Determination of the ion distribution function during magnetic reconnection in the versatile toroidal facility with a gridded energy analyzer

DETERMINATION OF THE ION DISTRIBUTION FUNCTION DURING MAGNETIC RECONNECTION IN THE VERSATILE TOROIDAL

FACILITY WITH A GRIDDED ENERGY ANALYZER

by

Jonathan H. Nazemi

B.S. Physics

Towson University, 2000

SUBMITTED TO THE DEPARTMENT OF PHYSICS

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN PHYSICS

AT THE

MASSACHUSETTS INSTITUTE OF TECHNOLOGY FEBRUARY 2003

© Massachusetts Institute of Technology, 2002. All Rights Reserved.

Author………

Department of Physics September 3, 2002

Certified by ………

Ambrogio Fasoli Assistant Professor of Physics Thesis Supervisor

Accepted by ………...

Thomas J. Greytak Professor of Physics Associate Department Head for Education

DETERMINATION OF THE ION DISTRIBUTION FUNCTION DURING MAGNETIC RECONNECTION IN THE VERSATILE TOROIDAL

FACILITY WITH A GRIDDED ENERGY ANALYZER by

Jonathan H. Nazemi

Submitted to the Department of Physics on September 3, 2000 in Partial Fulfillment of the Requirements for the Degree of

Master of Science in Physics

ABSTRACT

A gridded energy analyzer (GEA) diagnostic and associated electronics are designed and built to explore the evolution of the ion distribution function during driven magnetic reconnection in the Versatile Toroidal Facility. The temporal evolution of the ion characteristic is measured at different locations throughout the reconnection region, for a number of magnetic field configurations. The measured ion characteristics are found to be in excellent agreement with a theoretical fit constructed from a double Maxwellian distribution, from which the temperatures and drift velocities are found as functions of space and time. It is found that the ion temperature of each Maxwellian exhibit minor temporal variations during reconnection which are negligible in terms of ion heating. Additionally, the temperatures do not significantly change with varying radial position, or magnetic cusp strength. The drift velocities are observed to evolve in time, scale with magnetic cusp strength, and to depend on the exact location throughout the reconnection region. The ions are thus subject to acceleration due to the electric field induced by the ohmic drive. The appearance of a double Maxwellian distribution during the reconnection drive is hypothesized to be due to double ionization of argon atoms. Repetition of the experiment with a hydrogen plasma verified that this scenario is most probable.

Thesis Supervisor: Ambrogio Fasoli Title: Assistant Professor of Physics

Table of Contents

Index of Figures and Tables...4

1. Introduction...5

2. Driven Magnetic Reconnection in the Versatile Toroidal Facility...7

2.1 Driving Mechanism...8

2.2 Run Procedure...8

2.3 VTF Magnetic Field Characterization...9

2.4 Plasma Behavior ...10

3. Gridded Energy Analyzer Theory...12

3.1 Theory of operation...12

3.2 Interpretation of Characteristic...14

3.3 Determination of Temperature and Drift Velocity...15

3.4 Determination of Uncertainty in Ic...17

3.5 Interpretation of Measurements from Aperture, Grid 1, and Langmuir Tip...18

4. Gridded Energy Analyzer Design...19

4.1 Mechanical Design ...19

4.2 Electrical Design...20

4.3 Data Acquisition Control...23

5. Experiment...24

5.1 Experimental Configuration...24

5.2 Experimental Procedure...25

5.3 Example Results ...25

5.4 Necessity of Plotting Icollectorvs. (fret- ffloating)...27

6. Results ...28

6.1 Initial Observations of the Argon Characteristic...28

6.3 Target Argon Plasma Characterization ...30

6.5 Modification of Initial Fit...32

6.6 Testing of Modified Fit ...33

6.7 Application of Modified Fit to Raw Data...34

6.8 Hypothesis for the Existence of a Double Maxwellian Distribution Function...38

7. Discussion...41

8. Conclusion...42

Index of Figures and Tables

1. Introduction

Figure 1. 1 Illustrating a magnetic reconnection event...5

2. Driven Magnetic Reconnection in the Versatile Toroidal Facility Figure 2. 1 Cross section of VTF. . ...7

Figure 2. 2 Cross section of VTF showing the ohmic coils, and the induced electric field...8

Figure 2. 3 Typical order of events for one shot...9

Figure 2. 4 Example of data showing the plasma response to ohmic pulse...11

Table2. 1 Typical VTF parameters. ...7

3. Gridded Energy Analyzer Figure 3. 1 Demonstrating the operation of the gridded energy analyzer...12

Figure 3. 2 Example of theoretical chracteristic...14

4. Gridded Energy Analyzer Design Figure 4. 1 Cross section of gridded energy analyzer. ...19

Figure 4. 2 Side view of GEA. ...20

Figure 4. 3 Current to voltage converter. ...20

Figure 4. 4 Complete schematic of GEA electronics ...21

Figure 4. 5 LabView acquisition and display interface. ...23

5. Experiment Figure 5. 1 Explicitly shows the probe position for each value of R considered...24

Figure 5. 2. Example of raw data collected from one shot...26

6. Results Figure 6. 1 Five characteristics in time throughout the ohmic pulse. lo= 9m, R = 3cm...28

Figure 6. 2 Same form as Figure 6.1, with .lo= 4m and 15m, R = 0.972m...29

Figure 6. 3 Plot of Ar characteristic from raw data with lo= 9m, R = 0.0 cm ...30

Figure 6. 4 Plots of fit paramters vs. lofor all R...31

Figure 6. 5 Argon characteristic for lo= 7m, R = 0cm, at t = 80ìs after the ohmic pulse...32

Figure 6. 6 Stairstep characteristic with double hump distribution function theoretical fit...33

Figure 6. 7 Surface plot showing the evolution of the distribution function ...34

Figure 6. 8 Array of distribution surfaces for argon...35

Figure 6. 9 Histogram plot of temperature for the primary distribution ...36

Figure 6. 10 Histogram plot of temperatures for the secondary distribution ...36

Figure 6. 11 Four plots in time, each showing vdriftvs. lofor the primary distribution...37

Figure 6. 12. Three plots in time, each showing vdriftvs. lofor the secondary distribution...37

Figure 6. 13 Array of distribution function surface plots for H2(compare to figure 6.8)...39

Figure 6. 14 Histogram plot of the hydrogen ion temperature throughout the ohmic pulse...39

1. Introduction

When two opposing magnetic field lines tear and reconnect (figure 1.1), there is a rearrangement of the topology of the magnetic field, called magnetic reconnection.

Figure 1.1 Illustrating a magnetic reconnection event: (a)before reconnection (b)during reconnection, formation of the diffusion region (c)resultant rearrangement of the magnetic field topology.

Like many processes in the physical world, the picture of magnetic reconnection is elegantly simple. Nonetheless, when one tries to unravel the underlying nature of the process, a formidable task is realized.

