Experimental Studies of Magnetic Flux Tubes
Dissertation zur Erlangung des Grades
Doktor der Naturwissenschaften
in der Fakultät für Physik und Astronomie
der Ruhr-Universität Bochum
von
Holger Stein
aus Arnsberg
Bochum 2011
1. Gutachter: Prof. Dr. H. Soltwisch
2. Gutachter: Prof. Dr. J. Winter
Datum der Disputation: 19.05.2011
2
Contents
1 Laboratory Simulations of Solar Flares
5
2 Theory
9
2.1
2.2
Magnetohydrodynamics . . . . . . . . . . . . .
2.1.1 Ideal MHD . . . . . . . . . . . . . . . .
2.1.2 Resistive MHD . . . . . . . . . . . . . .
2.1.3 Classication of the experiment . . . . .
2.1.4 Kink and sausage instabilities . . . . . .
Titov-Démoulin model . . . . . . . . . . . . . .
2.2.1 Topology of the magnetic eld . . . . . .
2.2.2 Downscaling of the parameters . . . . . .
2.2.3 Simulated magnetic eld geometry: Solar
3 Experimental setup
3.1
3.2
3.3
3.4
Chamber setup . . . . . . . . . . .
3.1.1 Fast gas valve . . . . . . . .
3.1.2 Capacitor bank . . . . . . .
First plasma source . . . . . . . . .
3.2.1 Course of the experiment . .
Titov-Démoulin plasma source . . .
3.3.1 Pulse forming network . . .
3.3.2 Course of the experiment . .
Characterization of the experiment
3.4.1 Electrical parameters . . . .
3.4.2 Magnetic parameters . . . .
4 Diagnostics
4.1
4.2
4.3
Rogowski coil . . . . .
Pick-up coil . . . . . .
ICCD cameras . . . . .
4.3.1 Operating mode
4.3.2 PI-MAX . . . .
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of an ICCD
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model vs.
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plasma source
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3
Contents
4.4
4.3.3 hsfc pro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Dynamic range and image quality . . . . . . . . . . . . . . . . . . .
Triple probe, emission spectroscopy and interferometer . . . . . . . . . . .
5 Results and discussion - rst plasma source
5.1
5.2
5.3
5.4
Imaging the ux tubes . . . . . . . . . . . . . . . . . . . .
5.1.1 Plasma ignition . . . . . . . . . . . . . . . . . . . .
5.1.2 Tearing . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Inuence of the magnetic guiding eld . . . . . . .
5.1.4 Comparison to the experiment at Caltech . . . . . .
Spatial evolution of the ux tube . . . . . . . . . . . . . .
5.2.1 Parameterisation of the shape of the ux tube . . .
Magnetic eld measurements . . . . . . . . . . . . . . . . .
5.3.1 ϑ-component of the magnetic eld . . . . . . . . . .
5.3.2 Current density prole . . . . . . . . . . . . . . . .
5.3.3 ϕ-component of the magnetic eld . . . . . . . . . .
5.3.4 Observations after the tearing of the ux tube . . .
Comparison with calculations . . . . . . . . . . . . . . . .
5.4.1 Calculation of the current-generated magnetic eld
5.4.2 MHD simulation . . . . . . . . . . . . . . . . . . .
6 Results and discussion - Titov-Démoulin plasma source
6.1
6.2
Titov-Démoulin plasma source with magneto-static guiding
6.1.1 Expansion velocity of the neutral gas . . . . . . . .
6.1.2 Flux tube expansion velocity . . . . . . . . . . . . .
Argon . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Images of the ux tube . . . . . . . . . . . . . . . .
6.2.2 Inuence of the external magnetic guiding eld . .
6.2.3 Magnetic eld measurements . . . . . . . . . . . . .
7 Summary and Outlook
4
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117
1 Laboratory Simulations of Solar
Flares
Solar ares are of particular interest in solar research and observed by several satellite
missions. Sometimes they result in spectacular eruptions like eruptive prominences and
coronal mass ejections (CMEs). In gure 1.1 one of these eruptive prominences, captured
by the Solar Dynamics Observatory (SDO), is presented.
(a) Sun with eruptive prominence (it protrudes
about 2.6 ·105 km over the solar surface) on
March 30th, 2010
(b) Enlarged image of the same prominence a short time later; for comparison, the approximated size of the earth
is added
Figure 1.1: Images of a solar are, captured by the Solar Dynamics Observatory
[nasa](courtesy of SDO (NASA) and the AIA, EVE and HMI consortium)
Fundamental questions concerning the coronal heating are connected to solar ares: The
corona temperature (up to 107 K) is much higher than the temperature of the solar surface
(5800 K). Therefore, a non-thermal process is needed to transport energy from the sun's
core to its corona.
One possible source for the needed energy are solar ares. When they rise they take
5
1 Laboratory Simulations of Solar Flares
magnetic ux (frozen to their plasma) from deeper layers of the sun with them. Due to
magnetic reconnection the energy stored in the magnetic eld can be converted to kinetic
and thermal energy.
To investigate solar ares independently of observations of the sun an experiment has
been designed to generate magnetic ux tubes which are similar to them. The rst plasma
source employed at the experiment presented in this work has been used to reproduce and
extend studies of Bellan et al. [BH98].
The second one has been constructed to mimic a model which was proposed by Titov and
Démoulin [TD99] to investigate twisted magnetic congurations in solar ares. It was used
by Török and Kliem [TK05] to investigate the stability of magnetic ux tubes. Figure 1.2
shows the comparison of images of a solar are observed by the TRACE satellite (left-hand
side) with a simulation of it using the Titov-Démoulin model (right-hand side). Through
the choice of the starting parameters the simulation corresponds to the observation.
Using the Titov-Démolin plasma source it is tried to meet these starting parameters of the
model calculations and to produce a ux tube corresponding to the simulation (and thus
to the satellite observation). Thus, a tool to study the behaviour of solar ares can be
made available.
A brief introduction of the here-used MHD description of plasmas and the Titov-Démoulin
model for solar ares is given in chapter 2. The experimental setup of the pulsed power
supply and both of the plasma sources is presented in chapter 3. It is followed by the
introduction of the applied diagnostics in chapter 4. Chapter 5 and 6 present the results of
both of the plasma sources and their interpretation. The last chapter gives the summary
and outlook of this work.
6
Figure 1.2: Evolution of a solar are observed by the TRACE satellite [nasb] (May 27,
2002) and the numerical simulation of this solar are by Török and Kliem
[TK05] based on the model of Titov and Démoulin
7
2 Theory
This chapter will give an overview of the theoretical concepts which are employed to describe the ux tubes observed in this experiment. In general a ux tube is a tube-like
cylindrical surface of constant magnetic ux. In this work a plasma tube containing a
constant magnetic ux is called a ux tube.
Starting with Maxwell's equations and the equation of motion for ions and electrons, a
short introduction to the MHD equations will be given. They are the basis of simulations
carried out by Lukas Arnold [Arn08]. In these simulations, it was tried to reproduce the
evolution of the magnetic ux tubes as observed at the rst plasma source of this experiment (c.f. chapter 5).
Based on the MHD equations, further physical phenomena are introduced, as frozen magnetic ux, magnetic reconnection and plasma instabilities (e.g. the sausage and the kink
instability), which may occur in this experiment.
In the second part of this chapter a theoretical model, proposed by Titov and Démoulin
[TD99], is introduced. It describes the evolution of solar ares. Numerical simulations
based on this model, performed by Török and Kliem [TK05], are in good agreement with
observations of the TRACE project [nasb].
This model is also the basis of the second plasma source design, the Titov-Démoulin plasma
source. The experimental setup is introduced in chapter 3.3. In chapter 2.2 the model itself
as well as the scaling calculations for the plasma source and the comparison of the model
and the plasma source are presented.
For the fundamental understanding of the physics presented in this chapter, the books
of Chen [Che06], Boyd & Sanderson [BS03], Bellan [Bel00] and Priest & Forbes [PF00] are
recommended.
9
2 Theory
(a) Coordinates of the plasma sources
(b) Coordinates of the ux tube; the
toroidal components are marked green,
the poloidal ones blue
Figure 2.1: Determination of the coordinates
Determination of the coordinates
For the description of the plasma sources and the ux tube dierent coordinate systems
are used. The plasma sources are described in Cartesian coordinates as shown in gure
2.1 a) and the ux tube is described in toroidal coordinates based on the notation of a
tokamak (shown in gure 2.1 b).
In the coordinate system of the plasma source, the x- and the y-axis are located in the
electrode plane. The z-axis is perpendicular to this plane.
The z-expansion of the ux tube is dened as the distance of the ux tube to the electrodes and the y-expansion as the largest expansion in y-direction of the ux tube (c.f.
gure 2.1 a). A further illustration of this coordinate system is shown in gure 3.10 a) in
which the plane of the ux tube is indicated.
In the coordinate system of the ux tube, the ux tube is located along the toroidal (ϕ)
axis, with the poloidal (ϑ) axis pointing around it. The major radius R of the ux tube
corresponds to the z-expansion in Cartesian coordinates and the minor radius r corresponds
to the half of the diameter of the ux tube.
10
2.1 Magnetohydrodynamics
Maxwell's equations
In this chapter
Maxwell's equations are used:
~
~ ×E
~ = − ∂B
∇
∂t
~ ×B
~ = 0 µ0 ∂E + µ0~j
∇
∂t
ρq
~
~
∇·E =
0
~ ·B
~ =0
∇
(Faraday's law)
(2.1)
(Ampère's law)
(2.2)
(Gauss's law)
(2.3)
(Gauss's law for magnetism)
(2.4)
in which ~j is the current density and ρq is the charge density.
2.1 Magnetohydrodynamics
In magnetohydrodynamics (MHD) the plasma is described as a single uid. Hence, several
parameters of the two-uid description have to be combined to parameters of the singleuid description.
The resulting denitions of the mass density %, the mass velocity ~u and the current density
~j , for a quasineutral plasma with singly charged ions (n = ni = ne ) and the total pressure
p can be written as
(2.5)
% ≡ ni M + ne m ≈ n (M + m)
1
M~ui + m~ue
~u ≡ (ni M~ui + ne m~ue ) ≈
%
M +m
~j ≡ e (ni~ui − ne~ue ) ≈ e · n (~ui − ~ue )
(2.6)
p ≡ pi + pe
(2.8)
(2.7)
MHD equations
The equation of continuity can be obtained from the sum of the equations of continuity for
the single uids
∂(nj mj )
~ · (nj mj ~uj )
= −∇
(2.9)
∂t
with j = i for ions and j = e for electrons. Using the denitions for the mass density
(equation 2.5) and the mass velocity (equation 2.6) the equation of continuity can be
written as
11
2 Theory
∂%
~ · (%~u)
= −∇
(2.10)
∂t
The equation of motion can be derived the same way. The summation of the equation of
motion for ions and electrons
∂~ui ~
~
~
~ i + P~ie
M ni
+ ~ui · ∇ ~ui = eni E + ~ui × B − ∇p
∂t
∂~ue ~
~ + ~ue × B
~ − ∇p
~ e + P~ei
+ ~ue · ∇ ~ue = −ene E
mne
∂t
(2.11)
(2.12)
substituting the mass density (equation 2.5), mass velocity (equation 2.6), current density
(equation 2.7) and the total pressure (equation 2.8), gives the single-uid equation of
motion
~
~ ~
∂~u
~ ~u + j × B − ∇p
= − ~u · ∇
∂t
%
%
(2.13)
where the electric eld and the collision terms have canceled out (P~ei = −P~ie) and the
~ ~ucharge density is assumed to be zero (eni − ene = 0). In the derivation of the ~u · ∇
term ~u is assumed to be so small that the quadratic term is negligible.
To obtain the equation of induction, a simplied version of Ohm's law (2.23) and Ampère's
law (2.2) are equated and rearranged to
~ = 1 ∇
~ ×B
~ − ~u × B
~
E
(2.14)
µ0 σ
in which σ is the plasma conductivity and the displacement current is neglected due to the
assumption that the charge density is zero.
This term is substituted in Faraday's law (2.1):
∂B
1 ~
~ ×B
~ +∇
~ × ~u × B
~
=−
∇×∇
∂t
µ0 σ
1 ~2~
~ × ~u × B
~
=
∇B + ∇
µσ
|
{z
}
| 0 {z }
advection
term
diusion term
(2.15)
(2.16)
In ideal MHD, only the advection term of equation 2.16 is considered. As it is shown later
in this chapter, it leads to a bonding of the plasma to the magnetic eld lines: The plasma
can move along the eld lines but not perpendicular to it.
12
2.1 Magnetohydrodynamics
The diusion term of equation 2.16 is neglected in ideal MHD, but it needs to be considered
in resistive MHD. It implies the time scale (the resistive skin time) in which magnetic eld
variations can diuse.
In summary, the two uid equations are reduced to this set of single-uid equations
∂%
~ · (%~u)
= −∇
∂t
~
~ ~
∂~u
~ ~u + j × B − ∇p
= − ~u · ∇
∂t
%
%
~
∂B
1 ~2~ ~
~
=
∇ B + ∇ × ~u × B
∂t
µ0 σ
(equation of continuity)
(2.17)
(equation of motion)
(2.18)
(equation of induction)
(2.19)
Also the generalized Ohm's law should be mentioned in this section. It can be derived by
multiplying the equation of motion for ions (equation 2.11) with the mass of the electrons
and the equation of motion for electrons (equation 2.12) with the mass of the ions and
subtracting the latter from the former (with n = ni = ne because of quasineutrality):
M mn
∂
~ + en(m~ui + M~ue ) × B
~
(~ui − ~ue ) = en(M + m)E
∂t
~ i + M ∇p
~ e − (M + m)P~ei
−m∇p
(2.20)
Using equations 2.5, 2.6 and 2.7, and the collision term P~ei = ηe2 n2 (~ui − ~ue ), this results
in
M mn ∂ ~
~ + e%~u − (M − m)~j × B
~
j = e%E
en ∂t
~ i + M ∇p
~ e − %eη~j
−m∇p
in which η is the
(2.21)
plasma resistivity.
With the approximation M m, this equation can be rearranged to
~ + ~u × B
~ =
E
1 ~ ~ 1 ~ m ∂~
~
ηj +
∇pe
j×B +
j
−
2
|{z}
|en {z
}
|e n{z∂t }
|en {z }
resistive
Hall term
electron inertia gradient term
term
term
(2.22)
13
2 Theory
The terms on the right-hand side are related to dierent inuences, which have to be considered for specic cases.
In ideal MHD all of these terms are neglected. As a consequence, the magnetic ux is bound
to the plasma. In resistive MHD only the rst term on the right-hand side is considered.
The magnetic ux is no longer bound to the plasma and a rearrangement of magnetic eld
lines can occur. These cases will be discussed later in this chapter.
The Hall term has to be accounted for the case that ions and electrons decouple and move
separately. The electron inertia term can be neglected when the variation is slow compared
to other timescales (e.g. the cyclotron gyration). The gradient term can be neglected when
the quotient of the ion Larmor radius over the typical length scale is much smaller than
the quotient of the plasma velocity over the ion acoustic speed.
For the conditions considered in this work (c.f. table 2.1), the simplied form of Ohm's
law
~ + ~u × B
~ = η~j
E
(Ohm's law)
(2.23)
is sucient.
MHD parameters
Several parameters are necessary to estimate the range of validity of the used MHD equations and whether the plasma is susceptible to instabilities.
An important parameter for the magnetic connement of a plasma is the plasma beta.
It is dened as the ratio between the thermal pressure of a plasma and the magnetic
pressure surrounding it.
2µ0 p
(2.24)
B2
In a plasma with β 1, the magnetic pressure is dominant and the thermal pressure can
be neglected.
β=
Alfvén waves are hydromagnetic waves, traveling along the direction of the magnetic eld.
A displacement of the magnetic lines of force which are frozen in the plasma causes a
displacement of the plasma itself. Due to the inertia of the plasma, its displacement leads
to an oscillation of the magnetic lines of force and the plasma. The phase velocity of this
oscillation is called Alfvén velocity.
B
vA = √
, with % = ne M
µo %
14
(2.25)
2.1 Magnetohydrodynamics
The Spitzer resistivity η and the Spitzer conductivity σ are used for fully ionized plasmas.
They only depend on the plasma temperature and not on the plasma density (apart from
a very weak electron density dependence via the Coulomb logarithm) and are given by
√
1
πe2 m
η= ≈
ln(Λ)
σ
(4π0 )2 (kB Te )(3/2)
(2.26)
magnetic diusivity η∗ = ηµ0 and is used instead
In some textbooks, η is denoted as the
of the electrical resistivity.
The characteristic time for a magnetic eld to diuse across a plasma is the
skin time.
µ0 L20
τR =
η
resistive
(2.27)
in which L0 is a typical length scale of the ux tube.
The magnetic Reynolds number is related to the Reynolds number for uids. It is proportional to the ratio of the advection term and the diusion term in the equation of
induction (2.19) and dened as:
Rm =
µ0 L0 v0
η
(2.28)
This parameter will be used later in this chapter to determine whether a plasma has to
be analysed in terms of ideal or resistive MHD. High magnetic Reynolds numbers indicate
highly conducting plasmas, while low magnetic Reynolds numbers indicate more resistive
plasmas.
Besides the plasma resistivity η , the typical length scale L0 of the ux tube and the typical
velocity v0 of the plasma are necessary to determine Rm .
Substituting the Alfvén velocity in the magnetic Reynolds number, the Lundquist number
is obtained:
S=
µ0 L0 vA
τR
=
τA
η
(2.29)
with τA = vLA0 as the Alfvén transit time. This value is used to estimate the reconnection
rate in dierent models. The maximum reconnection rate for spontaneous reconnection
(according to the Sweet-Parker model) Msp is proportional to S −1/2 and for driven reconnection (according to the Petschek model) Mdr to (ln S)−1 . The dierence between both
models will be explained later in this chapter.
15
2 Theory
A summary of derived values for these parameters characterizing this experiment can be
found in table 2.1.
2.1.1 Ideal MHD
In ideal MHD all dissipative terms are neglected. In the limit η → 0 the diusion term of
equation 2.19 is inexistent and the MHD equations take their simplest form.
For a laboratory plasma a theoretically assumed innite conductivity is a reasonable simplication, when the resistive skin time τR is larger than any other relevant time constant.
As mentioned above, the magnetic Reynolds number indicates whether the plasma can be
described by the equations of ideal MHD or whether its resistivity has to be taken into
account.
The ratio of both terms on the right-hand side of the equation of induction (2.19) is in the
range of the magnetic Reynolds number:
~ × (~u × B)|
~
σµ0 |∇
≈ µ0 σuL0 = Rm
~
|∇2 B|
(2.30)
If Rm 1 for the observed plasma, the advection term is dominant and the diusion
term can be neglected. The resistivity tends to zero. This results in the disappearance of
electric elds, because every potential dierence is instantly balanced. Consequently, the
displacement current in Ampère's law (2.2) can be neglected.
(a) Frozen magnetic ux parameters,
taken from [BS03]
(b) Sketch of a ux tube
Figure 2.2: Frozen ux parameters and ux tube
16
2.1 Magnetohydrodynamics
Frozen magnetic ux
A magnetic ux tube (or just ux tube) is an open-ended cylindrical magnetic surface
~ is perpendicular to the normal ~n of the magnetic surface (see gure 2.2 b).
where B
One consequence of ideal MHD is the frozen ux theorem. Alfvén showed [Alf63], that the
magnetic ux through a surface (e.g. the cross section of a ux tube) stays constant when
the resistivity of the plasma tends to zero:
D
Dt
Z
~ · dS
~=
B
S
Z
S
~
∂B
~+
dS
∂t
I
~ · ~u × d~l
B
(2.31)
C
Leibnitz's theorem (equation 2.31, for the nomenclature see gure 2.2 a) states that the
total change of the magnetic ux is composed of the change of the magnetic ux itself and
the change of the area of the surface.
With Stoke's theorem for a vector F~
Z I
~
~
~
∇ × F · dS =
F~ · d~l
S
(2.32)
C
this equation can be written as
D
Dt
Z
~ · dS
~=
B
S
Z
S
~
∂B
~
~ × ~u × B
−∇
∂t
!
~=0
· dS
(2.33)
Since the integrand of equation 2.33 adds up to zero (equation of induction for ideal
MHD, c.f. equation 2.19), also the integral itself accounts to zero. Thus, magnetic ux
conservation is proven.
Bennett's relation for a z-pinch
In a z-pinch the surface of the plasma is surrounded by the azimuthal magnetic eld which
is produced by the axial current. Due to the rise of the current, also the value of the
azimuthal magnetic eld increases. In doing so, the magnetic eld lines are trying to
contract the plasma.
If the requirements of ideal MHD are fullled, the plasma is frozen to these magnetic
eld lines and moves along with them. For a static equilibrium, the magnetic pressure
B2
, trying to compress the ux tube, must have the same value as the thermal pressure
2µ0
opposing it.
Deduced from this equilibrium, the necessary current through the ux tube for a static
equilibrium can be calculated with Bennett's relation for a z-pinch:
8π
Ti
I =
kB Te +
Ne∗
µ0
Z
2
(2.34)
17
2 Theory
where Z is the atomic number and Ne∗ the number of electrons per unit length of the plasma
column
Z a
∗
2πrne (r)dr
(2.35)
Ne =
0
2.1.2 Resistive MHD
In the astrophysical and fusion context ideal MHD is widely employed. However, in many
laboratory experiments the plasma resistivity cannot be neglected.
In resistive MHD, a nite resistivity of the plasma as the only dissipative term of the MHD
equations is considered. As a result, the diusion term of equation 2.19 has to be taken
into account if the condition Rm 1 is not satised.
As a consequence, the plasma can move across magnetic eld lines, because the magnetic
ux is no longer perfectly frozen into it. The resistive skin time gives the time scale, on
which the magnetic ux can diuse across the plasma.
