design of a thermal probe for the plasma diagnostics

Contrib. Plasma Phys. 44, No. 7-8, 677 – 682 (2004) / DOI 10.1002/ctpp.200410100

Design of a thermal probe for the plasma diagnostics

H. Matsuura∗ and K. Michimoto

Graduate School of Engineering, Osaka Prefecture University, Sakai, 599-8531 Osaka, Japan

Received 15 April 2004, accepted 15 April 2004Published online 29 October 2004

Key words Thermal probe, Berkeley code, heat transmission factor, sheath.PACS 52.40.Kh

When a solid material is inserted into plasma, it receives heat flux and electrical current according to its biasvoltage. The heat flux-voltage characteristic is more complicated than the current-voltage characteristic anddepend on ion temperature as well as electron temperature. Thermal probe method uses this characteristic todetermine plasma parameter. We propose here to use it to deduce ion temperature and show the results onrelated work.

c© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The ion temperature is an important parameter for fusion or process plasma. But spectroscopic method is difficultto apply to measure low temperature plasma, since a very high resolution spectrometer is needed to determine theDoppler broadening of line spectra. In [1], ion sensitive probe method was proposed to measure ion temperature.As this method uses the difference of ion and electron cyclotron radius, it can not available in weak magneticfield. Even in strong magnetic field, if its direction varies spatially, the proper setting of the probe position isdifficult.

When a solid material is inserted into plasma, it receives heat flux and electrical current according to its biasvoltage. Langmuir probe method is based upon measuring the current into a probe tip as a function of the probebias potential, and gives us electron temperature and plasma potential. On the other hand, the heat flux into theprobe has more complicated characteristic than the current-voltage one. So we think it is possible to use this”thermal probe” method [2], which measures this heat flux-voltage characteristic, in order to determine the iontemperature.

In section 2, according to the analytical sheath theory, we present the formula, which describes the biascharacteristic of current and heat flux flowing into the sheath. Then we propose how to determine ion temperaturefrom the thermal probe measurement. For precise ion temperature measurement, we need to study adiabaticexponent of plasma and Bohm velocity at sheath edge. In section 3 we calculate sheath heat flux with PIC code(Berkeley code) [4,5] and study the effect of ion temperature on the heat flux-voltage characteristic. In section 4,we show the design concept of the prototype thermal probe for DC-glow plasma in our experiment.

2 Background theory

The dependence of ion temperature on the heat flux flowing into a probe has not been fully understood yet.Takizuka [3] pointed out that the heat flux in the floating condition not only depends on parallel ion temperature( Ti‖ ) but also on perpendicular ion temperature ( Ti⊥ ) . We extend his prediction to the arbitrary value of theprobe voltage.

Current I and heat flux Q are functions of normalized sheath potential drop φ = e(Φsheath − Φprobe)/kBTe,where subscript “sheath ” indicates the boundary between plasma and sheath. If φ > 0, ion particle flux Γ isconstant. We omit here the orbital effect for cylindrical or spherical probes. I and Q are proportional to Γ.

∗ Corresponding author: e-mail: [email protected], Phone: +81 72 254 9226, Fax: +81 72 254 9904


678 H. Matsuura and K. Michimoto: Design of a thermal probe

-15

-10

-5

0

5

0 5 10 15 20sheath pot. differ.

Sheath current transmission factor

a)0

5

10

15

20

0 5 10 15 20sheath pot. differ.

Sheath heat transmission factor

b)

Fig. 1 Example of CTF(a) and HTF(b) for He plasma with Te = 10eV. Five lines correspond to the theoritical curves forTi = 0, 5, 10, 15, 20eV respectively. Horizontal axis is normalized sheath drop φ. In ion dominant region( φ > 5), thoughCTF is not depend on Ti, HTF depends upon it Ti clearly.

