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
2
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 Icollector vs. (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
References .......................................................................................................................................................... 43
3
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. lo for 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 vdrift vs. lo for the primary distribution................. 37
Figure 6. 12. Three plots in time, each showing vdrift vs. lo for 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
Figure 6. 15 Plots of vdrift vs. lo for two positions for hydrogen....................................................... 40
4
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.
Diffusion region
(a)
(b)
(c)
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
h 2
+ Ñ ´ ( v ´ B) =
Ñ B,
¶t
mo
(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 (
h 2
Ñ B ) is the magnetic Reynolds number Rm.
mo
With Ñ = 1/L, v = L/t, Rm is given by
Rm =
m o L2
.
ht
(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
¶B
+ Ñ ´ ( v ´ B) = 0 .
¶t
(1. 3)
For small Rm (diffusion dominates), the plasma is allowed to break free of the magnetic field, giving
equation 1.1 as
¶B h 2
=
Ñ B.
¶t m o
(1. 4)
5
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-Parker3 model. 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 introduced 5. 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.
6
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).6 The 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.
Geometry/B-Field
Parameters
Major radius
Height
width
Bcusp
Btor
0.94
1.08
0.68 m
0-60 mT
0-200 mT
Thermodynamic Properties
Te
Ti
ne
base pressure
Table2. 1 Typical VTF parameters.
~20 eV
0.2-10 eV
~1017-1018 m-3
~10-7 Torr
Plasma Properties
For Btor=0, B=10 mT, n=1018 m-3
& singly ionized Ar:
Alfvén speed vA
3.5 104 m/s
vthe / vthi
1.4 106 / 1.5 103 m/s
fpe
9 GHz
fce / fci
560 MHz / 15 kHz
0.04 / 3 cm
re /rI
3 mm / 0.5 m
de=c/wpe / dI = c/wpi
~1000
Lundquist number S
~100 ms
Sweet-Parker time, tSP
7
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 ExBcusp drift. 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 ExBcusp drift drives the fields together, resulting in magnetic reconnection.
E
Iohmic
ExBcusp
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 (
red line.
dI ohmic
~ constant ) — is represented as the area bounded by the vertical
dt
8
DAQ
I O hmic
Klystr on
-5.5 -4.2
0
6.8
t(ms)
12.1
16.8
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 l o =
B toroidal
ÑB cusp
. 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 Vcusp in this experiment range
from 7V( lo~20m), to 17V(l0~ 7m ).
9
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.
10
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.
11
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.
Grid 1. Biased
negative to repel
electrons
Grid 2. Sweeps
positive to select
ions.
Grid 3. Biased
negative to repel
secondary electrons
Collector. Biased positive
with respect to grid 3 to
reabsorb secondary
electrons.
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 n =
¥
ò f ( v )dv , where n is the density, the
-¥
current measured by the collector is expressed as
I = ZeA
¥
ò vf ( v )dv .
v min
(3. 1)
12
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 onedimensional nature of the analysis, giving for the energy of an ion in the plasma,
1
mv 2 + Zef p ,
(3. 2)
2
where f p is the plasma potential. From (3.2), it is found that v dv = dE/m, which when substituted
E=
into (3.1) gives,
ZeA
I=
m
¥
2( E - Zef p ) ö
f æç
÷dE
m
è
ø
E min
Taking the derivative with respect to Emin, with f( ) = 0,
ò
(3. 3)
dI
ZeA æ 2( E min - Zef p ) ö
=fç
(3. 4)
÷.
m
dE min
m è
ø
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
æ
speed ç c s =
ç
è
E min =
ZTe
mi
ö
÷ . Therefore, ions just outside the analyzer have a minimum energy
÷
ø
1
mv 2min + Zef p ,
2
(3. 5)
and ions at the retarding grid have a minimum kinetic energy =
ZTe
1
mc s2 =
, giving
2
2
ZTe
+ Zef r .