In a vacuum, the process is trivial. In the presence of plasma, however, magnetic reconnection occurs via the violation of the ‘frozen-in-flux’ condition. ‘Frozen-in-flux’ is a statement which simply implies that plasma is fixed to the magnetic field. Consider the magnetic diffusion equation, B B v B 2 o ) ( t +Ñ´ ´ =mh Ñ ¶ ¶ _{, (1. 1)}

where v is the fluid velocity, and h is resistivity. A convenient parameter given by the ratio of the convective term (Ñ´(v ´B)) to the diffusive term ( 2B

o Ñ

mh ) is the magnetic Reynolds number Rm. With Ñ = 1/L, v = L/t, Rmis given by ht m = o 2 m L R . (1. 2)

When the convective term dominates (large Rm), then the plasma is ‘frozen’ to the magnetic field

lines, moving with the field, and equation 1.1 becomes 0 ) ( t +Ñ´ ´ = ¶ ¶B _{v B . (1. 3)}

For small Rm (diffusion dominates), the plasma is allowed to break free of the magnetic field, giving

equation 1.1 as B B 2 o t =mh Ñ ¶ ¶ _{. (1. 4)} Diffusion region (a) (b) (c)

The consequence of equation 1.4 is the existence of a diffusion region as shown in figure 1.1(b). In this region, ÑxB ¹ 0 and hence gives rise to a current sheet which flows perpendicular to the plane of the magnetic field. Generally, the spatial scale of the diffusion region and the rate at which it forms (called the reconnection rate), are the subject of magnetic reconnection research.

Magnetic reconnection presents itself in nature mainly in the astrophysical world. In fact, it was in the examination of astrophysical phenomenon that magnetic reconnection was first truly considered. It was later found that reconnection exhibited itself in solar flares, the solar corona, and in the earth’s magnetosphere and geomagnetic tail, making it responsible for the aurora.1

Additionally, in Tokamak fusion devices, reconnection is thought to be the mechanism behind disruptions which cause saw-tooth oscillations.2

There have only been a few models put forth in the attempt to describe magnetic reconnection, the most well known of which is the Sweet-Parker3model. Although the Sweet-Parker

reconnection time is much greater than those found in nature, the model is straightforward and its solutions are relatively easy to derive, making it a favored pedagogical model for magnetic reconnection. The Petschek4 _{model was a reworking of Sweet-Parker that produced a shorter}

reconnection time. However, due to the complexity of the theory, few truly explored its consequences when it was first introduced5. Both of these are MHD models, which ignore terms in

the generalized Ohm’s law such as the Hall, electron inertia, and pressure gradient, leaving only the resistive MHD ohm’s law, E+v´B=hJ. It is generally thought that ignoring these terms leads to the discrepancies between the observed and theoretical values.

The experimental study of magnetic reconnection at the Massachusetts Institute of Technology is performed with the Versatile Toroidal Facility (VTF), which is described in detail in chapter 2. Diagnostics such as Langmuir probes, magnetic probes, and Rogowski coils have been used to characterize the plasma during magnetic reconnection in VTF. From these, the plasma density, current density, electron temperature, and magnetic field structure can be reconstructed over time. However, questions still exist as to the kinetic behavior of ions in VTF during the reconnection event. Of particular interest is the evolution of the temperature of the ions, which until now has yet to be measured in detail. Therefore, the motivation for measuring the ion distribution function simply lies in the uncertainty of ion characteristics not only during driven reconnection, but in the target plasma as well.

In efforts to further characterize the ions, a gridded energy analyzer (GEA) is introduced. The GEA diagnostic allows for the determination of the ion distribution function, and hence the temperature of the ions. This thesis details the design and construction of the gridded energy analyzer diagnostic and related electronics, the method of data collection and analysis, and results.

2. Driven Magnetic Reconnection in the Versatile Toroidal Facility

The experimental study of collisionless magnetic reconnection at the Massachusetts Institute of Technology is performed with the Versatile Toroidal Facility (VTF).6The VTF is a toroidal device

with a rectangular cross section, which is configured to provide a poloidal cusp field, and a toroidal guide field (fig.2.1). Using 2.45GHz microwaves at up to 50kW, the plasma is produced via electron cyclotron resonance heating, with a resonant magnetic field of 87.5 mT.

Figure 2. 1 Cross section of VTF. Ohmic coils not shown. R is measured from center of device. Some typical parameters of the target plasma produced in VTF for the study of magnetic reconnection are given in table 2.1. The plasma produced has a mean free path ~100m, which is much larger than the size of the device, thus providing a collisionless plasma.

Table2. 1 Typical VTF parameters.

Geometry/B-Field

Parameters Thermodynamic Properties

Plasma Properties

For Btor=0, B=10 mT, n=1018m-3 & singly ionized Ar:

Major radius Height width Bcusp Btor 0.94 1.08 0.68 m 0-60 mT 0-200 mT Te Ti ne base pressure ~20 eV 0.2-10 eV ~1017_-1018_m-3 ~10-7_Torr Alfvén speed vA vthe/ vthi fpe fce/ fci re /rI de=c/wpe / dI= c/wpi Lundquist number S Sweet-Parker time, tSP 3.5 104_m/s 1.4 106_{/ 1.5 10}3_m/s 9 GHz 560 MHz / 15 kHz 0.04 / 3 cm 3 mm / 0.5 m ~1000 ~100 ms

2.1 Driving Mechanism

In order to experimentally examine magnetic reconnection, a mechanism which drives opposing magnetic fields together is needed. By producing a toroidal electric field, cusp magnetic fields are driven together by invoking an ExBcuspdrift. To produce this electric field, an ohmic drive

consisting of 25 turns of 500 MCM copper wire forming a solenoid on the inside wall of the vessel is used. By discharging a capacitor through the solenoid, a toroidal electric field is produced inside the vessel. The ExBcuspdrift drives the fields together, resulting in magnetic reconnection.

Figure 2. 2 Cross section of VTF showing the ohmic coils, and the induced electric field. The mechanism (E x B) which drives magnetic reconnection is shown.

2.2 Run Procedure

Figure 2.3 shows a typical example of the VTF run procedure. In a run sequence (a ‘shot’), the toroidal and poloidal fields are activated approximately 1-3 seconds before the klystron. Once the klystron is on, a plasma is produced, and the data acquisition begins (DAQ). During data acquisition the ohmic pulse is triggered. The region of interest occurs with the formation and dissipation of a current sheet, and is approximately 0.1ms in duration7_{. This region — within which the induced}

electric field is constant ( ~constant dt

dIohmic _{) — is represented as the area bounded by the vertical}

red line.

I

E

-5.5 -4.2 0 6.8 12.1 16.8 Klystron

I Ohmic DAQ

t(ms)

Figure 2. 3 Typical order of events for one shot. The area of interest occurs at the start of the ohmic pulse, is approximately 0.1ms in duration, and lies within the vertical red line.

2.3 VTF Magnetic Field Characterization

The necessity of characterizing the configuration of magnetic fields in VTF stems from significant differences in the plasma behavior observed with varying cusp field strength relative to toroidal field strength. 8 To characterize the magnetic configuration, a parameter is introduced to

define the relative values of the toroidal and cusp field strengths. This qualitative parameter is a length lo defined by cusp toroidal o B B l Ñ

= . In VTF experiments, lo is considered an independent variable,

with values typically ranging from 0 to 30 meters. In practice, the voltage applied to the cusp coils, Vcusp, is the experimental parameter that controls lo. Typical values for Vcuspin this experiment range

2.4 Plasma Behavior

Figure 2.4 shows an example of the plasma response to an ohmic pulse. The data was taken with p~10-5torr, and l₀ = 7m in an hydrogen plasma3. A Langmuir probe array was used to

reconstruct the density and potential profiles (left and right columns), whereas data from a magnetic coil array was used to reconstruct the current sheet (center column). Time is given in ms, thus the time scale for the creation and dissipation of the current sheet is ~60ms.

