Plasma and Fusion Research: Regular Articles

Volume 14, 2401086 (2019)

Eﬀect of High Ion Temperature on the Polytropic Coeﬃcient in theEnd Region of GAMMA 10/PDX∗)
Kunpei NOJIRI, Mizuki SAKAMOTO, Naomichi EZUMI, Satoshi TOGO, Takaaki IIJIMA,Seowon JANG, Akihiro TERAKADO, Yosuke KINOSHITA, Toshiki HARA,Tomonori TAKIZUKA1) , Yuichi OGAWA2) and Yousuke NAKASHIMA
Plasma Research Center, University of Tsukuba, Tsukuba 305-8577, Japan
Graduate School of Engineering, Osaka University, Suita 565-0871, Japan2)
Graduate School of Frontier Sciences, University of Tokyo, Kashiwa 277-8568, Japan1)(Received 30 September 2018 / Accepted 9 January 2019)
The eﬀect of high ion temperature on the ion polytropic coeﬃcient (γ) was experimentally investigated in
the end region of the GAMMA 10/PDX tandem mirror. The ion temperature parallel to the magnetic field (T i|| )was varied using the ion cyclotron range of frequencies (ICRF) waves, which heat the plasma in the central cell,and by applying an additional gas puﬀ in the central cell. Assuming the ion sound speed, γT i|| was evaluated bya Langmuir probe, and T i|| was evaluated in an ion energy analyzer. These parameters were combined to evaluate
γ. The time evolutions of γT i|| obtained from the probe and T i|| obtained from the energy analyzer were similar.
The plasma in the end region can be regarded as collisionless, but γ was lower with ICRF heating than without
the heating. A high T i|| component was identified in the heated case. As mentioned in a previous numerical study[B. Lin et al., Phys. Plasmas 23, 083508 (2016)], it is experimentally suggested that γ can be reduced by the highT i|| component resulting from the ICRF heating.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: ion temperature, ion polytropic coeﬃcient, sound speed, Langmuir probe, ion energy analyzer
DOI: 10.1585/pfr.14.2401086

1. Introduction
The ion sound speed (Cs ) [1–3] is an important parameter in boundary plasma physics related to sheath phenomena. It is a function of the electron temperature
T e , the ion temperature parallel to the magnetic field T i|| ,and the ion polytropic coeﬃcient γ. Specifically, Cs ={(kT e + γkT i|| )/mi }1/2 , where k is the Boltzmann constant[2,3]. Theoretical and numerical studies using plasma fluid
models have shown that the sheath characteristics are affected by ion temperature T i [4–7] and γ [8]. In a onedimensional (1D) particle model, the authors of [9] also
showed that the hot T i|| component, which is higher thanT e , imparts strong kinetic eﬀects and reduces the γ in collisionless plasma below the widely used value in 1D adiabatic flow, where the temperature is anisotropic and γ = 3in the flow direction. However, as experimental investigation of this phenomenon is lacking, investigating the eﬀectof high T i|| on γ and Cs is important.
Langmuir probes (LPs) are widely used for measuring basic plasma parameters such as T e , the electron density (ne ), and ion particle flux. The drift speed of ionsflowing into the LP reaches Cs and the ion saturation current reflects the value of Cs . Therefore, when T i|| is nonauthor’s e-mail: nojiri_kunpei@prc.tsukuba.ac.jp∗) This article is based on the presentation at the 12th International Conference on Open Magnetic Systems for Plasma Confinement (OS2018).

negligible compared to T e , there is a possibility to evaluate
γT i|| using the LP.
GAMMA 10/PDX is a tandem mirror device in which
the plasma T i exceeds the plasma T e [10, 11]. In a typical discharge, T i exceeds T e by one order of magnitude ormore. Furthermore, T i can be varied by applying the heating system using the ion cyclotron range of frequencies(ICRF).
In this study, we attempt to evaluate the eﬀect of highT i|| on γ using an LP and an ion energy analyzer in the end
region of GAMMA 10/PDX.