I = eZΓg(φ), Q = TeZΓf(φ), (1)

where Z is ion charge. ( In most cases, we can assume it to be unity. ) Hereafter, we call functions g(φ) and f(φ)as current transmission factor( CTF ) and heat transmission factor ( HTF ) respectively. For single ion plasma,CTF is very simple.

g(φ) = 1 − exp(φf − φ) (2)

where φf is the normalized potential drop of the floating probe. If we neglect secondary electron effect,

φf =12

log{

mi

2πme(1 + γaτ‖)

}(3)

is given, where τ‖ = Ti‖/Te is normalized ion temperature along the magnetic field, mi and me are ion andelectron mass, γa is adiabatic exponent. In [3], flow velocity and sound velocity (

√(Te + γaTi‖)/mi ) are

compared and γa ∼ 3 is proposed for the floating condition(φ = φf ).HTF is more complicated than CTF.

f(φ) ={

1Z

(τ⊥ + γaτ‖) +12

+ φ

}(1 − RiE) +

Erec

ZkbTe+ 2 exp(φf − φ) (4)

where RiE is energy reflection coefficient and Erec is effective recombination energy. Eq.(4) is the extendedversion of Eq.(B11) in [3] to any sheath potential drops and also includes heat flux of neutralized ions.

Erec must be determined from ionization energy of ion, dissociation energy, particle reflection coefficient,work function of the probe surface, and so on. Though examples of these values can be found in text books likeStangeby’s and Erec may be estimated for single-ion plasma, our method will be limited to relative measurementof Ti for more complex component plasma. In this paper, we assume that plasma has only one ion spices and thatErec is equal to ionization energy of ion.

For the ion dominant bias region ( φ > φf ), the third term of right hand side of Eq.(4) can be neglected andHTF becomes a linear function of φ. Examples of CTF and HTF theoritical curves for He plasma are shown inFig. 1a) and Fig. 1b). Though CTF is less dependant on ion temperature (we assume here that Ti = Ti‖ = Ti⊥),


Contrib. Plasma Phys. 44, No. 7-8 (2004) / www.cpp-journal.org 679

0

10 ion

electron~ Ti/Te

Estimate of Ti from HTF

Sheath potential drop[/Te]

He

at

Tra

nsm

isio

n F

acto

r

5 10

Fig. 2 Schematic drawing of how to determine Ti from thermalprobe data (Solid line). By extrapolating of HTF in ion dominantregion to no sheath potential drop limit (φ → 0) and subtractingthe effective recombination energy, you get ((1 + γa)Ti/Te +0.5)(1 − Rie) from the intercept (closed circle).

HTF depends upon Ti clearly. If we obtain the same HTF curves for two different plasmas’ measurement, Ti

must be the same.The procedure to determine ion temperature from the thermal probe data ( I and Q) is shown in Fig. 2. Firstly,

we determine Φsheath and Te from current in electron dominant bias region ( just same as ” Langmuir probe ” ).Secondly, we obtain HTF or eQ/TeI from absolute measurement of I and Q in ion dominant bias region. If weextrapolate this value to φ = e(Φsheath −Φprobe)/kBTe → 0, the gradient of the extrapolation line is (1−Rie)and the intercept (closed circle in Fig. 2) is ((1+γa)Ti +0.5Te)(1−Rie)+ErecTe. If the recombination energyErec can be subtracted, we obtain (1 + γa)Ti or, more exactly Ti⊥ + γaTi‖.

In order to obtain absolute value of Ti, the adiabatic exponent γa is also important. But there is no conclusivetheory on γa. In most simple sheath theory, it is often assumed that ion is “cold ”( ie. γa = 0 ). For isothermalmodel, we must use γa = 1. Takizuka studied this problem with particle simulations and said γa = 3 is properfor a floating target [3]. In the next section, we also study γa for a ion-dominantly( that is negatively ) biasedtarget.

