. Broadly, the plots indicate that in the
frequency band situated below fce, j? is always a positive
quantity. On the other hand as the resonance frequency
approaches the critical frequency, f ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f2
pe þ f2
ce
q
, one
observes the value of j? 0, and hence no propagation is
possible. This is the condition for the upper hybrid reso-
nance. Under this condition, the wave number goes to zero
and the wave undergoes reflection in the plasma.
The observation of the second resonance fr2 suggests the
slowing down of the wave phase velocity in magnetized
plasmas for frequencies that fall below fce. The behavior of
FIG. 14. Plot of fr vs fce where both X and Y axis are normalized by fo where
~k 90 ~B, fo ¼ 2.28 GHz, and fpe ¼ 0.73 GHz.
FIG. 12. ne vs discharge power when ~k 90 ~B. Operating conditions:
fo ¼ 2.48 GHz, 6.4 Â 10À3
mbar, and fce ¼ 2.80 GHz.
FIG. 13. ne (using fr1) vs discharge power. For a given condition ~k 90 ~B,
these data are obtained for two different cases when hairpin’s strongest
E-field component between the wires is parallel and normal to the strong
B-field. Operating conditions are 6.4 Â 10À3
mbar, fo ¼ 2.53 GHz and
fce ¼ 2.24 GHz.
FIG. 11. Resonance signals for the case when ~k k ~B and fo fce where
fo ¼ 2.68 GHz and fce ¼ 3.14 GHz.
123510-6 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)
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8. fr2 is typical of the response in a dielectric medium such as
in the case of solid and liquid dielectrics.
On the other hand, the monotonic increase in fr1 for fre-
quencies that are greater than fce suggests that the cyclotron
motion of electrons has minimal effect on large time-
averaged cycles. Therefore the wave-phase velocity contin-
ues to be greater than the speed of light as similar to the
non-magnetized plasma. Therefore the frequencies falling
above the upper hybrid resonance, such as fr1 can propagate
through the plasma and seem to respond to the changes in
the ambient ne.
In light of the above facts, it can be concluded that j? is
the effective permittivity in regard to the application of hair-
pin in magnetized plasma. Ideally the electron densities cal-
culated from the two resonances fr1 and fr2 must be the same.
However, by following Eq. (8) we found a discrepancy
between the ne calculated from the second resonance fr2 by
over a quarter as compared to fr1 as shown in Fig. 12.
The cause for the underestimated value of ne may indi-
cate possible losses resulting in the fall of electron density
around the hairpin. In the case when the resonance frequency
is close to fce, it may lead to resonant transfer of energy from
the oscillating electric field to the electrons as fr $ fce.
Localized electron heating can result in enhanced cyclotron
orbits while the conservation of particle flux in a given vol-
ume between the pins will result in the reduction of ne within
the volume. The pondermotive forces can also exist due to
~EðtÞ Â ~B fluctuations in the neighborhood of the probe. A
signature of the existence of strong interaction between the
oscillating electrons with the ambient magnetic field can be
evinced by inspecting the quality of resonance signals in
Figs. 8–11. In these figures, when fr1 or fr2 are relatively
closer to fce, we observe significant attenuation of the reso-
nance signal. These effects are not predicted from the analyt-
ical model of the resonance frequency based on j?. In view
of practical application of the resonator in magnetized plas-
mas, therefore one should select the hairpin dimensions in
such a manner that the vacuum resonance frequency should
be kept well above the fce.
It can also be seen from Fig. 8 that the lower value of
resonance frequency fr2 does not vary with density and tends
to saturate close to fce ¼ 2.34 GHz. On the other hand, in
Figs. 9 and 11, when fce fo we observe fr2 moving away
from fo in response to the changes in the ambient density. In
order to explain the observed behavior, we plotted the
expected values of ne as a function of fr which is normalized
with respect to the vacuum frequency fo as shown in Fig. 16.
In one case, the chosen value of fce fo (red curve) while in
the other case fce fo (black curve). Broadly the analytical
curves exhibit the following features.
In Fig. 16, the segment of the curves PQ and QR for
which the value of ne 0 is defined as the exclusion fre-
quency range between fce and fo. This is non-physical condi-
tion thus no resonances can be experimentally observed. The
plots also show the existence of dual resonances for the same
density ne. For the red-curve, the lower limit of the exclusion
region is fce as indicated by P while the upper limit at Q cor-
responds to the vacuum resonance fo. The situation inter-
changes for the black curve as the lower limit of the
exclusion region becomes fo while the upper limit at R corre-
sponds to fce. However, if fce ¼ fo the same condition is satis-
fied at a single point Q.
Physically, fr ¼ fo implies no plasma electrons are pres-
ent between the pins. This is point Q on the graph. At points
P and R, which correspond to fr ¼ fce, the value of ne is zero.
This is a paradoxical case because the existence of fce must
be associated with the presence of electrons around the
probe. However, if the electrons are strongly magnetized
such that fpe ( fce then the cyclotron motion is dominated
over the electron plasma frequency and violates the basic
definition of plasma. In order to observe a shift in fr2, there
must be sufficient density of electrons present in the volume
adjacent to the hairpin such that fpe fce. This is also evident
from Fig. 8 which shows the left hand branch of the reso-
nance frequency exhibits no apparent shift in fr2 as the back-
ground plasma density is increased. In contrast to this the
right hand branch fr1 shows a monotonic increase with the
density as the discharge power is raised. When the value of
fo is chosen to be lower than fce, we find that the lower
branch fr2 has a larger domain of variation with ne. This ob-
servation is consistent with Figs. 9 and 11, as both resonan-
ces fr1 and fr2 respond to the changes in the background
density.
FIG. 16. Electron density (ne) as a function of fr normalized to fo for a given
magnetic field strengths. The exclusion frequency at which ne 0 stretches
between fce and fo. These are indicated between the points PQ and QR,
respectively, for the red and black curves.
FIG. 15. Perpendicular component of the plasma permittivity in magnetized
plasma.
123510-7 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)
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9. VI. SUMMARYAND CONCLUSIONS
We have presented a detailed behavior of the resonances
observed by the hairpin probe when immersed in magnetized
plasma. In particular, we have shown that the response of the
lower branch of the resonance fr2 that occurs below the vac-
uum frequency is critical to the choice of the physical dimen-
sion of the hairpin. If the chosen value of resonance
frequency in vacuum fo is lower than the cyclotron frequency
fce, then the lower branch tends to saturate near fce, hence no
apparent shift is registered on increasing the ne. The density
inferred from the lower branch fr2 is also shown to be less
than the density calculated using upper resonance fr1. A qual-
itative interpretation is given which points towards strong
interaction of the oscillating electrons with the external B-
field. This may possibly lead to resonant heating associated
with the fall in electron density adjacent to the hairpin.
Based on this experimental investigation, we can arrive
at a reasonable conclusion that j? is a valid approximation
for obtaining the electron density formula for the hairpin
probe. However, one must be careful in choosing the physi-
cal dimension of the hairpin such that the expected resonan-
ces should avoid the cyclotron frequency. In conclusion, the
quarter wavelength transmission line presents a good model
of the theory of plasma dielectric in magnetized plasma.
ACKNOWLEDGMENTS
The research work was supported by Enterprise Ireland
under Grant No. TD/2007/335 and Science Foundation
Ireland under Grant No. 08/SRC/I1411. Dr. Gogna likes to
thank Dr. Paul Swift for proofreading the manuscript.
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