The magnetic probe array* consists of 40 – 40 turn copper coils, orientated so 20 coils face

the R-direction, and 20 coils face the Z-direction (refer to figure 2.1). The probe is approximately 20cm in length and is oriented horizontally lengthwise in the R-direction. The operation of the probe is solely based on Faraday’s law, for when the magnetic flux through the coil changes, a voltage is measured across the coil. Due to the dynamical nature of VTF experiments, this type of probe provides an excellent way to measure the magnetic fields. The probe is scanned vertically approximately 30 cm through the plasma, resulting in a measurement of dB/dt in space and time. When integrated in time, B(t,R,Z) is recovered. Taking the curl of B reveals toroidal direction currents as shown in figure 2.4.

* _{The author significantly contributed to the construction of the magnetic probe array, and design and}

implementation of the data acquisition system. Details are not presented as they are beyond the scope of this thesis.

Figure 2. 4 Example of data showing the response of a hydrogen plasma to an ohmic pulse. A Langmuir probe array was used to measure density and potential. Data from a magnetic coil array was used to reconstruct the current sheet.

3. Gridded Energy Analyzer Theory

3.1 Theory of operation

The ideal gridded energy analyzer (GEA) operates by first rejecting all incoming electrons, allowing a current consisting only of ions to enter the analyzer. This ion current, which consists of a distribution of energies, is then subjected to a retarding grid which is biased to allow only ions with sufficient energy to pass the barrier. The resulting current is then measured by the collector. By measuring the collector current as a function of the retarding potential, a characteristic curve can be constructed. Analysis of this relationship yields the energy (or velocity) distribution function of the ions, from which the moments of different order, such as drift velocity and temperature, can be directly calculated.

In practice, however, secondary electrons are emitted from the grids as they are struck by ions. These are eliminated with the introduction of a secondary electron repeller grid (grid 3 in figure 3.1) between the retarding grid (grid 2 in figure 3.1) and the collector. The secondary electron repeller is biased sufficiently negative to repel secondary electrons. The collector is also biased negative, but positive with respect to the secondary electron repeller. This insures that any secondary electrons emitted by the collector are returned to the collector, and do not contribute to the current flow.

Figure 3. 1 Demonstrating the operation of the gridded energy analyzer. Grid 1.) electron repeller. Grid2.) retarding grid. Grid 3.) secondary electron repeller. Collector.) measures resulting ion current.

The GEA used in this experiment has an aperture grid before grid 1, which is grounded to the vessel wall. The vessel wall provides the ground reference for which all potentials are measured.

For a one dimensional distribution given by

ò

¥

-= (fv)dv

n , where n is the density, the current measured by the collector is expressed as

ò

Collector. Biased positive with respect to grid 3 to reabsorb secondary electrons.

Where Z is the degree of ionization, e is charge, A the area of the entrance (aperture), and v the velocity. The measurements are made parallel to the magnetic field, thus enforcing the one-dimensional nature of the analysis, giving for the energy of an ion in the plasma,

p 2 _Ze mv 2 1 E= + f , (3. 2)

wherefpis the plasma potential. From (3.2), it is found that v dv = dE/m, which when substituted

into (3.1) gives,

ò

Taking the derivative with respect to Emin, with f( ) = 0,

÷ø ö çè æ -= 2(Emin_m-Zef ) min p f m ZeA dEdI . (3. 4)

A sheath will exist when the plasma potential is significantly larger than the potential of the aperture, demanding that all ions entering the sheath have a minimum velocity corresponding to the sound speed ÷÷ø ö ççè æ = i e s ZT_m

c . Therefore, ions just outside the analyzer have a minimum energy

p 2

min min ₂1mv Ze

E = + f , (3. 5)

and ions at the retarding grid have a minimum kinetic energy =

2 ZT mc 2 1 2 e s = , giving r e min ZT₂ Ze E = + f . (3. 6)

Conservation of energy requires that equation 3.5 equals equation 3.6, giving for vmin,

i s r p r e i min 2_mZe T_2e ( ) 2Ze(_m ) v ÷ = f -f ø ö ç è æ _{+ f -f} = (3. 7) where e 2 Te p s =f

-f is the sheath potential. At this point it can be seen that any data points taken for when fr <fs, are not valid for the analysis of the distribution function. From (3.6),

r min Zed

dE = f , giving for equation (3.4) ) v (f mA Ze ddI min 2 r = -f . (3. 8)

The ion distribution function can now be expressed as a function of the retarding potential, dropping the subscript on vmin,

r 2_{A d}dI Zem ) v (f f -= . (3. 9)

Experimentally, there is an additional factor that must be included in eq.(3.9), this is the transparency of the grid material, T. In the ideal analyzer, the transparency of each grid is 100%, however, this is not the case in practice. The current I must be corrected for by considering that the actual current at the collector is reduced by TN where N is the number of grids, assuming each grid

has the same transparency. In this experiment, the GEA has 4 grids, therefore, the collector current Icis related to the current at the aperture by

Ic= T4I. (3. 10)

Using (3.10) and (3.9), the distribution function is finally expressed as

r c 2 4_{e A d}dI ZTm ) v (f f -= . (3. 11) 3.2 Interpretation of Characteristic

Figure 3.2a shows an example of an ideal characteristic calculated for ions with a Maxwellian distribution, Ti = 2eV, Te = 10eV, andfp =20V. The solid line represents I (normalized) vs. fret,

and the dashed line represents f(v) (normalized) vs. fret.

14 15 16 18 20 22 24 26 28 30 0 0.2 0.4 0.6 0.8 1 f_ret I vs. f_ret f(v) vs. f_ret 0 0.2 0.4 0.6 0.8 1 f_retµ v2 I vs. f_ret f(v) vs. f_ret 0 851 1702 2554 3405 4256 5107 5958 6809 7661 8512 0 0.2 0.4 0.6 0.8 1 v (m/s) I vs. v f(v) vs. v

Figure 3. 2 . a.)Plot of normalized Icvs. fret(solid), with normalized f(v) vs. fret(dashed). Ti=2eV,

Te=10eV, fp = 20V. b.) Same as (a), with tick marks explicitly showing the relationship

between retarding potential and velocity. c.)Plot of data versus velocity, revealing the Maxwellian shape of the distribution function.

a.)

b.)

Equation 3.11 could be interpreted such that the plot of dIc/dfretvs. fretyields a Maxwellian

distribution. Doing so leads to serious error in the estimation of the temperature, as can be seen in figure 3.2a. Assuming that figure 3.2a is Maxwellian might lead one to visually estimate an ion temperature ~ 4eV (determined from distribution to the right of maximum), although it is known that Ti = 2eV. For reference, figure 3.2b explicitly shows the relationship between fret and velocity.