Magnetic reconnection
One consequence of resistive MHD is the possibility of magnetic reconnection. Here, only
the basic principle of this eect will be introduced. A complete overview of recent work on
this topic can be found in a publication by Yamada et al. [YKJ10] and the textbook on
this topic by Priest & Forbes [PF00].
Figure 2.3: Reconnection according to Sweet and Parker; the magnetic eld lines (red)
reconnect in a thin current layer (grey rectangle); the primal and reconnected
eld lines are separated by the separatrix (grey dotted line); the reconnected
eld lines relax, whereupon they take plasma with them (blue lines)
A detailed model describing this eect was introduced independently from each other by
Sweet [Swe58] and Parker [Par57]. It became known as the Sweet-Parker model of magnetic reconnection:
18
2.1 Magnetohydrodynamics
When two magnetic ux tubes with opposing magnetic elds (red lines) approach each
other (see gure 2.3), a thin current layer can be formed (grey rectangle in the center of
gure 2.3), which has a nite resistivity and a high current density.
The magnetic eld adds up to zero in this current layer. This allows the magnetic eld lines
entering it from top and bottom to reconnect. In doing so, the magnetic energy stored in
the tension of the eld lines is reduced and the magnetic eld lines are leaving the current
layer on the right and the left side. The surface, separating the primal and reconnected
eld lines, is called separatrix (grey dotted line).
Only in the current layer the nite resistivity is of importance. Outside of it the plasma
can still be treated as if it were frozen to the magnetic eld lines. The plasma enters the
current layer together with the magnetic eld lines along its length l and leaves it along
its thickness δ , again frozen to the reconnected eld lines (the plasma motion is indicated
by the blue arrows in gure 2.3).
The reconnection rate is limited by the velocity of the leaving plasma and the thickness of
the current layer. The leaving plasma is bound to the relaxing magnetic eld lines. Their
velocity is limited by the Alfvén velocity vA and with it the velocity of the leaving plasma.
The thickness of the current layer has to be small to gain a high current density.
Because the plasma, leaving the current layer, has the same volume like the plasma, entering the current layer (Bernoulli's law), the velocity of the entering plasma is given by
δ
vR =
vA
l
which is also the
(2.36)
reconnection velocity.
The theory of magnetic reconnection is motivated by observations of the sun: The heating
of the solar corona and the origin of the kinetic energy of solar ares cannot be explained
by ohmic dissipation. However, the resulting life time, calculated for a solar are (107 s)
using the Sweet-Parker model is much longer than their observed life time (103 s).
To accelerate the reconnection speed, Petschek [Pet64] proposed an improved model that
includes slow shocks, which speed up the velocity of the leaving plasma and with it the
reconnection velocity.
2.1.3 Classication of the experiment
Taking a closer look at the parameters measured and calculated for this experiment (c.f.
table 2.1), a magnetic Reynolds number of Rm = 11 to 31.6 and a resistive skin time of
τR = 14.1 to 39.6 µs depending on the minimum (marked red) and maximum (marked
blue) of the measured electron density and temperature can be derived.
19
2 Theory
quantity
electron density
electron temperature
max. magnetic ux densities
measured at the apex
typical length scales:
ux tube diameter
ux tube length
Alfvén velocity
Spitzer resistivity
resistive skin time
magnetic Reynolds number
Lundquist number
safety factor
value
ne
Te
Bϑ
Bϕ
=
=
=
=
Ld =
Ll =
vA =
η=
τR (Ld ) =
Rm (Ld ) =
S(Ll ) =
q(r) =
3 · 1021 - 1 · 1022 m−3
5 - 10 eV
0.15 T
0.05 T
0.03 m
0.15 m
3.3 · 104 - 6.0 · 104 m/s
2.9 · 10−5 - 8.0 · 10−5 Vm/A
14.1 · 10−6 - 39.6 · 10−6 s
11 - 32
141 - 216
0.21
Table 2.1: Plasma parameters (rst plasma source, hydrogen-helium gas mixture, ± 3 kV
charging voltage, determined at the apex of the ux tube); the values calculated
from the minimum of the electron density and temperature are marked red, the
values calculated from the maximum are marked blue
Both these quantities depend on the resistivity of the plasma. For the Spitzer resistivity,
the electron temperature is the only variable input parameter (apart from a very weak
(−3/2)
electron density dependence via the Coulomb logarithm), with η ∝ Te
.
The values used here for Te and ne are determined at the apex of the ux tube (c.f.
chapter 4.4). The temperature and density at other points of the ux tube will depart
from this values. Hence, it is just a rough estimation.
Apart from that, the condition Rm 1, as it is required for ideal MHD, is not everywhere
fullled. But the resistive skin time is longer than the life-time of the observed ux tube
itself.
As a consequence, the observed ux tube is in the intermediate domain between ideal and
resistive MHD.
2.1.4 Kink and sausage instabilities
Two ideal MHD instabilities which can occur in magnetic ux tubes are the kink and the
sausage instability. Both of them can be derived from the stability analysis of a cylindrical
ux tube [Bat78]. To describe this conguration cylindrical coordinates are used with Bϑ as
the poloidal magnetic ux component (produced by the current I through the plasma) and
Bϕ as the toroidal magnetic ux component (magnetic guiding eld along the current I).
20
2.1 Magnetohydrodynamics
At rst the ux tube is in a stable equilibrium (c.f. gure 2.4 a). The sausage instability
occurs if the radius r becomes smaller (due to a perturbation) in a section of the ux tube.
The magnetic ux density which is given by
µ0 I
(2.37)
2πr
increases at the surface of this section, resulting in a pinching of the ux tube (c.f. gure 2.4 b). Because an innite conductivity of the plasma is presumed in ideal MHD the
current through the decreasing cross section of the ux tube is not limited and the ux
tube pinches until it tears apart.
The ux tube can be stabilized with respect to the sausage instability by a magnetic guiding
eld along the ux tube which has to exceed a threshold [BS03] of
Bϑ (r) =
1
Bϕ2 > Bϑ2
2
(2.38)
This criterion is not sucient to avoid the kink instability. If the ux tube is kinking (due
to a perturbation, c.f. gure 2.4 c) the poloidal magnetic ux density at the inside of the
kink increases while the magnetic ux density on the outside of it is decreasing. Due to
this the magnetic pressure on the inside of the kink is larger than on the outside of it and
the kinking of the ux tube increases.
To avoid a kinking of the ux tube the Kruskal-Shafranov stability criterion (q(r) > 1) has
to be fullled. The safety factor q(r) is given by
2πrBϕ (r)
LBϑ
in which L is the length of the ux tube and r its radius.
q(r) =
(2.39)
21
2 Theory
(a) Stable ux tube with radius r and length L
(b) Flux tube with sausage instability
(c) Flux tube with kink instability
Figure 2.4: Schematic drawings of the sausage and kink instabilities
22
2.2 Titov-Démoulin model
2.2 Titov-Démoulin model
Figure 2.5: Sketch of the Titov-Démoulin model, taken from [TD99]
Titov and Démoulin presented a magnetic eld conguration (c.f. gure 2.5) which is modeled by a force-free circular ux tube with the total current I, a pair of magnetic charges
± q and a line current I0 . R is the major radius of the ux tube, a is its minor radius, d
the distance of the magnetic charges and the line current beneath the solar surface and
± L the position of the magnetic charges along the line current. Below the photosheric
plane z = 0 this conguration has no real physical meaning: It is used only to construct
the proper magnetic eld.
The force-free condition of the cross section of the ux tube is called the internal equilibrium. The external equilibrium corresponds to the force balance of a ring current in an
axisymmetric potential eld; the greater the toroidal eld Bϕ of the line current I0 , the
more stable is the equilibrium of the tube.
The temporal evolution of the ux tube begins in the state of the external equilibrium.
It becomes unstable when the number of eld line turns (Nt ) is changed (due to a rising
major radius R in the model and due to a rising total current I at the plasma source).
Then the magnetic eld energy can be transfered into kinetic energy in form of a coronal
mass ejection.
23
2 Theory
Force balance
The external equilibrium depends on the interaction of two opposing forces only: The force
due to the strapping eld by the magnetic charges
Fq = −
2qLI
(2.40)
(R2 + L2 )3/2
and the hoop force by the total current
µ0 I 2
F~I =
4πR
R
3 li
ln + ln 8 − +
a
2 2
~n
(2.41)
where li is the internal self-inductance.
The total current for the equilibrium can be calculated to
I=
8πqLR(R2 + L2 )−3/2
µ0 [ln(8R/a) − (3/2) + li /2]
(2.42)
In [TD99] the toroidal magnetic eld is denoted as Bϑ , the poloidal magnetic eld as BI
and the magnetic eld of the monopoles as Bq . To be consistent with the notation of
the rst plasma source, the toroidal magnetic eld is denoted in this work as Bϕ , the
poloidal magnetic eld as Bϑ and the magnetic eld of the horseshoe magnet (replacing
the monopoles) as the magnetic strapping eld Bst (c.f. schematic drawing of the plasma
source in gure 3.3).
2.2.1 Topology of the magnetic eld
In gure 2.6 a) the superposition of the z-component of Bϑ , Bϕ and Bst is plotted for the
x-y-plane at z = 0 (photosheric plane). The values for the calculation of the ux densities
are taken from table 2.2.
The line current ows parallel to the y-axis (with x = 0 Mm) at a distance d = 50 Mm
beneath the photosheric plane. The magnetic charges are placed at y = ± 50 Mm along
the line current. Along the x-axis, the ux tube breaks through the photosheric plane with
its foot points (black circles) centered at x = ± 85 Mm.
The dipole eld of the magnetic charges is distorted by the magnetic eld produced by the
line current (Bϕ ) and the total current (Bϑ ). The contour borderlines of ± 10, ± 20 and
± 40 mT are lled depending on their eld density with lighter or darker grey colors.
The line which separates the positive Bz values (lighter grey colors) from the negative
24
2.2 Titov-Démoulin model
parameter
model
plasma source
L
d
q
I0
R
a
Nt
I
50 Mm
50 Mm
100 T · (Mm)2
-7·1012 A
85 - 100 Mm
31 Mm
5
3.7 ·1012 A
40 mm
18 mm
1.28 ·10−4 T · m2
20 kA
47 mm
15 mm
5
10.2 kA
Table 2.2: Parameters of the Titov-Démoulin model describing solar ares at the sun and
for the Titov-Démoulin plasma source at the state of equilibrium; the blue
marked parameters are given by the dimension of the electrode ange
Bz values (darker grey colors) is called inversion line (IL, thin black line in gure 2.6).
The white line corresponds to the cut of the z-plane with the separatrix of the ux tube.
A separatrix is the boundary dividing magnetic eld lines which are allocated to dierent
magnetic structures (i.e. the primal and reconnected eld lines in gure 2.3).
The intersection of the inversion line and the separatrix is called a bald patch [TPD93] (BP,
thick black line). Crossing the inversion line around the bald patch the magnetic eld lines
are pointing from the corona of the sun downwards to the photosphere, touching these
parts and return again to the corona. In contrary to this behaviour, the magnetic eld
lines touching the bald patch are pointing upwards and return afterwards to the convective
zone.
When the major radius R of the ux tube increases a bifurcation of the bald patch occurs
(see gure 2.6 b), R = 98 Mm). With increasing distance between both bald patches they
shrink and disappear (c.f. results of Matlab calculations in gure 2.9).
In this model, the bald patches are important for the development of solar ares. They
correspond to regions where current sheets can develop. These current sheets have a nite
conductivity which is a necessary condition for magnetic reconnection.
The upwards directed eld lines touching the bald patches promote the rising of the ux
tube while the downwards directed eld lines around the bald patch retain the foot points.
2.2.2 Downscaling of the parameters
Here the downscaling of the Titov-Démoulin model to the laboratory experiment dimensions is shown. The plasma source design is proposed being consistent with the boundary
conditions implied by the model.
25
2 Theory
(a) R = 85 Mm
(b) R = 98 Mm
Figure 2.6: Topology of the Bz -component, taken from [TD99]; the contour borderlines of
± 10, ± 20 and ± 40 mT are lled depending on their eld density with lighter
(positive polarity) or darker (negative polarity) grayscale, corresponding to
positive and negative magnetic eld direction; parameters taken from table 2.2
For the downscaling two assumptions of the Titov-Démoulin model are used. First, the
number of eld line turns in a torus (Nt ), which is given by the ratio between the poloidal
(Bϑ ) and toroidal (Bϕ ) magnetic ux density, is kept constant. Second, the equilibrium
between the hoop force and the force due to the strapping eld is preserved.
The values of the parameters used in the Titov-Démoulin model are shown in table 2.2,
as well as the values for the Titov-Démoulin plasma source. This plasma source is evolved
from the rst plasma source design and several parameters are adopted (marked blue in
table 2.2).
The spatial parameters are the position of the magnetic poles of the horseshoe magnet
along the x-axis (± L), the distance of the magnetic poles and the line current to the plane
of the electrodes (d), the major radius of the ux tube at equilibrium (R) and the minor
radius (a). The line current I0 is given by the dimension of the pulse forming network.
26
2.2 Titov-Démoulin model
Figure 2.7: Stretched surface of a cylinder (as an approximation of the surface of a torus);
correlation of the number of eld line turns and the magnetic ux densities
Ratio between line current and total current
The number of eld line turns along a torus depends on the ratio between the toroidal
magnetic ux density Bϕ produced by the line current I0 and the poloidal magnetic ux
density Bϑ produced by the total current I. For the model a value of Nt = 5 is assumed.
If the surface of the torus is approximated as the surface of a cylinder (see gure 2.7), the
ratio of the toroidal and poloidal magnetic ux can be written as
Bϕ
2πR
µ0 I0 2πa
·
≈
≈
Bϑ
(2πa)Nt
2πR µ0 I
R2 I
⇒ Nt ≈ 2 ·
a I0
a2
⇒ I ≈ 2 Nt · I0
R
(2.43)
(2.44)
(2.45)
With the values for the plasma source of table 2.2 the total current can be calculated to
10.2 kA.
Equilibrium of the hoop force and the force due to the strapping eld
To calculate the magnetic ux which has to be provided by the magnetic charges (horseshoe magnet), the equilibrium between the outwards pointing hoop force and the inwards
pointing force due to the strapping eld is used.
The force due to the strapping eld is given by
Fq = −
2qLI
(R2 + L2 )3/2
(2.46)
27
2 Theory
With li = 1/2 and the values of table 2.2, the hoop force 2.41 can be written as
µ0 I 2
FI ≈
4πR
R
µ0 I 2
ln + 0.83 ≈
a
2πR
(2.47)
In the state of equilibrium both the forces have to be balanced. The magnetic ux can be
written as
q≈
µ0 (R2 + L2 )3/2
·
·I
4π
LR
(2.48)
For the plasma source a magnetic charge of q = 1.28 ·10−4 T · m2 can be calculated.
The resulting magnetic ux density at the apex of the ux tube is given by
Bq =
(R2
2L · q
µ0 I
≈
2
(3/2)
+L )
2πR
(2.49)
and can be calculated to 43.4 mT for the values of the plasma source.
To achieve this magnetic ux density, a horseshoe magnet is assembled from a pair of
permanent magnets to replace the monopoles of the model. The single permanent magnets
have a dimension of 4 cm x 4 cm x 6 cm and a magnetization of about 1200 kA/m
(c.f. chapter 3.3).
2.2.3 Simulated magnetic eld geometry: Solar model vs. plasma
source
In this section the model calculations of the magnetic eld structure of the plasma source
are compared to model calculations following the approach of Titov and Démoulin. These
calculations are carried out with Matlab and Comsol Multiphysics.
The Matlab calculations were performed by Henning Soltwisch. They show the values
of the Bz -component at the x-y-plane with z = 0 (which corresponds to the photospheric
plane) and the arrows are the projection of the magnetic eld on the x-y-plane. The magnetic ux density in z-direction reaches from -40 mT (-250 mT) of the contour borderlines
lled dark blue to +40 mT (250 mT) of the contour borderlines lled dark red for the solar
parameters (the plasma source parameters).
An analytic solution for the magnetic eld of bar magnets ([EHH05] and [HK95]) is used
for the Matlab simulation of the horseshoe magnet of the plasma source. Because the bar
magnets of the analytical solution are cuboidal, the permanent magnets are assembled of
several bar magnets and the superposition of their magnetic eld is calculated. However,
a simulation of the soft iron yoke of the horseshoe magnet with Matlab was not possible.
28
2.2 Titov-Démoulin model
The Comsol Multiphysics software is shortly introduced in chapter 3.4.2. It calculates
the quantities of interest on the nodes of a mesh. Hence, the resolution of the calculated
magnetic ux density (c.f. gure 2.10 depends on the resolution of the mesh which in turn
is limited by the memory requirements of the software.
With this software it is possible to simulate the permanent magnets of the horseshoe magnet and its soft iron yoke. The results of these simulations are used to check the results of
the Matlab simulations and to estimate the inuence of the soft iron yoke.
Magnetic monopoles versus horseshoe magnet
(a) Simulation of the magnetic eld topology for
the solar parameters produced by magnetic
monopoles and the line current
(b) Simulation of the magnetic eld topology for
the plasma source produced by the horseshoe magnet and the line current
Figure 2.8: Results of the Matlab simulations of the magnetic eld; the Bz -component is
shown as a contour plot (x-y-plane at z = 0, blue contours have negative Bz , red
contours have positive Bz ), the arrows are the projection of the magnetic eld
on the x-y-plane; the black line shows the inversion line where the direction of
the z-component inverts
To get an impression of the change of the magnetic eld due to the replacement of the
magnetic monopoles at the solar model by the horseshoe magnet at the plasma source,
simulations of both congurations are shown in gure 2.8. The magnetic eld consists only
of the guiding eld (toroidal component Bϕ ) and the strapping eld.
The poles of the horseshoe magnet have a nite elongation, the monopoles are point charges.
They are in a distance d beneath the plane of the electrodes (18 mm at the plasma source,
c.f. gure 3.3) and the photosphere (50 Mm at the solar model, c.f. gure 2.5) respectively.
With increasing distance between the magnetic sources and the electrode plane and the
29
2 Theory
(a) Magnetic eld topology for the solar parameters produced by the magnetic monopoles,
the line current and the ux tube
(b) Magnetic eld topology for the plasma
source produced by the magnetic
monopoles, the line current and the
ux tube
(c) Zoom into gure 2.9 b); the crossing of the
inversion line by the small arrows indicates
the 3-dimensional magnetic eld structure
(big arrows)
(d) Here only the inversion line and the projection of the magnetic eld are plotted; the
thick blue arrows indicate a crossing of the
inversion line from lower to higher Bz , the
thick red arrows from higher to lower Bz
Figure 2.9: Results of the Matlab simulations of the complete magnetic conguration; the
Bz -component is shown as a contour plot (x-y-plane at z = 0, blue contours
have negative Bz , red contours have positive Bz ), the arrows are the projection
of the magnetic eld on the x-y-plane; the foot points of the ux tube in the
x-y-plane are covered (black ellipse)
30
2.2 Titov-Démoulin model
Figure 2.10: Result of a Comsol Multiphysics simulation of the plasma source with the
parameters of table 2.2; the x-y-plane 18 mm above the poles of the horseshoe magnet shows the Bz -component with values from -255 mT (dark blue
coloured) to +255 mT (dark red coloured); the outlines of the horseshoe magnet, the ux tube and the line current are shown in addition (c.f. gure 3.6)
photosphere respectively the dierence of both the magnetic eld topologies decreases and
the course of the inversion line (black line) is kept.
Complete magnetic eld conguration
In gure 2.9 Matlab simulations are shown. They validate the scaling of the theoretical
model down to the dimensions of the plasma source. Figure a) shows the calculation of the
magnetic eld with the solar parameters and gure b) the calculations with the parameters
of the plasma source where the magnetic monopoles were replaced by a horseshoe magnet
(c.f. table 2.2). On this scale the magnetic eld proles of both congurations are looking
quite similar.
To be consistent with the conditions implied by the model the existence of a bald patch
has to be veried. A zoom into the region between the foot points of the ux tube is shown
in gure 2.9 c). In addition to gure b), the inversion line (IL) and an indication of the
three dimensional prole of the magnetic eld lines are shown.
In the center of this gure, the projection of the magnetic eld crosses the inversion line
(Bz = 0) from the right side (positive Bz ) to the left side (negative Bz ). This means the
magnetic lines of force are coming from beneath the plane of the electrodes on the right
31
2 Theory
side of the inversion line and are heading back on the left side of it.
Below and above this part of the inversion line it is crossed by the projection of the magnetic eld in the opposite direction. This means the magnetic lines of force are coming from
above the plane of the electrodes on the right side of the inversion line and are heading
back on the left side of it.
In gure 2.9 d) only the projection of the magnetic eld (with a higher resolution) and the
inversion line are plotted to clarify the topology of the magnetic eld.
These three dierently shaped parts of the magnetic eld are separated from each other
by a separatrix. Its intersection with the inversion line forms a bald patch. Depending on
the radius of the ux tube a single bald patch or a bifurcation of it occurs as it can be seen
in gure 2.6.
The shape of the magnetic lines of force coming from beneath the plane of the electrodes
favor a rise of the ux tube while the magnetic lines of force coming from above this plane
are holding the foot points of the ux tube back.
In addition to the calculations with Matlab, Comsol Multiphysics has been used to simulate the new plasma source in order to check the Matlab calculations and to estimate
the inuence of the soft iron yoke. Also the analytical method of the calculation of the
magnetic ux density of the permanent magnets could be conrmed.
The result of one of these Comsol Multiphysics simulations is shown in gure 2.10. It
agrees quite well with the values and the shape of the magnetic ux density previously
shown for the results of the Matlab simulations.
The inuence of the soft iron yoke is relatively small. If it is omitted from the model
calculations the maximum magnetic ux density (as shown in gure 2.10) is reduced from
255 mT to 235 mT.