2
Conservation of energy requires that equation 3.5 equals equation 3.6, giving for vmin,
E min =
v min =
2Ze æ Te
ö
+ ( f r - f p )÷ =
ç
m i è 2e
ø
where f s = f p -
2Ze( f r - f s )
mi
(3. 6)
(3. 7)
Te
is the sheath potential. At this point it can be seen that any data points taken
2e
for when f r < fs , are not valid for the analysis of the distribution function. From (3.6),
dE min = Zedf r , giving for equation (3.4)
dI
Ze 2 A
=f ( v min ) .
(3. 8)
df r
m
The ion distribution function can now be expressed as a function of the retarding potential, dropping
the subscript on vmin ,
f(v ) = -
m dI
.
Ze 2 A df r
(3. 9)
13
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
Ic is related to the current at the aperture by
Ic = T4 I.
(3. 10)
Using (3.10) and (3.9), the distribution function is finally expressed as
f(v) = -
dI c
m
.
4 2
ZT e A df r
(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, and f p = 20 V . The solid line represents I (normalized) vs. fret,
and the dashed line represents f(v) (normalized) vs. fret.
1
I vs. f
ret
f(v) vs. f
0.8
a.)
0.6
ret
0.4
0.2
0
14
15
16
18
20
1
22
24
f ret
26
30
I vs. f
ret
f(v) vs. f
0.8
b.)
28
0.6
ret
0.4
0.2
0
f ret µ v2
1
I vs. v
f(v) vs. v
0.8
0.6
c.)
0.4
0.2
0
0
851
1702
2554
3405
4256
5107
v (m/s)
5958
6809
7661
8512
Figure 3. 2 . a.)Plot of normalized Ic vs. 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.
14
Equation 3.11 could be interpreted such that the plot of dIc/dfret vs. fret yields 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
I c µ exp( -Z ef ret Ti ) .
(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,
Zef ret
+ constant .
(3. 13)
Ti
Finally, to solve for Ti, one could do a linear fit to (3.13), or take the derivative with respect to the
ln( I c ) = -
retarding potential,
-1
æ d ln( I c ) ö
÷÷
.
(3. 14)
Ti = -Zeçç
è df ret ø
Inevitably, there exists some offset in the measurement of the characteristic which is not easily
eliminated, meaning that the current will more likely follow
I c = A exp( - Zef ret Ti ) + I offset ,
(3. 15)
where A is a constant. This will lead to no linear relationship between Ic and the retarding potential.
Consider the log of equation 3.15,
I
æ
æ
öö
ln( I c ) = ln çç A exp( -Z ef ret Ti )ç 1 + offset exp( Zef ret Ti ) ÷ ÷÷
A
è
øø
è
Zef ret
I
æ
ö
= ln(A) + ln ç 1 + offset exp( Z ef ret Ti ) ÷
Ti
A
è
ø
with
I offset
exp( Zef ret Ti ) << 1 , the natural log can be expanded, giving,
A
I offset
Zef ret
exp( Z ef ret Ti ) + ln( A ) + 1 ,
A
Ti
the derivative of which gives,
ln( I c ) »
(3. 16)
d ln( I c ) ZeI offset
Ze
=
.
exp( Z ef ret Ti ) df ret
ATi
Ti
(3. 17)
15
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 Ic would give,
æ dI c ö
æ AZe ö
Zef ret
÷÷ = ÷÷ ,
lnçç + lnçç
Ti
è Ti ø
è df ret ø
revealing a linear relationship from which Ti is easily found.
(3. 18)
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 c2 test.
The functional form of Ic is given by equation 3.1, I = ZeA
Maxwellian ions, f(v) is given as
æ - m( v - v o ) 2
f ( v ) = expçç
2T
è
ö
÷.
÷
ø
¥
ò vf ( v )dv .
v min
Assuming
(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
I fit
v min
æ
ö
ç
÷
vf ( v )dv
ç
÷
0
÷
= Cç 1 ç
æ v min
ö÷
maxç vf ( v )dv ÷ ÷
ç
ç
÷÷
ç
è 0
øø
è
ò
(3. 20)
ò
The fit is performed by first supplying ranges for the values of Ti, and vo. A function S of these
values found from
S( Ti , v o ) =
å (I ( k ) - I
N
k
c
fit ( v ( k ), Ti , v o
))2 ,
(3. 21)
where Ic is the measured current, results in a ‘surface’. The minimum of S is the point where that
particular combination of Ti, and vo provide the best fit. The sum goes over the range of where the
fit is desired.