Finally, figure 3.2c shows f(v) versus velocity, recovering the Maxwellian shape of the distribution function.

3.3 Determination of Temperature and Drift Velocity

Assuming a Maxwellian distribution, the relationship between collector current and retarding potential can be expressed by

) T e Z exp( Ic µ - fret i . (3. 12)

It must be stated, however, that equation (3.12) is only valid in the region to the right of the inflection point of the characteristic. Taking the natural log of equation (3.12) gives,

constant Ti Ze ) I ln( ret c =- f + . (3. 13)

Finally, to solve for Ti, one could do a linear fit to (3.13), or take the derivative with respect to the

retarding potential, 1 ret c i Ze d_dln(I ) T -÷÷ø ö ççè æ f -= . (3. 14)

Inevitably, there exists some offset in the measurement of the characteristic which is not easily eliminated, meaning that the current will more likely follow

offset i

ret

c Aexp( Ze T ) I

I = - f + , (3. 15)

where A is a constant. This will lead to no linear relationship between Icand the retarding potential.

Consider the log of equation 3.15,

÷ ø ö ç è æ _{+ f} + f -= ÷÷ø ö ççè æ _÷ ø ö ç è æ _{+ f} f -= ) T e Z exp( A I 1 ln T Ze ln(A) ) T Ze exp( A I 1 ) T e Z exp( A ln ) I ln( i ret offset i ret i ret offset i ret c with exp(Ze T ) 1 A I i ret

offset _{f << , the natural log can be expanded, giving,}

1 ) A ln( T Ze ) T e Z exp( A I ) I ln( i ret i ret offset c » f - f + + , (3. 16)

the derivative of which gives,

i ret offset c) ZeI _{exp(Ze T )} Ze I ln( d _{= f - . (3. 17)}

Therefore, one finds an equation of the same form of equation 3.15, and is in no better position of finding the ion temperature. Of course, taking the derivative of equation 3.15 first would eliminate the offset, then taking the natural log of the negative derivative of Icwould give,

÷÷ø ö ççè æ + f -= ÷÷ø ö ççè æ f -i i ret ret c T AZe ln T Ze ddI ln , (3. 18)

revealing a linear relationship from which Tiis easily found.

Unless Ic is sufficiently smooth, taking the derivative may lead to a very erratic function.

Additionally, if Ic is not monotonically decreasing, there exists the possibility of taking the log of a

number which is equal to or less than zero. As these problems are often the case with the data taken in this experiment, a ‘brute force’ method of finding Ti and other fit parameters is utilized. This

method is simply to assume a functional form for Ic, and use a least squares test to find the

parameters which provide the best fit. The fit can then be checked by performing a c2test.

The functional form of Ic is given by equation 3.1,

ò

Maxwellian ions, f(v) is given as

÷÷ø ö ççè æ- -= T 2 ) v v ( m exp ) v (f o 2 _{. (3. 19)}

From a numerical analysis point of view, it is more convenient to use a normalized form for the collector current, and then scale this current by the experimental measurements. If this scaling factor is given by C, then the collector current can be expressed as

÷÷ ÷ ÷ ÷ ÷ ø ö çç ç ç ç ç è æ ÷ ÷ ø ö ç ç è æ -=

ò

ò

The fit is performed by first supplying ranges for the values of Ti, and vo. A function S of these

values found from

(

)

å

where Ic is the measured current, results in a ‘surface’. The minimum of S is the point where that

particular combination of Ti, and voprovide the best fit. The sum goes over the range of where the

To test the fit, consider the c2test,

( )

å

where dIc is the uncertainty in measuring Ic. If c2 £N, then the fit is acceptable. If c2 >>N, then

the fit is unacceptable, and is rejected. In practice, a fit is rejected if _c2 _³2N_.

3.4 Determination of Uncertainty in Ic

As will be seen in chapter 4, the collector current is determined by measuring the voltage of a sense resistor, and applying Ohm’s law, Ic= V/R. Solving for the uncertainty in Icas the magnitude

of the differential gives,

2 c 2 c c I_V V I_R R I ÷ ø ö ç è æ _d ¶ ¶ + ÷ ø ö ç è æ _d ¶ ¶ = d (3. 23)

where dV and dR are the uncertainties in voltage and resistance measurements. Finally,

2 c 2 c _R1 V I_R R I ÷ ø ö ç è æ_{- d} + ÷ ø ö ç è æ d = d (3. 24)

is the uncertainty in Ic which is used to calculate c2 in eq. (3.22). dV is determined from the

resolution of the data acquisition device, which is an 8-bit digitizer with a range of 10.24V, giving dV = 10.24/28~0.04V. dR is determined from the resolution of the setting of the ohmmeter which is

used to measure R, hence dR ~ 50W. In this experiment, a sense resistance of R = 10200W is used. The relative uncertainty of R is thus 0.5%.

3.5 Interpretation of Measurements from Aperture, Grid 1, and Langmuir Tip

Measurements from the aperture and grid 1 can also provide useful information when used in conjunction with a floating potential measurement. Considering Langmuir probe analysis, the region between the ion and electron saturation currents is described by

(

)

where Iisis the ion saturation current. For the aperture, fprobe= 0V, from equation 3.25 this gives for

the aperture current Iap

÷÷ø ö ççè æ f -÷÷ø ö ççè æ p -= e plasma e i is ap _T e exp m 2 Zm 1 I I (3. 26)

Given that the plasma and floating potentials are related through ÷÷ø ö ççè æ p + f = f e i e float p T_2eln _Z2m_m , (3. 27)

the aperture current becomes

(3. 28) The potential on grid 1 is sufficiently negative that the measured current can be considered the ion saturation current. Therefore, Iis= Igrid1. Using this in equation 3.28, and solving for ffloatgives

÷ ÷ ø ö ç ç è æ -= f 1 grid ap e float T_e ln _II . (3. 29)

Since the ffloat, Iap, and Igrid1 are all measured, then the electron temperature can be found from

equation 3.29. ÷÷ø ö ççè æ f -» ÷÷ø ö ççè æ f -÷÷ø ö ççè æ _- p -= p ÷÷ø ö ççè æ f -÷÷ø ö ççè æ p -= ÷ ÷ ø ö ç ç è æ ÷÷ø ö ççè æ _p + f -÷÷ø ö ççè æ p -= e float is ap e float i e is i e e float e i is i e float e e i is ap T e exp I I T e exp mm 2 Z 1 I mm 2 Z T e exp m 2 Zm 1 I mm 2 Z ln Te exp m 2 Zm 1 I I

4. Gridded Energy Analyzer Design

In designing the GEA, factors concerning its geometry — such as grid spacing and analyzer diameter — must be considered in relation to the parameters of the plasma. Other factors to consider are those which limit the electronics, such as the magnitude of the measured signal relative to noise and the resolution of the digitizer.

