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! ELECTRON!DENSITY!IN!A!HELIUM!DISCHARGE! PLASMA! ! ! BACHELOR!THESIS! ! SUBMITTED!BY! ! CHRISTIAN!KÜCHLER! ! DECEMBER!2013!! ! ! ! ! ! ! ! ! ! ! ! PRIMARY!EXMINER:!PROF.!DR.!JOACHIM!JACOBY! SECONDARY!EXAMINER:!PROF.!DAVID!Q.!HWANG,!PHD! ! ! ! ! ! ! ! Contents Abstract 4 Chapter 1. Introduction 5 Chapter 2. Theory of discharge plasma and their diagnostics 2.1. Definition of Plasma and important Properties 2.2. Ionization and Recombination in a Discharge Plasma 2.3. Breakdown Condition for a Discharge Plasma 2.4. Interferometry 7 7 8 9 11 Chapter 3. Experimental setup 3.1. Interferometer 3.2. Plasma source 3.3. Vacuum system 14 14 17 19 Chapter 4. Data Processing and Analysis 4.1. Reconstruction of line-integrated densities 4.2. Abel Inversion 4.3. Estimation and Evaluation of Uncertainties 21 21 24 29 Chapter 5. Results and Evaluation 5.1. Reproducibility 5.2. Results and Evaluation of Abel Inversion 5.2.1. Density profile 5.2.2. Electron density over time 5.3. Interpretation and Evaluation of Experimental Results 32 32 34 35 37 41 Chapter 6. Outlook 43 Chapter 7. Acknowledgements 45 Bibliography Erklärung/Declaration 46 48 3 Abstract The thesis at hand describes the experimental investigation of the electron density in a plasma source. The source has been developed to study the current drive by crossing two lasers within a plasma. The plasma is created by a helium discharge within a small vacuum chamber containing the two spherically shaped electrodes (about 1 cm apart). The energy of 11.3 J necessary is stored in a capacitor of capacity 0.15 µF. The electron density was measured at pressure in the range between 500 Pa and 1350 Pa. To investigate the electron density a heterodyne Michelsoninterferometer is used, which gives the line-integrated density along the interferometer beam. To reconstruct the density as a function of time and position, the process of Abel inversion is used. The overall goal is to find a time and position at which the density is most likely to be in the range of (2 ± 0.2) · 1015 cm 3 . The plasma was found to form at a diﬀerent position at each discharge in contrast to initial expectations. Therefore the results can only show the average density profile of the source over many discharges. This eﬀect also causes a moderate reproducibility at the center of the setup and a low reproducibility at the outer positions of the setup. In most cases the electron density had a peak between 0.5 and 1.5 cm away from the center of he source. Nevertheless some conclusions about radial density profile over time can be drawn and suggestions on the adjustment of the experimental setup are made. 4 CHAPTER 1 Introduction A plasma is a partially or fully ionized gas. Plasma are a integral part of our daily lives, yet mostly hidden in natural phenomena or everyday products. A major part of the world’s artificial lighting is based on gas-discharge plasma, and the earth’s major natural source of light, the sun, is a burning plasma by itself. While these plasma are completely diﬀerent in their physical properties, they are important for much of plasma-related research. The social, environmental and economic impact of modern energy production is becoming larger due to the fading supply of fossil fuels and the eﬀects of global warming. Nuclear fusion - the physical eﬀect that drives the energy production in the center of the sun - could be the basis of a future energy source. Current research in fusion science can be separated into two groups. In magnetic confinement fusion (MCF) a plasma of more than 10 keV is confined using external magnetic fields. In inertial fusion the conditions necessary for such a process are created by compressing a plasma very quickly and thereby creating very high densities and temperatures. A somewhat intermediate approach is the so-called magneto-inertial fusion. This method is based on inertial fusion, but exploits magnetic fields to increase confinement eﬃciency and thereby requires less extreme conditions. One method to achieve this could be to spherically distribute several plasma guns around a plasma target. When ignited, the plasma guns accelerate a plasma jet to about 200 km/s. When these jets merge, they form a shell around the initial target, which collapses quickly, ideally to fusion conditions. Confining the initial target by a magnetic field reduces losses due to heat and decreases the implosion velocity needed, leading to a higher target mass and therefore higher energy gains . The magnetic fields have to be very strong (⇠ 103 T), which cannot be created directly from the outside of a potential fusion reactor. A possible solution is to use the compression of the target itself to increase an initial seed magnetic field. If magnetic fields are compressed adiabatically, the magnetic flux is conserved, leading to the expression  Z ~ = const., ~ · dS = B (1.0.1) S ~ denotes the magnetic flux density through where S~ is a plasma surface area and B that surface. For the sake of simplicity the magnetic field has been estimated to be constant over the area. Decreasing the area A by a large amount will therefore result 5 1. INTRODUCTION 6 in an increase of the magnetic flux density B . It has been shown  that up to 3 kT can be achieved using this technique. The initial magnetic field can be formed by driving a current with a plasma wave. This current may be created by exploiting the phenomenon of Landau damping. It is based on the motion of charged particles due to the field of an electromagnetic wave. When the energy transfer by the electromagnetic wave W is averaged over initial positions and a distribution of initial velocities f (v) in the collisionless limit, the expression ⇡q 2 E 2 ! @f (v) ⇣ ! ⌘ W = (1.0.2) 2m|k| k @v k can be derived , where E is the electric field amplitude, q the particle charge, m their mass and ! and k the waves’ frequency and wave vector, respectively. The term @f /@v is essential for the current drive due to Landau damping. If the initial velocity of a particle is smaller than !/k , the particle will gain energy; if it is larger, energy will be lost. The ratio !/k can therefore determine the number of particles gaining energy and is responsible for driving the desired current. One way to create such a wave exploits the phenomenon of beating, i.e. the interference of two electromagnetic waves of diﬀerent frequencies. If the diﬀerence between those frequencies is approximately equal to the plasma frequency, Landau damping can cause a resonant excitation of the plasma and thus create a current and consequently a magnetic field. The experiment described in this thesis was carried out on a plasma source created to study this method of distant magnetic field creation. It is described in  and summarized as follows: Two CO2 lasers are crossed inside the plasma at a pre-defined point. The lasers may have a frequency diﬀerence of 30 GHz to 3.6 THz, which can excite plasma with densities of 1013 cm 3 to 1017 cm 3 . In order to study the beatwave process, a density of 2 · 1015 cm 3 has been chosen, corresponding to a plasma frequency of 2.5 THz. As the lasers have to be crossed at a pre-defined position and time within the plasma source, it is essential to know the time-resolved density distribution. The aim of this thesis is to experimentally investigate the electron density within a newly developed plasma source, ending with a conclusion as to which conditions, positions and times are suitable for studying the beat-wave process. CHAPTER 2 Theory of discharge plasma and their diagnostics This chapter will introduce the concept of plasma in a general sense and give a definition of the term. This definition will cover an extremely broad range of physical conditions that need to be characterized by certain parameters. Within a plasma several atomic processes take place and the most important of them are discussed briefly in the second part of this chapter. In the final part the condition necessary for a breakdown in a gas is derived, as the experiment treated in this thesis is based on a gas discharge. 2.1. Definition of Plasma and important Properties A plasma is most commonly defined as an ionized gas, i.e. a gas consisting of free electrons and ions, but without an overall charge (quasi-neutrality). This includes partially and fully ionized plasma, but a degree of ionization of 10 4 is regarded as the minimum amount of ionization . This definition covers a very wide range of physical conditions. In nature, the most fundamental plasma properties, electron density and electron temperature may have values of (10 6 - 1039 ) cm 3 and (104 - 108 ) K, respectively. Owing to this large range of values, it is essential to introduce additional parameters to characterize the physical conditions present within a plasma. In a plasma every charge carrier is surrounded by several charge carriers of opposite charge, resulting in a shielding of the initial charge and in a quasi-neutral plasma. The distance from a charge carrier at which the electron thermal energy Wth equals the electric potential energy W pot is defined as Debye length D . From ~ (r) /r, where E ~ (r) represents Poisson’s equation r2 V (r) = ne e/"0 and r2 V (r) ' E the electric field vector, it follows that ~( |E D) | = ne e "0 Using the condition Wpot = Wth and Wpot = e D = r R (2.1.1) D. D 0 E (e) dr = ne e D /(2"0 ) "0 k B T . ne e 2 A plasma of dimensions L is considered as quasi-neutral, if the condition satisfied . 7 , (2.1.2) D ⌧ L is 2.2. IONIZATION AND RECOMBINATION IN A DISCHARGE PLASMA 8 A similar parameter is the landau length L , which is the characteristic length for collisions between two charged particles that may have charges larger than e and move at a kinetic energy of kB T . It is derived by proposing Wpot ( L ) = kB T and using Wpot ( L ) = Z e2 / (4⇡"0 L ), which results in L = Z e2 . 