Occurrence of instability
The magnetic eld conguration shown above is forming the equilibrium condition of the
ux tube for a prescribed value of the number of eld line turns (Nt ). This conguration
gets unstable if Nt (c.f. gure 2.7) changes. However, the solar model and the plasma
source dier in their reason why Nt is changing.
In the solar model the major radius R increases and the ux tube rises. Both the currents
I and I0 remain constant but the distance between the line current and the ux tube is
increasing. Accordingly, Bϕ decreases while Nt increases (c.f. equation 2.43).
At the plasma source a capacitor bank is used to provide the total current I. Here, the
total current is increasing while the line current I0 is kept constant. This leads also to an
increase of Nt .
32
3 Experimental setup
In this chapter the general setup of the experiment is presented. It consists of a vacuum
chamber (gure 3.1) and two dierent plasma sources. Outside the chamber a fast gas
valve and the power supply are placed. Both components are independent of the mounted
plasma source. However, the course of the experiment depends on the used plasma source
and will be described in the respective chapter.
The plasma forms a ux tube, which only exists for a few microseconds. The whole cycle
of the experiment is called a shot and can be repeated every ve minutes.
Figure 3.1: Vacuum chamber with mounted plasma source on the front end (left side)
33
3 Experimental setup
3.1 Chamber setup
The vacuum chamber is a cylinder with 66 cm in length and 68 cm in diameter. This results
in a volume of about 250 liters. The discharge takes up only a small part of this volume.
This makes plasma-wall interactions negligible, except at the plasma source ange, which
is mounted on the top end of the chamber.
The plasma source is completely replaceable with only small eort. The rst plasma source
is described in chapter 3.2. It was constructed according to a model of Paul Bellan [BH98].
A completely new plasma source, based on theoretical considerations by Titov and Démoulin [TD99], was also constructed and will be presented in chapter 3.3.
The chamber provides several anges which are used to mount the vacuum pumps, gauges
and the diagnostics (chapter 4). The base pressure which can be reached is about 10−5 Pa.
Directly after a shot with standard parameters (voltage and pressure applied to the valve,
see chapter 3.1.1), the pressure goes up to 14 Pa. Hence, approximately 3500 Pascal-liters
of gas are injected into the chamber during a shot.
3.1.1 Fast gas valve
For this experiment a short gas pu in front of the electrodes is needed. Therefore, a fast
gas valve [THH+ 93] was constructed.
Figure 3.2: Schematic drawing of the fast gas valve
Inside the valve (see gure 3.2) a small aluminium plate is pressed onto an O-ring seal by a
spring to separate the vacuum chamber from the gas supply. Opposite the spring two little
coils are placed. When the valve is triggered a short current pulse provided by a capacitor
is owing through these coils. Their magnetic eld induces another magnetic eld with
34
3.1 Chamber setup
inverse orientation in the aluminium plate. Therefore, the plate is lifted and the valve is
opened.
The amount of gas can be adjusted by two parameters; the charging voltage of the capacitors and the pressure of the gas in the gas supply. The current through the coils is
proportional to the charging voltage of the capacitor. With a higher current pulse the
aluminium plate is lifted up higher and the valve opens a bit longer. This eect stops
when the range of the spring is exceeded and the aluminium plate bounces back. In this
case the gas volume, which is pued into the chamber, is no longer reproducible.
Gas
For most of the measurements which are presented in this work hydrogen with a small
amount of helium (10 percent) was used. The helium is added due to spectroscopic reasons,
because more spectral lines are available to evaluate the plasma density and temperature.
If another gas is used, it will be mentioned explicitly.
Indeed, the operation of the valve depends on the used gas (mass and speed of sound of it).
This has an eect on the point in time when the thyristor is triggered (see chapter 3.2.1)
and on the amount of gas injected into the vacuum chamber. De-mixing processes due to
dierent expansion speeds of dierent gas components are neglected.
3.1.2 Capacitor bank
To ignite the plasma a high voltage dierence is applied to both electrodes of the plasma
source. The cathode and the anode are charged between 1 kV and 3 kV negative, respectively positive, to ground. So the eective voltage between the electrodes has a range from
2 kV to 6 kV.
To provide these voltages a capacitor bank (c.f. gure 3.3) is used. It consists of two 60 µF
capacitors, connected in parallel, for each electrode. They are connected to the electrodes
by a thyristor. This kind of high current switch is used because of its reproducible switching behavior and is described later in this chapter. The capacitors are charged by two high
voltage power supplies, one for each polarity.
When the thyristor is activated and the plasma is ignited, the capacitors for the anode and
the cathode are connected in series and the capacitance of the whole capacitor bank adds
up to 60 µF.
This capacitance together with the inductance of the wires and the plasma creates an
LC-oscillator and a sinusoidal current shape can be observed. The period of the current is
around 55 µs and the amplitude reaches 35 kA (with 6 kV of eective voltage).
At the link between the thyristor and the electrodes, the current and the voltage drop of
the discharge can be monitored by means of a Rogowski-coil and a high voltage probe.
35
3 Experimental setup
Figure 3.3: Circuit diagram of the capacitor bank
Thyristor
A thyristor consists of four layers of P-type and N-type semiconductors (gure 3.4a) in the
order of P-N-P-N. The rst P-type semiconductor is connected to the anode (A), the last
N-type semiconductor to the cathode (C). The second P-type semiconductor acts as the
gate (G). Between these four layers there are three junctions (J1 , J2 and J3 ).
Without a voltage applied at the gate the thyristor is insulating. The rst and the third
junction are forward biased and conducting, but the second junction is reverse biased, it
is insulating. When a control current is applied at the gate, the second junction, and
therewith the thyristor, becomes conducting.
Compared to a transistor, it is a special characteristic of a thyristor that once it has
switched on, it remains in this state until the gate current and the current through the
thyristor have stopped. To understand this behavior a thyristor can be split into a pair
of two transistors, a PNP-transistor and a NPN-transistor (gure 3.4b). The gate of the
rst transistor is also the source of the second one, and the drain of the rst one is also
the gate of the second one. When the thyristor is in the open state, the current T1 keeps
the rst transistor open and the current T2 the second one.
When a thyristor is switched, the current is initially not distributed to the whole junction
J2 . It takes some time to distribute the current over the whole area of the junction. If
the current density is too high, the semiconductor can melt and the thyristor is destroyed.
36
3.2 First plasma source
(a) Schematic drawing
of a thyristor
(b) Equivalent circuit diagram; the thyristor is replaced by two transistors
Figure 3.4: Operating mode of the thyristor
Hence, at this experiment a thyristor, which consists of about 100 channels, on which the
load is distributed is used.
The here used thyristor aords a maximum current of 40 kA and a maximum voltage of
±3 kV. The slope of the current is limited to 10 kA
and the repetition rate to 5 minutes
µs
(due to the cooling of the semiconductor).
3.2 First plasma source
The construction of the initial plasma source was motivated by a work of Bellan [BH98].
It consists of two copper electrodes with the shape of semi-circles and a radius of 11 cm
(gure 3.5a). They are separated by a removable teon spacer to avoid direct breakdowns.
Each electrode has a gas inlet (distance of 8 cm to each other) embedded. The electrodes
are galvanically isolated from the grounded vacuum chamber by polyacetal spacers. In
these spacers, directly beneath the gas inlets of the electrodes, the gas lines and the poles
of a horseshoe magnet are placed (gure 3.5b). The gas lines are connected to the fast gas
valve, the electrodes to the thyristor.
Magnetic eld congurations
The external magnetic eld in this conguration is used as a guiding eld for the ux
tube. One component is realized by a horseshoe magnet. It consists of 23 magnets with
a diameter of 15 mm and a height of 8 mm on each arm. The arms are connected by a
soft-iron yoke.
To amplify the eld strength another component was added: A single magnet with a diameter of 20 mm and a height of 5 mm is placed directly under each electrode (in the same
37
3 Experimental setup
(a) Top view
(b) Cutaway view
Figure 3.5: Drawing of the rst plasma source
magnetic alignment as the horseshoe magnet).
Several combinations of the permanent magnets have been used at this experiment. Measurements were made with both magnetic components and with only one of them present.
The eect of the direction of the magnetic eld parallel and antiparallel to the direction of
the discharge current was also studied.
Examples for the prole of the magnetic eld, measured by means of a Hall probe, are
shown in chapter 3.4.2.
3.2.1 Course of the experiment
After the capacitors have been charged they are disconnected from the high voltage power
supply and the fast gas valve is opened. The gas ows through the gas lines and pus in
front of the electrodes. One millisecond after this, before the maximum density in front of
the electrodes is reached, the thyristor and the diagnostics are triggered.
The switching of the thyristor and the ignition of the plasma takes 1.5 µs. Then the
voltage at the electrodes breaks down and the discharge current begins to increase. This
is monitored by a Rogowski coil (see chapter 4.1) which is placed at one of the two power
supply lines connecting the thyristor with the electrodes.
The ux tube ascends and begins to pinch. After a few µs, depending on the discharge
current and therefore on the charging voltage, it begins to kink. As a result of the kinking
the inductance of the ux tubes rises, just as its impedance and a short circuit directly
between the electrodes takes over the main part of the current. As a result of this, the ux
tube gets unstable and tears o. This will be further discussed in chapter 5.1.
38
3.3 Titov-Démoulin plasma source
(a) Top view
(b) Cutaway view
Figure 3.6: Schematic drawing of the Titov-Démoulin plasma source
3.3 Titov-Démoulin plasma source
Following the work of Titov and Démoulin [TD99] a second plasma source was constructed.
In their work a model is introduced to describe the behavior of a certain class of solar ares
(see chapter 2.2).
This plasma source diers from the rst one in essential points. The surface of the electrodes is much smaller and these electrodes are embedded into a ceramic disc (with a
thickness of 15 mm) which insulates them galvanically from each other. Thus, direct break
downs between the electrodes are avoided and the base of the ux tube is strictly located.
The major modications were done to the external magnetic eld conguration. An additional strapping eld is installed. This strapping eld is part of the theoretical model (see
chapter 2.2) and is realized by a strong horseshoe magnet. The permanent magnets of the
guiding eld are replaced by a line current, which generates a magnetic eld, independent
in ϕ-direction. Supplementary, this eld declines slower 1r than the dipole eld of the
horseshoe magnet r13 .
The electrodes have a diameter of 2 cm. In their center the gas inlets are situated (see
gure 3.6) and they are 8 cm apart from each other. Behind the ceramic disc, outside
the vacuum chamber, a line conductor is placed. It is orthogonal to the connecting line of
the electrodes and has a length of 15 cm. With a high and constant current pulse (20 kA,
15 µs) the guiding eld is generated. Along the line conductor the horseshoe magnet, which
provides the strapping eld, is arranged.
39
3 Experimental setup
(a) Scheme of the PFN
(b) Scheme of the
spark gap
Figure 3.7: Pulse forming network and spark gap
Permanent magnets
For the strapping eld three horseshoe magnets were constructed. Each consists of two
magnet stacks which are connected by a yoke. The magnetic stacks have a base area of 4 cm
times 4 cm and dier in their length, which amounts to 2 cm, 4 cm and 6 cm. Respectively,
this results in three dierent magnetic eld strengths of the strapping eld which can be
used at the plasma source and which inuences the rising of the ux tube. Measurements
and calculations of this magnetic eld component are shown in chapter 3.4.2.
Line current
The guiding eld is produced by a rectangular current pulse with a duration of 13 µs and
a magnitude up to 25 kA which is owing through the line conductor. The line conductor
is placed between the electrodes (see gure 3.6).
Compared to the guiding eld of the rst electrode system this eld is constant on a given
radius around the line conductor. Compared to the rst electrode system it is easier to
change the eld strength, because it is directly proportional to the current through the line
conductor. The magnitude of this current depends on the charging voltage of the pulse
forming network described in 3.3.1.
3.3.1 Pulse forming network
To provide a high current of several kA, which is nearly constant for some µs, a pulse
forming network (PFN, gure 3.7a) was designed [Rei08]. It consists of ve high voltage
capacitors, each with a capacitance of 2.2 µF and a maximum charging voltage of 30 kV. The
charged electrodes of the capacitors are connected in parallel by coils with an inductance
40
3.4 Characterization of the experiment
of about 1 µH each, the grounded electrodes are directly connected. Between the charged
and the grounded electrodes of the capacitors the spark gap, the line conductor as one part
of the plasma source, and the dump resistor with a resistance of 0.65 Ω, are placed.
Due to the capacitance of the capacitors and the inductance of the coils, ve LC-oscillators
are created which are connected in series. The inductances and the dump resistor are
adjusted in that way that the resulting current of the PFN adds up to a rectangular
current pulse, shown in gure 3.9.
In practice, the rst coil is omitted, because the inductance of the remaining conductors
of the circuit has nearly the same amount.
Spark gap
The spark gap consists of three electrodes (c.f. gure 3.7b). The rst one (I) is connected
to the grounded electrodes of the capacitors, the second one (II) is connected with the
charged electrodes of the capacitors. The distance between them is high enough to avoid
a Paschen breakdown of the spark gap at the given charging voltage.
To trigger a breakdown, the third electrode (III), which is part of the second one, can be
provided with a voltage pulse of 15 kV in addition to the applied charging voltage. The
resulting potential dierence is large enough to ignite the spark gap and a high current,
up to 20 kA, is owing through the line conductor.
3.3.2 Course of the experiment
Although the complete plasma source was changed, only the charging and triggering of
the PFN has to be added. The PFN and the main capacitor bank are charged at the
same time. The PFN is triggered between the gas valve and the thyristor, so that the line
current - and with it the related magnetic eld - has reached its plateau when the ux tube
starts to be formed.
3.4 Characterization of the experiment
One of the initial aims of this experiment was the characterization of the electrical parameters of the discharge and the inuence of the external magnetic eld to the ux tube.
To this end, the temporal evolution of the current and the voltage and the shape of the
external magnetic eld were studied.
41
3 Experimental setup
(a) Current and voltage during the life time of
a ux tube
(b) Current and light intensity during the discharge of the capacitor bank; the light intensity was measured by a photo diode
Figure 3.8: Electrical parameters at a charging voltage of ± 3 kV
3.4.1 Electrical parameters
Power source
Due to the fact that the capacitor bank and the thyristor are connected in the same way
to the electrodes at both plasma sources, the electrical parameters, measured at each of
them, are comparable. The voltage, measured with a high voltage probe, and the current,
measured with a Rogowski coil, are both taken at one of the electrodes.
In gure 3.8 a) the evolution of the current and the voltage in the life time of a ux tube is
shown. At point a the thyristor is triggered and needs 1.3 µs to become conductive (point b).
Hence, the electrodes charge up and the potential dierence between the electrodes is rising
until the plasma ignites at 1.7 µs at point c. The current begins to rise and, therefore, the
voltage drops down. At point d, 7 µs after the thyristor is triggered, the ux tube tears o
which can also been seen at the ICCD camera pictures. At the same time a short circuit,
directly between the electrodes, seems to become the main current path.
Due to the capacitors and the inductance of the wires and the plasma, the discharge shows
the characteristics of a LC-oscillator. In gure 3.8 b) the damped oscillation of the current
is plotted, together with the brightness of the plasma, measured by a photo diode. In every
alternation of the current, a plasma is ignited, but a ux tube is formed only at the rst
ignition of the plasma. Afterwards, the short circuit takes over the current.
In the life time of the ux tube (1.7 µs to 7 µs) the current rises up to 20 kA while the
peak current of the damped LC-oscillator adds up to 35 kA (with a period of 60 µs).
42
3.4 Characterization of the experiment
Figure 3.9: Comparison between measurement and calculation (charging voltage, capacitance and inductance of the setup are used for the calculation)
Pulse forming network
An exemplary current pulse of the PFN (charging voltage of 22 kV) is shown in gure 3.9.
It was measured with a high voltage probe, monitoring the voltage drop over the dump
resistor of the PFN. The measurements are compared to calculations, performed with
MATLAB. The calculations had been used in [Rei08] to design the PFN.
The resulting current pulse consists of the ve individual currents of each capacitor. Due
to the inductance of the coils which connect the charged electrodes of the capacitors, the
individual current pulses are delayed and have dierent pulse lengths.
3.4.2 Magnetic parameters
As described before in this chapter, the plasma sources dier in their magnetic eld conguration. Permanent magnets are used in both of them, but the Titov-Démoulin plasma
source has an additional magnetic eld generated by a line current.
The magnetic eld congurations of the permanent magnets of both plasma sources were
measured by means of a Hall probe. These measurements are compared with simulations,
made with the Comsol Multiphysics software. This software can simulate physical phenomena by solving a system of dierential equations and their boundary conditions; material
properties can be attached to CAD-drawings and a solver calculates the values of interest
on a mesh (see gure 3.10 b) by nite element analysis.
The aim of these simulations is to estimate the magnetic eld structure of the plasma
source. Hence, changes of the conguration of the permanent magnets can be simulated
43
3 Experimental setup
(a) Notation of the coordinates,
used at the plasma source
(b) Simulation of the magnet stack
of the rst electrode system
with adaptive mesh
Figure 3.10: Notation of the coordinates and screen shot of the simulation
before they are implemented. This makes it sucient to verify the simulations with Hall
probe measurements at a few points.
Permanent magnets of the rst plasma source
For the following description the coordinate system shown in gure 2.1 and 3.10 a) are
used. The gas inlets are at y = ±4 cm and the z-axis is perpendicular to the plane of the
electrodes and indicates the distance to them.
The poles of the horseshoe magnet are placed directly behind the gas inlets and have a
distance of 8 cm to each other. In z-direction, the single magnet on each side is arranged
10 mm below the surface of the electrodes, followed by the horseshoe magnet after another
5 mm.
Figure 3.11 a) shows Hall probe measurements of By in the x-z-plane between the electrodes
(y = 0) and gure 3.11 b) shows measurements of Bz in a plane parallel to the electrodes at
z = 19 mm. One prole of both contour plots is compared to the simulation of the respective
component of the magnetic eld (c.f. gure 3.11 c and d). The simulation is, with regard to
its shape and amplitude, in good agreement with the measurements. Therefore predictions
can be made for modied magnetic stack congurations.
44
3.4 Characterization of the experiment
(a) Measurement of the By -component in
the x-z-plane
(c) Comparison between measurement and
simulation (x = y = 0)
(b) Bz -component, measured 19 mm in front
of the electrodes
(d) Comparison between measurement
and simulation (x = 0, z = 19 mm)
Figure 3.11: Measurements of the magnetic ux density and comparison with simulations
for the rst plasma source
Permanent magnets of the Titov-Démoulin plasma source
For this plasma source the gas inlets are also lying along the y axis at the points y = ±4 cm.
However, the horseshoe magnet is now placed along the x-axis (see gure 3.6). The center
of the poles are lying at x = ±4 cm and their magnetic eld is used as a strapping eld.
Here the horseshoe magnet with the smallest magnetic eld strength (stack length = 2 cm)
is shown. Figure 3.12 a) shows Hall probe measurements of By in the x-z-plane and
gure 3.12 b) shows measurements of Bz in a plane parallel to the electrodes with z = 40 mm.
Again, one line of both contour plots is compared to the simulation of the respective
component of the magnetic eld (c.f. gure 3.12 c and d). These simulations are, regarding
to their shape and amplitude, in good agreement with the measurements.
45
3 Experimental setup
(a) Measurement of the By -component in the
x-z-plane
(c) Comparison between measurement and
simulation (x = y = 0)
(b) Bz -component, measured 40 mm in front of the
electrodes
(d) Comparison between measurement
and simulation (y = 0, z = 40 mm)
Figure 3.12: Measurements of the magnetic ux density and comparison with simulations
of 2 cm stack of the Titov-Démoulin plasma source
Magnetic eld of the line current
The guiding eld is produced around the line conductor by the line current. The line
conductor is integrated into the plasma source. It has a length of 150 mm and is connected
to the PFN by two feedings (see gure 3.6 b). As the magnetic ux density at a given
radius is constant, the ux density at the apex of a forming ux tube is theoretically the
same ux density as at its foot points in front of the electrodes.
Figure 3.13 a) shows measurements of the magnetic ux density along the x-axis (y = 0,
z = 40 mm), which is parallel to the line conductor. The feedings are placed at 75 mm and
-75 mm and are pointing in negative z direction. The measurements (red asterisks) are
compared to a calculation of the magnetic ux density (black line), which consists of the
magnetic eld of the line conductor itself and the magnetic eld of the both feedings.
46
3.4 Characterization of the experiment
(a) Magnetic ux density of the line current
(13.5 kA, z = 40 mm)
(b) Magnetic ux density 8 mm in front of an
electrode
Figure 3.13: Evolution of the magnetic ux density of the line current
Decreasing of the magnetic eld in front of the electrodes
The fast rising line current induces an eddy current in the electrodes. This eddy current
produces a magnetic eld, opposing to the original one produced by the line current. Hence,
the magnetic eld directly in front of the electrodes is decreasing.
In order to minimise this eect dierent electrode designs were tested. In gure 3.13 b)
measurements of the magnetic ux density (blue and red line) directly in front of the
electrodes are shown in contrast to a undisturbed measurement (black line) without the
electrodes present.
The rst electrode design (blue line, diameter of 40 mm, thickness 6 mm) is decreasing the
magnitude of the magnetic ux density from 75 mT, measured without electrodes present,
to only 10 mT. In comparison, the slit electrodes (red line, diameter of 20 mm, thickness
2 mm), with a radial cut in the area of the electrode in order to reduce eddy currents, are
decreasing the magnetic ux density to 45 mT.
The better behaviour of the slit electrodes can be explained by reduced eddy currents and
their reduced area (decrease by a factor of 4). The thickness of the electrodes has only
a small inuence on the behaviour, because the skin depth of the induced eddy currents
adds up to about 0.1 mm (2 µs rising time of the line current, permeability and resistivity
of copper), which is much smaller than the thickness of the electrodes.
47
4 Diagnostics
To determine the plasma parameters of this experiment dierent diagnostics are employed.