16
To test the fit, consider the c2 test,
2
æ I c ( k ) - I fit ( v( k ), Ti , v o ) ö
çç
÷÷
(3. 22)
dI c (k )
ø
k =1 è
where dI c is the uncertainty in measuring Ic. If c 2 £ N , then the fit is acceptable. If c 2 >> N , then
c2 =
å
N
the fit is unacceptable, and is rejected. In practice, a fit is rejected if c 2 ³ 2 N .
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 Ic as the magnitude
of the differential gives,
2
2
æ ¶I
ö æ ¶I
ö
dI c = ç c dV ÷ + ç c dR ÷
è ¶V
ø è ¶R
ø
where dV and dR are the uncertainties in voltage and resistance measurements. Finally,
2
(3. 23)
2
ö
æ1
ö æ I
dI c = ç dV ÷ + ç - c dR ÷
(3. 24)
èR
ø è R
ø
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%.
17
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
æ
mi
I = I is ç 1 ç
Z2 pm e
è
(
ö
æ
÷ expç e f probe - f plasma
çT
÷
è e
ø
)ö÷÷ ,
(3. 25)
ø
where Iis is the ion saturation current. For the aperture, fprobe = 0V, from equation 3.25 this gives for
the aperture current Iap
æ
æ ef
ö
mi ö
÷ expç - plasma ÷
I ap = I is ç 1 ç
ç
÷
Z2 pm e ø
Te ÷ø
è
è
Given that the plasma and floating potentials are related through
Te æ m i ö
÷,
lnç
2e çè Z2 pm e ÷ø
the aperture current becomes
f p = f float +
æ
I ap = I is ç 1 ç
è
æ
= I is ç 1 ç
è
æ
= - I is ç 1 ç
è
mi
Z 2 pm e
mi
Z 2 pm e
Z 2 pm e
mi
æ ef
I ap » - I is exp çç - float
Te
è
(3. 27)
æ
ö
æ
÷ exp ç - e f float + ln ç Z 2 pm e
ç
÷
ç Te
mi
è
ø
è
ö
æ
÷ exp ç - ef float
ç
÷
Te
è
ø
ö
÷÷
ø
(3. 26)
ö Z 2 pm e
÷÷
mi
ø
ö
æ
÷ exp ç - ef float
ç
÷
Te
è
ø
öö
÷÷
÷÷
øø
ö
÷÷
ø
(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 ffloat gives
f float = -
Te æç I ap
ln
e ç I grid1
è
ö
÷.
÷
ø
(3. 29)
Since the ffloat, Iap, and Igrid1 are all measured, then the electron temperature can be found from
equation 3.29.
18
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
grid
spacer
ceramic
9.55 mm
Langmuir probe
9.2 mm
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 lDebye ~ 0.39mm. With the average grid spacing of 1.35mm(~4lDebye), 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
19
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.
to feed-through
1.275cm
Figure 4. 2 Side view of GEA.
3.45cm
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.
current: I
grid
Rsense
Rin
Vout
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 Vout =
Vin
R sense = IR sense . Evidently, the
R in
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.
20
20000
1
1
18240
Grid 3
-15V
LF356
2000
2
INA117
2
470pF
3
3
Vout
2
2000
10200
15V
2
Collector
6180
-15V
LF356
4
2000
INA117
1N2974A 10W
3
10V Zener
470pF
4
5
Vout
3
2000
3238
15V
5
3238
Grid 1
-15V
LF356
6
2000
INA117
6
-205V
470pF
7
30
7
30
Vout
6
2000
15V
9
Aperture
-15V
LF356
2000
1N3003 10W
82V Zener
INA117
470pF
Vout
10
2500
24W
2000
9
15V
-15V
Langmuir Probe
1E6
2000
INA117
1N2974A
470pF
Vout
3245
10
+9V Battery
332
2000
15V
-9V Battery
Grid 2
LF356
-15V
2000
INA117
+9V Battery
Grid 2
variable bias
-9V Battery
Figure 4.4 . Schematic of entire circuit used for the GEA.