4⇡"0 kB T (2.1.3) A quasi-neutral plasma will not remain in this exact state for a long time, but will perform microscopic oscillations that lead to a charge seperation and therefore to ~ (r) = a modified electron density n0e = ne + ne . Gauss’ law reveals rE ne e/"0 and by using the electron velocity ve , the equation of motion becomes me @~ve = @t ~ (r) = eE ne e 2 , "0 (2.1.4) where the equation of continuity @ ( ne ) = @t ne r~ve r ( ne~ve ) ' ne r~ve (2.1.5) makes it possible to solve 2.1.4 by applying r on both sides me r ✓ @~ve @t ◆ = @ 2 ( ne ) = @t2 e 2 ne ne . "0 me (2.1.6) Such an equation of motion is immediately identified as that of an harmonic oscillator of frequency s e 2 ne , (2.1.7) !P e = " 0 me which is defined as plasma electron frequency . 2.2. Ionization and Recombination in a Discharge Plasma The definition of a plasma as an ionized gas as given in Sec. 2.1 requires atomic processes of electron-atom-separation within an initially unionized neutral gas. The ionization and recombination processes within a plasma of the type under investigation are given as follows, based on  and . As described in Sec. 2.3, the ionization is generally initiated by an electron from an natural or artificial source. This mostly happens in a process described by A + e 1 ! A+ + e 2 + e 1 , where the electron e1 is slowed down as some of its kinetic energy has been used to ionize the atom A, releasing a free electron e2 . The electron e2 will then cause secondary ionizations and an avalanche may develop. Although a large portion of 2.3. BREAKDOWN CONDITION FOR A DISCHARGE PLASMA 9 subsequent ionizations will happen by the process of electron ionization, these secondary processes are more diversified. One modification of this process is stepwise ionization, where a previously excited atom (often in a metastable state) is ionized by an electron. This process is important for Helium discharges as the gas has two metastable states: 23 S1 (lifetime ⇠ 8000 s  ) and 21 S0 (lifetime 20 ms) . Another process involving metastable states is ionization by excited neutrals, also known as Penning-Eﬀect. This process can be modeled as A⇤1 + A2 ! A1 + A+ 2 +e The initially excited atom A1 has lost its excitation energy to the initially neutral atom A2 , ionizing the latter . Important ionization processes happen at the plasma surface as well. The electric field decreases the electrode atom’s electric potential energy and makes electron tunneling more likely. This is called field ionization. When energetic electrons, ions, and excited neutrals hit the metal surface, they may release electrons from it by various ionization procedures similar to those stated above. The three processes stated above are not the only existing ionization processes, but they are the most common ones in helium discharge plasma of the type examined in this thesis. In a plasma the ionizing processes experience competing processes called recombination. As the name suggests, ions catch free electrons and thereby lose some or all of their overall charge. Electron attachment and therefore ion-ion-recombination does almost never occur in a helium plasma due to this element’s atomic structure. Volume recombination therefore occurs only in form of a two-body collision (A+ + e ! A) and three-body collision (A+ + e1 + e2 ! A + e2 ). Surface loss of charge may happen by normal collisional recombination, absorption of electrons by metallic surfaces or the charging of dielectric surfaces. 2.3. Breakdown Condition for a Discharge Plasma In order to create a plasma in which the beat-wave process can be studied, a gas discharges was used. Gas discharges are the most commonly found types of low temperature plasmas and can be divided into discharges by the Townsend mechanism, streamers and microdischarges and discharges due to alternating electric fields . As the Townsend mechanism was responsible for the creation of the plasma used for this experimental work, a more detailed description of that process is given, based on  and . At the beginning of a Townsend discharge a free electron is provided by an external source between two parallel electrodes of separation d at some potential diﬀerence V . This electron is almost always provided naturally (e.g. cosmic background radiation, photoionization, natural redioactivity), but may also be created artificially. The electron is accelerated towards the cathode. As soon as it has enough kinetic 2.3. BREAKDOWN CONDITION FOR A DISCHARGE PLASMA 10 energy, this electron is capable of ionizing atoms or molecules in the gas, creating further electrons. Therefore, while moving towards the cathode for a distance z , the number of electrons ne (z) increases exponentially: ne (z) = e↵z , where ↵ is the number of ionizations per electron and length. This so-called first Townsend parameter is based on the mean free pathlength of an electron e , which is inversely proportional to gas pressure p, i.e. e p = C . It further depends on the ionization energy Wionization as well as on the absolute electric field strength given by E = V /d. The energy gained by an electron between two collisions is W = e · E . Therefore the parameter ↵ can be derived as follows: ✓ ◆ 1 Wionization ↵= exp W e ✓ ◆ Wionization pd = Cp exp (2.3.1) V Besides producing a free electron, an ionization process leaves behind a positively charged ion. This ion is accelerated towards the anode. Due to their much greater mass, ions will cause hardly any ionization by their own. They will however generate secondary electrons at the anode, which will then create further ionizations on their way towards the cathode. The number of free electrons dissolved from the anode per incoming ion is called the third Townsend coeﬃcient . One initial electron can therefore create nion = exp (↵d) 1 (2.3.2) ions, which in turn create ne,sec = (exp (↵d) 1) (2.3.3) secondary electrons at the cathode. The electron density at the cathode is ne = n0 1 exp (↵d) (exp (↵d) 1) (2.3.4) If the denominator of this equation approaches zero, the current density - which is directly connected to electron density - approaches infinity and the discharge condition is fulfilled. This condition is given by ✓ ◆ 1 ↵d = ln +1 (2.3.5) 2.4. INTERFEROMETRY 11 280 Breakdown Voltage (V) 260 240 220 200 180 160 140 120 2.260 5.320 7.980 10.640 13.300 15.960 18.620 21.280 23.940 26.600 pd (m Pa) Figure 2.3.1. Paschen curve for Helium using parameters from  Using 2.3.1, the Paschen-law determining the breakdown voltage at given pressure and electrode separation may be derived: Vbreakdown = Wionization · pd ⇣ ⇣ ⌘⌘ ln (Cpd) ln ln 1 + 1 (2.3.6) Wionization and C are parameters that depend on the specific experimental conditions, e.g. the type of gas used and the initial gas temperature. A typical Paschen curve is given in 2.3.1. 2.4. Interferometry The beat-wave plasma experiment will initially be performed in a small, pulsed apparatus, described in Sec. 3.2. For purposes of comparison with theoretical models and for further experiments, it is important to know the plasma electron density as a function of time and position. To measure the electron density, interferometry was used in the experiment at hand. In this chapter the physical and experimental basics of interferometry as described in  are summarized. Interferometry is an experimental technique based on the comparison of the phase of the two parts of a previously splitted beam of light using the interference of those parts. Several experimental setups have been developed based on this idea. The most common types are the Michelson, FabryPerot and the Mach-Zehnder interferometers. All of them use one or more beamsplitters, which let the lightwave travel through one or more optical media and then use mirrors to achieve interference. Therefore, interferometers are devices that can be chosen to investigate a variety of physical phenomena that cause a phase-shift of a lightwave. Examples are high-precision distance measurements and material properties, such as refractive index or density. 2.4. INTERFEROMETRY 12 If two lightwaves with the same angular frequency ! and amplitude A, but different phases 1 and 2 interfere with each other, the resulting intensity is I = |E1 (t) + E2 (t)| 2 =A2 · (cos (!t + + 2 cos (! t + 1) + cos2 (!t + 1 ) cos (! t + 2) 2 )) (2.4.1) This equation describes the physical meaning of the output of an interferometer operating at a single frequency (homodyne). The quantity that reveals physical informa2 2 tion is the phase diﬀerence | 1 2 |. To obtain it, time averages of A cos (! t + 1 ), A2 cos2 (! t + 2 ) and 2A2 cos (! t + 1 ) cos (! t + 2 ) have to be measured with high accuracy. These are absolute quantities and therefore very sensitive to noise and changes in output power and alignment. In practice it cannot be assumed that the initial phase shift of the beams vanishes, adding a phase term 0 to each cosineargument. Due to the high frequencies the resulting wave consists of, homodyne mixing causes an output signal that changes on a timescale that cannot be easily handled by analog electronics. Therefore further signal processing has to be employed, which is always subject to additional noise and amplification of existing noise. Furthermore, the phase diﬀerence has to be obtained mathematically and calculations involving amplitudes can increase the uncertainty of a measurement by a large amount and should therefore be avoided when possible. These diﬃculties can be avoided when the beam frequency of one interferometer arm is changed slightly, which is called heterodyne interferometry. When using two slightly diﬀerent frequencies !1 and !2 in the two interferometer arms the above equation becomes  1 2 A (cos (2!1 t + 1 ) + cos (2!2 t + 2 )) 2 + A2 (cos ((!1 + !2 ) t + ( 1 + 2 )) + cos ((!1 I =A2 + !2 ) t + ( 1 2 ))) (2.4.2) The parts of the signal that are not in the frequency-range of !1 !2 can be filtered out easily. Therefore a signal can be used that has a phase equal to the phase diﬀerence of the two interfering beams. This eﬀect is used in heterodyne interferometers, which operate at slightly diﬀerent frequencies in each arm in contrast to homodyne interferometers that operate at the same frequency in both arms. The experimental technique of interferometry is characterized by a negligible influence on the plasma measured. The light traveling through the plasma does not change its parameters significantly and the rest of the diagnostics reside outside the plasma source. The additional advantages of heterodyne interferometry have been stated above. Despite its several advantages, interferometry involves the diﬃculty of only measuring the phase-shift caused by the whole plasma along the path of the laser beam, resulting in the integrated density along that line of sight. To reconstruct the local plasma density, the process of Abel inversion will be used, as discussed in Sec. 4.2. 