Contents 1 Motivation 2 2 Theory Motion in a Penning trap Motional excitation

Transcription

1 Collective oscillation of an electron cloud conned in a Penning trap Diplomarbeit von Tristán Valenzuela Salazar Institut für Physik Johannes GutenbergUniversität Mainz Mainz, Februar 2001

2 Contents 1 Motivation 2 2 Theory Motion in a Penning trap Motional excitation Experimental Setup and Method Vacuum chamber Trap Detection circuit Excitation circuit Control and data acquisition Timing Results Motional spectrum Labeling the oscillations Collective and individual oscillation Frequency shift Asymmetry Threshold Discussion and Outlook 22 List of Figures 24 Bibliography 25 Acknowledgements 26

3 Chapter 1 Motivation In 1936 F.M. Penning [Pen36] was working on vacuum gauges and thought that by using a proper combination of electric and magnetic elds it may be possible to extend the path of electrons inside the gauge in order to improve the sensitivity. Then in 1956 J.R. Pierce [Pie54] mentions that with the combination of hyperbolic electrodes and a magnetic eld charged particles can be conned. Since then the technique of conning particles has been developed. And now the physicists use several kind of traps. In this work I will talk about the trap based on the ideas of J.R. Pierce and F.M. Penning, the so called Penning Trap. APenning trap is a device to conne charged particles to a nite volume in space by a combination of electric and magnetic elds. It is used in several elds of experimental atomic physics, like mass measurements or optical spectroscopy and plasma connement, due to the possibility of conning particles for a very long time, days or even months. For mass measurements it is suitable due to the fact that the proper motion in the trap depends on the charge over mass ratio and is easy to measure. On the other hand for spectroscopy it is important that a magnetic eld is used for the connement and this magnetic eld produces a Zeeman splitting. For all these purposes it is very important to conne particles without any loss. In order to avoid losses the connement should be achieved in a range of magnetic and electric eld amplitudes, this is the so called the stability area or, if plotted, stability diagram. In the frame of an experiment with a negatively charged gold ion cluster in a Penning trap, L. Schweikhard et al. [Sch95] observed an instability near the border of the stability area for a Penning trap. Also, in the run of an experiment about high precision spectroscopy in Ca + ions in a Paul trap, by Th. Goudjons et al. [Gud96] unpredicted connement instabilities were observed. This observation lead to a much more detailed study of the stability properties of Penning and Paul traps. With this aim some experiments were performed. One of these experiments, by K. Hübner et al. [Hüb97], lead to the observation of some instabilities of ion connement in a Penning trap within the theoretical stability region. In an experiment, by R. Alheit et al. [Alh97], they observed some instabilities within the stability region, in the motion of an ion cloud in a Paul trap which could be assigned to deviations from the ideal trapping potential. In a dierent experiment they observed also an excitation of the axial oscillation of the ion cloud that was interpreted as an oscillation of the whole ion cloud due to the independence of the excitation frequency on space charge in the trap. This independence is very useful because it makes possible very accurate charge-to-mass ratios measurements with no dependence on the number of ions in the trap. In this way we became interested in the observation of this kind of instabilities in apenning trap, because we knew that many detected instabilities in a Paul trap occurred also in a Penning trap. Then we began a more detailed investigation of the motion of an electron cloud in a Penning trap in order to improve our understanding of the trapping processes. These investigations are the subject of the present work.

4 Chapter 2 Theory 2.1 Motion in a Penning trap An ideal hyperbolical Penning trap [Fig. 2.1] is a device consisting of three hyperbolic electrodes that generates a quadrupolar electric potential [Fig. 2.2] (2.1). By a proper Figure 2.1: Penning trap choice of the applied voltage it connes charged particles in the direction along the z-axis. In addition a magnetic eld B = B z connes the particles in the radial direction. The electrostatic potential in the ideal trap is: = V (x2 + y 2, 2z 2 ) (2.1) Here V 0 is the applied potential between the electrodes and 0 is the minimal radius of the ring electrode. The motion of a charged particle in this electromagnetic eld can be studied in two parts [Bro86]. On one hand the axial motion is decoupled from the magnetic eld and is a simple harmonic motion given by: On the other hand we have the radial motion that is given by: z +! z 2 z =0 (2.2) That equation leads to: m ~ = e[ E ~ _~ +( c ~ B)] (2.3) ~, ~! c _ ~, 1 2 ~! z 2 ~ =0 (2.4)