4.1 Mechanical Design

The GEA (figure 4.1) consists of four grids, a collector, and a Langmuir probe, each of which are biased relative to the vessel wall. Its housing is constructed from 99.8% pure aluminia

Figure 4. 1 Cross section of gridded energy analyzer.

ceramic tubing with ID 12.57mm , and OD 15.88mm. The grid material is 150 lines per inch tungsten mesh, with line thickness 24.5mm, and grid spacing 144mm, giving a transparency of 73%. The grids are held with one washer on each side, which are soldered together around the joint. The washers are steel shortening shoulder screw shims with ID 9.55mm, OD 12.2mm, and thickness 0.508mm. The grids are separated by ceramic spacers with ID 9.55mm, OD 12.2mm, and average thickness 0.74mm.

To avoid cross-talk between adjacent grids, the grids must be spaced a few Debye lengths apart. This insures that there is enough space for the plasma to shield adjacent potentials. With Te~20eV, and n~1017m-3, the Debye length ~ 0.33mm. Inside the probe, however, the density is

reduced due to size of the sampling orifice, and transparency of the grids. For an estimate of the Debye length inside the probe, consider that after the first grid of transparency T, the Debye length is increased by T-1/2, giving l_Debye~ 0.39mm. With the average grid spacing of 1.35mm(~4l_Debye), the

grids are spaced sufficiently apart to allow for shielding of the bias potentials.

The Langmuir tip — used to measure the floating potential— is a stainless steel ball of diameter 2.03mm, located 4.55mm from the end of the probe face. The ball is mounted on the end of a ceramic tube, which is Torr-Sealed onto the bottom of the GEA housing. A wire soldered to the ball provides the electrical connection, which is guided through the ceramic tube, and up into the

grid ceramic spacer

9.55 mm

9.2 mm Langmuir probe

housing of the GEA. All electrical connections are then guided through a larger ceramic tube (figure 4.2), approximately one meter long, to a vacuum tight electrical feed-through.

Figure 4. 2 Side view of GEA.

The entire probe arm is mounted on a mechanical feed-through which allows for a wide range of motion. The probe is able to be rotated about its axis, raised and lowered, and pivoted radially relative to the torus.

4.2 Electrical Design

The goal of the electronics is to construct a circuit allowing for the biasing, and measurement of current for each grid. The unique nature of each grid being biased to a different potential presents a challenge, for the desirable design utilizes as few power supplies as possible. The final circuit uses one power supply which biases the aperture, grids 1 and 3, and the collector. It also provides the power for the op-amps used to measure the current from each grid. There is an additional power supply which is used to supply the variable voltage on grid 2. The fundamental circuit is the current to voltage converter shown in figure 4.3.

Figure 4. 3 Current to voltage converter.

The power supply at the non-inverting input supplies a bias at the inverting input, and hence to the grid. The input and output voltages are related through sense sense

in in

out _RV R IR

V = = . Evidently, the output voltage will be offset by the bias voltage. Therefore, an instrumentation amplifier is used to measure Vout, eliminating the bias voltage. The complete circuit is shown in figure 4.4.

3.45cm 1.275cm to feed-through Vout grid Rsense current: I Rin

1 Grid 3 1N2974A 10W 2 LF356 2 3 10V Zener 10200 3 2 4 6180 Collector -205V LF356 3 1N2974A 4 5 ₃₂₃₈ 6 5 3238 Grid 1 7 LF356 6 7 1N3003 10W 24W 30 9 30 Aperture LF356 10 10 82V Zener 2500 9 INA117 2 2000 2000 470pF INA117 3 2000 2000 470pF INA117 6 2000 2000 470pF INA117 2000 2000 470pF 1 20000 18240 -15V -15V -15V -15V 15V 15V 15V 15V 3245 -9V Battery 332 Grid 2 LF356 +9V Battery INA117 2000 2000 470pF -15V 15V +9V Battery -9V Battery INA117 2000 2000 470pF -15V 15V 1E6 Langmuir Probe Grid 2 variable bias Vout Vout Vout Vout Vout Vout

The aperture, grids 1,3 and the collector are all biased via the zener diode chain, which also provides the power to the LF356 op-amps. The zener diodes provide the +/- 10 volts to power the LF356’s relative to whatever the bias voltage is on that particular grid. The power supply is a Kepco BOP 500M bipolar operational power supply/amplifier set to –204 volts. This gives an aperture bias of 0V, grid1 bias -148V, grid3 bias -189V, and collector bias -178V.

Grid 2, because it has a variable bias, has a separate bias supply, and is powered by two 9 volt batteries. The bias supply is a Kepco BOP 100-2M bipolar operational power supply/amplifier, providing a max retarding potential of 100V. To increase the range, two Kepco BOP 100-2M’s are connected in series, providing up to 200V of retarding potential. This potential is controlled with a Sony Tektronix AFG310 function generator via GPIB.

The Langmuir probe is connected to a 1MW potentiometer, assuring that very little current will flow, yielding a measurement of the floating potential.

The outputs of all measurements are sent to a series of INA117 instrumentation amplifiers, powered by a separate supply providing +/-15V. These eliminate the common mode — the bias voltage — and output only the differential voltage. They were chosen for their maximum common mode rejection of +/- 200V. The inputs of each INA117 have a configuration of two 2kW resistors, and one 470pF capacitor which provide a low pass filter with cutoff ~100kHz.

One of the most common problem with electronics on the VTF is the existence of ground loops. Extensive troubleshooting was carried out to eliminate large ground loop signals due to the ohmic pulse that often were larger than the actual data measurement. Measurements on the GEA are on the order of mA, meaning that large sense resistors (~10kW) are required to see ‘healthy’ signals. Several steps were taken which resulted in the complete elimination of all ground loop signals from the main signal. The reduction of 4 power supplies (one for each grid) to 2 power supplies helped the problem somewhat, but the signal was still not acceptable. Introduction of the battery power supply for grid 2 electronics, and the use of INA117 instrumentation amplifiers to separate grounds from the probe electronics to the data acquisition system were the two main solutions which resulted in the elimination of ground loops.

4.3 Data Acquisition Control

After passing through the electronics, the current measurements from each grid and the Langmuir tip are sent to a LeCroy 2264 8-bit waveform digitizer, sampling at 500kHz. The sampling pulse is provided externally with a LeCroy 8501 programmable clock. The low pass filter in the electronics assures that sampling is performed only on data below the Nyquist frequency. As the digitizer has 8KB of memory, only 16ms of data can be sampled in one shot. The digitized data is read back from the Camac crate via GPIB into a standard consumer desktop PC.

LabView software — combined with built-in Matlab functionality — is used to control all data acquisition and storage. A virtual instrument interface was designed and programmed to measure data from all grids and present the data as shown in figure 5.2. The interface has a great degree of flexibility as a stand-alone interface, and can be used easily by a user with no prior LabView programming experience. The interface is shown below in figure 4.5. In addition to its acquisition routines, the interface also remotely controls the Sony function generator which controls the retarding grid voltage.

5. Experiment

The focus of this experiment is to examine the evolution of the distribution function of ions during magnetic reconnection in time and space, and to explore its relationship with the magnetic field configuration. The parameter lo, which describes the relationship between the guide and cusp

magnetic field strengths, is the defining variable for the magnetic field configuration, and hence several values of loare considered.