2. Experimental Device and Diagnostics
2.1 GAMMA 10/PDX tandem mirror
Figure 1 shows schematics of (a) part of the GAMMA
10 tandem mirror and (b) its magnetic field. The device
consists of a central cell, anchor cells, plug/barrier cells
and end regions. The device is 27 m long and the volumeof the vacuum vessel is 150 m3 . The main plasma production, ICRF heating of the plasma, and hydrogen gas puﬀing are applied in the central cell. The anchor cells have aminimum-B configuration and maintain the stability of theplasma. The present experiment employed two types ofICRF heating systems: one that produced the plasma andheated it in the anchor cells, and another that heated the2401086-1

c 2019 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

Volume 14, 2401086 (2019)

plasma in the central cell. Hereafter, the former and latterare referred to as RF1 and RF2, respectively. The standard discharge in GAMMA 10/PDX employs both RF1and RF2.
In the west-end region, an LP and an ion energy analyzer named end loss ion energy analyzer [10, 12] are installed. The analyzer is hereinafter referred to IEA. The LP
is installed near the roof target of the divertor simulation
experimental module (D-module [10, 11]; see Fig. 1 (c)).
The vertical position of the D-module and the probe areadjustable. The tungsten LP electrode is 1.5 mm in diameter and 1.1 mm long. The IEA is schematized in Fig. 1 (d).
Ions with lower energy than the eﬀective potential energy
at the ion repeller grid are reflected and collected at the collector plate. In this study, T i|| was evaluated by Maxwellian
fitting of the energy spectra measured by the IEA. Whentwo components of the Maxwellian distribution were identifiable, we calculated the eﬀective T i|| from the relative
densities of the components.

2.2

Evaluation of γT i|| using a Langmuirprobe

The ion and electron saturation currents (Iis and Ies ,respectively) of the hydrogen plasma are respectively givenbykT e + γkT i
1,(1)Iis = Sene2mi
8kT e
1,(2)
Ies = Sene
4πmeto the upstream density derived from the conservative formof the momentum equation [13]. The values of Iis , Ies andne can be obtained from the LP current-voltage characteristics. In this study, ne was evaluated from Ies and T e using
Eq. (2). Combining Eqs. (1) and (2), γT i|| is calculated asfollows:⎡⎤ ⎢⎢⎢ 2mi Iis 2⎥⎥⎢− 1⎥⎥⎥⎦ T e .
(3)
γT i = ⎢⎣πme Ies

3. Experimental Results and Discussion
3.1 T i variation by ICRF heating and additional gas puﬀ in the central cellIn the experiment, the T i|| of the end-loss plasma wasvaried by applying RF2 and an additional hydrogen gas
puﬀ in the central cell. The plasma discharge was conducted with or without RF2 (RF1 was applied in both discharges). The additional gas in the central cell was supplied from time t = 160 to t = 240 ms at a plenum pressureof 70 Torr. Figure 2 shows typical time evolutions of thediamagnetism (DM CC ) in the central cell (a), electron line
density (NLCC ) in the central cell (b), and the T i|| (c) andparticle flux (d) measured by the IEA (T i||,IEA and Γi||,IEA ,which mean T i|| and Γi|| at the end plate, respectively). The
temporal evolutions of DM CC and NLCC were those of atypical discharge. The T i||,IEA and Γi||,IEA data were aver-where S is the eﬀective probe surface area, and mi and me
are the ion and electron masses, respectively. The surface
areas of Iis and Ies are assumed identical. The coeﬃcient
1/2 of Eq. (1) is the ratio of the density at the sheath edgeFig. 1 Schematics of (a) part of the GAMMA 10/PDX tandem
mirror, (b) magnetic field distribution in part (a), (c)Langmuir probe installed near the roof target, and (d)end-loss ion energy analyzer.

Fig. 2 Time evolution of (a) diamagnetism and (b) electron line
density of plasma in the central cell; (c) ion temperature
parallel to the magnetic field T i|| and (d) ion particle flux
Γi|| at the end plate, measured by the IEA in the west end
region. Red lines and circles: with RF2; blue lines andcrosses: without RF2.

2401086-2

Plasma and Fusion Research: Regular Articles

Volume 14, 2401086 (2019)

Fig. 3 Typical current-voltage characteristics of the LP.

aged over three similar discharges. The DM CC was higher
with the RF2 than without the RF2, indicating a higher T iof the central plasma in the former operation. Supplying
additional gas reduced the DM CC in the setup with RF2.
The NLCC was also higher with the RF2 than without theRF2. In both discharges, the additional hydrogen gas increased the NLCC . In the west-end region without the additional gas, T i||,IEA and Γi||,IEA with the RF2 were higher thanthose without RF2. The T i||,IEA with RF2 was higher than
without RF2 by virtue of a high-temperature component
that was not identified in the absence of RF2. The details
are described in Sec. 3.3. The gas supply in the central cell
reduced the T i||,IEA and Γi||,IEA in the RF2 operation. In theno-RF2 operation, the additional gas slightly decreased theT i||,IEA and increased the Γi||,IEA . Both values became similar to those with the RF2. In the absence of the gas puﬀ, theT i||,IEA values with and without the RF2 were ∼240 eV and∼140 eV, respectively. When the gas was supplied, T i||,IEAwas ∼130 eV in both the RF2 and no-RF2 operations.