5 2.1 Motion in a Penning trap 4 q 2qV 0! c = q m B and! z = md 2 (d 2 = 1 (z )) are, respectively, the free space cyclotron 2 frequency and the axial frequency. The solution of this equation of motion can be found Figure 2.2: Electric and Magnetic elds in a Penning trap easily by making the proper coordinates transformation, ~u () = _ ~,! ^z ~. With this transformation the solution is two uncoupled harmonic oscillators with frequencies!. Then the motion of the charged particles in an ideal Penning trap is the combination of three uncoupled harmonic oscillators, one in the z direction and two in the radial plane. The frequencies of these three harmonic oscillators are the axial frequency (! z ) and: r! + =! 0 c =! c!c 2 2 +,! z r!, =! m =!!c 2 c 2,,! z (2.5) (2.6)! 0 c is the so-called reduced cyclotron frequency and! m is the magnetron frequency. So the motion has a shape like shown in Fig Figure 2.3: Motion in a Penning trap

6 2.1 Motion in a Penning trap 5 In a real trap there may be some imperfections (electrode misalignments, imperfect hyperbolical shape of the electrodes, holes or slits in the electrodes) that lead to non quadrupolar contributions to the potential. The potential can always be expanded in a series of spherical harmonics. Assuming cylindrical symmetry the spherical harmonics reduce to Legendre polynomials P n. Then we have the next expression for the potential in the trap: V = V 0 2 1X lpl C l (cos ) (2.7) 0 l=0 and are the spherical coordinates. With this expression for the electrical potential and neglecting higher order terms, l>4, the equations of motion in Hamiltonian formulation can be written as follows:, _p x = (eb)2 m, _p y = (eb)2 m, _p z = (eb)2 m _x = p x m + qeb 2m y x 4, py 2eB + y 4, px 2eB h + n (n+1) 2 2z, 3C 3 _y = p y m, qeb 2m x h n (n+1) 2, x, C 3 3zx + C 4 h 0 n (n+1) 2, y, C 3 0 3zy + C 4 0 z 2, x2 2, y2 2 _z = p z m 0 2 (3x 3 +3xy 2, 12xz 2 ) 2 (3y 3 +3yx 2, 12yz 2 ) 0 i + C 4 2 (8z 3, 12zx 2, 12zy 2 ) 0 i i (2.8) (2.9) n =! + =!,. These equations cannot be analytically solved but it can easily be seen that the three oscillations get coupled due to the anharmonic components of the potential. We performed a simulation, by solving numerically these equations and we got trajectories like in g Figure 2.4: Simulated trajectory in the x-y plane of a real Penning trap including an octupole term in the potential with coecient C 4 =10,3 for the two cases n =! + =!, =8:7 and n =! + =!, =9 It can be demonstrated that the three eigen-frequencies are related by the so-called invariance theorem [Bro86]:! z 2 +! + 2 +!, 2 =! c 2 (2.10)

7 2.2 Motional excitation Motional excitation As discussed in the previous section, the motion of charged particles in a Penning trap is the composition of three oscillators. Due to imperfections in the trapping potential these three degrees of freedom are coupled. These oscillations can be excited by an external eld. When this external eld gets in resonance with one of the motional modes of the particles they can be ejected from the trap. In this way the motional spectrum can be analyzed experimentally. Due to the coupling of the motional degrees of freedom it should be possible to observe not only the three fundamental oscillations,! z,! m and! 0 c, but also some combinations, so-called side bands, of the form! =! + l! m k! z, with l; k 2 N. This analysis can be done by observing the number of particles, which remain in the trap, as a function of the external driving frequency.

8 Chapter 3 Experimental Setup and Method Our setup can be divided into ve sections: vacuum chamber,trap, detection circuit, excitation circuit and control and data acquisition. They will be explained in more detail in the following sections. 3.1 Vacuum chamber In order to avoid instabilities due to collisions with other particles the electrodes are placed in an ultra high-vacuum (UHV) chamber [Fig. 3.1] in which we achieve a pressure of the order of 10,10 mbar. The system was evacuated by aturbo-molecular pump. After initial pumping the pressure is maintained by an ion-getter pump. As the trap was used previously for laser spectroscopy, the chamber has several glass windows which are not used for the present work. Figure 3.1: Vacuum chamber