2.4. INTERFEROMETRY 13 This requires assumptions and calculations involving experimental data, which are both undesirable. An alternative and widely used method for determining plasma density is the use of Langmuir probes, which can provide time-resolved information about electron temperature, plasma potential and electron density at a single point. While this method overcomes many disadvantages of interferometry, it involves the placement of an additional apparatus inside the plasma source at various positions. This is impractical, as the plasma source had to be disassembled, reassembled and evacuated for each change in position. Furthermore placing an apparatus inside the source might alter the overall behavior of the plasma significantly. These disadvantages of Langmuir probes have made interferometry the best choice to measure the plasma electron density under the specific experimental circumstances. CHAPTER 3 Experimental setup In this chapter a thorough description of the experimental setup will be given, divided into three parts. First, the interferometric apparatus will be presented, providing an overview of the whole setup, followed by a description of the plasma source and the circuit and in the last part a short overview of the vacuum system will be given. 3.1. Interferometer The interferometer is a heterodyne Michelson-Interferometer using a Uniphase 1125P He-Ne-Laser, built in 1992, as light source. He-Ne lasers operate at 632.8 nm, 1152.3 nm and 3392.2 nm, produced by Ne transitions from 20.66 eV to 18.70 Mirror Plasma Mirror Beam Expander Polarizer Polarizer Beam Expander AOM HeNe-Laser Beam Splitter Figure 3.1.1. The Interferometer: A Michelson-Interferometer was used. The polarizers and beam expanders were necessary to improve the mixing process and to align the setup. The AOM (Acousto-Optical Modulator) causes the frequency-shift necessary to take advantage of heterodyne interferometry. 14 3.1. INTERFEROMETER 15 Figure 3.1.2. Photograph of the interferometer beam splitting/mixing section showing the laser on the right and subsequent components as in Fig. 3.1.1 eV, from 19.78 eV to 18.70 eV and from 20.66 eV to 20.30 eV, respectively. These transitions are induced by collisions between the Ne atoms and the metastable He states 21S and 23S , which transfer energy to the Ne atoms and create a population inversion . The transition from 20.66 eV to 18.70 eV creates the red-colored beam of wavelength 632.8 nm, that can be detected by the naked eye and is used in the present experiment. Alongside with choosing a low-powered laser, this reduces laser-related dangers. The laser used for the experiment has an output power of below 6 mW. The laser beam is separated into the two interferometer arms by a beam-splitter. The circular polarizers have polarization angles that are adjusted to make the mixing process by the phase detecting electronics as eﬃcient as possible. The beam then penetrates the plasma and is reflected by a mirror. The key component used to achieve the frequency shift is a Newport N24080 Acousto-Optical Modulator (AOM) that is placed in the reference path. AOMs change the frequency and direction of propagation of monochromatic light using the principle of Bragg refraction and the Doppler-eﬀect [19, 20]. In an AOM sound waves of wavelength s travel through a crystal. These sound waves periodically cause regions of higher or lower refractive indices. The laser light is scattered at regions of high refractive indices, which are separated by s . Therefore the acoustic wave in the crystal behaves like a Bragg diﬀraction grating, creating an interference pattern if light travels through it. The locations of the intensity-maxima are given by the well-known equation n l = 2 · s sin(✓), 3.1. INTERFEROMETER 16 Figure 3.1.3. Photograph of the plasma source and trigatron. The lower electrode can be clearly seen, while the upper electrode is partly covered by the source walls. The trigatron is on top of the source. where n is the order of the maximum, l denotes the wavelength of the light, s is the wavelength of the acoustic wave, traveling through the crystal and ✓ represents the angle, at which the light leaves the Bragg cell. Moreover, while interfering with the wavefronts, an exchange of momentum and energy between the photons of the incident light and the phonons of the wavefront occurs. When the light is refracted at a wavefront, the energy of one phonon is transferred to each of the incoming photons. The order n is not only the order of the maximum, but represents the number of refractions at acoustic wavefronts. Thus the energy of the photons is altered by an amount EP hoton = n · EP honon . As the energies of photons and phonons are related to the frequency of the corresponding waves, the Bragg cell causes a frequency shift of the incoming light of fl = n · fs (3.1.1) By placing an AOM into the reference arm of the interferometer, the eﬀect described in Section 2.4 and by Equation 2.4.2 is realized, resulting in a signal containing only the phase-diﬀerence between the two arms of the interferometer. In the specific experiment a frequency shift of 160 MHz was introduced. The signal leaving the interferometer is then sent through a circuit that transforms it into a sine and cosine signal. This signal is recorded by two oscilloscopes, one storing the sine- and cosine values at a rate of about 2 ms per sample, the other storing the data at a rate of 100 ns per sample. These sampling rates are required in order to reconstruct the phase diﬀerence during data analysis, which is explained in greater detail in Sec. 4.1. 3.2. PLASMA SOURCE 17 2.54 cm 7.6 cm 1 cm 5.1 cm 15.2 cm Figure 3.2.1. Side-View of the plasma source. The plasma forms between the two electrodes and the laser beam enters the chamber through the window. The source walls are made of acrylic. 3.2. Plasma source The plasma source built for this experiment consists of two copper electrodes, (1.0 ± 0.1) cm apart. The electrodes have a semi-spherical shape with a diameter of 2.54 cm and are placed within an acrylic vacuum chamber of dimensions 15.2 ⇥ 15.2 ⇥ 7.6 cm. The laser beam enters the vacuum chamber through windows of diameter 5.1 cm. The discharge is triggered by a three-electrode spark-gap switch, known as a trigatron. Trigatrons of the type used (R.E. Beverly III and Associates SG-111 M75C) are capable of switching high currents of 1-100 kA in short pulses and high voltages of 2-20 kV, which electromagnetic switches cannot achieve. A trigatron consists of three electrodes, two main electrodes and a trigger electrode. The distance between the two main electrodes is typically chosen just above the breakdown limit of the gas at operating conditions, as determined by the Paschen law. When triggering, a small spark between the trigger electrode and one of the main electrodes is created, resulting in an initial ionization. Light emitted from this ionization process releases photo-electrons from the cathode and thereby the breakdown-condition between the two main electrodes is satisfied. An arc between these electrodes establishes the desired electric connection between anode and cathode . The trigatron was operated with nitrogen at atmospheric pressure, which makes it capable to operate at voltages between 8 kV and 30 kV, at a maximum peak current of about 100 kA and a breakdown delay of less than 50 ns. The electrode separation is 3.81 mm. (All data taken from ). 3.2. PLASMA SOURCE 18 Figure 3.2.2. Electric circuit: The capacitor C1 is charged by a power supply through resistors R1 and R2. It is discharged over R4 when the switch is closed and trigger and High-Voltage Trigger Transformer initiate a discharge in the trigatron. This creates an electric connection to the plasma source and the discharge is triggered. The voltage across the trigatron and the plasma is measured by the voltmeter V. The Resistor R4 was changed to 50 ⌦ for some experiments. The trigatron itself is triggered by a R. E. Beverly III THD-01B trigger head and trigger transformer. When initiating a discharge, the trigger head produces a short pulse of about 800 V. This pulse is then transformed into a 40 kV-pulse, which is provided to the trigger electrode of the trigatron. The trigger head itself receives its signal from a manual external switch connected by a optical fiber. The switch can be operated from outside the experiment and is used as the main control for initiating the discharge. It is designed to give an optical as well as an electrical signal at the same time, allowing it to trigger the discharge and the oscilloscope storage simultaneously. The trigatron anode is connected to the cathode of the plasma source, 3.3. VACUUM SYSTEM 19 supplying the necessary potential diﬀerence between the two electrodes reliably and with a small delay. The cathode of the trigatron is connected to a voltmeter, which measures the charging voltage across the capacitor. This voltage was kept at (12.3 ± 0.1) kV during all experiments A 200 M⌦ resistor is connected in parallel to the trigatron and the current through this part of the circuit is measured using a Rogowski coil. The coil is connected to one of the oscilloscopes, adding further information to each dataset. The voltage responsible for the discharge is provided by a 0.15 µF high-power-capacitor. The capacitor is charged by a high-voltage power supply, which can create an output voltage of up to 40 kV. Using the operating voltage and the capacitance of 0.15 µF, the overall energy Wtot stored by the capacitor was estimated as Wtot = 1 CV 2 = (11.3 ± 0.2) J 2 The circuit elements in use have nonzero impedance and capacitance and consequently act as a RLC-Oscillator. This causes an unwanted oscillation in the currentmeasurements, which is avoided by using a resistor in series with the plasma source. This resistor was chosen to introduce a critical damping on the RLC-oscillation, which is the case for a resistance of 8 ⌦. For high pressures (1064 and 1330 Pa) a higher resistance of 50 ⌦ was chosen to apply overdamping on the circuit, which is expected to decrease the chaotic oscillations in the plasma. The eﬀect of repeated discharges on the electrodes and the plasma formation was considered when choosing a discharge repetition rate. Previous experiments had shown, that a repetition rate of about 1 discharge per minute or less does not influence the experimental results significantly. This repetition rate was therefore used in the experiment at hand and proved to be a good choice. 