470pF
Vout
2000
15V
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.
22
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.
Figure 4. 5 LabView acquisition and display interface.
23
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 lo are 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.
0.6
0.4
0.2
Z
0
-0.2
-0.4
-0.6
0.5
1
1.5
2
R
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).
24
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
minimum value.
25
f ret =0V
f ret =45V
f ret =180V
1000
500
I(mA)
0
Aperature
-500
-1000
400
200
I(mA)
Grid1
0
-200
-400
100
I(mA)
Grid2
50
0
-50
-100
100
50
I(mA)
Grid3
0
-50
-100
100
50
I(mA)
0
Collector
-50
-100
50
Volts
0
Langmuir
-50
0
5
time(ms)
10
0
5
time(ms)
10
0
5
time(ms)
10
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.
26
5.4 Necessity of Plotting Icollector vs. (fret - ffloating)
In a plasma, all potential measurements are relative to the plasma potential. Referring to
equation 3.7, v min =
2Ze æ Te
ö
+ ( f r - f p ) ÷ , this fact is made explicit. In this experiment, however,
ç
m i è 2e
ø
it is not the plasma potential which is measured, but the floating potential. These two potentials are
related through
Te æ m i
lnç
2e çè Z2 pm e
giving for vmin,
f p = f float +
v min =
If ffloat
ö
÷÷ ,
ø
(5. 1)
æ mi
T æ
2Ze æç
f r - f float + e ç 1 - ln çç
ç
m i çè
2e è
è Z2 pm e
ö ö ö÷
÷÷ ÷ .
(5. 2)
÷
ø ø ÷ø
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
different for each shot to warrant plotting Icollector vs. (fret- ffloat).
27
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.
70
60
1
2
3
4
5
50
I(mA) 40
30
20
10
0
-20
0
20
40
60
80
100
f ret - f floating
40
I(mA)
1
20
0
-1
-0.8
-0.6
-0.4
-0.2
120
140
160
180
3
2 4
5
0
time(ms)
0.2
0.4
0.6
0.8
1
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 lo 4m and 15m. One can immediately see how changing lo can 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
include the point of the knee of the characteristic and its slope.
28
50
45
40
1
2
3
4
5
35
30
I(mA) 25
20
15
10
5
0
-20
0
20
40
60
80
100
f ret - f floating
120
140
160
180
60
40
1
I(mA)
20
0
-1
-0.8
-0.6
-0.4
-0.2
3
2 4
5
0
time(ms)
0.2
0.4
0.6
0.8
1
Figure 6. 2 a. lo = 15m, R = 0.972m.
70
60
1
2
3
4
5
50
I(mA) 40
30
20
10
0
-20
0
20
40
60
80
100
f ret - f floating
100
I(mA)
140
160
180
45
50
0
-1
120
3
2
1
-0.8
-0.6
Figure 6.2.b. lo = 4m, R = 0.972m.
-0.4
-0.2
0
time(ms)
0.2
0.4
0.6
0.8
1
29
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.
16
14
12
10
I
8
6
4
2
0
0
20
40
60
f
80
ret
-f
100
120
140
160
float
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 v o ( l o , R ) , which are then
plotted vs. lo for 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
increase with lo.
30
4
R=
R=
R=
R=
R=
3
0cm
3cm
6cm
9cm
12cm
Ti (eV) 2
1
0
2
4
7
9
l
o
12
15
17
Figure 6.4a. Plots of Ti( l o , R ) vs. lo for each R for the target argon plasma.
10
9
8
v (km/s) 7
o
6
R=
R=
R=
R=
R=
5
4
2
4
7
9
l
12
15
0cm
3cm
6cm
9cm
12cm
17
o
Figure 6. 4b Plots of v o ( l o , R ) vs. lo for each R for the target argon plasma.
31
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.