They range from electrical measurements monitoring the electric supply (Rogowski coils
and high voltage probes) and diagnostics with direct contact to the plasma which may
inuence the plasma (triple probe and magnetic pick up coils) to optical measurements
which do not disturb the plasma (ICCD images, emission spectroscopy and laser interferometry).
The number of diagnostics is limited by two characteristics of the experiment: the short
timescale of the discharge (the ux tube exists only a few microseconds) and the high
voltage and current which is used to ignite and form the plasma (up to 6 kV and 35 kA).
Hence, the diagnostics should have a temporal resolution better than 100 ns and, in case
they are in contact with the plasma, galvanic insulation.
4.1 Rogowski coil
(a) Drawing of a Rogowski coil
(b) Calibration of a Rogowski coil
Figure 4.1: Drawing and calibration of a Rogowski coil
A Rogowski coil [RS12] is a simple tool to measure a high, time varying current. The
current in a power line produces a magnetic eld, which is proportional to the current
49
4 Diagnostics
itself. When the current changes, the varying magnetic eld induces a voltage in the
Rogowski coil, which can be measured by an oscilloscope. If the signal variation is not too
fast, the integral of the measured voltage is proportional to the current through the power
line, because then the inductance of the Rogowski coil itself can be neglected.
The Rogowski coil is placed around the line current. It needs no direct connection to the
conductor, so it can be easily galvanically isolated. There are several designs of Rogowski
coils [Köp61], which dier in their sensitivity to stray elds. A drawing of a standard
version of a Rogowski coil is shown in gure 4.1 a).
Calibration
To calibrate the Rogowski coil it is placed at one of the feedings of the pulse forming
network (chapter 3.3.1). During a shot of the PFN, the voltage, induced in the Rogowski
coil, is measured. At the same time the voltage drop across the dump resistor of the PFN
is measured with a high voltage probe. Since the resistance of the resistor is known, the
current pulse of the pulse forming network can directly be calculated from Ohm's law. Now
the induced voltage of the Rogowski coil is integrated. From the relation of both signals
the calibration factor is obtained (gure 4.1b).
4.2 Pick-up coil
Figure 4.2: Pick-up coil in a ceramic tube
A Pick-up coil measures a single component of a magnetic eld. The simple setup, used
in this experiment, is shown in gure 4.2. It consists of a xed coaxial cable, with about
two cm of the cable core dismantled. The cable core is bent to a loop, and its end is soldered
to the shielding of the coaxial cable. The loop has a diameter of 5 mm and is placed in a
ceramic tube to galvanically separate it from the plasma.
In analogy to the Rogowski coil, a voltage is induced, which is proportional to the change
of the magnetic ux through the loop of the pick-up coil (equation 4.1). To calculate the
magnetic eld, averaged over the area of the loop, the induced voltage is integrated and
multiplied with a calibration coecient.
Z
∂
~ A
~
Uind = − φmag , φmag =
Bd
(4.1)
∂t
A
50
4.2 Pick-up coil
(a) Frequency dependence of the pick up coil
(b) Calibration coecient
Figure 4.3: Calibration of the pick up coil
The signal level depends on the frequency of the signal, the magnetic eld strength and the
area of the loop. The rst two parameters are given by the experiment. The last parameter
can be changed by varying the area of the loop or the number of loops.
Also pick-up coils, measuring more than one eld component at once [Rah07], were tested.
This causes perturbations of the measurements, because the probes for the dierent directions were interacting and the data could not be evaluated.
Calibration
The calibration coecient equates to the reciprocal of the loop area c = A1 . It can be
determined by directly measuring the area, which is not very accurate (with r = 2.5 mm,
c = 51000 m12 ). Alternatively, it can be determined by measuring the magnetic eld of a
Helmholtz coil c = 38500 m12 . With the latter method, which is used for analysis, also the
frequency dependence of the pick-up coil can be observed.
BR=0
32
IH · N
4
= µ0 ·
·
5
RH
(4.2)
The magnetic eld in the centre of a Helmholtz coil can be calculated by equation 4.2. The
sinusoidal current through it (IH ) is supplied by a waveform generator. For the calibration,
the frequency is scanned from 200 kHz (minimal measurable induced voltage) to 12 MHz
(limit of the waveform generator). This frequency range is larger than the range of the
alteration of the magnetic eld, produced by the ux tubes.
The current through the Helmholtz coil and the voltage, induced in the pick-up coil, are
measured in 200 kHz steps.
51
4 Diagnostics
At higher frequencies, the impedance of the pick-up coil leads to a damping of the measured
voltage and limits the temporal resolution of the probe. For probes, used in this work,
such a behaviour cannot be observed in the measuring range of the experiment.
The induced voltage, divided by the magnetic eld strength is plotted in gure 4.3 a) versus
the frequency. The value of U/B increases linearly with the frequency, as can be expected
from equation 4.1. In gure 4.3 b) the value of the calibration coecient is plotted over
the frequency and stays constant over the whole frequency range.
4.3 ICCD cameras
Two dierent types of ICCD cameras are used in this experiment. In the beginning single
shot cameras (Princeton Instruments PI-MAX [Pri]) were used. Since October 2008, a
multi frame camera (hsfc pro [pco]) is available.
These cameras are used to take images of the ux tube and as detectors for spectroscopic
measurements. Some qualitative characteristics of the ux tubes can be directly observed,
like the reproducibility of the discharge and the formation of instabilities. Also geometric
information, like the expansion velocity and the width of the ux tube can be determined.
4.3.1 Operating mode of an ICCD
An ICCD camera consist of a CCD (charge-coupled device) and an intensier. A CCD
is a line or a matrix of small photoactive regions (pixel) of a semiconductor. When a
photon hits a pixel it is absorbed and excites an electron (inner photoelectric eect). The
electron is stored into the pixel; their number is proportional to the light intensity on the
photoactive region. An analogue-to-digital converter reads the CCD out. To avoid thermal
noise, the CCDs of the hsfc-pro and PI-MAX cameras are cooled by a Peltier device down
to -10 ◦ C and to -20 ◦ C, respectively.
The ICCD cameras used in this work have exposure times down to 3 ns. In this short
time period only a few photons are collected by the objective of the camera. Hence, it
is necessary to use an intensier. Additionally it is used as an electronic shutter. The
intensier can be separated into three parts: a photocathode, a micro-channel plate and a
phosphor screen.
Photocathode
The material of the photocathode determines the spectral sensitivity of the camera. It is
coated with a photosensitive compound with a low work function. Hence, electrons can be
released easily when they are hit by a photon of sucient energy.
In the closed state, a small positive voltage, with respect to the MCP, is applied to the
52
4.3 ICCD cameras
Figure 4.4: ICCD camera in the open state
photocathode. So the electrons remain there and recombine. When the camera is triggered,
the intensier switches to the open state for the time period of the exposure time (see
gure 4.4). Now a negative voltage of about 200 V is applied to the photocathode, with
respect to the MCP, in order to accelerate the electrons towards the MCP.
Micro-channel plate
The MCP (micro-channel plate) consist of millions of micro channels, with a typical width
between 6 and 25 µm. As shown in gure 4.4, between the left-hand side of the MCP,
which is grounded, and the right-hand side, a voltage, up to 1 kV, is applied.
When the electrons reach the MCP they enter the micro channels and hit their walls.
Further electrons are released and accelerated by the applied voltage. Multiple collisions
with the walls follow, and every time more electrons are released. The number of electrons
depends on the applied voltage; the voltage is therefore used as adjustable gain factor
[Kun09].
Phosphor screen
When the electrons leave the MCP, they are further accelerated by a constant voltage (up
to 8 kV), applied between the MCP and the phosphor screen. Then the electrons hit the
phosphor screen and they excite it to uorescence. The uorescence light is guided to the
CCD by bre optics.
53
4 Diagnostics
4.3.2 PI-MAX
Two PI-MAX cameras with dierent CCD-detectors (image resolution of 512x512 pixel vs.
1024x1024 pixel) were used. They both use a generation II intensier, P43 phosphor and
a pixel size of 13 µm x 13 µm. A benet of these cameras is the given dynamic range of
the CCD of 16 bit.
The minimal exposure time is 2 ns, which is shorter than the typical exposure time for
this experiment (between 20 ns and 100 ns). Only one image per shot can be taken with
these cameras. Due to the reproducibility of the ux tubes, images, taken at dierent time
steps and shots can be merged to a description of the evolution of a ux tube. This is not
sucient for a detailed observation of instabilities, but it is sucient for a basic overview
of the evolution of the ux tubes.
4.3.3 hsfc pro
In contrast to the PI-MAX cameras, the hsfc pro can take up to 8 images of a ux tube per
shot. It consists of 4 single camera modules which can be exposed a second time (double
shutter mode), at least 500 ns after the rst exposure.
To this end the CCD and the image intensier are modied. The readout of the image from
the CCD to the computer takes much longer than 500 ns. Instead, the image information
of each pixel is transfered to a dedicated dark pixel, lying next to the active pixel. The
given dynamic range of the camera modules is 12 bit.
The gain of the intensier is limited by two factors: charging of the micro-channel plate
and a nite decay time of the phosphor screen. Due to its design, the micro-channel plate
has a relatively large capacitance, and it is not possible to charge it up within the small
time period between the both exposures of the camera module. Hence, the charged microchannel plate has to last for both exposures, and the grey scale value is cut into halves.
For the phosphor screen a phosphor with a shorter decay time is chosen (P46). This results
in a decay time of 300 ns instead of 1 ms (P43) for a light intensity decay from 90% to
10%. The rest light intensity can be segregated by the camera software.
The four camera modules share the same optics. 22 % of the incoming light is distributed
to each of the four camera modules by beam splitters. The optical path length from the
objective to each camera module is for all camera modules the same. Between the beam
splitters and the camera modules suitable lters can be placed for spectral range selection.
The minimal exposure time for this camera is 3 ns in single shutter mode (one image per
camera module) and 20 ns in double shutter mode (two images per camera module). Two
or more camera modules can be triggered at the same time or with a delay of at least 5 ns.
54
4.3 ICCD cameras
(a) PI-MAX camera, t = 3.6 µs, 80 ns exposure time
(b) hsfc pro camera, t = 3.6 µs, 60 ns
exposure time
Figure 4.5: Dierences between the used cameras; both images taken at the same time and
shot
4.3.4 Dynamic range and image quality
The used cameras dier in their dynamic range and image quality. An example is given
in gure 4.5: both images are taken at the same time and the same shot, from opposite
sides of the vacuum chamber. In the image of the hsfc pro camera (b), the electrodes are
obscured by a part of the ange.
Here, the reasons of these dierences, are discussed.
Dynamic range
The dynamic range denes the grey scale value of an ICCD camera. It is limited by several
components of the camera, like the micro-channel plate, the CCD chip or the analogue-todigital converter. In the booklets of the manufacturers, the theoretical dynamic range of
the analogue-to-digital converter is given. However, this is not the only value, which limits
the grey scale value of the camera.
As mentioned before, the micro-channel plate cannot be recharged in the short time period
of a single exposure. Therefore, the number of electrons, multiplied by the micro-channel
plate, is limited by the voltage, applied to it. The number of these electrons is a rst limit
of the grey scale.
When a photon strikes a pixel of the CCD, it excites an electron, which is stored into
the volume of the pixel. This volume, and with it the number of electrons which can be
55
4 Diagnostics
collected, depends on the area of a pixel. The area of a pixel of a double shutter camera
(6.7 µm x 6.7 µm in this case) is much smaller than the area of a single shot camera
(13 µm x 13 µm in this case), since the electrons of the rst exposure have to be stored into
a dark pixel, situated next to the active pixel.
Hence, the number of electrons, which can be stored into a pixel, limits the number of
voltage levels, measured by the analogue-to-digital converter. The width of a voltage level
depends on the readout noise of the analogue-to-digital converter. A higher readout noise
causes a larger voltage level width, which results in a lower dynamic range of the camera.
The readout noise partly depends on the readout frequency of the analogue-to-digital converter; a higher frequency is associated with a higher readout noise.
In summary, the dynamic range of the signal, converted by the analogue-to-digital converter, is much smaller than the theoretical dynamic range of the converter itself. As well,
the analogue-to-digital converter does not achieve the declared dynamic range, due to the
value of its readout noise.
Image quality
The quality of the images depends not only on the dynamic range of the camera. Additionally, the resolution of the image intensier and the behaviour of the phosphor screen
have to be considered. As described in chapter 4.3.1, the micro-channel plate consists of
a matrix of micro tubes, which limits the resolution of the image to the resolution of the
micro-channel plate.
For a single shot camera a phosphor (P43) with a relatively long decay time of 1 ms is
used. Therefore, it oers a resolution, which is at least at the same level as the resolution
of the CCD, as well as an eciency of 20 emitted photons per impinging electron and kV
(acceleration voltage, applied between multi-channel plate and phosphor screen).
In contrast, for double shutter cameras a faster phosphor is needed (see the description
of the hsfc pro camera). This results in a phosphor (P46, decay time of 300 ns) which is
coarse-grained which reduces the resolution of the images. Also its eciency drops down
to 6 emitted photons per impinging electron and kV.
Consequently, the double shutter feature of the hsfc pro camera modules, which is needed
at this experiment, results in images with a noticeable loss of quality. Particularly, this
eect can be observed in gure 4.5. The resolution of the CCD of both cameras is approximately the same, but the image, taken with the PI-MAX camera (a), seems to have
a higher resolution as the image, taken by the hsfc pro camera (b).
56
4.4 Triple probe, emission spectroscopy and interferometer
(a) Setup of the triple probe
(b) Equivalent circuit of
the triple probe
Figure 4.6: Triple probe, taken from [Mac09]
4.4 Triple probe, emission spectroscopy and
interferometer
These diagnostics are operated by other members of the working group, but their results
are mentioned in this work and used for interpretation. Here they are shortly presented.
Triple probe
A triple probe is a electrostatic probe (gure 4.6 a), which can measure the plasma temperature and density. The one used in our work was constructed by Felix Mackel in the
frame of his diploma thesis [Mac09]. The triple probe is in direct contact to the plasma
and needs to be galvanically insulated. It consists of three tungsten tips, each acting like
a Langmuir probe with a xed voltage applied.
A Langmuir probe sweeps a voltage ramp, with at least 100 values, to measure the voltagecurrent characteristics of a plasma sheath, which is formed around its tip. To sweep the
voltage ramp takes at least 1 ms and is much to slow for our purpose.
Even in thermal equilibrium, the velocity of the electrons is much higher than the velocity
of the ions. Hence, more electrons than ions are reaching the tip and a current can be measured. This current, plotted over the applied voltage, results in the probe characteristics.
From this probe characteristics, the electron temperature and density can be determined.
In contrast to the Langmuir probe, the tips of the triple probe are biased to each other
with constant potential dierences (gure 4.6 b). The currents, owing between the three
tips, are measured. So the triple probe is limited by the adjustment speed of the plasma
sheath around the tips, which results in a temporal resolution better than 100 ns. This
allows time dependent measurements in the life time of a ux tube (about 5 µs). However,
57
4 Diagnostics
Figure 4.7: Laser interferometer with Michelson setup, taken from [KMS+ 10b]
only three voltage values are included and a higher measurement error, with respect to the
Langmuir probe, has to be accepted. A detailed introduction to triple probes can be found
in [Mac09] and [CS65].
Emission spectroscopy
An optical bre is placed into the vacuum chamber to guide the light, emitted by the
plasma of a ux tube, to a 1 m grating spectrograph. As detector, a single shot ICCD
camera is used. On a shot to shot basis, measurements can be made at dierent points in
time.
A short introduction in plasma spectroscopy is given in [Fan06] and a detailed one in
[Kun09]. For our purpose, the Stark broadening of line proles could be used to determine
the plasma density. Additionally, the line ratio of hydrogen atoms and helium ions could
be used to determine the plasma temperature, and again the plasma density.
Laser interferometer
To measure the electron density of the plasma of the ux tube a CO2 laser interferometer
has been set up [PSS05] in a standard Michelson conguration [KMS10a] (c.f. gure 4.7).
The laser beam is split before entering the interferometer to monitor the laser intensity
during operation.
The test beam crosses the vacuum chamber at a xed distance of 11 cm in front of the
58
4.4 Triple probe, emission spectroscopy and interferometer
Figure 4.8: Flux tube moving across the laser beam, taken from [KMS+ 10b]
electrodes. The ux tube passes this position several µs after the ignition of the discharge
depending on its expansion velocity (c.f. gure 4.8).
The plasma of the ux tube changes the optical path length of the test beam and results
into a phase change (∆φ) between the test beam and the reference beam. It is given by
Z
∆φ = re λ ne dl
(4.3)
L
in which re is the classical electron radius, λ the laser wavelength and L the optical path
length in the plasma.
The phase change is related to the intensity measured at the detector of the interferometer
according to
I(t) − B
∆φ(t) = arcsin
− φ0
(4.4)
A
in which A is the amplitude, B the oset and φ0 the initial phase of the interferometer
signal, which are taken from calibration measurements.
Just the line-integrated electron density is provided by the interferometer. To calculate
the local electron density an asymmetric Abel inversion is used [KMS+ 10b].
Not only the electron density can be measured with the laser interferometer. Also the
width of the ux tube can be estimated.
59
5 Results and discussion rst plasma source
A detailed overview of the observations and measurements at the rst plasma source
[SKM+ 10] will be given in this chapter.
To get a rst impression of the experiment the rst part starts with a selection of ICCDcamera images showing typical discharge observations with this plasma source. The behavior of the ux tube is compared to measurements of the discharge current and the voltage
at the electrodes (c.f. chapter 3.8). In this context details of the evolution, such as the
ignition and the tearing of the ux tube is discussed as well.
Figure 5.1: Image taken with a single-frame ICCD camera (PI-MAX camera, 6 µs after
trigger, 20 ns exposure time, ± 3 kV charging voltage)
In the second part of this chapter the spatial evolution of the ux tube is analysed. It
is shown that the expansion velocity in z-direction (perpendicular to the electrodes) is
constant for a given charging voltage of the main capacitor bank while the y-velocity
61
5 Results and discussion - rst plasma source
working gas
gas delay
total injected gas volume
charging voltage
max. current
life time of the ux tube
z-expansion velocity
diameter of the ux tube
exposure time ICCD cameras
H2 /He gas mixture
850 µs
1500 Pa · l
± 3 kV
35 kA
5 µs
2.4 cm/µs
3 - 4 cm
10 - 40 ns
Table 5.1: External parameters and geometric properties employed at the rst plasma
source
increases during the life time of the ux tube. Possible reasons for this behaviour will be
discussed.
The temporal and spatial evolution of the magnetic eld of the ux tube is studied in the
third part of this chapter. It is split into the discussion of the poloidal (ϑ) and the toroidal
(ϕ) component. Bϑ is generated by the current along the ux tube and Bϕ by the magnetic
ux of the guiding eld frozen into the plasma of the ux tube.
In the last part of this chapter calculations carried out by Lukas Arnold in the frame of his
doctoral thesis are presented. They are adapted to the external parameters and geometric
properties of the rst plasma source. Their results are compared with observations and
measurements of the ux tubes in the experiment.
External parameters
During the early phase of this experiment external parameters have turned out to ignite a
reproducible ux tube. Table 5.1 gives a brief overview of these parameters and of some
geometric properties.
A hydrogen-helium gas mixture is used as working gas for all shots with the rst plasma
source. For a reproducible ignition of the plasma a delay between the triggering of the fast
gas valve and the thyristor of the capacitor bank of 850 µs is used.
For most of the here presented measurements the capacitor bank is charged up to ± 3 kV.
This results in a maximum current of 35 kA and an expansion velocity of the ux tube
of 2.4 cm/µs. Expect where explicitly mentioned, the magnetic guiding eld is present as
described in chapter 3.2.
62
5.1 Imaging the ux tubes
5.1 Imaging the ux tubes
To begin the discussion about the ux tubes observed in this experiment a rst impression
of their evolution and behavior is given in this section. In addition to the evolution of a
ux tube in a shot with standard discharge parameters, also its ignition and its detachment
from the electrodes (which is called tearing in this work) are described.
The image shown in gure 5.1 was taken with a single-frame ICCD camera in a shot with
standard discharge parameters. The gas inlets and the electrodes can be clearly identied
on the left-hand side of the image (c.f. gure 3.10 a).
The temporal evolution of a ux tube taken with the multi-frame ICCD camera is shown
in gure 5.2. Due to the dierences of the image quality of the used ICCD cameras (see
chapter 4.3.4), the electrodes cannot be identied here.
To analyse the evolution of these images, the raw data of the Rogowski coil (and the
resulting current) measured simultaneously are shown in gure 5.3. The time in the discharge evolution where each image was taken is tagged beneath the respective gure. A
brief overview of a shot sequence of the rst plasma source has already been given in
chapter 3.2.1.
At the time of the rst image (gure 5.2 a, 1.3 µs after the trigger), the ux tube has not
yet formed. The thyristor has switched and the absolute values of the voltages applied to
the electrodes are rising (c.f. gure 3.8 a). At the cathode (upper electrode in the images
of this chapter), the gas begins to ignite and the rst plasma can be seen.
At this moment a current ow cannot yet be recognized by the Rogowski coil. The current
ow starts very reproducible after 1.7 µs (as it can be observed later at the Titov-Démoulin
plasma source also for gases other than the hydrogen-helium mixture used here).
The next image at t = 2.3 µs shows the forming ux tube. The absolute value of the
voltages applied to each of the electrodes drops down from 3 kV to 700 V (again, c.f. gure 3.8 a) and the current begins to rise.
Until t = 6.3 µs (gure 5.2 f), the ux tube is expanding and the current through it
is rising up to 16 kA. Simultaneously the voltage applied to each of the electrodes increases to 900 V.