3.3. Vacuum system In order to operate the experiment, two gas supplies are necessary. The trigatron uses nitrogen as operating gas at atmospheric pressure with a flow rate of (5.6 ± 0.6) m3 /h. The nitrogen is regulated using a simple valve, flows through the trigatron and is let out into the room. The plasma was created in helium at various pressures. In order to get reliable and reproducible results, the pressure was held within about ±1% of the desired value. This is achieved using two valves. A rough valve is opened just enough to create a suﬃcient flow of helium through the plasma. A second valve is then carefully closed to reduce the pressure in the plasma to the desired value. This valve is located outside the experimental area, so the pressure can be adjusted during the experiment without entering an area of high voltages. After passing the fine valve, the gas flows through the plasma container. As the pressure is below atmospheric pressure for all experiments (between 532 Pa (4 Torr) and 1333 Pa (10 Torr)), a Varian Tri-Scroll vacuum pump is used to maintain that pressure. In addition to a 3.3. VACUUM SYSTEM 20 continuous monitoring during the experiment by a capacity-based pressure-sensing instrument, commonly known as Baratron, pressure was also recorded using one of the oscilloscopes. CHAPTER 4 Data Processing and Analysis In this chapter the process of density calculation and data analysis will be described. In Sec. 4.1 the process of reconstructing the line-integrated densities from the raw data stored by the oscilloscopes is given. Afterwards a brief summary of the theory of Abel inversion is presented followed by an evaluation of methods to apply this theory to experimental data. The chapter will conclude with an estimation of errors, providing a basis for the decision in favor of one Abel-inversion method. 4.1. Reconstruction of line-integrated densities As described in Sec 3.1, the initial data consists of arrays representing the sineand cosine-values of the phase diﬀerence between the two arms of the interferometer. Sine low time-resolution (A) Cosine low time-resolution (B) 1 1 x-Signal y-Signal 1.5 0.5 0 0.5 2 0 1 2 Time (s) Sine high time-resolution (C) 1 0 1 2 Time(s) Cosine high time-resolution (D) 0.4 0.35 0.3 0.25 0.2 0.15 1 2 1 x-Signal y-Signal 0 0.5 1.5 1 0.95 0.5 0.9 0.85 phase diﬀerence y-Signal 5 5 10 15 5 0 5 10 15 0 ⇥10 5 ⇥10 5 Time(s) Time(s) Data Points and Circular Fit (E) phase-diﬀerence without plasma (F) 0.35 1 0.5 0 0.3 0.25 0.5 1 1 0.5 0 0.5 x-Signal 1 0.2 5 4 3 2 time 1 0 ⇥10 5 Figure 4.1.1. Overview of data analysis: A and B show the raw data of the interferometer at a low time resolution, used to determine radius R and center coordinates (xC ,yC ) of the circle shown in E. C and D show the data used to determine the actual phase shift based on R and (xC ,yC ). The phase-diﬀerence before the discharge (F) is necessary to determine the phase diﬀerence due to the discharge plasma. 21 4.1. RECONSTRUCTION OF LINE-INTEGRATED DENSITIES 22 One of those arrays has a high time-resolution, but records only a short period of time and will be referred to as short-timescale data. The other array records a longer period of time, but has a much lower time-resolution and will be referred to as long-timescale data. An example of this raw data is given in Fig. 4.1.1 (A-D). The sine- and cosine-data can be seen as y- and x-values of a circle with radius R and central coordinates (xc , yc ), as shown in Fig 4.1.1 (E). In order to extract the phase information from the data, these values have to be known, because otherwise any equation recovering the phase information will be undetermined and to calculate the density of the plasma, it is therefore necessary to find these values. The purpose of taking the long-timescale data is to determine xc , yc and R, as it will provide a more reliable basis for extracting this information (see Fig. 4.1.1(E)). Fitting a circle to a given set of data is a well-studied problem in data analysis. The method used in the procedure described here was introduced in 1976 by Kàsa . A summary of it is given as follows. Fitting a circle relies on the premise of minimizing the diﬀerence between the datapoints and the equation of the circle obtained during fitting. If the fitted circle has central coordinates (xc , yc ) and radius R, this requirement reduces to u= N q X (xi 2 xc ) + (yi yc ) 2 R2 = min (4.1.1) i=1 The datapoints to which the best-fit circle is to be found for are denoted by xi and yi . This minimum can be found by diﬀerentiating with respect to xc , yc and R and setting the resulting equation to zero. These three equations can then be simplified to a linear system of equations by introducing the new variable c = R2 x2c yc2 : P P 2 x c x i + 2 yc yi + c N P P P 2 xc x2i + 2 yc xi yi + c xi P P P 2 xc xi yi + 2 yc yi2 + c yi P 2 = x + yi2 P 3i = x + xi yi2 P i2 = xi yi + yi3 (4.1.2) This is a system of linear equations of the form M̂ · ~a = ~b, which can be solved by a computer for ~a = (xc , yc , c). Once the numerical values of xc , yc and R are known, the data-points are written as vectors v~i = (xi , yi ) and are normalized, i.e. each point undergoes a transformation (x~i x~c ) /R. The rest of the procedure involves only short-timescale data. First, the coordinates of the normalized circle are converted into phases using the formula = arctan (yi /xi ) and a computational procedure to take into account the discontinuities at = ⇡/2: The first two x- and y -values of the long-timescale data provide an initial phase . ~i form a circle and two consecutive vectors on this circle obey the 0 The vectors v relations kvi+1 ~ ⇥ v~i k = kvi+1 ~ k · k~ vi k · sin ( i+1 ) (4.1.3) 4.1. RECONSTRUCTION OF LINE-INTEGRATED DENSITIES 23 and kvi+1 ~ ⇥ v~i k = xi+1 yi (4.1.4) yi+1 xi , which can be combined to xi+1 yi yi+1 xi = kvi+1 ~ k · k~ vi k · sin ( i+1 ) . (4.1.5) For geometric reasons |~a| = |~b| = R, which is known from the circular fit. Therefore each datapoint is given by ✓ ◆ xi+1 yi yi+1 xi (4.1.6) i = arcsin R2 It is now straightforward to calculate the phase at each point by n = 0 + n X n (4.1.7) i=1 To calculate the phase shift due to the plasma, the phase shift present without any plasma nP has to be known. The oscilloscopes constantly take data and are set up to store it for a given time before the actual trigger signal (t = 0). This data (t < 0) is now exy tracted and a linear fit procedure is being applied to it (see 4.1.1 (F)). Large fluctuations in vi+1 ~ electron density before plasma creation are not likely and it v~i has been verified using the longtimescale data, that a linear 0 R x baseline is reasonable for the electron density before plasmacreation. This linear baseline is extrapolated to times after the trigger-signal. Now the diﬀerence in phase shifts can easily be calculated for each time usFigure 4.1.2. Phase recovery procedure ing P = nP . The final step towards the calculation of the plasma density is now to make use of basic electrodynamic theory. The refractive index of a plasma is given by  ⌘= r 1 !P2 lasma ⇡1 !2 !P2 lasma 2! 2 (4.1.8) 4.2. ABEL INVERSION 24 Within the plasma the wave vector is therefore kP l = ⌘ · !P2 lasma = k0 2c! ! ! = c c !P2 lasma 2c! (4.1.9) The resulting electric field of the laser penetrating a plasma between y0 and +y0 is thus 0 1 Zy0 Eres = E0 @exp (k0 iy) + exp(ik0 y + kP l (y) dy 2k0 y0 )A y0 0 0 Eres = E0 eik0 y @1 + exp @ i Zy0 !P2 lasma (y) 2c! y0 11 dy AA , (4.1.10) where the standard formula for electromagnetic waves and Eq. 4.1.9 have been used. The phase-shift is given by the absolute argument of the second exponential: = Zy0 y0 Zy0 !P2 lasma (y) e2 dy = 2c! 4⇡"0 me c ne (y) dy y0 This is rearranged to give Zy0 ne (y) dy = y0 Introducing the new variable Ne (y) = Ne (x) = ✓ R y0 ✓ y0 4⇡"0 me c e2 ◆ (4.1.11) · ne (x) dx, this becomes 4⇡"0 me c e2 ◆ · . 