3 Simulation result

The two dimensional PIC simulation code( Berkeley Code: XOOPIC) [4, 5] was used to investigate the effectof target potential and ion temperature on heat flux flowing into the target plate. Since XOOPIC could not beused relatively large geometry, we selected smaller size target than our thermal probe design and modified inputparameters to give larger density and smaller Debye length λD. The size of calculation geometry( Lx × Ly

) is about 100λD × 100λD. Plasma electrons and ions are injected from three boundary as shown in Fig. 3.The potential of these boundary is set to be 0[V]. Target (Thermal probe) was 1.5 × 10−4[m] long and set atx = 4 × 10−4[m]. Magnetic field is applied perpendicular to the target surface (ie. By = Bz = 0). More than105 particles are used in simulation. The plasma potential in the most region is −20 ∼ −10[V].

We divided calculation geometry into set of small meshes and estimate the particle and heat flux flowing toj–th mesh with

Γj =1

∆Vj

Nj∑vx, Qj =

1∆Vj

Nj∑ 12m(v2

x + v2y + v2

z)vx (5)

where the summation is carried for Nj particles which is in the j–th mesh. ∆Vj is the volume of the j–thmesh. Temperature is also calculated with the similar formula. Though these value is two dimensional function,we restrict our treatment to volume-averaged temperature and flux integrated over Thermal probe surface forsimplicity.



� � � �

� � � �

�

� � � �

� � � �

� � � � � �

� � � � �

� � � � � � � �

� � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � �

� � �

Fig. 3 Schematic drawing on calculation geometry. Plasma electron and ion are injected from three boundary. The potentialof these boundary is set to be 0[V]. Target (Thermal probe) was 1.5 × 10−4[m] long and set at x = 4 × 10−4[m].

50 100 1500

50

100

150

(Te)X(HTF)

Te

Initial electron energy 10[eV]

–Target potential[V]

Te a

nd

Qi/F

[e

V]

Fig. 4 Horizontal axis is the bias voltage applied to target plate. Verticalaxis is Te (open circles) and Q/Γ (closed circles). As the Te is almostconstant, the latter is correspond to HTF.

One result of our simulation is shown in Fig. 4. Horizontal axis is the bias voltage applied to target plate.Vertical axis is the electron temperature( open circles ) and the ion heat flux normalized with the particle flux (Closed circle). As the Te is almost constant in these simulations, the latter is correspond to Te× HTF, which showthe linear dependence ( y ∼ 5.13 + 0.94x) on the target potential(that is Φprobe). These result is well correspondto the dash line in Fig. 2. The method which proposed previous section gives us Ti which is nearly equal to oneobtained in the simulation within the uncertainty of Erec and γa.

In order to study the adiabatic exponent γa, we must make simulation with different ion temperature. InXOOPIC, however, temperature is not a input parameter, but the simulation result. So we change the initialion energy at plasma emitting boundary.( see Fig.3. ) We show the result in Fig. 5a). Ion temperature( closedcircles ) and flow energy ( open circles ) change drastically with keeping electron parameter ( triangles ) nearlyconstant. As shown in Fig. 5b), heat flux dependence on the initial ion energy seems to be similar as particle flux.This means that HTF is a week function of ion temperature Ti and, considering the theoretical prediction ( HTF∼ (1 + γa)(1 − RiE)Ti ), there exists a contradiction with γa = 3 proposed in [3].


Contrib. Plasma Phys. 44, No. 7-8 (2004) / www.cpp-journal.org 681

101 102100

101

102Initial electron energy 10[eV]

Initial ion energy[eV]

Tem

p. and K

.E.[eV

]

a)100 101 102106

107

108

10–10

10–9

10–8Initial electron energy 10[eV]

Initial ion energy[eV]

He

at

flu

x[a

rb.]

Pa

rtic

le f

lux[a

rb.]

b)

Fig. 5 Example of simulation results with different Ti. In a), ion temperature (closed circles), ion flow energy (open circles),electron temperature (closed triangles), and electron flow energy (open triangles) are plotted. In b), ion heat flux is closedcircles and particle flux is closed squares.

One reason may be plasma instability. For low initial ion energy case( to say Ti < 10eV ), plasma is not stableand simulations hardly converge. For further study on γa, moreover it may be needed to consider spatial variationof temperature and flux and to check the velocity distribution of ion in more detail.