From that time, the ux tube becomes unstable. This can be observed at the raw signal of
the Rogowski coil (c.f. gure 5.3). Its sinusoidal shape is disturbed by a dip which implies
a structural change of the ux tube. At the deepest point of this dip (at t = 7.3 µs) the
ux tube seems to detach from the cathode (upper electrode at gure 5.2).
Thereupon, the intensity of the emitted light of the plasma is reduced and the ux tube
becomes blurred.
63
5 Results and discussion - rst plasma source
(a) 1.3 µs
(b) 2.3 µs
(c) 3.3 µs
(d) 4.3 µs
(e) 5.3 µs
(f) 6.3 µs
(g) 7.3 µs
(h) 8.3 µs
Figure 5.2: Evolution of a ux tube (main capacitor bank ± 3 kV, 70 % of the magnetic
guiding eld, hsfc pro multi-frame camera, 100 ns exposure time)
64
5.1 Imaging the ux tubes
Figure 5.3: Rogowski coil measurement: raw data (black line) and resulting current (red
line) simultaneously to the images shown in gure 5.2
Images from other perspectives
Due to the setup of the vacuum chamber, only the projection of the ux tube in the y-zplane could be observed. Because anges in a 45◦ -angle to this plane do not exist, images
can be taken additionally only from the top (side view of the ux tube) and the right side
(top view of the electrodes) of the vacuum chamber (c.f. gure 3.1).
For images from these positions a problem occurs, because some parts of the ux tube
superimpose. Figure 5.4 a) shows the top view of a ux tube. The light emitted by its
arms is integrated over their length, so they occure much brighter than the rest of the ux
tube.
Figure 5.4 b) shows the side view of a ux tube. Here, both arms are one behind another
and the emitted light cannot be distinguished.
5.1.1 Plasma ignition
Independent of the value of the magnetic guiding eld (even without it), shots with both
plasma sources have shown that the igniting plasma forms an arc-shaped structure (c.f.
gure 5.5 c, rst plasma source with magnetic guiding eld present, and gure 5.5 d, TitovDémoulin plasma source without magnetic guiding eld present).
If the ignition of the plasma would follow the electric eld, the plasma would ignite much
closer to the electrodes. Hence, the ignition of the plasma seems to depend on the gas
density distribution in front of the electrodes.
65
5 Results and discussion - rst plasma source
(a) Top view with ries
(b) Side view with Newton's rings
Figure 5.4: Images from other perspectives, taken with the PI-MAX single-frame camera;
at the side view image Newton's rings can be observed while the ries at the
top view image are a feature of the plasma
Figure 5.5 a) shows a sketch of the Titov-Démoulin plasma source with the assumed neutral
gas cloud in front of the electrodes. The neutral gas cones of both electrodes are merging
in a distance of about 4 cm in front of the electrodes.
Measurements of the neutral gas density (see gure 5.5 b) support the assumption that
there is no neutral gas between the electrodes. The ionisation gauge, used for this measurements, is inaccurate in front of the gas inlets (at y = ± 40 mm): The high neutral gas
velocity at these points aects the temperature of its hot cathode.
As can be inferred from the ICCD camera images, the path along which the plasma ignites
has a length of about 115 mm. Figure 5.5 e) shows a Comsol Multiphysics simulation of
the electric eld of the rst electrode design of the Titov-Démoulin plasma source. The
colouring encodes the electric eld strength in z-direction at a charging voltage of ± 3 kV.
The arrows indicate the electric eld prole.
Because the eld strength varies along the path of ignition, its maximum value (in zdirection directly in front of the electrodes, E = 1.2 × 105 V/m) and its minimum value
(in y-direction at the apex of the igniting plasma, E = 1.2 × 104 V/m) are chosen for
further estimations.
In gure 5.5 f), the Paschen curve for hydrogen [Bel00] is shown. In addition, the estimated voltage drops along the path length of the igniting plasma (d = 115 mm) for the
minimum and the maximum electric eld (blue dotted lines) are plotted.
66
5.1 Imaging the ux tubes
(a) Schematic of the assumed neutral gas
pu in front of the electrodes
(b) Ionisation gauge measurements (z =
10 mm, x = 0 mm)
(c) Image of the ignition of the plasma;
rst plasma source with magnetic
guiding eld present; PI-MAX singleframe camera
(d) Image of the ignition of the plasma;
Titov-Démoulin plasma source without the magnetic strapping and guiding elds; hsfc pro multi-frame camera
(e) Comsol Multiphysics simulation of the electric
eld before the ignition of the plasma
(f) Paschen curve for H2 , d ≈ 115 mm
Figure 5.5: Ignition of the plasma
67
5 Results and discussion - rst plasma source
The intersection points of the Paschen curve with the estimated voltage drops along the
igniting ux tube are marked by red asterisks, as well as the minimum of the Paschen
curve itself. With the assumption of a homogeneous neutral gas density the neutral gas
pressure is calculated for these points.
The measurements of the neutral gas density are excluding the points on the right-hand
side of the minimum of the Paschen curve, because the pressure at these points is far too
high. For the intersection points on the left-hand side of the Paschen curve a minimum
pressure along the path of ignition of 10 Pa can be calculated.
Based on the minimum of the Paschen curve, the minimum of the electric eld strength
necessary to ignite an arc-shaped plasma in front of the electrodes could be estimated to
E = 3000 V/m. For this value (according to the Comsol Multiphysics simulation) a charging voltage of about ± 700 Volts can be calculated. This corresponds to the experimental
minimum of the charging voltage to ignite a ux tube.
5.1.2 Tearing
The tearing of the ux tube occurs reproducibly at the cathode of the plasma source at
about t = 7 µs. At this time, the current measured at one of the feedings of the electrodes
amounts to 17 kA. A dip can be observed in the raw signal of the Rogowski coil (c.f.
gure 5.3). From then on a ux tube can be no longer observed but the current continues
to rise.
The evolution of the tearing on a coarse time scale is shown in gure 5.6. In the rst
image (t = 5.3 µs) the whole ux tube can be seen. The second image shows the region in
front of the cathode (highlighted in the rst image). The ux tube is still connected to the
cathode, but the rst part of it (about the rst three centimeters) is a bit brighter than
the apex or the other arm of the ux tube.
The next image shows the same region of the ux tube 1 µs later. The current has increased
from 12 kA to 15 kA. In this region, the ux tube is much brighter than in the apex or in
the other arm and pinching can clearly be recognised.
The structure of this part of the ux tube reminds of the sausage instability, mentioned
in chapter 2.1.4. The poloidal magnetic eld pinches the whole ux tube. Additionally,
the toroidal magnetic eld is very strong in front of the electrodes (due to the horseshoe
magnet behind them) and keeps this part of the ux tube also together. The current
density increases because a rising current ows through a contracting cross-section.
Thus, the poloidal magnetic eld increases even more and pinches the ux tube further
until it tears, as seen at the last image at t = 7.3 µs.
68
5.1 Imaging the ux tubes
(a) 5.3 µs, image of the whole ux tube
(b) 5.3 µs, I = 12 kA, zoom to the cathode
(c) 6.3 µs, I = 15 kA, zoom to the cathode
(d) 7.3 µs, I = 17.3 kA, zoom to the cathode
Figure 5.6: Tearing of the ux tube; diameter of the gas inlet amounts 4 cm; ± 3 kV
charging voltage, hfsc pro multi-frame camera, 20 ns exposure time
69
5 Results and discussion - rst plasma source
The magnetic ux density Bϑ at the surface of the ux tube is given by the Biot-Savart
law. For a part of the ux tube in front of the electrodes (as seen in gure 5.6 c), with a
radius of ρ = 0.5 cm, a length of 3 cm and a current through it of 15 kA) and under the
assumption that this part of the ux tube is cylindrical Bϑ (ρ) can be calculated to 0.57 T.
Bennett's relation (equation 2.34) can be used to estimate the thermal pressure and the
electron density which is necessary to prevent the ux tube from a further contraction
due to the poloidal magnetic eld. For a current through the ux tube of 15 kA at t =
7 µs the product of electron temperature and electron density would need to be 40 times
larger than the values measured (Te = 10 eV and ne = 1022 m−3 ) at the apex of the ux tube.
The same process as in gure 5.6 can be observed in gure 5.7 for another shot, with
a higher temporal resolution. The delay between the images is only 100 ns. In the rst
image (at t = 7.0 µs) the ux tube is already tearing from the cathode. Only a thin plasma
lament can be recognised between them. This thin plasma lament is hardly observable
in the next image, 100 ns later.
At t = 7.2 and 7.3 µs, the ux tube seems to have lost its connection to the cathode.
Instead, a thin plasma bulge, which can be seen in the upper left corner of every image of
gure 5.7, is increasing in brightness.
As mentioned above, the tearing of the ux tube is very reproducible. It always takes
place at the cathode at about 7 µs (± 200 ns). That it occurs directly in front of one of
the electrodes can be explained by the magnetic guiding eld and the neutral gas density
distribution.
The magnetic guiding eld is produced by a horseshoe magnet, mounted behind the electrodes (c.f. gure 3.5), and its magnetic ux density decreases very fast with increasing
distance to the electrodes.
It supports the stabilization of the plasma. Hence, a higher current density is distributed
to a smaller cross-section. This promotes the pinching of the ux tube in front of the
electrodes. The magnetic pressure resulting from the magnetic guiding eld is not large
enough to prevent the pinching of the ux tube (as estimated earlier in this chapter).
5.1.3 Inuence of the magnetic guiding eld
For the hydrogen-helium gas mixture, the magnetic guiding eld is necessary to form a ux
tube. Without the guiding eld present the plasma does not pinch, as shown in gure 5.8.
In gure 5.9 several congurations of the magnetic guiding eld were employed. About
30 images from dierent shots are averaged for every time step and conguration. For all
congurations, the main capacitor bank is charged to ± 3 kV. Additionally, the averaged
images are showing the reproducibility of the ux tubes, as the plasma structure is still
70
5.1 Imaging the ux tubes
(a) 7.0 µs
(b) 7.1 µs
(c) 7.2 µs
(d) 7.3 µs
Figure 5.7: Tearing of the ux tube, high time resolution; ± 3 kV charging voltage, hfsc
pro multi-frame camera, 20 ns exposure time
71
5 Results and discussion - rst plasma source
(a) 5.0 µs
(b) 7.5 µs
Figure 5.8: Images of the ux tube without magnetic guiding eld present; ± 1 kV charging
voltage, PI-MAX single-frame camera, 1 µs exposure time, the Newton's rings
are not a feature of the ux tube
clearly visible and only slightly blurred.
In the rst column both parts of the magnetic guiding eld are present, the horseshoe
magnet and the single magnets directly behind the electrodes (c.f. chapter 3.2). The
magnetic ux density in z-direction has a maximum of 110 mT at the surface of the
electrodes.
The electric and the magnetic eld are antiparallel. With this conguration, the ux
tubes last about 1 µs longer than for the conguration of the second column. As the only
dierence to the rst column, the direction of the magnetic eld was reversed. The electric
and magnetic eld are now parallel and the magnetic helicity has changed its sign.
In the third column the horseshoe magnets are removed. The magnetic ux density in
front of the electrodes amounts to 70 percent of its former value. Changes compared to
the images of the second column cannot be recognized.
The velocity in z-direction is independent of the orientation of the elds, relative to each
other (parallel or antiparallel) and of the strength of the magnetic guiding eld. This
indicates that the tension force of the magnetic eld lines has little or no inuence on the
expansion of the ux tube.
5.1.4 Comparison to the experiment at Caltech
As mentioned before, the construction of the rst plasma source was motivated by an experiment operated by Bellan at Caltech [HTB04]. However, the setup of both experiments
72
5.1 Imaging the ux tubes
Figure 5.9: Images of the ux tube at dierent congurations of the magnetic guiding
~ antiparallel to B
~ , second column E
~ parallel to B
~ and third
eld; rst column E
~ parallel to B
~ (magnetic eld strength reduced to 70%); for each of
column E
the shown images about 30 single images taken with the hsfc pro multi-frame
camera were averaged
73
5 Results and discussion - rst plasma source
Figure 5.10: Image of a ux tube at Bellan's experiment, taken from [HTB04]; resulting
charging voltage 6 kV, 2 µs after ignition
diers in several points. The specications, published in the thesis of Hansen [Han01], are
used to compare both systems.
The experiments dier in the evolution of the discharge current. At the same capacitance
and charging voltage, the inductance of the capacitor bank of Bellan's experiment is much
smaller. As a consequence, the current rises twice as high in half the time (74 kA maximum after 6.5 µs) in comparison to our experiment presented here (35 kA maximum after
13.5 µs). Hence, the evolution of the ux tube is much faster in Bellan's experiment (c.f.
gure 5.10).
Another dierence is given by a much stronger magnetic guiding eld at Bellan's experiment. The magnetic ux density at the foot points of the ux tube is 580 mT (110 mT
for our plasma source) and at the apex of the ux tube at its ignition 80 mT (10 mT for
our plasma source). The ratios of the magnetic ux densities at the foot-points and in the
apex dier from each other because the horseshoe magnets are placed at dierent distances
behind the electrodes.
74
5.2 Spatial evolution of the ux tube
(a) Distance between the electrodes and the
apex of the ux tube; for the pick-up coil
measurements the HWHM is shown as error
bars
(b) Expansion of the ux tube in in y-direction
with quadratic t (red line); this information is obtained from ICCD-camera images
Figure 5.11: Expansion of the ux tube at ± 3 kV charging voltage
5.2 Spatial evolution of the ux tube
The geometric expansion of the ux tube is shown in every employed diagnostic. The
multi-frame ICCD camera looking side-on on the ux tube allows the taking of images at
dierent time steps of a shot. The ceramic tube (∅ 12 mm) which isolates the pick-up coil
from the plasma and the gap between the gas inlets of the electrodes (8 cm) can be used
as a scale to estimate the distance between the electrodes and the apex of the ux tube
from camera images.
The pick-up coil is positioned at dierent locations along the z-axis. The analysis of the
data obtained with it allows the determination of the spatial evolution of the ux tube.
Two independent components of the magnetic eld are measured: the Bϑ -component and
the Bϕ -component. An estimate of the current density can be obtained from the data of
the Bϑ -component. A complete overview of the pick-up coil measurements is given in the
next section.
Three dierent quantities are measured: First, the expansion of the ux tube in z-direction
which is equivalent to the major radius R of the ux tube. Second, the expansion of the
ux tube in y-direction. Third, the minor radius r of the ux tube.
75
5 Results and discussion - rst plasma source
(a) One example for the determination of the
ux tube velocity: z-position of the apex of
the ux tube with line of best t (± 3kV)
(b) z-velocity of the ux tube, depending on the
charging voltage; the velocity increases linear over the charging voltage
Figure 5.12: Measurement of the ux tube velocity, depending on the charging voltage
z-expansion
The position of the apex of the ux tube in z-direction, taken from the magnetic eld
measurements and ICCD-camera images, is plotted versus time in gure 5.11 a). Additionally, the half-width of the ux tube (taken from the magnetic eld measurements, c.f.
chapter 5.2) is plotted for each measuring point of the pick-up coil as error bars.
Comparing the positions obtained from the ICCD camera and the pick-up coil one can see
that they are shifted 10 to 15 mm with respect to each other. This originates from the
structure of the ux tube which is simplied for the analysis of the pick-up coil measurements. This eect will be discussed in detail in chapter 5.3.1.
z-velocity
The distance of the apex of the ux tube to the electrodes rises linear with time. Constant
z-velocities in the range from 1.5 cm
at a charging voltage of ± 1 kV up to 2.4 cm
at a
µs
µs
charging voltage of ± 3 kV were measured.
Figure 5.12 b) shows the dependency of the expansion velocity on the charging voltage.
The rst measuring point is disregarded, because a voltage of ± 750 V is just above the
limit to ignite a plasma for this plasma source and working gas (c.f. chapter 5.1.1).
Only ICCD-camera images have been analysed for the values of gure 5.12 b). The velocities for ± 1 kV, ± 2 kV and ± 3 kV have been conrmed by measurements with the
pick-up coil and the triple probe.
An example for the determination of the z-velocity from a series of images is shown in
76
5.2 Spatial evolution of the ux tube
Figure 5.13: Illustration of the hoop force: plasma torus with toroidal current and poloidal
magnetic eld
gure 5.12 a). The images were taken at a single shot with the multi-frame ICCD-camera.
The slope of the line of best t (1.1 cm/µs + 0.44 cm/(µs · kV) · U [V]) corresponds to
the velocity of the ux tube.
The z-velocity is proportional to the charging voltage of the capacitor bank. Also the peak
current of the discharge depends on the charging voltage (∼ 12 kA at ± 1 kV, ∼ 24 kA at
± 2 kV and ∼ 36 kA at ± 3 kV).
y-expansion
For the analysis of the y-expansion of the ux tube ICCD-camera images were used, because
further measurements with other diagnostics are only available for the apex of the ux tube.
In gure 5.11 b) the expansion in y-direction is shown for a charging voltage of ± 3 kV. A
non-linear increase of the expansion is observed which is assumed to be quadratic.
A quadratic t (red line in gure 5.11 b) of the y-expansion yields a maximum velocity of
5 cm/µs for the time just before the ux tube tears.
Cause of the expansion
The z- and the y-expansion show a strongly dierent behaviour: The velocity of the zexpansion is constant, the velocity of the y-expansion grows quadratically.
At rst the expansion velocity of the neutral gas which is pued in front of the electrodes
was estimated. It is limited by the speed of sound of the used gas. In hydrogen it amounts
to 1270 m/s (0.127 cm/µs) and in helium to 970 m/s (0.097 cm/µs). Due to collisions
between the gases, the speed of sound for the hydrogen-helium gas mixture is estimated
77
5 Results and discussion - rst plasma source
to be between these values.
As part of a bachelor thesis [Mac10] velocity measurements of the neutral gas expanding
into the vacuum chamber were carried out. Although the reliability of these measurements
are restricted they can be used as a rough estimation. They have shown a velocity for
the hydrogen-helium gas mixture of about 670 m/s (0.067 cm/µs). Hence, the expansion
velocity of the ux tube is at least larger by a factor of 20 compared to the expansion
velocity of the neutral gas.
One approach to explain the expansion of the ux tube is the hoop force (equation 2.41).
If a toroidal plasma carries a toroidal current, the current produces a poloidal magnetic
ux. The value of the magnetic ux on the inside (ψi ) and the outside (ψo ) of the torus is
the same (c.f. gure 5.13).
However, the magnetic ux on the inside of the torus passes through a smaller area than
the magnetic ux on the outside of the torus. This leads to a higher magnetic ux density
on the inside than on the outside of the torus and a net force (FI , equation 2.41) which
aims radially outwards.
The hoop force depends on the toroidal current, the amplitude of which is proportional to
the charging voltage of the capacitor bank. This could be used to explain the dependence
of the z-velocity on the charging voltage, as shown in gure 5.11 b). However, a force
should result in an acceleration which is not observed for this direction.
There are two possible explanations for this behaviour: First, there is a counter force which
compensates the hoop force in z-direction after the ux tube is accelerated to its measured
z-velocity (during the rst µs of the life-time of the ux tube).
The tension force of the magnetic guiding eld (which is frozen into the plasma) and the
friction force between the plasma and the neutral gas were taken into account as counter
forces. But the estimated values of these forces are much smaller than the calculated hoop
force of the ux tube. However, the hoop force is increasing (due to the increasing discharge current) which means the counter force would have to increase as well. For this
behaviour no suitable force could be found.
Second, the hoop force is not acting in z-direction, because the part of the ux tube parallel
to the electrode surface (around the apex) has no counter part (c.f. gure 5.17). In the
rst µs of its life time the ux tube is accelerated to its measured z-velocity by an other
eect.
The second explanation is supported by the course of the y-expansion of the ux tube.
Both arms of the ux tube are perpendicular to the electrode surface, the current through
the rst one is antiparallel to the current through the second one. And as it can be expected, the hoop force provides a quadratic increase of the y-expansion versus time.
The question remains which eect can accelerate the ux tube in z-direction in the rst µs
after it is formed.
78
5.2 Spatial evolution of the ux tube
(a) Prole of the ux tube, measured by means
of the pick-up coil and the triple probe,
11 cm in front of the electrodes
(b) Diameter of the ux tube, depending on the
distance of the apex to the electrodes; the
dotted line marks the tearing of the ux
tube
Figure 5.14: Diameter of the ux tube
Diameter of the ux tube
The diameter of the ux tube depends on the balance between the thermal pressure of
the plasma and the magnetic pressure of the poloidal magnetic eld. If the magnetic eld
strength increases the ux tube is pinched, if the thermal pressure increases the ux tube
expands.
The pick-up coil and the triple probe are employed to measure the diameter of the ux
tube. Also the ICCD-camera images give an impression of the diameter but this method
is not very accurate due to the limited dynamic range of the ICCD-cameras.
The proles of one triple probe (conductivity) and two pick-up coil (Bϕ , current density j) measurements are shown in gure 5.14 a). They were measured at dierent shots
of the plasma source at the same distance (11 cm) in front of the electrodes. The rst
peaks of these proles occur at almost the same time and correspond to the rst ux tube
(c.f. chapter 5.3.4).
The form and the width of these peaks dier from each other. The rise of the peak of the
Bϕ -prole starts directly after the ignition of the plasma at t = 2.1 µs, the rise of the peaks
of the other proles 2 µs later. Their decrease occurs simultaneously at all proles. The
early rise of the Bϕ peak can be assigned to the frozen ux phenomenon and is discussed
in chapter 5.3.3.
The FWHM of the peaks of these proles is taken as the diameter of the ux tube. Due
to the early rise of the Bϕ peak its FWHM is overestimated. The current density and the
conductivity prole show similar values for their FWHM.
79
5 Results and discussion - rst plasma source
Figure 5.14 b) shows the temporal evolution of the diameter of the ux tube determined
from Bϕ and from the conductivity prole. Although their absolute values dier signicantly from each other (as it can be expected), both diameters show a similar behaviour.