4.2. Abel Inversion Due to the symmetry of the experimental setup, it has been assumed that the plasma shows cylindrical symmetry at least when averaged over many discharges. Under this assumption it is possible to construct a radial density profile of the plasma using an inverse Abel transformation. Abel transformation was first introduced by Niels Henrik Abel in 1923 when he wished to calculate the path an object will take on a friction-free path f (y) when starting at a height y0 and traveling to the origin due to gravity in a time t(y0 ) . Using simple Newtonian dynamics, this problem reduces to t (y0 ) = Zy0 0 p f 0 (y) 2g (y0 y) dy (4.2.1) 4.2. ABEL INVERSION 25 This is a special case of the general Abel transform which can be written as  g(y) = 2 Z1 y p f (r) r2 y2 rdr (4.2.2) and is in turn a special case of the first-order Volterra integral equation. This equation is solved analytically for f (r) by  f (r) = 1 ⇡ Z1 r dg(y) 1 ·p dy dy y2 r2 (4.2.3) In many experiments it is only possible to measure integrated quantities, such as the integrated luminosity of a light-emitting probe. In the experiment described in this thesis, the density of a plasma integrated along the interferometer beam is being measured. It can easily be seen that the measured line-integrated density Ne (x) is given in terms of the actual density at radius r by Ne (x) = 2 ZR y n (r) p e rdr, r2 y2 (4.2.4) which is immediately identified as an Abel transform. Inverting this formula according to eq. 4.2.3 gives ne (r) = 2 ⇡ ZR r dNe (x) 1 ·p dy. 2 dy y r2 (4.2.5) Under certain assumptions, it is therefore possible to construct the radial density profile of the plasma by measuring the density distribution perpendicular to its axis. However, several diﬃculties arise when applying Abel inversion to real physical data. First, the data will not be in the form of a smooth curve, but given in discrete data points, which are subject to uncertainties. These uncertainties will then be amplified when taking the derivative of Ne (y) or an equivalent. In the past several methods (e.g. , , ,  and ) have been developed to overcome these problems and make Abel inversion useful for data analysis. In order to find the most appropriate method for this experiment, two diﬀerent approaches will be applied to a simulated set of data. An obvious method is using standard interpolation methods on the data and analytically solve the integral in eq. 4.2.3. Polynomials of order 3 have been found to give suﬃcient results. If the inverse Abel transform is being rewritten using 4.2. ABEL INVERSION 26 y Lkl l n3 n5 n4 n2 n1 x k r Figure 4.2.1. Matrix-Method of Abel Inversion: The plasma is divided into several sheaths of constant density and constant width r. At each position the laser beam penetrates a sheath for a length given by Lkl . k depends on the position of the beam, while l is determined by the sheath radius. The unknown overall radius can be taken into account by increasing the width of the outermost sheath. integration by parts as  ne (r) = 0 2 @ ne (r) ne (R) p ⇡ 12 r 2 ZR r ne (r) (y 2 ne (R) 3 r2 ) 2 1 dy A , (4.2.6) the diﬀerentiation of the interpolation-function is avoided. This method is easy to implement into a program and produces an analytical solution with reasonable errors. It assumes however, that the density is suﬃciently well-behaved and does not fluctuate significantly between the data points. Furthermore, the shape of the plasma examined in this experiment is not expected to be modeled by polynomials well enough. It is instead expected that the electron density is decreasing with increasing distance from the center of the plasma, which will not be the result of a polynomial fit and a following analytical Abel inversion. Fitting another type of curve through the data involves further diﬃculties. Not every function can be 4.2. ABEL INVERSION 27 Abel-inverted analytically. As there is no theoretical forecast of the plasma profile, fitting a curve other than a polynomial to the data cannot be justified suﬃciently well. Along with the practical diﬃculties this has led to the decision of not using a interpolation- or fitting-method for the inverse Abel transform. A computationally easier method was developed by Nestor and Olsen in 1960  and is based on the assumption that the derivative of the function Ne (y) is constant over small intervals and eq. 4.2.3 can be simplified to give ne,i = = 2 ⇡· x 2 ⇡· x M X1 ✓ i=k M X1 ✓ i=k Ni+1 Ni (i + 1)2 i2 Ni+1 Ni (i + 1)2 i2 ◆ p (i + 1)2 · ◆ k2 2i + 1 p i2 k2 · Ai,k Subtracting two points of data always involves an amplification of uncertainties and should therefore be avoided. This can be realized by substituting Ai,k by Bi,k using the following transformations: 8 < A i=k k,k Bi,k = , (4.2.7) :Ak,i 1 Ai,k i k + 1 which leads to: ni (r) = 2 ⇡· x M X1 i=k Ni (x) · Bi,k (4.2.8) This method is solely based on manipulating data points in a discrete way and therefore does not need any further assumptions. However, by adding data points very often, initial errors are amplified. A third method uses linear algebra to solve the inversion problem. As illustrated in Fig. 4.2.1, the situation can be approximated as a circular shaped plasma, that has m rings of constant density nl , each having a width of r, which is equal to the step width of the laser-positions. At a certain position x, the laser beam penetrates some of these rings for a certain length Lkl . The integrated density at a position xk is therefore given by l X Ne (x) = Lkl nl , (4.2.9) k=1 where Lkl is an entry of a matrix, which entries can be calculated by the following geometric considerations: Considering Fig. 4.2.1, it becomes clear that the k -th laser beam penetrates the l-th ring for a length of 8 ✓ ◆ q p > 2 2 <2 r 2 2 l (k 1) (l 1) (k 1) l k m k+1 Lkl = (4.2.10) > :0 else 4.2. ABEL INVERSION 28 With this matrix, the inversion problem reduces to ~e (x) . L̂ · n~e (r) = N This equation can be easily solved by a computer. The obvious method is to invert the matrix L̂. However, this matrix is not square, because the corresponding system of linear equations is overdetermined and therefore can only be inverted in a least-squares sense using the Moore-Penrose-Inverse, or pseudo inverse. This inverse is based on the minimization of the quantity =k L̂·~x ~y k2 , which can be expanded as  ⇣ ⌘T ⇣ ⌘ = ~y L̂ · ~x ~y L̂ · ~x (4.2.11) =~y T ~y ~y L̂~x ~xT L̂T ~y + ~xT L~T ~x Minimization can be achieved by diﬀerentiating this result with respect to ~x: 2 L̂T L̂~x = 2 L̂T ~y , which yields the solution for ~x: ⇣ ⌘ ~x = L̂T L̂ 1 ~y (4.2.12) ⇣ ⌘ 1 where L̂T L̂ is called pseudo- or Moore-Penrose-inverse. This method has been used in the calculation of errors, but has proven to result in negative densities when applied to the actual data. Because this is not physically reasonable, the leastsquares-estimate has been modified by adding the constraint of nonnegative densities. The two methods considered suitable for this thesis (matrix inversion and NestorOlsen method) have been tested with the analytically invertible test function f (x) = 1/(x2 + 1) and a computer-generated random error. The results of this simulation are summarized in Fig. 4.2.2. The analytically-inverted function is given by Fa (r) = 1 2 (1 + r2 ) 3/2 (4.2.13) Both methods show a good approximation of the analytically inverted function in their calculated values, the matrix-method being a little closer to the actual function. There is however a large diﬀerence in their eﬀect on errors. While the method proposed by Nestor and Olsen increases the error by very large amounts, the matrix method is causing a much smaller increase in uncertainties. This is due to their diﬀerent handling of the data. A detailed description of the error estimations used in both methods is given in Sec. 4.3. A feature both methods share is their bad approximation of the analytically-inverted function at large values of x. This is due to the limited range of the simulated data. The upper integration limit of 1 in the analytic Abel inversion can only be performed correctly if f (x0 ) ' 0 and x0 is 4.3. ESTIMATION AND EVALUATION OF UNCERTAINTIES 0.8 29 Nestor-Olsen-Method Matrix-Method Fa (r) 0.7 0.6 0.5 F(r) 0.4 0.3 0.2 0.1 0 0.1 0 0.2 0.4 0.6 0.8 1 x 1.2 1.4 1.6 1.8 2 Figure 4.2.2. Simulation of diﬀerent Abel-inversion methods. While both methods approximate the analytic result well, it can be seen that the NestorOlsen method causes a larger amplification of initial uncertainties than the matrix method. included in the data. This eﬀect can be avoided when using the matrix method by increasing the radii of the outer shells in Fig. 4.2.1, moving them closer to r = 1. 4.3. Estimation and Evaluation of Uncertainties Any scientific experiment is subject to uncertainties and errors in its results. It is a very important task of the investigator’s to thoroughly examine these uncertainties and consider them when drawing conclusions from measurements. The relevant raw data can be interpreted as coordinates of a circle. These coordinates suﬀer from instrumental noise, which causes an uncertainty in the radius and the center coordinates of the circle. Using these values, the data is normalized, leading to a propagation of errors. Considering Fig. 4.1.1 it is however clear that these uncertainties will be rather small as the data very clearly forms a circle with only small noise. This is a positive eﬀect of heterodyne interferometry. Then the normalized data is used to recover the absolute phase shift, which amplifies the errors as well. To find the phase shift without a plasma, a linear model is fitted on the pre-discharge-data. This fit results in coeﬃcients of determination (R2 ) near 1, indicating the linear model to be a good estimate and justifying the extrapolation to later times of the plasma formation. Finally, eq. 4.1.11 leads to the line-integrated density at each position y . The position was measured using a micrometer, resulting 4.3. ESTIMATION AND EVALUATION OF UNCERTAINTIES 30 in a measurement error of ±0.0025 cm, which is very small compared to the other uncertainties involved in the experiment and can therefore be neglected. Most scientific experiments are prone to statistical errors, i.e. repeated experiments resulting in diﬀerent outcomes. This phenomenon is especially important in the experiment at hand, as a plasma is a complex, chaotic many-particle-system and therefore not very reproducible. To account for this source of error, the electron density is measured ten times at each position under the same experimental conditions and the mean of this sample is used as an estimate. The statistical error is then given by the standard error of the mean: i = tcN p i M (4.3.1) i denotes the standard deviation of the sample and M is the number of measurements taken. The Parameter tcM is given by the Student’s t-function, which determines the size of the standard error at a given confidence level c and sample size N . The confidence level is the probability that a repetition of the experiment under the same conditions will give a result within the calculated boundaries. The experiment proved to be not very reproducible leading to large statistical errors. Due to this, the very small noise in the raw data may be neglected during further data analysis as well. The overall goal of the experiment was to determine a configuration of pressure, position and time, at which the electron density is likely to be (2.0 ± 0.2) · 1015 cm 3 . It is not expected to find a configuration for which this is true in more than 50% of all cases. Choosing a large confidence interval of 90% or higher would result in large