9 3.2 Trap Trap The trap, as mentioned above, is a Penning trap consisting of three hyperbolic electrodes with an internal radius 0 = 20mm. The center of the trap is located in the magnetic eld center where the magnetic eld is homogeneous. The magnetic eld is supplied by 2 Helmholtz coils made of an aluminum plane band with a maximum current of 10 Amperes. The eld in center is in the range of 0-60 Gauss for a current up to 9 Amperes. The calibration of the eld was made both by using a Hall probe and by measuring the cyclotron frequency of stored electrons for several coil currents giving the magnetic eld strength as a function of the coil current as shown in Fig The result of the eld/current calibration is: B z [G] =(7; 21 0; 04) I coil [A], (0; 58 0; 24) Gauss (3.1) A measurement of the eld homogeneity in the center of the trap was also made, by measuring the magnetic eld for dierent orientations of the Hall probe with respect to the z-axis, giving the results shown in [Fig. 3.2]. The expected dependency of the eld with Figure 3.2: Homogeneity of the magnetic eld the angle is B / cos() that ts very well with the measured eld. From the deviation can be said that the eld in-homogeneity is small enough (< 1%) to treat the magnetic part of the trapping potential as ideal. The electron creation is performed by means of a heated wire that is placed outside the trap, just above the upper endcap. In oder to allow the electrons to get into the trap the upper endcap consists of a mesh [Fig. 3.3]. Because of a previous use of the trap for laser spectroscopy the ring electrode has two holes of 4mm diameter, and the lower endcap has two small slits that were used to allow ions evaporated from two laments to enter the trap. The radius of the ring electrode is 2cm. The electric potential is provided by the computer by means of the ADC card. But for achieving the potentials that we need, the working points are in the range of Volts, we have to amplify the signal given by the ADC. For this purpose we use a home-made amplier designed by S. Stahl that also generates the detection slope. The calibration of the applied voltage displayed in [Fig. 3.10] gives: V 0 =(0; ; 0003) V comp +(7; 56 0; 04) Volts (3.2)

10 3.3 Detection circuit 9 Figure 3.3: Electrodes 3.3 Detection circuit In order to investigate the particle motion we detect the number of conned electrons. This is done by an electronic, non destructive method. This mechanism is based on the fact that the electrons are moving in the trap with a harmonic oscillation in the z direction. See [Fig. 3.4] the drawing of the circuit. Figure 3.4: Detection Scheme Since the electrons are moving between the electrodes they induce an image charge on them. If we connect both electrodes there is a induced current that can be detected. To increase the detection sensitivity we couple a LC circuit parallel to the trap. Then we feed this circuit with an external a.c. voltage with the resonance frequency! 0 of the circuit. In our case the resonance frequency of the circuit is set to 17 MHz. Since the axial frequency of the electron's motion is related to the ring potential (V 0 )we can change this oscillation frequency in order to achieve! z =! 0. This happens in our case at V 0 =12:9V olts.wedo it by scanning the potential in the ring electrode [See detection time in Fig. 3.7]. When the electrons oscillate with an axial frequency equal to! 0 we can observe an energy transfer

11 3.4 Excitation circuit 10 from the a.c. supply to the electrons. This energy transfer dampens the LC circuit and can be observed as a decrease of the a.c. amplitude across the circuit. After amplication and demodulation we observe a minimum [Fig. 3.5], so-called dip, Figure 3.5: Signal sample in the output voltage of the demodulator [Kla99]. The amplitude of the dip is proportional to the detected electron number as long as no saturation occurs. 3.4 Excitation circuit In our experiment we try to nd the motional frequencies of an electron cloud. In order to achieve this we feed the trap with an external radio-frequency eld (RF). We do it by means of a small lament [Fig. 3.6] that acts as antenna, placed in a small slit in the Figure 3.6: Microwave feed-through lower endcap. This external eld is applied during the storage time [Fig. 3.7]. When this RF is of the same frequency as the electrons they get excited by absorbing energy from the external eld and then we can observe a decrease in the electron number conned in the trap. From this observation we can study the motional spectrum of the electrons in the trap. 3.5 Control and data acquisition One important part of the setup is a personal computer that is used as data acquisition and trapping control system. This control and acquisition is made by means of a 12 bit ADC