The error bars are relatively large compared with the values of the diameter. This makes
it dicult to interpret its course. Notable are two parts of it: From 10 to 13 cm in front
of the electrodes the ux tubes seem to pinch, but it is unclear why it expands afterwards
(13 to 17 cm in front of the electrodes). And after the tearing of the ux tube the diameter
stays constant.
5.2.1 Parameterisation of the shape of the ux tube
For the calculations presented in chapter 5.4 the geometry of the ux tube is needed as
an input parameter. To obtain this information ICCD-camera images (e.g. gure 5.2)
were evaluated. In table 5.2 the z- and the y-expansion of the ux tube as well as the
corresponding discharge current are listed.
The expansion can be expressed by the following terms: The z-expansion can be written as
z(t) = −2.22 · 10−3 m + 2.37 · 104 m/s · t
(5.1)
dy (t) = 8.26 · 10−2 m − 2.07 · 104 m/s · t + 4.88 · 109 m/s2 · t2
(5.2)
and the y-expansion as
80
5.2 Spatial evolution of the ux tube
time
µs
z-expansion
mm
y-expansion
mm
current
kA
1.8
2.3
2.8
3.3
3.8
4.3
4.8
5.3
5.8
6.3
6.8
7.3
7.8
8.3
8.8
46.8
57.1
63.8
73.0
82.6
96.2
108.5
121.0
135.2
148.6
158.9
174.5
182.6
194.5
206.3
60.2
60.2
63.6
68.2
73.9
85.0
97.2
108.0
126.5
142.3
167.5
193.4
218.0
247.0
278.4
0.32
1.91
4.42
7.57
10.50
12.56
14.39
16.15
17.88
19.53
21.05
22.55
23.76
25.04
26.43
Table 5.2: Expansion of the ux tube with corresponding current taken from ICCD-camera
images; the blue values are interpolated, the red ones are extrapolated
81
5 Results and discussion - rst plasma source
5.3 Magnetic eld measurements
Figure 5.15: Flux tube with Bϑ and current at t = 5.2 µs
At the rst plasma source two components of the magnetic elds appear: The static magnetic eld of the permanent magnets built into the plasma source and the time varying
magnetic eld of the ux tube. The magnetic eld of the permanent magnets of the plasma
source can be measured by means of a Hall probe. Since it is static, it is not recognized
by pick-up coils, which are only able to measure the change of the magnetic ux. In
chapter 3.4.2, as part of the characterization of the plasma source, measurements and calculations of the magnetic eld, without a plasma present, are shown.
Bϑ and Bϕ are measured independently at dierent shots at the apex of the ux tube, on
a line perpendicular to the electrodes. The employed pick-up coil is placed in a ceramic
tube which is insulating it from the plasma. The ceramic tube (with a diameter of 12 mm)
can be seen on most of the ICCD-camera images (i.e. in gure 5.15). Its presence seems to
have no or just little inuence on the evolution of the ux tube, as far as it can be observed.
It has to be kept in mind that the overall picture of the magnetic eld measurements
presented here is a composition of single shots. To give a statement about the general
behaviour of the magnetic elds several measurements are averaged for every spatial measuring point.
The pick-up coil is moved in steps of 1 cm along the z-axis measuring the voltage induced
by the variation of the magnetic ux. To obtain the temporal evolution of the magnetic
ux density the induced voltage is integrated over time and multiplied with the calibration
coecient determined in chapter 4.2.
82
5.3 Magnetic eld measurements
(a) Bϑ , 10 cm in front of the electrodes
(b) 2-d plot of Bϑ , showing the whole evolution
of this component
Figure 5.16: Measurements of Bϑ at a charging voltage of ± 3 kV: the prole shown in
gure a) is taken from gure b) (position marked by white dots)
5.3.1 ϑ-component of the magnetic eld
The Bϑ -component is produced by the current ow through the ux tube. Figure 5.16 a)
gives an example of the temporal evolution of Bϑ . Here, the pick-up coil is placed 10 cm
in front of the electrodes. At point a) the plasma ignites, the discharge current begins to
increase and the ux tube is formed.
First, the pick-up coil is located outside the major radius R of the ux tube. The ux tube
begins to rise and its apex moves with a velocity of about 2.4 cm/µs towards the position
of the pick-up coil. Due to the decreasing distance between the apex of the ux tube and
the pick-up coil and the increasing current (i.e., see gure 3.8 b), the absolute value of Bϑ
also increases.
At point b) the apex of the ux tube crosses the position of the pick-up coil. Thereby, the
pick-up coil reaches into the major radius R of the ux tube. On this side of the ux tube
the magnetic ux points in the opposite direction and therefore Bϑ changes its sign. This
fact is used in chapter 5.2 to determine the position and the velocity of the apex of the
ux tube in z-direction.
The pick-up coil itself is smaller (diameter of 0.5 cm) than the width of the ux tube
(diameter of about 3 to 4 cm, see chapter 5.2) so that the crossing takes at least one µs
(depending on the z-velocity of the ux tube, see chapter 5.2).
83
5 Results and discussion - rst plasma source
Figure 5.17: Calculation of Bϑ of a ux tube (I = 5 kA)
Structure of the ux tube
After the ux tube has crossed the pick-up coil the absolute value of Bϑ is increasing to a
much larger value than before. This is partly due to the still increasing discharge current
but more important for this eect is the structure of the ux tube.
Bϑ is produced by the current, owing in ϕ-direction (from the perspective of the ux
tube). This current can be split into a y- and a z-component (from the perspective of the
plasma source).
Figure 5.17 shows a rough approximation of the magnetic ux density in ϑ-direction. When
the pick-up coil is placed outside the major radius R of the ux tube, the magnetic ux
produced by the y-component of the current is pointing in the opposite direction of the
magnetic ux produced by the z-component of the current. Thus, both magnetic ux
components cancel each other out partly.
This is a similar situation as described for the magnetic eld produced by the line current
of the Titov-Démoulin plasma source. In gure 3.13 the calculated magnetic eld of the
line conductor (blue line) is shown. To achieve the measured magnetic eld (red asterisks),
the magnetic eld of the feedings has to be subtracted.
When the pick-up coil is placed inside the major radius R the magnetic ux produced by
the y-component of the current changes its sign (from the perspective of the pick-up coil)
and points in the same direction as the magnetic ux produced by the z-component of the
current. Hence, the sum of both magnetic ux components inside the major radius R is
bigger than outside of it.
84
5.3 Magnetic eld measurements
The geometry of the ux tube has also an eect on the measured position of its apex:
due to the above described shift of the amplitude of the magnetic ux density also the zero
crossing of Bϑ is shifted.
This eect can be seen in gure 5.11 a) where the distance of the apex to the electrodes is
plotted versus time: The distance measured by the pick-up coil is shifted to the electrodes
compared to the distance measured by the ICCD camera.
Temporal and spatial evolution of Bϑ
Figure 5.18: Position of the apex of the ux tube, taken from the zero crossing of Bϑ at
± 3 kV
In gure 5.18 the position of the apex of the ux tube is plotted versus time. The linear
rising of this distance results in a constant velocity of 2.4 cm/µs. Even after the tearing
of the ux tube (at t ≈ 7 µs, marked by blue dots) a Bϑ prole can be measured which
is similar to proles before the tearing. The apex moves on with constant velocity. This
behaviour will be discussed later in chapter 5.3.4.
In gure 5.16 b) the spatial and temporal evolution of Bϑ is shown. The evolution of
Bϑ at a single position of the pick-up coil which is shown in gure 5.16 a) is marked by
white dots in it.
The colour transition from blue to green indicates the change of sign of Bϑ and with it the
position of the apex of the ux tube. After the tearing of the ux tube a decrease of the
absolute value of Bϑ inside the major radius R occurs whereas the discharge current is still
rising.
85
5 Results and discussion - rst plasma source
(a) Bϑ -component of the magnetic eld
(b) Bϑ , time axis converted to a spatial axis
(c) Current density versus time, maximum at
t = 5.4 µs
(d) Current density versus distance to the electrodes at t = 5.4 µs
Figure 5.19: Calculation of the current density, 11 cm in front of the electrodes
5.3.2 Current density prole
The measured Bϑ allows an estimation of the current density of the ux tube under the
crude assumption that the ux tube is a straight cylinder. In order to convert the temporally resolved data into spatially resolved data, the velocity of the ux tube is needed.
As described in [Erb64] one can use the dierential form of Ampère's law
~ ×B
~ = µ0~j
∇
(5.3)
~ = Bϑ to obtain the current density of a cylindrical ux tube
in which ~j = jz and B
jz =
86
1 ∂
(rBϑ ) ~ez
µ0 r ∂r
(5.4)
5.3 Magnetic eld measurements
Figure 5.19 illustrates the procedure to obtain the current density. As an example the
evolution of Bϑ (diagram a) in a distance of 11 cm to the electrodes is used. Since the ux
tube is not a straight cylinder, a strong amplitude asymmetry of Bϑ can be observed.
At point a) in this diagram, the plasma is ignited and the ux tube is formed. Point b)
shows the crossing of the ux tube over the pick-up coil. A second peak appears at point c)
which can be interpreted as a second ux tube, formed after the tearing of the rst one.
To calculate the current density, Bϑ has to be measured with high spatial resolution (with a
distance between the measuring points of about 1 mm). However, such an approach would
be impractical due to the amount of time needed and the diameter of the pick-up coil itself
(∅ 5 mm). Measurements with several probes operated in parallel were not successful (c.f.
chapter 4.2).
Therefore, another method was chosen which is slightly less accurate. Using the velocity
of the apex of the ux tube the time axis of a single measurement can be converted into a
spatial axis. The result of such a conversion is shown in gure 5.19 b).
The zero point of this axis has been set at the zero crossing of Bϑ which indicates the
center of the ux tube. So the spatial axis corresponds to the minor radius r. As long as
the minor radius stays negative the pick-up coil is outside the major radius R of the ux
tube. Afterwards, when the minor radius is positive the pick-up coil is inside the major
radius R.
Using equation 5.4 the current density can be estimated. In order to compare its evolution with other measurements it is plotted versus time again (gure 5.19 c). Here a
second peak of the current density can be recognized (at t = 10 µs). It can be interpreted
as a second ux tube which is formed after the tearing of the rst one (c.f. chapter 5.3.4).
For a further evaluation of the second peak of the current density Bϑ has to be analysed a
second time with the zero point of the minor radius r shifted to the time of its occurrence
(which should correspond to the center of the second ux tube).
The ux tube reaches the position of the pick-up coil at t ≈ 4 µs. The steep rise of
the current density indicates the transition from the neutral gas background to the plasma
of the ux tube. The maximum of the current density is reached at t = 5.4 µs which
indicates the position of the center of the ux tube. This information was already used in
the discussion of the evolution of the ux tube earlier in this chapter (c.f. gure 5.11 a).
In contrast to its steep rise the current density decays slowly. This can be attributed to the
structure of the ux tube (it is not cylindrical, c.f. chapter 5.3.1) and to the plasma between the ux tube and the electrodes (c.f. gure 5.15), which still carries a small amount
of the current. Due to the 1/r-dependency of the current density the measured signal gets
damped more strongly with increasing absolute value of the minor radius r.
87
5 Results and discussion - rst plasma source
(a) Comparison of the current density at dierent charging voltages 11 cm in front of the
electrodes
(b) Temporal and spatial evolution of the current
density; charging voltage ± 3 kV; the tearing
of the ux tube at 7 µs is marked by white
dots
Figure 5.20: Current density at dierent charging voltages and 2-d plot of the temporal
and spatial evolution at ± 3 kV charging voltage
In gure 5.19 d) the spatial distribution of the current density at a xed time (t = 5.4 µs) is
shown. For every pick-up coil position the corresponding current density is plotted. First
the pick-up coil is located between the electrodes and the ux tube until the maximum of
the current density is reached in a distance of 11 cm to the electrodes. After the maximum
the pick-up coil is located outside the ux tube.
In comparison to the single measurement shown in gure 5.19 c) the prole of the diagram
is reversed. First, the pick-up coil is located inside the major radius R and the current
density rises slowly. After its maximum the pick-up coil is located outside the major radius
R and the current density decays fast.
In gure 5.20 a) the current density for three dierent charging voltages of the capacitor bank is plotted. Again, the pick-up coil is located 11 cm in front of the electrodes. In
addition to the charging voltage the discharge current at the time of the maximum of the
current density is specied. Due to the dierent expansion velocities at dierent charging
voltages the maxima of the current density are shifted in time. Hence, the data is plotted
against the minor radius r of the respective ux tube.
The peak current density increases more than the discharge current at that time does.
Also the half-width of the signal is smaller for higher charging voltages and the associated
higher discharge current. Consequently a smaller half width (and a stronger pinching) of
the ux tube is observed.
88
5.3 Magnetic eld measurements
Integrating the current density over the cross-section of the ux tube the total current
can be estimated. The resulting currents from the current density proles shown in gure 5.20 a) can be compared with the total current measured at one of the feedings of the
electrodes at the time of the current density maximum.
For a charging voltage of ± 3 kV the current through the ux tube can be estimated to
14.3 kA (15.3 kA measured), for ± 2 kV to 8.6 kA (12.8 kA measured) and for ± 1 kV to
7.0 kA (9.4 kA measured). Despite the crude assumptions, the estimated current values
through the ux tube correspond to 70 to 90 percent of the total current.
Temporal and spatial evolution
The temporal and spatial evolution of the current density is shown in gure 5.20 b). As
mentioned above, the signal is strongly damped for values outside the width of the ux
tube which results in a uniformly colored background.
A linear expansion of the ux tube can be observed. The expansion velocity of 2.7 cm/µs,
resulting from this measurement is in good agreement with the previously measured velocities of the ux tube (c.f. chapter 5.2). Even after the tearing (t = 7 µs, marked by white
dots) a linear expansion can be observed.
A signicant decrease of the current density occurs after the tearing of the ux tube. As
discussed earlier in this chapter, this conrms the assumption that the current through the
ux tube is decreasing and a second ux tube in front of the electrodes is formed, taking
over the current. This behaviour will be discussed in chapter 5.3.4.
5.3.3 ϕ-component of the magnetic eld
In contrast to the observations of the ϑ-component of the magnetic eld (which can be
attributed entirely to the current owing through the ux tube) the contribution to Bϕ
(c.f. gure 5.21) can come from various sources. It can be produced due to a kinking of
the ux tube (c.f. chapter 2.1.4) or by the frozen-in magnetic ux of the guiding eld.
The result of two dierent measurements is shown in gure 5.21: The pick-up coil is
placed 8 cm in front of the electrodes. The black line shows the temporal evolution of Bϕ
with the magnetic guiding eld antiparallel to the discharge current.
After this measurement the orientation of the magnetic guiding eld was inverted. The
red line shows the temporal evolution of Bϕ now with the magnetic guiding eld parallel to
the discharge current. The prole of both signals shows roughly the same evolution, only
it's sign has changed.
89
5 Results and discussion - rst plasma source
Figure 5.21: Bϕ 8 cm in front of the electrodes; the orientation of the magnetic guiding
~ ↑↑ E
~ , black line B
~ ↑↓ E
~)
eld was inverted (red line B
This behaviour could partly be attributed to the following cause: If the ux tube kinks,
it leaves the y-z-plane (from the perspective of the plasma source) and the x-component
of the current through the ux tube creates a magnetic eld in ϕ-direction. The direction
of the kink (and with it the direction of the ϕ-component of the magnetic eld) depends
on the orientation of Bϑ and the magnetic guiding eld: If the magnetic guiding eld is
inverted this would be also true for Bϕ .
But in this case the direction of Bϕ would change when the ux tube crosses the position
of the pick-up coil (as it can be observed for Bϑ , c.f. chapter 5.3.1). However, this change
of sign cannot be observed at any Bϕ -measurement. Depending on the orientation of the
guiding eld Bϕ is either positive or negative (c.f. gure 5.21).
Also the ICCD camera images of the ux tube from the top and from the side (c.f. gure 5.4)
do not provide much information about the structure of the ux tube. A kinking of the
ux tube is not observable.
It cannot be excluded that a slight kinking of the ux tubes occurs. However, another
eect with a signicantly higher amplitude (which compensates the change of sign of a
current generated magnetic eld after the crossing of the position of the pick-up coil by
the ux tube) is necessary to produce the observed Bϕ .
Frozen magnetic ux
Another cause for the measured Bϕ could be the magnetic guiding eld of the plasma
source (c.f. chapter 3.2), frozen into the plasma as described in chapter 2.1.1. Under normal circumstances the magnetic guiding eld is static and thus cannot be detected by a
pick-up coil. In the experiment the ux tube can transport the magnetic ux with it when
90
5.3 Magnetic eld measurements
(a) Bϕ 11 cm in front of the electrodes; this prole is taken from gure b) (position is marked
by white dots)
(b) Temporal and spatial evolution of Bϕ ; the
ux tube tears at 7 µs (rst vertical black
dots), the maximum discharge current is
reached at 13 µs (second vertical black dots)
Figure 5.22: Measurements of Bϕ at a charging voltage of ± 3 kV
it expands. Then the magnetic ux at the location of the pick-up coil changes with time
and can be measured.
Frozen magnetic ux is an eect that occurs in ideal MHD. The important parameters for
it are the magnetic Reynolds number (c.f. equation 2.28) and the resistive skin time (c.f.
equation 2.27). Due to measurements of the electron density and temperature by means
of the triple probe and the laser interferometer (c.f. chapter 4.4) Rm can be estimated to
have a value of 11 to 32 and τR to have a value of 14.1 to 39.6 µs (c.f. table 2.1).
The magnetic Reynolds number is larger than 1 and the resistive skin time is larger than
the life time of the ux tube. Although these values are in the intermediate domain between ideal and resistive MHD, during the life time of the ux tube a signicant part of
the magnetic ux is frozen into its plasma.
Additionally, it must be noted that the electron density and temperature were measured
only in the apex of the ux tube and certainly vary in other parts of it. This applies also
to all derived parameters.
In this case the orientation of Bϕ depends directly on the orientation of the magnetic
guiding eld. If the direction of the magnetic guiding eld is inverted, also the direction
of Bϕ is inverted.
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5 Results and discussion - rst plasma source
(a) Comparison of Bϕ (pick-up coil) and the line
integrated electron density (laser interferometer) 11 cm in front of the electrodes
(b) Maximum of Bϕ of 150 single measurements
(black dots), the ux tube tears 15 cm in
front of the electrodes (vertical red dots); for
comparison the maximum values of Bϕ for
the reduced magnetic guiding eld (70%) are
shown (blue asterisks)
Figure 5.23: Comparison of the shape and evolution of the maximum amplitude of Bϑ
Temporal and spatial evolution
An example for the temporal evolution of Bϕ is presented in gure 5.22 a). The pick-up
coil is placed 11 cm in front of the electrodes and the plasma ignites at t = 1.8 µs. After
a slow rise of the amplitude of Bϕ a rst maximum occurs at t = 5.3 µs. Comparing
the position of the maximum with the z-expansion (major radius R) of the ux tube (c.f.
gure 5.11 a) and the measurement of the line-integrated electron density (by means of the
laser interferometer, c.f. gure 5.23 a) it can be assigned to the position of the ux tube.
The rising of the amplitude of Bϕ begins much earlier (at t = 2.1 µs) compared to the
amplitude of the current density and the conductivity (each at t = 4.1 µs). This behaviour
can also be explained by the frozen ux phenomenon. As well as the frozen-in magnetic
ux cannot leave the plasma the magnetic ux in front of the ux tube cannot penetrate
it. When the ux tube is rising it pushes the magnetic ux in front of it.
A second maximum can be observed at t = 9.7 µs, after the tearing of the rst ux tube
(c.f. chapter 5.3.4). Compared to the measurements of the conductivity (triple probe) and
the current density (pick-up coil) the amplitude of the second maximum of Bϕ is much
larger (c.f. gure 5.14 a). This observation will be discussed later in this chapter.
In gure 5.22 b) the temporal and spatial evolution of Bϕ is presented. The single prole
shown in gure 5.22 a) is marked by horizontal white dots, the tearing of the ux tube (at
92
5.3 Magnetic eld measurements
t = 7 µs) and the time of the maximum of the discharge current (at t = 13 µs) by vertical
black dots. Additionally, the linear z-expansion of the rst and the second ux tube is
marked by solid white lines. From these lines the velocities can be estimated to 2.7 cm/µs
for the rst ux tube and to 3.8 cm/µs for the second one.
The evolution of the ux tube is shown in the spatial range from 8 to 21 cm in front of the
electrodes. Closer to the electrodes the amplitude of the measured signal is much smaller.
Also a strong electrical disturbance due to the proximity of the probe to the electrodes can
be observed which reduces the signal to noise ratio further.
The amplitude of Bϕ increases until the tearing of the rst ux tube at t = 7 µs occurs. After this time an expansion with still constant velocity of the ux tube can be
observed but with a decreasing amplitude. This behaviour will be discussed in more detail
later.
Coinciding with the tearing of the rst ux tube a second one is formed which takes over
a part of the discharge current. In the beginning of the second ux tube the measured
magnetic ux density is much larger than the measured magnetic ux density of the rst
one.
This could be explained by the topology of the magnetic guiding eld: The rst ux tube is
formed with its apex in a distance of about 3.5 to 4 cm to the electrodes. At this position,
the magnetic ux density of the guiding eld has a value of 9 to 12 mT. The second ux
tube ignites much closer to the electrodes. For example, the magnetic guiding eld has a
ux density of 20 mT at a distance of 2 cm to the electrodes.
Amplitude of the magnetic ux density
Although the evolution of Bϕ can be explained, the value of its amplitude has to be considered in more detail. The maximum values of Bϕ of 150 single measurements with full
magnetic guiding eld present (black dots) in dierent distances to the electrodes is shown
in gure 5.23 b). The maximum values of 9 averaged measurements with reduced magnetic
guiding eld (70 %, blue asterisks) is added to this gure.