errors and decrease the significance of the result with respect to the initial experimental goal. Therefore the confidence level has been chosen at 80%. This will keep the final result interpretable, while providing a good estimate of errors. The uncertainties in the initial measurements have to be taken into account during the process of Abel-inversion. The Nestor-Olsen algorithm amplifies the initial uncertainties by a large amount, as it involves the summation of experimental data. Therefore the error in the i-th radial density at any time and pressure is given by ni (r) = M X1 2 Ni (y) · Bi,k . ⇡· y (4.3.2) i=k In the second inversion process, a nonnegative least-squares procedure has been used to find a physically reasonable solution to a matrix equation. This algorithm is very similar to the calculation of the Moore-Penrose pseudo-inverse, mentioned in Sec. 4.2. Thus the calculated values of the radial density-profile ni (r) are given in the form of M X ni (r) = Akl · Ni (y), (4.3.3) l=1 4.3. ESTIMATION AND EVALUATION OF UNCERTAINTIES 31 where Akl denotes the inverted matrix mentioned above. In this form, the well-known gaussian error-formula v uM ✓ ◆2 uX @ni (r) t ni (r) = · Ni (y) @Ni (y) l=1 can be applied, resulting in n~i (r) = Â N~i (y). (4.3.4) This results in a much smaller error amplification than introduced by the NestorOlsen method. Therefore the Matrix-Inversion-Method is the method of choice for all further analysis in this thesis. CHAPTER 5 Results and Evaluation 5.1. Reproducibility The line integrated density is the quantity that can be determined using the least amount of assumptions and calculations. It can therefore act as a first estimate of the regime of the electron density and the reproducibility of the plasma. Each graph of Fig.5.1.2 depicts ten measurements of integrated electron density at 532 Pa at several positions. In this case, the positions are the distances from the center of the setup at which the line-integrated density was measured. They show a good reproducibility at central positions (±0.254 cm, ±0.508 cm), but high deviations at the outer positions, an eﬀect that appears at all pressures. To examine the plasma location within the source, multiple long-exposure photographs of the discharge have been taken, three of which are shown in Fig. 5.1.1. It is obvious, that the overall plasma density at one point is diﬀerent for each discharge. A closer examination of the pictures gives rise to the assumption, that the plasma shapes of diﬀerent discharges are comparable, but the location of plasma formation is randomly distributed around the center of the plasma source. This thesis is supported by the low reproducibility of the outer positions, mentioned earlier. The center of the electrodes experiences about the same density regardless of the position of the discharge, as the location of formation is assumed to be equidistant and not very far away from the central point. From discharge to discharge, the line-integrated density will therefore vary less than at the outer positions. The latter will experience dramatically diﬀerent electron densities, depending on the location of plasma-formation, as they are either close or very far away from the center of the plasma. Figure 5.1.1. Photographs of the plasma indicating changing positions of formation 32 5.1. REPRODUCIBILITY ⇥1015 33 ⇥1015 -0.254 cm 2 6 Density (cm Density (cm 2 ) 8 ) 8 0.254 cm 4 2 0 5 0 ⇥1015 5 10 Time(s) 15 6 4 2 0 5 ⇥10 0 5 5 ⇥1015 -0.508 cm 15 ⇥10 5 ⇥10 5 ⇥10 5 ⇥10 5 0.508 cm 8 6 Density (cm Density (cm 2 2 ) ) 8 10 Time(s) 4 2 0 5 0 ⇥1015 5 10 Time(s) 15 6 4 2 0 5 ⇥10 0 5 5 ⇥1015 -0.762 cm 15 0.762 cm 8 6 Density (cm Density (cm 2 2 ) ) 8 10 Time(s) 4 2 0 5 0 10 Time(s) ⇥10 0 0 ⇥1015 -1.016 cm 5 10 Time(s) 15 1.016 cm 8 ) 2 Density (cm Density (cm 2 5 8 6 4 2 0 5 4 5 15 2 ) ⇥1015 5 6 0 5 10 Time(s) 15 6 4 2 0 5 ⇥10 5 0 5 10 Time(s) 15 Figure 5.1.2. Integrated density of several shots at 532 Pa. It can be seen that the electron density is moderately reproducible at the center of the setup, while reproducibility is very low at the outer position of the setup. 5.2. RESULTS AND EVALUATION OF ABEL INVERSION 34 i ri r0 Figure 5.2.1. Six discharges at diﬀerent angular positions, but at the same radius r0 , forming an average radial density profile 5.2. Results and Evaluation of Abel Inversion Performing an Abel inversion using the data is more complex under the assumptions given in Sec. 5.1 and the results will be less applicable. Still, some conclusions can be drawn: Instead of calculating the density distribution of a single shot, the average density distribution of many shots will be calculated. This results in a radial density profile, that can give an estimate of the mean plasma density at each point over many shots. This is possible, because the location of formation is assumed to be randomly distributed around the center of the electrodes at a constant radius r0 (see Fig. 5.2.1). At an individual discharge the conditions necessary to perform an Abel inversion are not satisfied as the plasma of a single discharge is not cylindrically symmetric around the center of the setup. However, the average over many discharges will show a symmetric behavior with an annulus of high electron density at r0 and lower densities at the outer and inner positions, as depicted by Fig. 5.2.1. To achieve this, the average of integrated densities of 10 consecutive discharges was used when applying the matrix method of Abel inversion. 5.2. RESULTS AND EVALUATION OF ABEL INVERSION 35 5.2.1. Density profile. By applying the Matrix-Inversion-Algorithm using a nonnegative least-squares procedure, the density profile of the plasma at several times could be calculated. The results are given in Fig. 5.2.2. In this case the position is the radial distance from the center of the setup, i.e. ri in Fig. 5.2.1. The profiles confirm the results taken from the photographs and plots of line-integrated electron density. The density is well-reproducible at the center of the apparatus, indicated by a small uncertainty at that point. The electron density then rises with increasing distance from the center, reaches a peak between 0.5 and 1.5 cm and then drops quickly. The peaks and the majority of the rest of the data show densities on the order of 1015 cm 3 with higher electron densities appearing at higher pressures. This general behavior is common to all pressures and times, but diﬀers in extent. While there is only a small radial variation in density at 665 Pa, there are large fluctuations at 789 Pa and 1064 Pa. However, as this is only an average of many discharges and the plasma has been found to form at random positions around the center, the uncertainty becomes larger at the outer positions of the plasma source, as the measurements are more sensitive to diﬀerent locations of formation at these points. The radial profiles at pressures of of 532 Pa and 789 Pa are similar in both magnitude and radial regime at all times shown in the graphs. At a pressure of 665 Pa there is a big diﬀerence in the density at a position of 1.4 cm. While the density increases sharply 130 µs after triggering, it remains more or less constant 170 µs after and decreases at times of 50 and 90 µs. In contrast, the density profiles of at 789 Pa are remarkably similar at each time shown. The density profiles at 1064 Pa are less similar at 65, 85, 105 and 145 µs after triggering, showing a broad peak between 0.7 and 1.4 cm. 145 µs after triggering this peak has separated into two clear peaks. In order to be able to find the best possible point of crossing the two lasers as described in the introduction, some measurements have been made replacing the 8 ⌦-resistor R4 in Fig. 3.2.2 by a 50 ⌦-resistor. The results show similar behavior over time, with 1330 Pa having a remarkably similar behavior at most times. While the electron density peaks at r = 1.3 cm and no data is available for larger radii, there is a clear and broad peak between 0.5 and 0.8 cm when a pressure of 1330 Pa was applied. The data suggests that a second peak of similar magnitude is present at some times at r = 1.3 cm, but the high uncertainties at these positions make this result disputable. The density at high pressures is naturally higher than at lower pressures. 5.2. RESULTS AND EVALUATION OF ABEL INVERSION ⇥1015 ⇥1015 Density Profile at 532 Pa 4 50 µs Density Profile at 665 Pa 5 50 µs 4 110 µs 3 170 µs 90 µs 110 µs Density (cm 3) Density (cm 3) 90 µs 3 130 µs 170 µs 2 1 130 µs 2 1 0 0 0 0.5 1 0 1.5 0.2 0.4 Position (cm) ⇥1015 ⇥1015 3 170 µs 1 1.2 1.4 1.6 25 µs 65 µs 10 Density (cm 3) Density (cm 3) 90 µs 110 µs 0.8 Density Profile at 1064 Pa 12 50 µs 4 0.6 Position (cm) Density Profile at 789 Pa 5 130 µs 2 1 85 µs 105 µs 8 145 µs 6 4 2 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.5 Position (cm) ⇥1015 ⇥1015 Density Profile at 1064 Pa, damped circuit Density (cm 3) 130 µs 2 110 µs 130 µs 2.5 170 µs 1.5 Density Profile at 1330 Pa, damped circuit 90 µs 3 110 µs 2.5 1.5 50 µs 3.5 90 µs 3 1 Position (cm) 50 µs 3.5 Density (cm 3) 36 170 µs 2 1.5 1 0.5 1 0.5 0 0 0 0.5 Position (cm) 1 1.5 0 0.5 Position (cm) Figure 5.2.2. Density Profiles at various pressures. The increase in uncertainty at outer positions, owing to the low reproducibility of plasma formation, can be seen. The position of highest density is between 0.7 and 1.3 cm in most cases. 