5.1 Experimental Configuration

Five different values of lo (4m, 7m, 9m, 12m, 15m) are examined, at 5 different probe

positions (R = r – ro= 0.0m, 0.03m, 0.06m, 0.09m, 0.12m, where ro= 0.942m, the radial distance to

the center of the machine ). Due to the relatively long probe arm, the Z displacement is not significant, as DZmax~1cm. Figure 5.1 shows graphically where the probe is positioned. Positions to the left of center (with respect to figure 5.1) were measured, however, it was determined that the signals collected were insufficient for proper analysis, as in the configurations considered in the present work, the plasma exists mainly in the region to the right of center.

0.5 1 1.5 2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 R Z

Figure 5. 1 Each probe position is shown, with the probe face drawn to scale as the red circles. The aperture faces clockwise from the top of the VTF (out of the paper). This direction is chosen such that the initial ohmic pulse results in ion flow into the probe. All measurements are taken in the toroidal direction. The constant parameters for the experiment are the gas pressure, (1.5E-5 torr), microwave power (15kW), and the electric field strength (~10V/m).

5.2 Experimental Procedure

The experiment is performed by first choosing a probe position and lo. A retarding potential

is set initially to zero, two shots are taken, then the potential is advanced by 5 volts to a final voltage of 180V. The parameter lo is then changed while the position is held constant. After scanning 5

values of lo, the position of the probe is changed, and the above procedure is repeated. In total,

~1850 shots are taken for the experiment. Logically, this procedure only amounts to a series of nested for loops which can be shown explicitly as:

for R = {0m 3m 6m 9m 12m} for lo= {4m 7m 9m 12m 15m}

for fret= 0 to 180 Volts (with DV = 5V)

take 2 shots next fret

next lo

next R

5.3 Example Results

Consider the configuration lo = 9m, and R = 3cm. Figure 5.2 shows a side by side

comparison of the results of three shots taken at different retarding grid potentials. Each column shows the current measured by each grid, and the floating potential measured by the Langmuir tip. The time is given with respect to the ohmic pulse. References to ‘target plasma’ refer to the plasma at a time before the ohmic pulse. At fret= 0V, the currents are at the maximum amount. At fret= 45V,

the collector current from the target plasma has already dropped off to its minimum amount, but the current spike still remains. At fret= 180V, it is observed that the current spike amplitude reaches its

-1000 -500 0 500 1000 I(mA) Aperature fret=0V -400 -200 0 200 400 I(mA) Grid1 -100 -50 0 50 100 I(mA) Grid2 -100 -50 0 50 100 I(mA) Grid3 -100 -50 0 50 100 I(_mA) Collector 0 5 10 -50 0 50 time(ms) Volts Langmuir fret=45V 0 5 10 time(ms) fret=180V 0 5 10 time(ms)

Figure 5. 2. Side by side comparison of raw data from three shots taken at fret= 0V, 45V, and 180V.

Data shown has parameters lo= 9m, and R = 3cm Time is given with respect to the ohmic

pulse. Each column of plots is the current measurements from the Langmuir probe and grids. It is observed how the collector current is reduced with increasing fret, as expected.

5.4 Necessity of Plotting Icollectorvs. (fret- ffloating)

In a plasma, all potential measurements are relative to the plasma potential. Referring to

equation 3.7, ÷ ø ö ç è æ _{+ f -f} = 2_mZe ₂T_e ( ) v e r p i

min , this fact is made explicit. In this experiment, however,

it is not the plasma potential which is measured, but the floating potential. These two potentials are related through ÷÷ø ö ççè æ p + f = f e i e float p T_2eln _Z2m_m , (5. 1)

giving for vmin,

÷ ÷ ø ö ç ç è æ ÷÷ø ö ççè æ ÷÷ø ö ççè æ p -+ f -f = e i e float r i min 2_mZe T_2e 1 ln _Z2m_m v . (5. 2)

If ffloat is not assuredly the same value from shot to shot, then it is not certain that the current

measured is truly that which relates to the particular choice of fret. In fact, ffloat is sufficiently

6. Results

6.1 Initial Observations of the Argon Characteristic

Figure 6.1 shows several characteristics of Argon over five different points in time throughout the ohmic pulse. The configuration is the same as for figure 5.2 (lo= 9m, and R = 3cm).

Plot 1 is during the target plasma period, and represents an ion temperature of approximately 2eV. In plot 2 the characteristic develops a ‘convex’ curvature, between 40V and 70V, as the corner of the characteristic moves outward. It will be shown that this curvature is due to the development of a double Maxwellian distribution function. Plots 3, 4, and 5 continue to show the evolution of the characteristic. -20 0 20 40 60 80 100 120 140 160 180 0 10 20 30 40 50 60 70 f_ret-f_floating I(mA) 1 2 3 4 5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 time(ms) I(mA) 1 2 3 4 5

Figure 6. 1 Plots showing five characteristics in time throughout the ohmic pulse, where t = 0 is the start of the pulse. lo= 9m, R = 3cm.

For comparison, figure 6.2 shows similar plots for the same position, at the same points in time, but with lo4m and 15m. One can immediately see how changing locan have a significant effect

on the characteristic. In figure 6.2.a., in which lo= 15m, it is observed how the characteristic evolves

in a less dramatic nature than in the lower limit of lo= 4m shown in figure 6.2.b. Obvious differences

-20 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 30 35 40 45 50 fret-ffloating I(mA) 1 2 3 4 5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 time(ms) I(mA) 1 2 3 4 5 Figure 6. 2 a. lo= 15m, R = 0.972m. -20 0 20 40 60 80 100 120 140 160 180 0 10 20 30 40 50 60 70 fret-ffloating I(mA) 1 2 3 4 5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 50 100 time(ms) I(mA) 1 2 3 45 Figure 6.2.b. lo= 4m, R = 0.972m.

6.3 Target Argon Plasma Characterization

Consider the case for argon plasma, lo= 9m, and R=0cm. Figure 6.3 below shows a plot of

the experimentally measured characteristic during the target period in red with gray error bars. The analytical fit from equation 3.20 is plotted in black. This shows that the first order moment of a Maxwellian distribution provides an excellent fit to the measured characteristic. The parameters for this fit are Ti= 1.3 eV, and vo= 7380m/s.

0 20 40 60 80 100 120 140 160 0 2 4 6 8 10 12 14 16 f_ret- f_float I

Figure 6. 3 Plot of Ar characteristic from raw data in red with lo= 9m, R = 0.0 cm. The analytical fit

is shown in black, with Ti = 1.3eV, and vo= 7380m/s. Error bars are shown in gray.

Analysis of the characteristics of target plasmas for all values of lo and probe position R

consistently produce acceptable fits as shown in figure 6.3. Fit parameters Ti(t, lo, R) and vo(t, lo, R)

are averaged over the target plasma period, returning Ti l(o,R)and vo l(o,R), which are then

plotted vs. lofor each probe position R. This is shown in figures 6.4a and 6.4b

In figure 6.4a, it is observed that the target plasma temperature decreases slightly with increasing lo. This is not the case for the drift velocity shown in figure 6.4b, which is observed to

2 4 7 9 12 15 17 0 1 2 3 4 l_o Ti (eV) R = 0cm R = 3cm R = 6cm R = 9cm R = 12cm

Figure 6.4a. Plots of Ti l(o,R)vs. lofor each R for the target argon plasma.