25
Primary characteristic
20
Ic
Secondary characteristic
15
10
5
0
-5
20
45
70
95
f ret - f float
120
145
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
æ - m ( v - v o1 ) 2 ö
æ - m( v - v o 2 ) 2
m
÷ + (1 - A ) m expç
expçç
÷
ç
2 pT1
2 T1
2 pT2
2 T2
è
ø
è
where A is a normalization factor which describes the relative amplitude
f(v ) = A
ö
÷
(6. 1)
÷
ø
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.
32
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.
25
20
Ic
15
10
5
0
-5
20
45
70f
1
- f float 95
Ic
0.5
0
-10
ret
0v (km/s)10
o
20
30
120
145
30
20
10
0
-1
-0.5
time(ms)
0
0.5
1
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.
33
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
in figure 6.8.
34
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 lo and R, this makes 100x5x5 = 2500 values for T1. All 2500
values of T1 are binned to show the percent that they occur during the time shown in figure 6.8. This
is shown in figure 6.9.
35
25
20
15
% occurences
10
5
0
0
1
2
3
4
Ti(eV) dT = 0.25eV
5
6
7
8
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
Ti =
å N ( Ti )Ti( j) ,
å N ( Ti )
j
j
s Ti =
å N ( Ti )( Ti( j) - Ti )
å N ( Ti )
j
j
2
.
(6. 2)
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.
15
10
% occurences
5
0
0
1
2
3
4
Ti(eV) dT = 0.25eV
5
6
7
8
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
36
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 lo is 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 lo that appears close to linear.
t = 0ms
30
t = 50ms
t = 100ms
t = 150ms
R = 0cm
R = 3cm
R = 6cm
R = 9cm
R = 12cm
20
v (km/s)
10
0
4
7
9
12
l (m)
o
15
4
7
9
12
l (m)
o
15
4
7
9
12
l (m)
o
15
4
7
9
12
l (m)
o
15
Figure 6. 11. Four plots in time, each showing vdrift vs. lo for 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 lo is shown.
t = 50ms
30
R = 0cm
R = 3cm
R = 6cm
R = 9cm 20
R = 12cm
v (km/s)
t = 100ms
t = 150ms
10
0
4
7
9
12
l (m)
o
15
4
7
9
12
l (m)
o
15
4
7
9
12
l (m)
o
15
Figure 6. 12 Three plots in time, each showing vdrift vs. lo for the secondary distribution.
37
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 lo and R as for argon (compare figure
6.8).
38
Figure 6. 13 Array of distribution function surfaces for H2.
15
10
% occurences
5
0
0
1
2
3
4
Ti(eV) dT = 0.25eV
5
6
7
8
Figure 6. 14 Histogram plot of the hydrogen ion temperature throughout the ohmic pulse, lo, and R.
The temperature is found to be Ti = 1.5 ± 0.9eV.
39
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 lo is evident.
t = 0ms
30
t = 50ms
t = 100ms
t = 150ms
R = 0cm
R = 3cm
20
v (km/s)
10
0
4
7
9
12
l (m)
o
15
4
7
9
12
l (m)
o
15
4
7
9
12
l (m)
o
15
4
7
9
12
l (m)
o
15
Figure 6. 15 Plots of vDrift vs. lo for two positions for hydrogen. Ion acceleration is observed, and the
relation between vDrift and lo is 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.
40
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 lo is 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 lo are 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 lo is 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.
41
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.
42
References
1
E. Priest. T. Forbes. Magnetic Reconnection Cambridge University Press 2000 pgs. 1-3
2
J. Wesson. Tokamaks. Clarendon Press, Oxford 1997 2nd edition pg. 343
3
E. N. Parker. The solar flare phenomenon and the theory of reconnection and annihilation of
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4
H.P. Petschek. AAS-NASA Symposium on Solar Flares (NASA, Washington, DC, 1964),
NASA SP50, p. 425
5
E. Priest. T. Forbes. Magnetic Reconnection Cambridge University Press 2000 pg. 135
6
J.Egedal, A.Fasoli. Plasma Generation and Confinement in a Toroidal Magnetic Cusp.
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A.Fasoli, J.Egedal, J.Nazemi. Laboratory Observation of Fast Collisionless Magnetic Reconnection
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8
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43