1 1.5 5.2. RESULTS AND EVALUATION OF ABEL INVERSION 37 5.2.2. Electron density over time. While in the previous part some information about the development of electron density over time has been given, it is important to examine the changes of electron density over time in greater detail. This will make it possible to find not only the conditions and positions creating the desired density of (2.0 ± 0.2) · 1015 cm 3 , but also the times at which this density is most likely to occur for a long time. To provide a good overview, figures for each pressure and radial position have been created. The lower and upper bounds of the confidence range are illustrated by black curves, while the red curves show the values derived by the abel inversion process. At 532 Pa the electron density is in the range of 1014 cm 3 at all times, but showing the typical time-development of a quick rise of density, followed by a local minimum and second peak at about 140 µs of equal intensity. The density at 0.254 cm shows an untypical behavior of a sudden drop shortly after the initial rise. This has not been seen in any other graph and the validity of the result is doubtable. At r = 0.508 cm the density does not develop a clear second peak, but remains in the range of (1.8 ± 0.8) · 1015 cm 3 for a long time while showing relatively small variations. This makes this position a candidate for crossing the two lasers creating a beat-wave. The same is true for the position of r = 0.762 cm, where the density stays in the required range for almost all the times it was measured. At a radius of 1.016 cm the graphs suggest a steady increase in density over time and the outermost position gives hardly any information, suggesting a zero average density at all times at varying but high uncertainties. This eﬀect occurs in other cases as well and is suspected to be due to the algorithm used to solve the least-squares-problem which is set up to avoid negative results. The electron density at 665 Pa shows a similar development over time and relatively small uncertainties at all radial positions. Except for r = 1.016 cm, all positions show the two peaks mentioned before more or less clearly and at similar times. Possible candidates for usage in the beatwave experiment are the positions of 0.245 cm, 0.762 cm and 1.016 cm. While the radial position of 1.27 cm shows densities in the desired range for a long time, it is prone to large uncertainties. At 789 Pa the electron density is showing smaller uncertainties than at a pressure of 665 or 532 Pa. While the density is in the range of 1014 cm 3 at the center of the plasma source, there are three radial positions at which the mean electron density has been calculated to be zero with varying values of uncertainty. The other radial positions show similar behavior of two peaks and an increase in uncertainties at outer positions. The density is in the range of 2 · 1015 cm 3 at 0.254 cm distance from the center between 3 and 6 µs after the discharge. This density is measured to occur longer at a radius of 0.508 cm between 14 and 20 µs. The position of r = 1.27 cm is remarkable due to its high and constant density over time while showing only small relative uncertainties. When the pressure within the plasma source has a value of 1064 Pa, the electron density is low at the center of the setup, similar to other pressures. At r = 0.254 5.2. RESULTS AND EVALUATION OF ABEL INVERSION 0 cm 3 0 5 10 15 Time(s) 0.508 cm 15 ⇥10 5 ⇥10 2 Density (cm Density (cm 3 3 ) ) ⇥10 3 1 0 1 0 5 10 Time(s) 15 ⇥10 ) 3 4 3 2 1 0 1 2 5 10 Time(s) 15 ⇥10 5 10 Time(s) 0.762 cm 15 0 5 10 Time(s) 15 ⇥10 5 ⇥10 5 ⇥10 5 ⇥10 5 ⇥10 5 ⇥10 5 3 2 1 0 1 ⇥1015 1.016 cm 0 0 15 5 Density (cm 3 ) ⇥1015 Density (cm 0.254 cm ) 6 5 4 3 2 1 0 1 ⇥1015 3 2.5 2 1.5 1 0.5 0 0.5 Density (cm Density (cm 3 ) ⇥1014 38 1.27 cm 5 4 3 2 1 0 1 2 0 5 5 10 Time(s) 15 (a) Electron Density over time at 532 Pa ⇥1015 ) 0 cm 3 4 Density (cm Density (cm 3 ) ⇥1014 2 0 0 15 ⇥10 0 ⇥10 3 3 Density (cm Density (cm 1 0 1 15 5 10 Time(s) 0.762 cm 15 15 5 10 Time(s) 1.016 cm 15 ⇥10 5 10 Time(s) 1.27 cm 15 5 10 Time(s) 15 3 2 1 0 1 0 5 ⇥10 15 3 ) ) ⇥10 3 1 0 5 2 0 Density (cm 4 Density (cm 3 2 ) 5 10 Time(s) 0.508 cm ) ⇥10 15 0.254 cm 2 0 0 5 10 Time(s) 15 ⇥10 5 2 0 2 0 (b) Electron Density over time at 665 Pa Figure 5.2.3. Red curves show the calculated density, black curves indicate confidence levels. 5.2. RESULTS AND EVALUATION OF ABEL INVERSION ⇥1015 3 4 3 2 1 0 1 0 ⇥10 5 10 Time(s) 15 15 0.508 cm ⇥10 0 ⇥10 3 ) ) Density (cm Density (cm 3 2.5 2 1.5 1 0.5 0 0.5 5 2.5 2 1.5 1 0.5 0 0.5 1 0 5 10 Time(s) ⇥1015 15 ⇥10 ) ) 3 Density (cm 10 5 Time(s) 15 ⇥10 0.762 cm 5 10 Time(s) ⇥1015 2 1.5 1 0.5 0 0.5 1 15 ⇥10 5 ⇥10 5 ⇥10 5 12 10 8 6 4 2 0 2 0 1.016 cm 0 5 10 Time(s) 14 5 Density (cm 3 0.254 cm ) 0 cm Density (cm Density (cm 3 ) ⇥1014 39 15 1.27 cm 6 5 4 3 2 1 0 1 0 5 5 10 Time(s) 15 (a) Electron Density over time at 789 Pa 3 4 2 0 10 15 ⇥10 4 2 0 5 ⇥10 5 0 5 Time(s) 0.762 cm 10 15 ⇥10 5 5 0 5 Time(s) 1.27 cm 10 15 ⇥10 5 5 0 5 Time(s) 10 15 ⇥10 5 15 3 ) 3 1 0 1 0 15 5 Time(s) 1.016 cm 10 15 ⇥10 4 2 0 5 ⇥10 15 ) ⇥10 3 8 6 4 2 0 Density (cm ) 5 Time(s) 0.508 cm 2 5 3 0 15 Density (cm Density (cm 3 ) ⇥10 0.254 cm ) 6 5 Density (cm ⇥1015 0 cm Density (cm Density (cm 3 ) ⇥1014 5 0 5 Time(s) 10 15 ⇥10 5 10 5 0 (b) Electron Density over time at 1064 Pa Figure 5.2.4. Red curves show the calculated density, black curves indicate confidence levels. 5.2. RESULTS AND EVALUATION OF ABEL INVERSION ⇥1014 ⇥1014 ) 3 3 ) 0 cm Density (cm Density (cm 6 4 2 0 0 5 10 Time(s) 0.508 cm 15 ⇥10 10 0 0 ⇥1015 ) 0 1 15 ⇥10 15 ⇥10 5 10 Time(s) 1.27 cm 15 5 10 Time(s) 15 ⇥10 5 ⇥10 5 ⇥10 5 ⇥10 5 ⇥10 5 ⇥10 5 1 0 ⇥1015 8 6 4 2 0 2 3 5 10 Time(s) 15 0 Density (cm 3 0 5 10 Time(s) 0.762 cm 1 5 3 2 1 0 1 Density (cm 2 ) ) ⇥1015 10 5 Time(s) 1.016 cm 3 1 3 Density (cm Density (cm 3 2 0 0.254 cm 20 5 ) ⇥1015 40 0 5 (a) Electron Density over time at 1064 Pa with overdamped circuit ⇥1014 ⇥1014 3 3 Density (cm 3 4 Density (cm ) ) 0 cm 2 1 0 0 5 10 Time(s) 15 0.508 cm ⇥10 5 ⇥10 5 3 0 ⇥1015 2 1.5 1 0.5 0 0.5 1 5 10 Time(s) 15 15 0.762 cm 0 ⇥1015 Density (cm 3 ) ) 1.016 cm Density (cm 3 ⇥10 5 4 3 2 1 0 1 5 10 Time(s) ) 5 4 3 2 1 0 0 15 Density (cm Density (cm 3 ) ⇥10 15 0.254 cm 25 20 15 10 5 0 5 0 5 10 Time(s) 15 ⇥10 5 5 10 Time(s) 15 1.27 cm 6 4 2 0 2 0 5 10 Time(s) 15 (b) Electron density over time at 1330 Pa with overdamped circuit Figure 5.2.5. Red curves show the calculated density, black curves indicate confidence levels. 5.3. INTERPRETATION AND EVALUATION OF EXPERIMENTAL RESULTS 41 cm the data suggest a very low electron density with the mean being at 0 cm 3 and an increase about 14µs after the discharge has been triggered. The other positions clearly show the two peaks mentioned before. While the uncertainties are relatively low, the density is fluctuating more over time. There are however two positions at which the plasma density is in the desired range for a very long time and with a relatively high probability: At r = 0.508 cm about 15-20 µs after discharge initiation and about 10-15 µs after triggering at a radius of 1.016 cm. At this pressure it is also clearly evident that the electron density is higher than at lower pressures as its peak values are in the range of over 6 · 1015 cm 3 , compared to peak values of 3 · 1015 cm 3 at pressures of 665 or 532 Pa. To examine the eﬀects of a overdamped circuit, the resistor R4 has been replaced by a 50 ⌦-resistor. On average the electron density was smaller than in the critically damped circuit although the same pressures were applied. The formation of a second peak at about 150 µs was clearer in this case as well. It is also remarkable that the majority of graphs indicate a decrease in electron density about 160 µs after the discharge had been triggered. Despite these interesting features, the overdamped circuit did not produce results suitable for the beat-wave experiment. At all times and positions the density was either below the desired range or uncertainties were so large that the corresponding condition could not be regarded as suitable. 5.3. Interpretation and Evaluation of Experimental Results The results further support the conclusions drawn from the photographs and the untransformed data. The plasma forms about 1 cm away from the axis of symmetry (r =0) of the experimental setup, but the formation occurs at random angles around the center. Therefore, the results can only provide the average radial density over many discharges at one position. The electron density is very low at the center and shows a low uncertainty there. The uncertainties increase with increasing radii caused by the random location of formation. This results in a very low reproducibility of the discharges at most positions. This is a very important result for the experiment, this work is meant to support as it limits the number of positions that can be used reliably to cross two lasers at a density of (2.0 ± 0.2) · 1015 cm 3 . It also limits the quality of the results of this experiment. Instead of a radial plasma profile, only some information about the average density distribution at a given radius, time and pressure over many successive discharges could be obtained. The most reproducible position was the center of the setup, which is expected if the geometry of the setup is considered and a relatively smooth decrease in density from the plasma center is assumed: The center of the plasma is located at about the same radius from the center of the source, but at a diﬀerent angle at each discharge. The center will therefore experience the same electron density regardless of the angle at which the plasma forms. This is however not a good choice for crossing the two laser beams in the beatwave experiment, the electron density is far below the desired range of (2 ± 0.2) · 1015 cm 3 . While the center of the plasma source was certainly 5.3. INTERPRETATION AND EVALUATION OF EXPERIMENTAL RESULTS Pressure Position Time Density 532 Pa 0.762 cm 150 µs - 185 µs (1.2 - 3.0)·1015 cm 3 532 Pa 0.508 cm 100 µs - 200 µs (1.0 - 2.6)·1015 cm 3 665 Pa 0.254 cm 130 µs - 150 µs (1.5 - 2.6)·1015 cm 3 665 Pa 0.762 cm (2.0 - 2.5)·1015 cm 3 665 Pa 1.016 cm 100 µs - 150 µs (0.9 - 3.2)·1015 cm 3 789 Pa 0.254 cm 3 0 µs - 30 µs 90 µs - 115 µs 42 (1.1 - 3.0)·1015 cm Table 1. Possible conditions for the beatwave experiment showed the most reproducible density-profile, other conditions have been found to be close to the desired density while still showing only a small shot-to-shot deviation. These conditions are summarized in Table 1. All of these combinations of pressure, time and position are possible candidates to use for the beat-wave experiment, the most promising one being the position of 0.254 cm at a pressure of 665 Pa about 130-150 µs after triggering the discharge. Using the density, an estimate of the degree of ionization can be made. If the gas is assumed to be ideal, the overall number of particles per