2 4 7 9 12 15 17 4 5 6 7 8 9 10 l_o v_o(km/s) R = 0cm R = 3cm R = 6cm R = 9cm R = 12cm

6.4 Argon Plasma Characteristic During Ohmic Pulse

It is observed that in some instances, as the characteristic evolves during the ohmic pulse, a secondary characteristic appears in the form of a ‘stairstep’ branching from the original. Consider the case of lo = 7m, R=0cm. At 80ìs after the ohmic pulse is initiated, the characteristic exhibits this

stairstep behavior. This is shown in figure 6.5. One could hypothesize that the most probable reason for this behavior is the existence of a double hump distribution function. Therefore, equation 3.20 needs to be modified to test this hypothesis.

-5 20 45 70 95 120 145 0 5 10 15 20 25 f_ret-f_float I_c Secondary characteristic Primary characteristic

Figure 6. 5 Argon characteristic for lo= 7m, R = 0cm, at t = 80ìs after the ohmic pulse is initiated.

The stairstep behavior of the characteristic is observed.

6.5 Modification of Initial Fit

To accommodate for a double Maxwellian distribution function, the form of f(v) becomes

÷÷ø ö ççè æ- -p -+ ÷÷ø ö ççè æ- -p = 2 2 2 o 2 1 2 1 o 1 2T ) v v ( m exp T 2m ) A 1 ( T 2 v ) v ( m exp T 2m A ) v (f (6. 1)

where A is a normalization factor which describes the relative amplitude of each distribution, and ranges from 0 to 1. Now, there are 5 fit parameters to consider. They are T1, T2, vo1, vo2, and A. Once

these parameters are chosen, equation 6.3 can be inserted into equation 3.20, giving the functional form of the characteristic.

6.6 Testing of Modified Fit

The application of the modified fit to the raw data must be performed to test the double hump distribution function hypothesis. Still considering the raw data from figure 6.5, the fit using equation 6.3 is shown below in figure 6.6. The fit parameters are T1 = 1.8eV, T2 = 0.9eV, vo1 =

8610m/s, vo2 = 18400m/s, and A = 0.25. These result in a distribution function as shown in the

lower left plot. The lower right plot shows the point in time where this characteristic is constructed.

-5 20 45 70 95 120 145 0 5 10 15 20 25 f_ret-f_float I_c -10 0 10 20 30 0 0.5 1 v_o(km/s) -1 -0.5 0 0.5 1 0 10 20 30 time(ms) I_c

Figure 6. 6 The double hump distribution function fit is shown in black against the raw data in red with gray error bars. The lower left plot is the resulting distribution function from the fit parameters, the lower right plot shows where this characteristic occurs in time. Fit parameters: T1= 1.8eV, T2= 0.9eV, vo1= 8610m/s, vo2= 18400m/s, and A = 0.25

It is shown that the double Maxwellian distribution hypothesis has produced a very acceptable equation of fit to the raw data. The application of this fit consistently applies to other occurrences of the stairstep characteristic.

6.7 Application of Modified Fit to Raw Data

Every 2ìs from -2ìs to 198ìs (relative to the ohmic pulse), a characteristic is constructed, and fit parameters are obtained. This produces a 3-D surface plot showing the evolution of the distribution function. Figure 6.7 shows this surface for the same data as considered thus far (lo=7m,

R=0cm).

Figure 6. 7 Surface plot showing the evolution of the distribution function for lo= 7m, R=0cm. Time

is relative to the ohmic pulse. The double hump evolution is clearly visible.

The appearance of a double Maxwellian distribution is clearly visible as it grows and decays between 60ìs and 150ìs. Data from all lo, and R are analyzed in this manner. The most enlightening way to

display the results of the analysis is to construct a figure consisting of a 5 surface by 5 surface array, whereby the surfaces can easily be compared to one another on the basis of lo and R. This is shown

Figure 6. 8 Array of distribution surfaces showing the cumulation of all data collection and analysis for Argon. The horizontal axis is velocity in km/s, and the vertical (perspective) axis is time in ìs. Each surface shows time: -2ìs to 198ìs, where 0ìs is the initiation of the ohmic pulse. Immediate observations of figure 6.8 indicate that the double hump behavior becomes more prevalent as lo decreases, and is seen most dramatically for R=3 and 6cm. An explanation for the

latter observation would be simply that the probe is moving through regions of varying plasma current density. The surfaces each exhibit similar initial temperatures during the target region, as previously stated Ti = 1.5 ±0.6eV.

The temperature of the primary distribution function does not dramatically change throughout the entire evolution. To show this, consider the temperature for all time, lo, and R. Each

surface consists of 100 points in time (200ìs), therefore there are 100 values for each fit parameter in each surface. As there are 5 values for loand R, this makes 100x5x5 = 2500 values for T1. All 2500

values of T1are binned to show the percent that they occur during the time shown in figure 6.8. This

0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 Ti(eV)dT = 0.25eV % occurences

Figure 6. 9 Histogram plot showing percent occurrences of temperature for the primary distribution during the time shown. This distribution exhibits Timean= 1.8, sTi= 0.9eV.

As illustrated above, the temperature of the primary distribution for all lo and R for ~ 22% of the

time is between 1 and 1.25eV. If N(Ti) is the number of bin elements between Ti and Ti + dTi, then the statistical average and standard deviation of Ti are given by

å

å

å

å

From the equations of 6.2, the temperature of the primary distribution is found to be Tip=1.8±0.9eV.

The same plot can be produced for the temperature of the secondary distribution, and is shown in figure 6.10. Using equation 6.2 for the secondary distribution gives Tis= 1.4 ± 0.9eV.

0 1 2 3 4 5 6 7 8 0 5 10 15 Ti(eV)dT = 0.25eV % occurences

Figure 6. 10 Histogram plot showing percent occurrences of temperature for the secondary distribution during the time that it exists. Ti = 1.4 ± 0.9eV

The ion temperatures are shown as histograms which incorporate all time, lo, and R to show

that there is no significant heating of ions during or after the ohmic pulse. If there were heating, then the histograms would not show a distribution localized about one value, but would rather be spread out amongst a wider range of temperatures.

Consider the drift velocity for both the primary and secondary distributions. Figure 6.8 demonstrates that in cases of longer lo, the primary distribution exhibits a drift when the ohmic pulse

is applied. As lois decreased, the predominant behavior is that the primary distribution tends to drift

less, as a secondary distribution begins to develop. Consider 4 points in time, t = 0, 50, 100, and 150ms. For each time, the drift velocity is plotted versus lo and shown in figure 6.11. The primary

distribution is found to drift with time, implying that ion acceleration occurs. The mechanism of the acceleration is the induced electric field of the ohmic pulse. The drift velocity exhibits a dependence on lothat appears close to linear.

4 7 9 12 15 0 10 20 30 l_o(m) t = 0ms v (km/s) 4 7 9 12 15 l_o(m) t = 50ms 4 7 9 12 15 l_o(m) t = 100ms 4 7 9 12 15 l_o(m) t = 150ms R = 0cm R = 3cm R = 6cm R = 9cm R = 12cm

Figure 6. 11. Four plots in time, each showing vdriftvs. lofor the primary distribution.

