Interpreting the behavior of a quarter-wave transmission line resonator in amagnetized plasmaG. S. Gogna, S. K. Karkari, and M. M. Turner Citation: Physics of Plasmas (1994-present) 21, 123510 (2014); doi: 10.1063/1.4904037 View online: http://dx.doi.org/10.1063/1.4904037 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/21/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Noise temperature improvement for magnetic fusion plasma millimeter wave imaging systems Rev. Sci. Instrum. 85, 033501 (2014); 10.1063/1.4866652 Ion diode performance on a positive polarity inductive voltage adder with layered magnetically insulatedtransmission line flow Phys. Plasmas 18, 053106 (2011); 10.1063/1.3587082 Diagnosis of plasmas in compact ECR ion source equipped with permanent magnet Rev. Sci. Instrum. 75, 1520 (2004); 10.1063/1.1691525 Electron Bernstein wave experiment in an overdense reversed field pinch plasma AIP Conf. Proc. 595, 346 (2001); 10.1063/1.1424207 Production of highly uniform electron cyclotron resonance plasmas by distribution control of the microwaveelectric field J. Vac. Sci. Technol. A 17, 3225 (1999); 10.1116/1.582046

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Interpreting the behavior of a quarter-wave transmission line resonatorin a magnetized plasma

G. S. Gogna,1,a) S. K. Karkari,2,b) and M. M. Turner1

1NCPST, School of Physical Sciences, Dublin City University, Dublin 9, Ireland2Institute 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.comb)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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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 INMAGNETIZED 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 ¼c

2 2lþ wð Þ ffiffiffiffiffijpp ¼ foffiffiffiffiffi

jpp ; (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? �jj� 0

jj� j? 0

0 0 jjj

0B@

1CA; (2)

where the plasma permittivity components are

jjj ¼ 1�f 2pe

f 2; (3)

j? ¼ 1�f 2pe

f 2 � f 2ce

; (4)

j� ¼fce

f

f 2pe

f 2 � f 2ce

: (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

f 4r � f 2

r ð f 2o þ f 2

ce þ f 2peÞ þ f 2

ce f 2o 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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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 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðF=2Þ6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðF=2Þ2 � f 2

o f 2ce

qr; (7)

where F ¼ f 2o þ f 2

ce þ f 2pe.

Further solving Eq. (1) in terms of fpe as a function of fr,fce, and fo we obtain

ne 1010cm�3ð Þ ¼ 1:23� 1� f 2ce

f 2r

!f 2r � f 2

o

� �: (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 f 2pe; ne ¼ pme

e2 f 2pe.

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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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� 10�3 mbar and 8.0� 10�3 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 d¼ 0

shows the location of the center of the magnetic axis.

FIG. 6. Resonance signals for the case when ~k 90� ~B under conditions

fo¼ 2.63 GHz, 200 W, 6.4� 10�3 mbar, and fce¼ 2.80 GHz.

123510-4 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)

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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

fo¼ 2.68 GHz, 200 Watt, 6.4� 10�3 mbar, and fce¼ 3.08 GHz.

FIG. 8. Resonance signals for the case when ~k 90� ~B and fo> fce, where

fo¼ 2.52 GHz and fce¼ 2.30 GHz.

FIG. 9. Resonance signals for the case when ~k 90� ~B and fo< fce where

fo¼ 2.64 GHz and fce¼ 2.80 GHz.

FIG. 10. Resonance signals for the case when ~k k ~B and fo> fce where

fo¼ 2.68 GHz and fce¼ 1.00 GHz.

123510-5 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)

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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 j�. 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 (2.0 GHz–8.0 GHz). For example, fr2¼ 1.82 GHz

if fce¼ 2.28 GHz and fpe¼ 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) fpe¼ fce, and (3) fpe> fce. The plots are scat-

tered across the opposing quadrant between the intersection

lines f¼ fce and f¼ 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 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 2pe þ f 2

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, fo¼ 2.28 GHz, and fpe¼ 0.73 GHz.

FIG. 12. ne vs discharge power when ~k 90� ~B. Operating conditions:

fo¼ 2.48 GHz, 6.4� 10�3 mbar, and fce¼ 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� 10�3 mbar, fo¼ 2.53 GHz and

fce¼ 2.24 GHz.

FIG. 11. Resonance signals for the case when ~k k ~B and fo< fce where

fo¼ 2.68 GHz and fce¼ 3.14 GHz.

123510-6 Gogna, Karkari, and Turner Phys. Plasmas 21, 123510 (2014)

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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 � ~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 fce¼ 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 fce¼ fo the same condition is satis-

fied at a single point Q.

Physically, fr¼ 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 fr¼ 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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VI. SUMMARY AND 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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