With increasing distance to the electrodes (which is equivalent to later time and an increasing discharge current) the maximum amplitude of Bϕ is also increasing until the ux
tube tears (15 cm in front of the electrodes). This can be explained by two dierent eects:
First, the pinching of the ux tube due to its poloidal magnetic eld. The toroidal magnetic eld is frozen into the plasma. When the ux tube pinches also the magnetic lines
of force are compressed and the magnetic ux density is increasing.
Second, the rising of the discharge current. Due to a slight kink of the ux tube the pick-up
coil detects also the current generated magnetic eld. Until the ux tube tears, the current
through it is increasing and with it the current generated magnetic eld.
93
5 Results and discussion - rst plasma source
(a) Cross-section of the ux tube divided into slices to calculate the
ux through it
(b) Magnetic ux through the cross-section
(∅ 4 cm) of a ux tube (red asterisks); the
black line indicates the magnetic ux through
the cross-section of the ux tube provided by
the guiding eld for dierent ux tube diameters (c.f. gure 5.25)
Figure 5.24: Calculation of the magnetic ux through the cross-section of the ux tube
After the tearing of the ux tube the amplitude of Bϕ drops. As it will be discussed in
chapter 5.3.4, the discharge current through the rst ux tube decreases from that point.
Also the conductivity of the plasma begins to decrease and the plasma parameters move
further away from the domain of ideal MHD. The magnetic ux is frozen less into the
plasma and diuses out of the ux tube.
If the magnetic guiding eld is reduced to 70% of its eld strength (measured at the electrode surface) also the measured maximum values of Bϕ are reduced. This can be explained
by a reduced magnetic ux which is frozen into the plasma.
In contrast to this observation, the maximum values of Bϕ should increase if this component is primarily produced by the discharge current. The guiding eld stabilizes the ux
tube and a decrease of its magnetic ux density should lead to a more kinked ux tube.
From Bϕ the magnetic ux through the cross-section of the ux tube can be estimated.
Due to the measured diameter of the ux tube (c.f. chapter 5.2) a constant value of 4 cm
is taken for the calculations. Furthermore, it is assumed that most of Bϕ is caused by
the frozen magnetic ux of the guiding eld. It should be noted that the here presented
measurements and calculations refer only to the apex of the ux tube.
The amplitude of the magnetic guiding eld decreases with increasing distance to the electrodes (magnetic dipole, c.f. gure 5.25). Hence, the cross-section of the ux tube is
divided into slices of constant magnetic ux density (c.f. gure 5.24 a). The measured
94
5.3 Magnetic eld measurements
Figure 5.25: Hall probe measurement of the magnetic guiding eld (c.f. chapter 3.4.2); the
provided magnetic ux is estimated using the position and the diameter of
the ux tube
value of Bϕ of every slice is multiplied with the height of the slice and then integrated
along the z-axis.
The result of these calculations is shown in gure 5.24 b). Just like the maximum amplitude of Bϕ the magnetic ux increases linearly until the ux tube tears and the magnetic
ux drops thereafter.
The magnetic ux of the guiding eld is calculated for several parameters to check whether
the magnetic ux through the cross-section of the ux tube can be provided by it or not.
As it can be observed in gure 5.2 the igniting ux tube is located in a distance of 3.5 to
4 cm in front of the electrodes. The basis of these calculations are the Comsol simulations
of the magnetic guiding eld which are based on Hall probe measurements. Again, the
cross-section of the ux tube is divided into slices of constant magnetic ux, every slice is
multiplied with the height of the slice and then integrated along the z-axis (c.f. gure 5.25).
Starting with a diameter of 4 cm for the ux tube the magnetic ux has only a value of
12.4 to 23.4 µWb (depending on the distance of the igniting ux tube of 3.5 to 4 cm to the
electrodes). This value is compared with the maximum of the magnetic ux of 54.0 µWb
(measured 13 cm in front of the electrodes) much too small. But it is enough magnetic
ux to explain the measured values closer to the electrodes (e.g. 22.6 µWb 8 cm in front
of the electrodes).
Assuming now that the ux tube pinches (and with it the magnetic lines of force) the collection area for the magnetic ux has to be increased. For a diameter of 6 cm the magnetic
ux has already a value of 30.4 to 44.6 µWb, for 7 cm a value of 43.8 to 57.2 µWb and for
8 cm a value of 60.0 µWb (only possible for a distance to the electrodes of 4 cm for the
igniting ux tube).
95
5 Results and discussion - rst plasma source
The lower limit for each diameter are marked by black lines in gure 5.24 b). With the
pinching of the ux tube and the resulting increase of the estimated magnetic ux within
the cross-section of the ux tube not only the values of the measured magnetic ux can
be explained. Also the increasing of the measured magnetic ux (and the magnetic ux
density) over time can be attributed to the pinching of the ux tube.
However, it should be considered that the value of the resistive skin time (14.1 ·10−6 39.6 · 10−6 s) allows a diusion of the magnetic ux within the life time of the ux tube.
This could be compensated by the magnetic eld produced by the discharge current of
a slightly kinked ux tube. Because the discharge current increases with time also the
current generated magnetic eld increases. This behaviour would be also consistent with
the measurements.
5.3.4 Observations after the tearing of the ux tube
The tearing of the ux tube is described in chapter 5.1.2. The statements in that chapter
are based on the images of the ICCD-cameras. Taking the pick-up coil measurements into
account the tearing of the ux tube can be considered in more detail.
The images in the gures 5.6 and 5.7 show that the ux tube becomes detached from the
electrodes at t = 7 µs. At the same time, a dip in the raw signal of the Rogowski coil can
be observed which suggests that the discharge current is changing its path through the
plasma. This assumption is supported by the decrease of the emission intensity and the
increasing width of the ux tube.
Additionally to these observations, the pick-up coil measurements show further information about the evolution of the ux tubes. The evolution of Bϑ (gure 5.16 b), the current
density (gure 5.20 b) and Bϕ (gure 5.22 b) are showing that the expansion in z-direction
goes on with a constant velocity even after the tearing of the ux tube. However, the
amplitude of all three signals is decreasing after t = 7 µs which can be attributed to a decreasing current through the ux tube (while the total discharge current is still increasing).
Nevertheless, it seems that a fraction of the current is still owing through the ux tube,
even if it is not understood how the ux tube is still connected to the electrodes. Since the
expansion velocity remains constant but the current through the ux tube is reduced, the
expansion velocity seems to be independent of the current (but not of the charging voltage
of the capacitor bank, c.f. gure 5.12 b).
As mentioned in chapter 5.1.2 a short circuit between the electrodes can be observed
after the tearing. The plasma of the short circuit seems to form a second ux tube. This
assumption is supported by several observations:
Every measured signal which suggest a second ux tube appears after the tearing of the
rst ux tube.
96
5.3 Magnetic eld measurements
A second peak can be observed in most of the probe measurements. An example of these
measurements is given in gure 5.14 b) where Bϕ , the current density and the conductivity of the plasma are compared. The second peak occurs in all three measurements at
t = 10 µs. Thus, it is a reproducible signal which can be measured by dierent diagnostics
at the same time of the discharge.
The course of a rst and a second maximum of Bϕ can be followed in gure 5.22 b). Just
like the rst maximum the second one expands with a constant velocity. This behaviour
indicates a reproducible structure which can be attributed to a second ux tube.
For the rst 2 cm of these measurements the amplitude of Bϕ of the second ux tube is
larger than the amplitude of the rst one. This might be attributed to the place where
the ux tube is formed: The rst ux tube is formed in a distance of 3.5 to 4 cm in front
of the electrodes. The magnetic guiding eld in that region has a ux density of 10 to
12 mT. Assuming that the second ux tube is formed closer to the electrodes (using the
direct short circuit between the electrodes as starting point, c.f. gure 5.2 h), it is formed
in a region where the ux density is larger (about 20 mT in a distance of 2 cm to the
electrodes). Therefore, more magnetic ux can be frozen into the second ux tube than
into the rst one.
Together with the pick-up coil measurements the attempt was made to take images of
the second ux tube. But even with the single shot ICCD-camera (higher dynamic range,
c.f. chapter 4.3.4) it was not possible to identify a corresponding structure on any image.
A reason for this could be the neutral gas density in the volume in front of the electrodes.
The rst ux tube has collected much of the neutral gas in front of the electrodes to ignite
its plasma.
In chapter 5.1.1 the necessary neutral gas density (H2 ) along the ignition path of the ux
tube is estimated to 10 Pa. This value corresponds to a neutral particle density (H2 ) of
2.7 ·1021 m−3 which provides an electron density of 5.4 ·1021 m−3 if the neutral gas is fully
ionized (H2 → 2 electrons). In comparison to this calculated value the measured peak electron density (by means of the laser interferometer) accounts to 1022 m−3 . Thus, a highly
ionized plasma can be expected.
The supply of neutral gas injected by the fast gas valve has only an expansion velocity
of 0.670 mm/µs (c.f. table 6.1). In the short time between both ux tubes (≈ 5µs) this
volume cannot be lled with neutral gas again. Therefore, a signicantly lower neutral gas
density and with it a limited number of charge carriers is available for the second ux tube
(while the discharge current is still increasing). So the plasma of the second ux tube may
be fully ionized and very little light would be emitted.
97
5 Results and discussion - rst plasma source
5.4 Comparison with calculations
(a) Example for the parameterisation
of the current channel (red dots) of
the ux tube; the current density
prole of the ux tube is shown for
the surface of the electrodes
(b) Comparison of the measured magnetic
eld components with the calculations at
t = 5 µs; the measured current through the
ux tube accounts to 15.5 kA, for the calculation a value of 14 kA is used
Figure 5.26: Simulation of the current channel
The calculations presented here were carried out by Lukas Arnold in the frame of his
doctoral thesis [Arn08] and all gures presented in this chapter are taken from it. They
are based on the experiment presented in chapter 3.
First, a model calculation is employed to describe the evolution of Bϑ and Bϕ . As input
parameters the time dependent discharge current and the geometry of the ux tube are
used (c.f. chapter 5.2.1). This approach neglects that the magnetic ux of the guiding eld
is frozen into the plasma of the ux tube.
Second, a numerical MHD simulation is carried out [ADG+ 08] to calculate the included
variables self-consistently.
The close cooperation between the simulation and the experiment is indicated by the logo
of the Research Unit 1048 which is also shown on the cover of this thesis. It shows on
the left side an image of the ux tube taken at the experiment which merges into the
simulation of one on the right side of it.
5.4.1 Calculation of the current-generated magnetic eld
Lukas Arnold performed model calculations ([Arn08], chapter 3.6) to describe the magnetic
eld produced by the current owing through the ux tube. Therefore, the temporal
98
5.4 Comparison with calculations
evolution of the shape of the ux tube taken by ICCD-camera images was parameterised
(c.f. table 5.2). Assumptions were made for the deformation of the apex of the ux tube
(kink instability, c.f. chapter 2.1.4) which could not be observed by the ICCD-camera.
Further adjustable parameters in this calculation are the half-width and the prole of the
current density.
Another parameter used in the model calculations is the current through the ux tube
which corresponds to the measured current in the experiment; a Gaussian current density
prole is assumed.
The magnetic guiding eld provided by the horseshoe magnet of the plasma source is
neglected in this model. Since this component of the magnetic eld is pointing in ϕdirection it has no inuence on Bϑ . In contrast to this model, in the experiment (c.f.
chapter 5.3.3) it is assumed that the magnetic ux of the guiding eld which is frozen into
the plasma of the ux tube is responsible for most of the measured Bϕ .
Figure 5.27: Comparison between the calculated Bϑ with and without the consideration of
the feedings of the electrodes
This calculation assumes that Bϕ is generated by the current owing through a kinked ux
tube. An example for a parameterised ux tube is shown in gure 5.26 a). The red dots
indicate the position of the ux tube and of the feedings of the electrodes. A kinking can
be observed in the apex of the ux tube.
Hence, the amplitudes of the calculated Bϑ and Bϕ depend on the degree of the kinking of
the ux tube: The more the ux tube is kinked, the higher the amplitude of Bϕ and the
lower the amplitude of Bϑ .
Figure 5.26 b) shows the spatial evolution of the measured and calculated magnetic elds
at t = 5 µs along the z-axis of the plasma source. The discharge current measured at the
experiment for this time accounts to 15.5 kA, for the calculated magnetic elds a value of
14 kA is used. Under these circumstances the shape and the amplitude of the measured
99
5 Results and discussion - rst plasma source
Bϑ (cross-signs connected by red line) is well reproduced by the calculation (blue line).
Although the calculation reproduces the amplitude of the measured Bϕ it does not reproduce its prole: The calculated Bϕ shows a similar prole as Bϑ . After the ux tube has
crossed the pick-up coil (in the calculation a point along the z-axis where the magnetic
eld is calculated for) Bϕ changes its direction.
This behaviour cannot be observed at the measured Bϕ : Their values stay either positive
or negative all the time, a change of sign does not occur (c.f. chapter 5.3.3). Thus, the
ux tube seems not to kink at the experiment.
The electrodes are connected to the thyristor by feedings (c.f. gure 3.5). It has turned
out that their contribution has to be taken into account to reproduce the measured prole
of Bϑ (c.f. gure 5.27).
The proles of Bϑ with (green line) and without (red line) consideration of the feedings
dier signicantly from each other. While the amplitude of Bϑ decreases in front of the
electrodes if the feedings are neglected it stays constant when the feedings are taken into
account.
As already shown in gure 5.26 b) the calculation with consideration of the feedings agrees
to the measured prole of Bϑ .
How much of the current ows through the ux tube?
The comparison of the measured and calculated Bϑ can be used to estimate how much
of the discharge current (measured at one of the feedings by means of a Rogowski coil)
is owing through the ux tube. It cannot be excluded that a fraction of the discharge
current is owing through a part of the plasma which cannot be observed by the ICCDcameras because it is hidden by the electrodes or it does not emit light.
However, the current needed for the calculation to produce the same amplitude of Bϑ
as the discharge current in the experiment depends also on the degree of the kinking of
the ux tube. For the proles shown in gure 5.26 b) the value of the current used for
the calculation amounts to 90% of the discharge current. If the calculation is carried out
without a kinking of the ux tube only 60% of the discharge current are necessary to
produce the appropriate amplitude of Bϑ .
5.4.2 MHD simulation
Besides the calculations presented above, the thesis of Lukas Arnold is focused on the threedimensional, time-dependent numerical MHD simulation of magnetic ux tubes [ADG+ 08].
This simulation is based on the simulation code racoon [DG05] which solves the ideal
MHD equations (c.f. chapter 2.1) on a numerical grid.
100
5.4 Comparison with calculations
Figure 5.28: Initial condition of the MHD simulation with already formed ux tube (green)
with typical grid layout
For the evolution of the ux tube a numerical box of 64 cm × 64 cm × 64 cm is provided.
Figure 5.28 shows a detail of this box with the initial conditions of the simulation. The ux
tube is indicated by the current density. Mesh-adaptive computations with local renement
are used to obtain a sucient spatial resolution. Key features of the simulated evolution
are the expansion of the ux tube, its pinching and the formation of a characteristic dip
in the late expansion phase.
Initial conditions
In the beginning of the simulation the ux tube is at rest, the entire numerical box is
lled with highly conducting plasma and the electrodes are located at the bottom of the
numerical box (plane with z = 0) at ±4 cm · ~ey .
Like the magnetic eld in the experiment, the magnetic eld in the simulation constitutes
of two parts: First, the magnetic guiding eld (realized by the horseshoe magnet in the
experiment) is modeled by two opposing magnets. These are placed under the electrodes,
just outside the numerical box.
Second, a vector potential is related to the current produced magnetic eld in the experiment. Contrary to the experiment this current is kept constant at all times.
All quantities of the simulation are normalized to dimensionless parameters, using the
magnetic ux density B0 = 0.3 T, the length scale L0 = 1.6 cm, the mass density %0 , the
√
Alfvén velocity vA = B0 / µ0 %0 = 6 · 105 m/s and the time scale t0 = L0 /vA = 2.6 · 10−8 s.
The plasma density is assumed to n0 ≈ 1020 m−3 .
101
5 Results and discussion - rst plasma source
The simulation is carried out using dierent density models for the mass transport. Therefore, the equation of continuity 2.17 was supplemented by a source term S resulting in
∂%
~ · (%~u) + S
= −∇
∂t
(5.5)
The following four density models were used:
Mass conservation:
The additional source term is neglected (S = 0) and as an initial
condition a homogeneous mass density (% = 1) is used.
Fixed density:
Here the mass density is kept constant during the whole simulation (% = 1).
Therefore, equation 5.5 is abandoned.
Fixed Alfvén velocity:
The mass density is continuously adjusted in that way that the
~ x, t)|2 ), resulting in a homogeneous
Alfvén velocity remains constant (%(~x, t) ∝ |B(~
communication of Alfvénic disturbances.
Ionisation/recombination model:
The source term is a simplied ionisation-recombination
2
~
balance: S = Γj j + Γr (% − %0 ), with Γj = 0.5, Γr = 5 and %0 = 1.
Further details of the simulation and the density models are given in [ADG+ 08] and [Arn08].
Simulation results
The simulation based on the xed Alfvén velocity density assumption produces a ux tube
which resembles most the ones observered at the experiment. Figure 5.29 shows the resulting current density distribution of one of these simulations which is interpreted as the
shape of the ux tube.
The side view shows the projection of the ux tube on the y-z-plane in the end of its
evolution. Simulation and experiment show the same features: The apex dips towards the
electrodes and the y-expansion is larger than the z-expansion.
In contrast to the side view, the top view of the simulated ux tube diers from the ICCDcamera images (c.f. gure 5.4 a). Such a clear S-shape as it can be seen in the simulation
cannot be seen on the images of the experiment. Furthermore, the magnetic ux density
Bϕ shows little or no indication of a twisting (c.f. chapter 5.3.3).
The MHD simulation also provides data about the evolution of the magnetic ux density. Figure 5.30 shows the temporal and spatial evolution of Bϑ and Bϕ . They are taken
along the same line perpendicular to the electrodes where the measurements are performed
(c.f. chapter 5.3).
102
5.4 Comparison with calculations
(a) Side view
(b) Top view
Figure 5.29: Snapshot of the MHD simulation with constant Alfvén velocity
The evolution of Bϑ shown in gure 5.30 a) diers from the measured evolution of the
magnetic ux density (c.f. gure 5.15) in the section between the electrodes and the ux
tube. In the simulation an increasing of Bϑ is observed while the measured Bϑ is already
on a constant value.
As shown earlier in this chapter (c.f. gure 5.27) the feedings contribute considerably to
the magnetic ux once the probe is within the major radius R. Those feedings are not
included in the simulation.
Figure 5.30 b) shows the evolution of Bϕ . This one also diers from the measured Bϕ . In
the simulation a change of sign can be seen in most of the calculations of Bϕ which is not
observed in the experiment. It can be caused by the current owing through the kinked
ux tube. Hence, Bϕ is produced by the current through the kink of the ux tube and
partly by the frozen in magnetic ux of the guiding eld.
Dierences between the experiment and the MHD simulation
The simulation is based on ideal MHD, therefore a perfectly conducting plasma is assumed.
As discussed in chapter 2.1.3, the plasma parameters of the experiment suggest that the
plasma is in the intermediate range between ideal and resistive MHD. Through this, the
simulation neglects a decoupling of the magnetic eld from the plasma and the possible
occurrence of magnetic reconnection. As a consequence, it would be necessary to extended
the simulation to describe ideal and resistive MHD eects.
103
5 Results and discussion - rst plasma source
(a) Bϑ
(b) Bϕ
Figure 5.30: Bϑ and Bϕ calculated by means of the MHD simulation for several times of
the MHD simulation
Another point is the ignition of the plasma. The MHD simulation begins with an already
formed ux tube which is at rest and a fully ionized background plasma. The uid description is not a suitable tool to describe the ignition of the plasma because it is based on
single particle eects. For the consideration of these eects a dierent approach would be
necessary, e.g. a PIC ( article n ell) simulation.
Even if the evolution of the simulated and experimentally observed ux tube is similar, the
time scale of the simulation is much faster. The life time of the ux tube in the experiment
corresponds to 5 µs and in the simulation to 20·t0 ≈ 0.5µs. By adapting the now measured
plasma parameters (c.f. table 2.1) a higher life time can be estimated. The simulation was
not carried out using these parameters but the life time of the ux tube would be in the
same order of magnitude as in the experiment.
Also the currents through the ux tube deviate from each other: The boundary conditions
of the simulation imply a constant discharge current while it is increasing in time in the
experiment.
P
104
I C
6 Results and discussion Titov-Démoulin plasma source
In the frame of this thesis only the rst steps on the way to the complete Titov-Démoulin
design can be presented. Due to technical diculties with the rst electrode design for
this plasma source (the magnetic eld in front of the electrodes provided by the PFN is
damped by eddy currents, see chapter 3.4.2) an intermediate step between both plasma
source congurations is used (c.f. gure 6.1): The magnetic eld produced by the line
current is replaced by the magnetic eld of permanent magnets.
The results of the measurements from this intermediate plasma source are presented in
this chapter. As a rst step the forming and expansion of ux tubes using several working
gases is investigated.
In this context it is noticed that an argon ux tube can be formed without any magnetic
guiding eld present. Hence, these ux tubes are investigated in more detail in the second
part of this chapter in order to analyse the correlation between the magnetic guiding eld
and the measured Bϕ .
105
6 Results and discussion - Titov-Démoulin plasma source
Figure 6.1: Setup of the Titov-Démoulin plasma source with static magnetic guiding eld,
c.f. gure 3.6
6.1 Titov-Démoulin plasma source with
magneto-static guiding eld
In order to provide a magnetic guiding eld the line current is replaced by permanent
magnets mounted behind the electrodes (see gure 6.1). The prole of this guiding eld is
roughly the same as in the case of the rst plasma source.