3.2

Langmuir probe analysis

Figure 3 shows the typical LP current (Ip )-voltage (Vp )characteristics measured at t = 120-140 ms during theshots of IEA in the RF2 operation. The I p- Vp characteristics were analyzed by the standard procedure describedin Ref. [14]. To avoid the error resulting from sheath expansion, Iis was derived from the linearly fitted current atVp = Vf , where Vf is the floating potential. Meanwhile,the electron current Ies was determined at Vp = Vs , whereVs is the space potential. Having determined Ies and T e , ne
was determined by rearranging Eq. (2). To compensate themagnetic field eﬀect on the collection area of the electrons(and consequently on ne ), S was taken as the projected areaof the magnetic field line on the probe surface.
Figures 4 (a)-(e) show the time evolutions of T e , ne , Iisand Ies , and (Iis /Ies )2 obtained from the LP. Each dataset
was measured during the shots of the IEA, and the error
bars were obtained from the fitting errors. In the period
without the additional gas puﬀ in the central cell, T e was∼30 eV with the RF2 and ∼20 eV without the RF2. During

Fig. 4 Time evolutions of (a) T e , (b) ne , (c) Iis , (d) Ies , (e)(Iis /Ies )2 , (f) γT i|| evaluated using a Langmuir probe. Redcircles: with RF2; blue crosses: without RF2.

the same period, ne was ∼2×1016 m−3 and ∼1×1016 m−3
with and without the RF2, respectively. In both discharges,T e decreased when the additional gas was supplied. At thattime, ne slightly decreased with the RF2 and slightly increased without the RF2, such that the two ne converged.
In the period without the gas puﬀ, Iis and Ies were both
higher in the RF2 than in the no-RF2 operation. When thegas was supplied, the Iis values in both discharges werealmost identical. The determination of γT i|| in Eq. (3) crucially depends on the parameter (Iis /Ies )2 . The square ofthe current ratio was lower with the RF2 than without theRF2 throughout the discharge. Substituting the above parameters in Eq. (3), we obtain γT i|| . Figure 4 (f) shows
the time evolution of γT i|| obtained from the LP (γT i||,LP ).
When the gas was supplied in the central cell, γT i||,LP decreased over time with the RF2, but was relatively stablewithout the RF2. Similar tendencies were observed in T i||
evaluated by the IEA. On the other hand, in the period
without the additional gas puﬀ (t = 100-160 ms), γT i||,LPwith the RF2 was lower than without the RF2. The val-2401086-3

Plasma and Fusion Research: Regular Articles

Volume 14, 2401086 (2019)

Fig. 5 Time evolutions of (a) evaluated ion polytropic coeﬃcient and (b) ne,LP /T i||,IEA2 , which is proportional to collisionality. Red circles: with RF2; blue crosses: without
RF2.

ues of γT i||,LP with and without RF2 were ∼440 eV and∼620 eV, respectively.

3.3

Evaluation of γ using the Langmuirprobe and ion energy analyzer

To obtain γ, we assumed that the T i|| values near theLP and IEA were nearly identical, and divided γT i||,LP by
T i||,IEA . Figure 5 (a) shows the time evolution of the evaluated γ. In the RF2 operation, γ was ∼1.8 in the first period,increasing to ∼2.5 during the gas supply. In no-RF2 operation, γ was approximately 4.4. Since there are large errors
in γ due to the fitting errors of the LP, we qualitatively analyze γ with and without RF2 in the period without theadditional gas.
First, we consider the characteristics of γ in the widelyused plasma fluid model [2,3,13]. In this model, γ depends
on the ion-ion collisionality L/λi-I ∝ n/T i2 . As n/T i2 increases, γ decreases from 3 to 1. Figure 5 (b) shows thetime evolution of ne,LP /T i||,IEA2 , the quotient of ne measured by the LP and T i||,IEA2 measured by the IEA. Thene,LP /T i||,IEA2 values were almost unchanged by the additional RF2 system. The collision times in both cases (withand without the RF2) were ∼400 ms or longer. As the collision time markedly exceeded the discharge time (200 ms),the plasmas in both cases were always collisionless. Therefore, the diﬀerent γ values between the RF2 and no-RF2
operations were caused by other factors.
Next, we consider the eﬀect of the high-temperaturecomponent of the ions on γ, mentioned in Ref. [9]. TheFig. 6 Normalized ion energy spectra (a) with RF2 and (b) without RF2, measured by the IEA. The normalization factor
was the peak value of the corresponding spectrum.