Similar plots are produced for the drift velocity of the secondary distribution, and are shown in figure 6.12. Ion acceleration and a drift velocity dependence on lois shown.

4 7 9 12 15 0 10 20 30 l_o(m) t = 50ms v (km/s) 4 7 9 12 15 l_o(m) t = 100ms 4 7 9 12 15 l_o(m) t = 150ms R = 0cm R = 3cm R = 6cm R = 9cm R = 12cm

These plots show that the secondary distribution has drift velocities which are approximately twice as much as the primary distribution. This ratio suggests that ions may be experiencing double ionization, which is further explored in section 6.8.

The dependence of the acceleration on lo is expected, as the physics of the formation and

evolution of the current sheet is related to lo.

6.8 Hypothesis for the Existence of a Double Maxwellian Distribution Function

Measured characteristics are successfully explained by considering a double Maxwellian distribution function. An interpretation for this unexpected behavior comes from the possibility that argon atoms be ionized more than once. As the application of an electric field (the ohmic pulse) would certainly push twice ionized argon atoms more than singly ionized argon atoms, it seems logical that a second velocity class of particles would emerge. To test this idea, the same experiment is performed with hydrogen. Since hydrogen can be ionized only once, the secondary distribution should not exist, proving that double ionization was the most likely cause of the double hump distribution function.

The result of the experiment and analysis is shown in figure 6.13. Note that the missing position, R = 12cm, is due to the extremely low signals measured in that region, making analysis unacceptable. Using equation 6.2, the temperature is found to be 1.5 ± 0.9eV (see figure 6.14), and is consistent throughout the ohmic pulse, i.e., negligible heating is observed. Turning back to figure 6.13, except for lo = 7m, R = 3cm, it is observed that a double hump distribution function is not

predominant in the evolution of the distributions for the same loand R as for argon (compare figure

Figure 6. 13 Array of distribution function surfaces for H2. 0 1 2 3 4 5 6 7 8 0 5 10 15 Ti(eV)dT = 0.25eV % occurences

Figure 6. 14 Histogram plot of the hydrogen ion temperature throughout the ohmic pulse, lo, and R.

Plots of the drift velocity vs. lo for R = 0, 3cm are shown in figure 6.15. Again, acceleration

is observed, and the relationship with lois evident.

4 7 9 12 15 0 10 20 30 l_o(m) t = 0ms v (km/s) 4 7 9 12 15 l_o(m) t = 50ms R = 0cm R = 3cm 4 7 9 12 15 l_o(m) t = 100ms 4 7 9 12 15 l_o(m) t = 150ms

Figure 6. 15 Plots of vDriftvs. lofor two positions for hydrogen. Ion acceleration is observed, and the

relation between vDriftand lois made explicit.

One major issue with performing the experiment with hydrogen is its susceptibility to be easily ‘dirtied’ by other particles. Since hydrogen is so light, any other impurities will exhibit themselves much more than in argon. At the time the experiment was performed, the VTF was considered fairly ‘dirty’, and the data from hydrogen was mostly reliable, but concerns were expressed about the purity of a hydrogen plasma. It could be argued that the data presented in figure 6.13 does not exhibit double ionization, and discrepancies are due to impurities.

7. Discussion

Observation of a ‘stairstep’ shape characteristic upon initiation of the ohmic pulse is found to be due to the evolution of a double Maxwellian distribution function. This phenomenon is only seen for lo<15m, and becomes more prevalent as lo is decreased. Additionally, it appears to be

spatially localized in a region of ~6cm. This spatial scale is in agreement with current sheet measurements (figure 2.4), which explicitly show the spatial scale in which toroidally flowing plasma is formed due to the ohmic pulse. Therefore, as the probe moves through this area, spatial variations of measured signal become evident. A theoretical equation of fit which is constructed from a double Maxwellian distribution function is found to be in excellent agreement with the raw data. From this fit, the temperature of each ‘hump’ of the double Maxwellian distribution is found.

The primary (original) distribution is observed to dynamically evolve from a single to a double Maxwellian distribution function upon execution of the ohmic pulse. Considering all time, R, and lo, minor variations in the primary distribution temperature are observed, however, these

variations are negligible in terms of ion heating. Additionally, the existence of a significant ion temperature gradient (in the 1 spatial direction within a 12cm range) can be eliminated. The temperature of the primary distribution in all time, R, and lois found to be Ti1= 1.8±0.9eV.

The appearance of the secondary distribution occurs with the application of the ohmic pulse. Like the primary distribution, temperature variations are observed, however are not significant to be labeled as ion heating. Furthermore, no substantial variations in the temperature as a function of R and loare observed. The secondary distribution exhibits a temperature of Ti2= 1.4±0.9eV.

Temporal variations in drift velocities for the primary and secondary distributions are seen, providing evidence for ion acceleration. An induced electric field from the ohmic pulse provides the acceleration mechanism. The drift velocity is found to scale with lo. This is expected, as the physics of

the formation and evolution of the current sheet is related to lo.

It is observed that for larger lo(lo>12m), application of the ohmic pulse leads to a drifting of

the primary distribution without the appearance of a secondary distribution. As lois decreased (Bcusp

increases), a double Maxwellian distribution becomes more predominant.

The appearance of the double Maxwellian distribution function is hypothesized to be explained by the existence of a significant percentage of argon atoms experiencing double ionization. This hypothesis is tested by performing the same experiment with hydrogen, eliminating the effect of double ionization. It is observed that a double hump distribution function is not predominantly observed for hydrogen. Therefore, it is concluded that it is most probable that there is double ionization occurring to argon atoms during magnetic reconnection for lo<15m.

8. Conclusion

A gridded energy analyzer was successfully designed, built, and used to measure the ion distribution function in the Versatile Toroidal Facility during magnetic reconnection. Ion temperatures as low as 0.1eV were successfully measured. Relying upon shot to shot plasma reproducibility, ion characteristic curves were successfully constructed in 2ms intervals. Variable positioning in the R and Z directions allow for a scannable area ~1440cm2. Additionally, the GEA

housing is capable of being rotated 360 degrees about the axis of the probe arm. The spatial resolution of the diagnostic is limited by aperture size to 0.72cm2_.

The experiment produced evidence that argon ions are ionized twice under the action of the induced electric field associated with driven reconnection for relatively low values of the ratio between guide and cusp magnetic field. To verify the double ionization hypothesis, the experiment was reproduced with hydrogen. All but one configuration confirms this hypothesis, in that no double hump distribution is observed. The case of a double hump distribution in hydrogen plasma could be related to the presence of impurities due to contamination from plasma facing materials and warrant additional experimental investigations.

It is observed that ion energization occurs via drifts without significant thermal heating during reconnection. Ion acceleration is seen for both primary and secondary distributions, and is due to the electric field induced by the ohmic pulse. The dynamics of the plasma response to the reconnection electric field is a major part of ongoing investigations on VTF. The GEA diagnostic contributes to the study by providing large amounts of information, from the ion distribution to its moments, such as drift velocity and temperature. These quantities, in particular their dynamical evolution and spatial structure, play a key role in the physics of reconnection.

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