4 Design concept

In order to demonstrate the possibility to measure ion temperature with thermal probe method, we design aprototype thermal probe. As previously reported in [6], we confirmed the dependence of bias voltage on the heatflux experimentally. In this experiment, we used a disk-shape target to monitor the heat flux. But this target istoo large( D = 50[mm] ) to use as a ”probe”. Moreover, its time constant of the temperature evolution is also toolarge( about 20 [min.] ) to use real time monitor. So the size of the thermal probe tip is reduced to 10× 20[mm],which enable us to setup it trough small ports. The thickness of thermal probe tip is below 0.2[mm]. We usecommercial Al-foil as the tip material. If heat loss through the support is same, the time constant is expected tobe shorter by order of magnitude.

The probe temperature can be easily measured with thermocouples. The heat flux is equal to the heat loss,which is proportional to the temperature difference between the probe and the background. So the calibrationof this proportional coefficient must be done before the thermal probe is applied to discharge plasma. We havealready proposed the methods to determine this coefficient [6] by measuring the time constant of the target platetemperature and estimated the heat flux to it. However, as the time constant is designed to be small, it mightbe difficult to determine the time constant of our new probe exactly. So we designed the thermal probe withauxiliary calibration heater. The schematic drawing of prototype thermal probe design is shown in Fig. 6. We usea Cu/CuNi44 thermocouple to monitor temperature and a NiCr heater to calibration. By monitoring the currentfrom the Al tip, it also can be used as a conventional Langmuir probe.

Though the contact of the thermocouple is covered with Al-foil, its wire is exposed to plasma due to the easeto make the device. This does not cause serious damage on it in our experimental condition. As long as Te iskept to be nearly constant, our thermal probe shows almost the same Q − Φ characteristic. Ti/Te deduced likeFig.2 is ∼ 0. As plasma is produced by DC-glow discharge [6], Ti seems to much less than Te in our device. Totest our propose on the thermal probe, other plasma such as divertor plasma of Tokamak or compact torus plasmamay be preferred.



Recorder

ceramic

Al

Cr

Rp

to Oscilloscope

thermocouple (Cu/Ni44)

Fig. 6 Prototype thermal probe design. We use a Cu/CuNi44 thermocouple to monitor temperature and a NiCr heater tocalibration. As the tip material, commercial Al-foil is used.

5 Conclusion

The results of this work are the following.

• We proposed theoretical formula of CTF and HTF for arbitrary bias voltage, and derived the procedure todetermine ion temperature form the data of current and heat flux.

• With PIC simulation code XOOPIC, we confirmed heat flux dependence on bias voltage. As for dependenceon ion temperature or adiabatic exponent, discrepancy exists between our result and [3]. More detailed studywill be needed.

• Comparing the target plate used in previous experiment [6], we design a prototype of the thermal probe. Inorder to calibrate heat flux, we prepare a small heater in this probe.

As mention already in Section 2, the effect of Erec on the absolute value of Ti is serious problem for theplasma with multi-spices ion ( including negative ion) . If the different ions have different temperature, theproblem becomes more complicated. Our thermal probe method, however, could give us the value closely relatedwith ion temperature. If the time response becomes sufficiently fast, it may be a useful monitoring method.

Acknowledgements The authors thank Plasma Theory and Simulation Group in UC-Berkeley, since they have developedexcellent PIC codes and made them available for scientist in many countries.

References

[1] N.Ezumi, Contri. Plasma Phys., 41, 488 (2001).[2] E.Stmate, Appl. Phys. Lett., 80, 3066 (2002).[3] T.Takizuka, Kakuyugo Kenkyu, in Japanese, 64, 255 (1990).[4] J.P.Verboncoeur et al., Comp. Phys. Comm. 87, 199 (1995).[5] http://www.eecs.Berkeley.edu/.[6] H.Matsuura, Proc. ESCAMPIG16/ICRP5 Joint Conf., 2, 75 (2002).


design of a thermal probe for the plasma diagnostics

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