authors of [9] suggested that a high T i component in collisionless plasma would reduce γ below its value in thewidely used model. Figures 6 (a) and (b) show the energy spectra obtained by the IEA in the RF2 and no-RF2operations, respectively. The spectra were collected fromt = 120 to 140 ms and normalized by their peak values.
Without RF2, the spectrum was a single Maxwellian distribution of T i|| ∼150 eV, whereas the RF2 spectrum wasthe convolution of two Maxwellian distributions of T i||∼130 eV and ∼570 eV. Together with the result in Ref.
[9], this result suggests that γ was lowered by a hightemperature ion component resulting from the ICRF heating.
Without the RF2, the evaluated γ exceeded 3.0, theupper limit in the plasma fluid model. In order to clarify
the reason for this discrepancy, it requires a more quantitative investigation. In this study, we assumed that the T i||values were almost the same at the LP and the IEA. If
there was a large T i|| gradient in the setup without RF2, theevaluated γ would change. The spatial distribution of T i||
can be investigated using numerical simulations. In addition, the measurements were performed in an expanding
magnetic field. In such a magnetic field, ion flow acceleration [15] and a diamagnetic property of electrons [16]have been confirmed in the lower T i plasma. Elucidatingthese eﬀects in the high T i|| plasma also requires a kinetic
simulation. These investigations are future works.

2401086-4

Plasma and Fusion Research: Regular Articles

Volume 14, 2401086 (2019)

4. Summary

Acknowledgments

We experimentally investigated the eﬀect of high T ion the ion polytropic coeﬃcient (γ). Utilizing the highT i plasma and ICRF heating system in the GAMMA 10tandem mirror, the T i|| was varied in the end region ofGAMMA 10, and its eﬀect on γ was evaluated. The γT i||
was determined using a LP, assuming that the ion drift
speed was the sound speed. The T i|| was evaluated by an
IEA installed downstream of the LP. Assuming almost
identical T i|| values at the LP and IEA positions, γ was evaluated by dividing γT i||,LP by T i||,IEA . The T i|| in the end region was varied by applying ICRF heating (RF2), and supplying an additional gas puﬀ in the central cell. The T i||,IEA
was higher with the RF2 than without the RF2, and its time
evolution was similar to that of the obtained γT i||,LP . Theevaluated γ was lower with the RF2 than without the RF2.
In a widely used plasma fluid model, γ depends on theion-ion collisionality; however, the plasmas in the present
study were collisionless in both the RF2 and no-RF2 operations. In the RF2 operation, the ion energy spectra obtained by the IEA had a high T i|| component. The valueof γ evaluated in this experiment might have been lowered
under the eﬀect of a high-temperature ion component resulting from ICRF heating, as also mentioned in [9].

This study was performed with the support of NIFS
Collaborative Research Program (NIFS16KUGM119).
[1] D. Bohm, The Characteristics of Electrical Discharges in
Magnetic Fields, edited by A. Guthry and R.K. Wakerling(McGraw-Hill, New York, 1949).
[2] E. Zawaideh et al., Phys. Fluids 29, 463 (1986).
[3] K.-U. Riemann, J. Phys. D: Appl. Phys. 24, 493 (1991).
[4] S. Robertson, Phys. Plasmas 16, 103503 (2009).
[5] H. Ghomi and M. Khoramabadi, J. Plasma Phys. 76, 247(2010).
[6] J. Ou and J. Yang, Phys. Plasmas 19, 113504 (2012).
[7] M.M. Hatami, Phys. Plasmas 20, 083501 (2013).
[8] J.I.F. Palop et al., J. Phys. D: Appl. Phys. 37, 863 (2004).
[9] B. Lin et al., Phys. Plasmas 23, 083508 (2016).
[10] Y. Nakashima et al., Nucl. Fusion 57, 116033 (2017).
[11] Y. Nakashima et al., Fusion. Sci. Technol. 68, 28 (2015).
[12] Y. Sakamoto et al., Rev. Sci. Instrum. 66, 4928 (1995).
[13] P.C. Stangeby, The Plasma Boundary of Magnetic Fusion
Devices (IOP, Bristol, 2000).
[14] I.H. Hutchinson, Principles of Plasma Diagnostics secondedition (Cambridge University Press, 2002).
[15] M. Inutake et al., J. Plasma Fusion Res. 78, 1352 (2002).
[16] K. Takahashi et al., Phys. Rev. Lett. 120, 045001 (2018).

2401086-5