Because the magnet stack with its yoke is missing and the single magnets are not directly
placed behind the electrodes (the thickness of the ceramic disc together with the electrodes
adds up to 21 mm), the magnetic eld strength is only 35 mT at the surface of the electrodes
(compared to 110 mT at the rst plasma source).
Supplementary to shots with the established hydrogen-helium gas mixture, discharges are
operated in hydrogen, deuterium, helium, argon and neon. Of particular interest is the direct comparison of the expansion velocity of deuterium discharges and hydrogen discharges.
From this, the mass dependence of the ux tube expansion can be investigated.
6.1.1 Expansion velocity of the neutral gas
The expansion velocity of the neutral gases into the vacuum is limited by their respective
speed of sound:
r
γ · kB · T
(6.1)
m
in which the adiabatic index is given by γ = CP , the ratio of the heat capacity at constant
CV
pressure (CP ) and the heat capacity at constant volume (CV ). Hence, at a given temperature and adiabatic index the speed of sound depends on the reciprocal square root of the
mass of the gas.
cgas =
106
6.1 Titov-Démoulin plasma source with magneto-static guiding eld
gas
mass
u
H2
D2
H2 /He
He
Ne
Ar
2.0158
4.0271
≈ 2.4
4.0026
20.1797
39.948
speed of
sound
m/s
expansion
velocity
m/s
pressure
(measured)
Pa
pressure
(corrected)
Pa
injected
gas
Pa · l
670
410
18
14
14
4
2
1.5
8
6
6
3
3
3
2000
1500
1500
750
750
750
1270
899
970
435
319
280
Table 6.1: Mass, speed of sound (both taken from [wik]) and total amount of the injected
gas into the vacuum chamber of the used working gases; the speed of sound of
deuterium is derived from the speed of sound of hydrogen using equation 6.1
This aects the timing of the experiment, because the time which is needed by the gas to
get from the valve into the volume in front of the electrodes varies. The delay between the
gas injection and the triggering of the capacitor bank has to be adapted (see chapter 3.2.1).
It can also be observed that the total amount of gas injected into the chamber and with it
the neutral gas density in front of the electrodes decreases with increasing mass number.
In table 6.1 all available data of the neutral gas, its expansion velocity and the injected
gas volume are collected. These values were measured in the frame of a bachelor thesis
[Mac10] by means of an ionisation gauge.
The pressure values given in this table are the values measured by means of a commercial
pressure sensor (Pfeier PKR-251) after the gas of a single shot has distributed evenly in
the vacuum chamber. Because the pressure sensor is calibrated for air, the values have to
be corrected for the corresponding gas.
The necessary calibration curves are available on the manufacturer's website. With the
corrected value of the gas pressure and the volume of the vacuum chamber (250 l) the total
amount of the injected gas can be calculated. It is given in the last column of the table.
6.1.2 Flux tube expansion velocity
For gases with a low mass number and charging voltages up to ± 2 kV the ux tube is
not pinching and becomes blurry due to the reduced magnetic guiding eld (compared to
the rst plasma source). Therefore only shots with a charging voltage of ± 3 kV could be
evaluated and their error bars are quite large (up to ±30% of their value).
Figure 6.2 shows the ux tube expansion velocities for the gases used here. A decrease of
the expansion velocity with increasing mass number of the ions is observed. It is noticed
107
6 Results and discussion - Titov-Démoulin plasma source
Figure 6.2: Expansion velocity of the ux tube as a function of the mass number; charging
voltage of the capacitor bank ± 3 kV
that the hydrogen-helium ux tube expansion velocity of both of the plasma sources is the
same (c.f. chapter 5.2).
The expansion velocity of a hydrogen-helium ux tube is 2.4 cm/µs and for a deuterium
ux tube 1.7 cm/µs. The ratio of both of these values is the square root of the mass
ratio of both the atom ions (if the helium fraction of the hydrogen-helium gas mixture
is neglected). This may indicate that the expansion velocity depends on the ion sound
velocity
r
cS =
γZkTe
∝
mi
r
1
mi
(6.2)
in which Z is the charge state and γ the adiabatic index.
In contrast, the ratios of the expansion velocities of helium, neon and argon ux tubes
do not follow this mass dependency. These ions dier not only in their mass from each
other but also in their adiabatic index, charge state and the electron temperature of their
plasma. The adiabatic index for the dierent ions is known and even if the same electron
temperature is assumed for them their charge state has to be estimated.
It should be mentioned that the ion sound velocity is not the only possible velocity depending on the reciprocal square root of the ion mass. Further investigations are necessary
to determine the cause of the here observed constant expansion velocity.
108
6.2 Argon
Figure 6.3: Discharge current with argon and the hydrogen-helium gas mixture as working
gas; the tearing of the ux tube is marked by an asterisk
6.2 Argon
Argon has the largest mass number of all gases employed so far. Argon ux tubes show a
very slow z-velocity (0.76 cm/µs) as compared to hydrogen-helium ux tubes (2.4 cm/µs,
c.f. chapter 6.1.2).
Thus, the whole evolution of the ux tube runs in "`slow motion"'. The evolution of the
discharge current (see gure 6.3) is similar to the hydrogen-helium case, just its amplitude
is lower. Because of this, the current through the ux tube is higher at the same spatial
expansion of the ux tube compared to the lighter gases.
In contrast to any other working gas a formation of a ux tube without magnetic guiding
eld present can be observed. This leads to new insights about frozen in magnetic ux
(c.f. chapter 6.2.3).
The evolution of the discharge current depends on the used working gas. In comparison
to a shot using the hydrogen-helium gas mixture (c.f. gure 6.3), the maximum discharge
current of a shot using argon is reduced from 30.9 kA to 28.2 kA. The argon plasma seems
to have a higher resistance than the hydrogen-helium plasma.
6.2.1 Images of the ux tube
Figure 6.4 shows the evolution of an argon ux tube taken with the single shot ICCD
camera. Every image shows a dierent plasma discharge. However, the evolution of the
ux tube is highly reproducible and the composition of the image series of single shots
cannot be recognized.
109
6 Results and discussion - Titov-Démoulin plasma source
(a) 12 µs, 15 kA
(b) 13 µs, 18.2 kA
(c) 14 µs, 21.2 kA
(d) 15 µs, 23.9 kA
(e) 16 µs, 26.2 kA
(f) 17 µs, 28.2 kA
(g) 18 µs, 29.6 kA
(h) 19 µs, 30.4 kA
(i) 20 µs, 30.7 kA
Figure 6.4: Evolution of a ux tube with argon as working gas (main capacitor bank ± 3 kV,
with magnetic guiding eld, 8 ns exposure time, PI-MAX ICCD camera)
110
6.2 Argon
The argon plasma is much brighter than the discharges in other gases. This requires an
exposure time of only 8 ns (in contrast to 20 to 40 ns for other working gases). Supplementary to the point in time the discharge current is given for every image.
The ignition of the plasma begins at t = 7.7 µs. This value consists of a 6 µs delay added
for the ignition of the spark gap of the PFN and the rising of its current to a constant
value (as mentioned before, the PFN is not used here; for details of the current pulse of
the PFN see chapter 3.3.2) and the switching time of the thyristor of the capacitor bank
(1.7 µs) which is observed at any conguration at both plasma sources. The switching time
is independent of the used working gas.
The rst three images (gures 6.4 a-c) show an uninterrupted ascending ux tube. The
ceramic tube which is used for the pick-up coil measurements and the copper electrodes
which are embedded into the ceramic disc can already be identied.
At t = 15 µs (gure 6.4 d) the lower foot point of the ux tube tears from the anode.
However, the evolution of the ux tube continues and at t = 16 µs (gure 6.4 e) the upper
foot point which is still visibly connected to the cathode begins to kink before it also tears
at t = 17 µs (gure 6.4 f).
The last three images (gures 6.4 g-i) show a structure which seems to be completely disconnected from the electrodes but is still rising further away from them. Only at the last
image the inner structure of the ux tube becomes blurred. At this point the discharge
current has already passed its maximum.
The behaviour of the argon ux tubes diers in many ways from hydrogen-helium ux
tubes observed at either of the plasma sources (e.g. expansion velocity, higher current at
the same spatial expansion).
Hence, the analysis of the argon ux tubes can help to understand the inuence of the
magnetic guiding eld on the evolution of the ux tubes and verify a frozen magnetic ux
and a twisting of the ux tube at higher discharge currents.
6.2.2 Inuence of the external magnetic guiding eld
The conguration used here makes it possible to observe the formation of ux tubes without
the presence of the magnetic guiding eld. For lighter gases this results in a blurred plasma
cloud which does not pinch. For this reason, neither the evolution of a ux tube nor the
appearance of instabilities could be observed for such a conguration.
At rst sight, the evolution of a ux tube without the magnetic guiding eld does not
dier from the evolution of a ux tube with the magnetic guiding eld present but their
life time is shorter. Figure 6.5 shows the tearing of a ux tube without magnetic guiding
eld present. It tears from the anode as it can be observed for ux tubes with the magnetic
guiding eld present (c.f. gure 6.4).
111
6 Results and discussion - Titov-Démoulin plasma source
(a) 12.6 µs
(b) 12.8 µs
(c) 13.0 µs
(d) 13.2 µs
Figure 6.5: Flux tube with argon as working gas; tearing from the anode; main capacitor
bank ± 3 kV, without magnetic guiding eld, 9 ns exposure time, PI-MAX
ICCD camera
112
6.2 Argon
Figure 6.6: Rogowski coil raw signal, with (red) and without (black) magnetic guiding eld
present
As already mentioned in the last chapter, the tearing of the ux tube is indicated by a
dip in the raw signal of the Rogowski coil measurements. Figure 6.6 shows the evolution
of an averaged raw signal with (red line) and without (black line) magnetic guiding eld
present. The tearing occurs 1.3 µs later with (t = 14.2 µs) than without (t = 12.9 µs)
it. This behaviour can also be seen in images of the ux tube: in gure 6.4 the ux
tube tears from the anode between 14 and 15 µs, in gure 6.5 between 12.8 and 13.2 µs.
Hence, the magnetic guiding eld seems to stabilize the ux tube with respect to its tearing.
But why is a ux tube formed without magnetic guiding eld present if argon is used
as working gas instead of the hydrogen-helium gas mixture? One reason can be the inertia
of the atomic ions, and with it their thermal velocity:
vT i =
r
kTi
mi
(6.3)
The ion temperature could not be measured at this experiment. For a rough estimation
of the thermal ion velocities their values are calculated for temperatures of 1 to 2 eV
to vT i,H = 9.8 - 13.8 mm/µs for hydrogen ions (the helium ions are disregarded for this
calculation) and to vT i,Ar = 1.5−2.2 mm/µs for argon ions. Both of these values are smaller
than the expansion velocity of the ux tube. Independent from the assumed temperature
of the ions, the thermal velocity of the argon ions is always smaller than the the thermal
ion velocity of the hydrogen ions by a factor of 6.3 (square root of their mass ratio).
Hence, the hydrogen ions move much faster than the argon ions before the current through
the ux tube is large enough to produce a sucient magnetic connement. The current is
113
6 Results and discussion - Titov-Démoulin plasma source
distributed across a much larger cross section and the current generated magnetic eld is
not able to form a ux tube. It is assumed that for a ux tube using the hydrogen-helium
gas mixture as working gas, the magnetic guiding eld slows down the movement of the
ions. They are kept together until the discharge current is large enough to form a ux tube.
6.2.3 Magnetic eld measurements
The evolution of the ϑ- and the ϕ-component of the magnetic eld were also measured
at this plasma source conguration with argon as working gas. These measurements are
analysed with special care because here the forming of a ux tube without the presence of
the external magnetic guiding eld can be observed.
Previous shots of the rst plasma source without the external magnetic guiding eld present
and with the hydrogen-helium gas mixture as working gas were not successful. Only a
blurred plasma cloud could be seen (c.f. gure 5.8). The pick-up coil measurements from
these shots do not provide useful information about the magnetic eld and the structure
of the plasma.
This intermediate plasma source conguration with argon as working gas was not investigated as extensively as the rst conguration. Hence, the database of these measurements
is much smaller than the database of the measurements at the rst plasma source. The
evolution of the magnetic ux density is measured at a smaller number of measuring points
and just three individual measurements per measuring point are averaged. This results in
an increase of the error bars of the measurements and only qualitative statements can be
made.
Figure 6.7 shows the temporal evolution of Bϑ and Bϕ for both plasma source congurations
(with and without magnetic guiding eld). The main capacitor bank is charged up to ± 3 kV
and three individual measurements are averaged for every measuring point. According to
the measurements of the magnetic ux at the rst electrode system (c.f. chapter 5.3) the
pick-up coil is placed along the z-axis midway between the electrodes.
ϑ-component
of the magnetic eld
Figure 6.7 a) shows the evolution of Bϑ with and without magnetic guiding eld present.
Here, the pick-up coil is placed in a distance of 7 cm to the electrodes. For this magnetic
ux component the presence of the magnetic guiding eld seems to have no inuence on
the shape and amplitude of the measured signal; the variation of both signals is smaller
than the error bars of the pick-up coil calibration.
Compared to the measurements of Bϑ taken at the rst plasma source using the hydrogenhelium gas mixture (see chapter 5.3.1) roughly the same signal shape is observed when the
ux tube moves across the probe.
114
6.2 Argon
(a) Bϑ with and without magnetic guiding eld
present; the pick-up coil is placed 7 cm in front
of the electrodes
(b) Bϕ without magnetic guiding eld present for
several distances to the electrodes; the tearing
of the ux tube at t = 12.9 µs is marked by
the dotted line
(c) Bϕ with magnetic guiding eld present for several distances to the electrodes; the tearing of
the ux tube at t = 14.2 µs is marked by the
dotted line
(d) Bϕ without magnetic guiding eld on a larger
timescale; additionally the discharge current
is plotted
Figure 6.7: Comparison of the components of the magnetic eld of a ux tube with argon
as working gas with and without magnetic guiding eld present (main capacitor
bank ± 3 kV, 3 single measurements are averaged)
115
6 Results and discussion - Titov-Démoulin plasma source
Due to the lower z-velocity of the argon ux tube a higher current is appearing at the same
expansion of the ux tube. As at the rst plasma source the zero crossing of Bϑ gives the
position of the ux tube. The presence of the magnetic guiding eld seems to have a rather
weak inuence on the expansion velocity of the ux tube. Thus, the magnetic guiding eld
can be neglected as source for a counter force for the expansion of the ux tube.
ϕ-component
of the magnetic eld
In contrast to Bϑ , the results of Bϕ show a dependency on the presence of the magnetic
guiding eld. Figure 6.7 b) shows the temporal evolution of Bϕ without the magnetic guiding eld present, measured at several distances to the electrodes. The dotted line indicates
the time of the tearing of the ux tube. Even after this (until t = 15 µs) the magnetic ux
density remains beneath a value of 10 mT for all distances.
However, neither the signal strength before t = 15 µs nor its rising afterwards can be
correlated to the position of the pick-up coil. But the evolution of Bϕ after t = 15 µs can
be attributed to the alteration of the discharge current (c.f. gure 6.7 d). This behaviour
indicates short circuits to the chamber walls or electrical disturbances of the probe itself
and cannot be interpreted as magnetic ux densities produced in any way by the ux tube.
In summary, without a magnetic guiding eld no interpretable ϕ-component of the magnetic eld can be observed.
In gure 6.7 c) Bϕ with the magnetic guiding eld is plotted for several positions of the
pick-up coil. In contrast to the measurements without the magnetic guiding eld, the single measurements here show a direct dependence on the position of the pick-up coil: The
maximum of the measured magnetic ux decreases and its position is shifted to later times
at larger distances. This observation suggests that Bϕ is caused by the magnetic ux of
the guiding eld frozen to the plasma.
Additionally, the measured Bϕ begins with a slightly negative value and changes its sign
before the main peak appears. This evolution of Bϕ reminds of the evolution of Bϑ and
may indicate a slight twisting of the ux tube.
The ux tube leaves the y-z-plane. This results in a measured magnetic ux which is a
combination of a current produced and a frozen in magnetic ux. Exactly this behaviour
was ruled out at the rst plasma source (c.f. chapter 5.3.3) because a change of sign was
not observed.
In summary, Bϕ shows a strong dependency on the presence of the magnetic guiding eld.
The measurements presented here indicate that the magnetic ux of the guiding eld is
partly frozen to the plasma.
Also, the magnetic guiding eld seems to be necessary for a twisting of the ux tube. Otherwise a current generated magnetic ux should have been observed at the measurements
without the magnetic guiding eld present.
116
7 Summary and Outlook
Two aims were set for this work: First, an extensive characterisation of the rst plasma
source and second, the construction of a new source based on a model proposed by Titov
and Démoulin [TD99].
The rst plasma source used at this experiment is able to produce arch-shaped ux tubes
which are reminiscent of solar prominences. The gross behaviour of them has been observed by ICCD camera images and pick-up coil measurements and it is noticed that the
formation and the spatial and temporal evolution of these ux tubes are very reproducible.
The images provide information about the expansion velocity of the ux tube and show a
reproducible tearing of the ux tube from the cathode. The here derived expansion velocity
was veried by further diagnostics.
Using the ϑ-component of the magnetic ux density a rough estimation of the current density prole of the ux tube was possible and the ϕ-component indicates that the magnetic
guiding eld is frozen in the plasma of the ux tube. Finally, the measured evolution and
magnetic ux density was compared with calculations and MHD simulations carried out
by Lukas Arnold [Arn08].
At this point some measurements remain which are not yet understood: First, the constant expansion velocity of the ux tube. Second, the magnetic eld measurements after
the tearing of the ux tube. And third, the occurrence of periodic striations along the ux
tube and ray-like tracers leaving the plasma in radial direction (c.f. gure 5.8) when the
magnetic guiding eld is not present.
These points will be the subject of forthcoming investigations.
The Titov-Démoulin plasma source is still under development. In chapter 2.2 the basic equations and the downscaling of the parameters were presented and in chapter 3.3 the
setup of the plasma source itself.
Due to technical diculties with the rst electrode design an intermediate step to the nal
plasma source design was made in this work, replacing the line current produced magnetic
guiding eld with the magneto-static eld of two permanent magnets.
With this conguration further working gases have been used and it could be observed that
the expansion velocity decreases with increasing mass number. From this working gases
argon stands out particularly. It is the heaviest gas used at this experiment. Hence, it is
the one with the slowest expansion velocity.
117
7 Summary and Outlook
Figure 7.1: Image taken with a single-frame ICCD camera of an argon ux tube (6 µs after
ignition, 10 ns exposure time, ± 3 kV charging voltage of the capacitor bank,
25 kV charging voltage of the PFN, without strapping eld present)
Using argon the formation of a ux tube with and without the presence of the magnetic
guiding eld is possible. Magnetic ux measurements for both of these ux tubes provide
further indications for a magnetic ux, frozen in the plasma of the ux tube.
At last another step to the nal plasma source design has been made: Figure 7.1 shows an
image of an argon ux tube using a revised electrode design and the line current generated
magnetic guiding eld. As last step the strapping eld has to be added to this conguration
and the equilibrium conditions of the model calculations have to be fullled.
118
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Danksagung
Als erstes danke ich Herrn Soltwisch für das Ermöglichen diese Arbeit in seiner Gruppe.
Zahlreiche gemeinsame Diskussionen über Kameraaufnahmen, Messdaten und deren Ursache brachten das hervor was in dieser Arbeit vorgestellt wurde.
Für eine tolle Zusammenarbeit möchte ich mich bei Lukas Arnold und Jürgen Dreher
bedanken, die für viele Fragen und Diskussionen immer Zeit gefunden haben.
Meinen Kollegen Jan Tenfelde, Felix Mackel und Philipp Kempkes danke ich für eine
groÿartige gemeinsame Zeit in der AG. Ihre Unterstützung bei der Auswertung und Interpretation von Messdaten sowie bei Arbeiten im Labor und bei Diskussionen im Kaeeraum
soll nicht unerwähnt bleiben.
Mein besonderer Dank gilt Ivonne Möller, Jan Tenfelde und Sarah Müller für ihre fachliche
und psychologische Betreuung, gerade in den letzten Monaten dieser Arbeit.
Ohne Frau Iris Nikas und dem EP5-Techniker-Team, bestehend aus Bernd Becker, Frank
Kremer und Thomas Zierow wäre diese Arbeit an der Bürokratie oder meinem handwerklichem Geschick gescheitert. Vielen Dank für Zeit, Geduld und Ideen bei kleineren und
gröÿeren Projekten.
Allen Kolleginnen und Kollegen von EP5 danke ich für ein hervorragendes Arbeitsklima,
welches die Tage an der Uni kurzweilig erscheinen lieÿ.
Ohne die Unterstützung meines Vaters Wolfram und meiner Schwester Julia wäre diese
Arbeit nicht möglich gewesen. Danke.
Lebenslauf
Persönliche Daten
Name:
Holger Stein
Anschrift:
Am Bleckmannshof 51
44799 Bochum
Geburtsdatum:
29. September 1978
Geburtsort:
Arnsberg
Familienstand:
ledig
Staatsangehörigkeit:
deutsch
Ausbildung und Abschlüsse
02/2006 - 01/2011
Wissenschaftlicher Mitarbeiter am Institut für
Experimentalphysik V an der Ruhr-Universität Bochum
02/2006
Beginn der Promotion an der Ruhr-Universität Bochum
01/2006
Abschluss des Studiums mit der Diplomarbeit "Aufbau eines
Messsystems zur spinpolarisierten Rastertunnelspektroskopie
an magnetischen Oberächen", in der AG Oberächenphysik
bei Prof. Dr. U. Köhler
10/2001
Vordiplom
10/1999
Beginn des Physikstudiums an der Ruhr-Universität Bochum
06/1998
Abitur
08/1989 - 06/1998
Besuch des Graf-Gottfried-Gymnasiums in Arnsberg
08/1985 - 06/1989
Besuch der Städtischen Gemeinschaftsgrundschule Mühlenberg
in Arnsberg