Volume is given by N/V = kB T /P . T is the temperature before plasma creation, i.e. room temperature. With this estimate the degree of ionization is in the range of 1%. The experiments revealed far-reaching properties of the plasma source, most importantly the discovery of random fluctuations in plasma position. This has however limited the applicability of the results as the density in the plasma is highly dependent on the position of formation. Still it was possible to extract a radial density profile showing the average density over many discharges along the radial lines from the center of the plasma source. The density profiles were subject to large uncertainties, especially at the outer positions of the plasma. These uncertainties and the low applicability of the data could have been avoided by first taking photographs and then measuring the density, after adjusting the plasma source as described in Ch. 6. Alternatively a photograph could have been taken at each discharge to determine the approximate position of the plasma at the given discharge and thereby reconstruct the radial profile of the actual plasma. Due to the specific conditions at the location of the experiment, it is impossible to take photographs while the interferometric diagnostic is installed. Because previous photographs at diﬀerent conditions had shown a highly reproducible plasma shape and due to the symmetry of the setup it was not regarded as necessary to take photographs of the plasma before installing the interferometer. Under these circumstances the maximum amount of reasonable information has been extracted from the experimental data. CHAPTER 6 Outlook An important result of this experiment is the inconsistency in the position of the plasma itself as described earlier in this chapter. The recommended conditions for coupling the beatwave into the plasma are the results of an average over many positions of formation. Therefore only a limited number of discharges will provide the necessary density of (2.0±0.2) · 1015 cm 3 and the experiments might have to be repeated many times until the plasma forms in the desired way. This is both ineﬃcient and a possible source of systematic errors. A possible solution to the problem of random locations of formation is to change the plasma source in order to cause the plasma to form at the same position at each discharge. If the results of the present work should be used further, it is crucial to change the plasma source as little as possible. A break of symmetry can be achieved by breaking the symmetry that causes the random fluctuations by changing the shape of one or both of the electrodes. The changes in electrode shape should be kept small, so that only the position of formation is changed and the later stages of plasma burning are only slightly influenced. This could keep the data acquired in this experiment usable. An alternative approach is to introduce a magnetic field from outside the source. This magnetic field has to be strong enough to eliminate the randomness in plasma formation, but should be small enough to maintain the usability of the results presented in this work. The main goal is to initiate the discharge at a certain location. At the beginning of plasma formation the number of free electrons is relatively low. Therefore the necessary magnetic field can be estimated using single-particle motions: The kinetic energy of a single electron when an accelerating voltage V0 is applied is given by 1 me ve2 = e · V0 . 2 Setting V0 ⇠ 5 kV this can be rearranged to give ve ⇠ 107 m/s. Being only a very rough estimate, the relativistic eﬀects at that speed may be neglected. The radius of gyration due to the magnetic field should be in the range of the characteristic dimension of the system, i.e. the distance between the electrodes of ⇠ 1 cm. The equation determining the gyration radius is rg = me v e , eB 43 6. OUTLOOK 44 which can be rearranged to give B= me v e . e rg The values necessary to calculate the minimum magnetic field derived above are plugged into this equation and result in magnetic field densities between 10 and 20 mT. As soon as the plasma has formed, a magnetic field of this order of magnitude will not change the conditions within the plasma significantly, because the magnetic field due to the massive flow of current through the plasma will override the eﬀects of the external magnetic field. Once the random fluctuations on plasma formation have been eliminated, the density could be measured again, giving not only a much better profile of plasma density at this particular plasma source, but also revealing some possibly important information about Abel inversion procedures: Applying Abel inversion to data of this experiment’s characteristics is not well-documented and comparing its results to those involving better initial data might be an interesting subject of further studies. Although the plasma density was not very reproducible in its dependence on position, the behavior over time was similar for most discharges. The clearest feature of this time dependence was the formation of a second peak in electron density about 130 µs after triggering the discharge. While the physical processes behind this phenomenon are of small relevance to the beat-wave experiment, it might be an interesting subject of future study on its own. A possible experimental way to investigate these processes could be a spectroscopic measurement of the plasma. The ionization- and recombination processes happening at this stage of plasma formation could be derived and a model of plasma behavior could be developed. CHAPTER 7 Acknowledgements This thesis and the associated experiment would have been impossible without the scientific and non-scientific support of various persons, which I would like to thank at this point. I thank Prof. David Q. Hwang for welcoming me in the CTIX work group in Livermore, providing important scientific support throughout the experiment and reading and grading this thesis. I thank Prof. Joachim Jacoby for accepting the experimental work as an external bachelor thesis and giving helpful opinions on the work. I am also very thankful for his help regarding formalities and for evaluating this thesis. Sean Hong has not only supervised my lab work, but discussions with him made the long and tedious data taking sessions in a boiling lab much more enjoyable. He always helped when things went wrong, gave very helpful thoughts and insights and contributed many ideas to the experiment. His support did not end at the lab doors, but also included help with visa procedures, housing recommendations, car rides and countless other small, but important favors. I am truly thankful for the great time! I thank Robert Horton, who was the person to ask for the exits to what seemed to be scientific dead-end streets. I am very thankful for having had the opportunity to learn from you. I also thank Ruth Klauser and Russ Evans for all the splendid lunch breaks, which were not only a culinary trip around the world, but most importantly gave deep insights into American culture. They were essential for coping with everyday life in this great country. I thank the DAAD for providing financial support, the great RISE program that established the contact to the CTIX workgroup and a huge amount of knowledge concerning all aspects of research in the US. I thank all my friends for all the great times we had during lectures, get-togethers, sport practices and so on. Being with you has always helped by moving physics out of my head and bringing life back in. A very warm thank you goes to my family in Bruchköbel. By being always there when needed and always on my side when even I was not, they provided an amount of support which is beyond words, which is why I do not even try further. 45 Bibliography  Y. C. Francis Thio, Charles E. Knapp, Ronald C. Kirkpatrick, Richard E. Siemon, and Peter J. Turchi. A physics exploratory experiment on plasma liner formation. Journal of Fusion Energy, 20(1/2), 2001.  C. M. Fowler, W. B. Garn, and R. S. Caird. Production of very high magnetic fields by implosion. 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The Fourier Transform and its Applications, chapter Relatives to the Fourier Transform, pages 262–265. McGraw-Hill, 1999.  A. Sainz, A. DÌaz, D. Casas, M. Pineda, F. Cubillo, and M. D. Calzada. Abel inversion applied to a small set of emission data from a microwave plasma. Appl. Spectrosc., 60(3):229–236, 2006.  O.H. Olsen and H.N. Nestor. Numerical methods for reducing line and surface probe data. SIAM Review, 2(3):200–207, 1960.  Moshe Deutsch and Israel Beniaminy. Derivative-free inversion of abel’s integral equation. Appl. Phys. Lett., 41(1):27–28, 1982.  L. Montgomery Smith and Dennis R. Keefer. Abel inversion using transform techniques. J. Quant. Spectrosc. Radiat. Transfer, 39(5):367–373, 1988.  Pablo A. Vicharelli and Walter P. Lapatovich. Iterative method for computing the inverse abel transform. Appl. Phys. Lett., 50(10):557–559, 1987.  Berthold K.P. Horn. Solving over- and under-determined sets of equations. ERKLÄRUNG/DECLARATION 48 Erklärung/Declaration Erklärung nach §28 (12) Ordnung des FB Physik an der Goethe-Universität für den BA- und MA-Studiengang Hiermit erkläre ich, dass ich die Arbeit selbstständig und ohne Benutzung anderer als der angegebenen Quellen und Hilfsmittel verfasst habe. Alle Stellen der Arbeit, die wörtlich oder sinngemäß aus Veröﬀentlichungen oder aus anderen fremden Texten entnommen wurde, sind von mir als solche kenntlich gemacht worden. Ferner erkläre ich, dass die Arbeit nicht - auch nicht auszugsweise - für eine andere Prüfung verwendet wurde. Declaration according to §28 (12) of the rules of the department of physics at the Goethe-University for BA and MA courses I hereby declare that I have written this thesis by myself without using other sources of information or help than those cited. All parts of the thesis that have been taken from other texts literally or analogously have been indicated as such. I further declare that this thesis or parts of it have not been used for another examination. Christian Küchler Bruchköbel, den 12. Dezember 2013