12 3.5 Control and data acquisition 11 Figure 3.7: Measurement Cycle card with 4 analog outputs and 16 single-ended inputs (8 dierential) with a maximum amplication factor of The potential limits are for the output V,5V,10V and for the input the card is protected up 60V. The ADC card is controlled by ac ++ program, written by S.Trapp, that controls the voltages, timing and data acquisition Timing We call timing to the time dependency of ring voltages. This dependency as told is controlled by the PC. The time cycle [Fig. 3.7] is composed mainly of 5 sections. We call creation time the time when electrons are created and during this time the potential in the ring (creation potential) is smaller than the one at the lament. The electrons are then accelerated into the trap. After this time we apply a cooling potential during the so-called cooling time. This potential tries to control the number of electrons in the trap by allowing the "hottest" ones to escape from the trap. Then begins the storage time, where a storage potential tries to avoid that the electrons escape. During this time all the manipulations, like excitation with the external driving eld, are made. At the end of the storage time begins the detection ramp where the ring potential goes from the storage value to some value under 0 Volts crossing the 13 Volts point where the axial frequency of the electrons gets in resonance with the detection circuit and after that the electrons are kicked out of the trap. After some time, waiting time, the cycle begins again.

13 3.5 Control and data acquisition 12 Figure 3.8: Experimental setup

14 3.5 Control and data acquisition 13 Figure 3.9: Calibration of the Magnetic eld

15 3.5 Control and data acquisition 14 Figure 3.10: Calibration of the Potential in the ring

16 Chapter 4 Results 4.1 Motional spectrum In our work we want to study the motion of charged particles in a Penning trap. In particular we are interested in the motional spectrum, i.e. the motional frequencies, of electrons in a non-ideal Penning trap. As a rst step, we scanned a whol e range of frequencies, covering all the eigen-motions, i.e.! z,! + and!,.we observed the spectrum shown in [Fig. 4.1]. Apart from the fundamental frequencies many other resonances can be observed. 4.2 Labeling the oscillations The rst thing to do is identifying the oscillations. We know very well the axial frequency! z / p V 0 because the ring potential was calibrated. An other easy assignment is the pure cyclotron! c =! + +!, because it is the only one that does not depend on the ring potential. By observing the motional spectrum (in the cyclotron area) for dierent potentials [Fig. 4.2] these two oscillations can be labeled. Then from the measured values for this two frequencies the other two main frequencies,! + and!,, and several combinations can be extracted. If we look carefully to the gure 4.2 it can be observed that the strongest resonance is not the reduced cyclotron! + but a side band of it at! +,!,. This is due to the fact that our excitation, that has no well dened mode, has a strong quadrupolar component. This is shown in g The parametric quadrupolar excitation was performed by applying the external excitation directly to the ring electrode. In this gure [4.3]it can be seen that the quadrupolar eld mainly excites the axial and the! +,!, oscillations. From there also can be explained why the 2! z oscillation is so prominent in our measurements. 4.3 Collective and individual oscillation Coming back to the gure 4.1. If looked in detail it can be seen that several resonances have some structure. We decided to study in more detail the 2! z oscillation. Studying it with higher resolution [Fig. 4.4]. We observed a double structure, consisting in a broad asymmetric part and a narrow symmetric feature on the high frequency side. It remind us to a similar structure previously seen [Alh97] in a Paul trap. Such a structure was already in 1975 seen in a Penning trap by D. Wineland and H. Dehmelt [Win75] but without further investigation. In the experiment in a Paul trap [Alh97] it was proved that this structure is due to the fact that the motion of an ion cloud in a Paul trap has two components, the incoherent motion of each ion and the coherent motion of the whole cloud as one particle

17 4.3 Collective and individual oscillation 16 Figure 4.1: Full frequency scan

18 4.3 Collective and individual oscillation 17 Figure 4.2: Frequency scan for dierent storage potentials. The dots are the centers of the resonances

19 4.3 Collective and individual oscillation 18 Figure 4.3: Comparison of the antenna excitation mode to a parametric quadrupolar mode

20 4.3 Collective and individual oscillation 19 located in the center of mass. Here we have investigated if we nd a similar behavior in a Penning trap. Figure 4.4: 2! z resonance. The simulation of the non-collective resonance (dashed line) assumes an octupole contribution to the trapping potential of strength C 4 =10, Frequency shift We have studied how the resonance behaves as a function of the electron number. In order to control the number of electrons in the trap we introduced in the measurement cycle a so called "cooling time" [Fig. 3.7]. By decreasing the potential well during the cooling time some electrons are lost and so the number of electrons can be reduced. We obtained an electron number as a function of the cooling potential as shown in gure 4.5. We have measured how the resonances behave under changes in the electron number and we found [Fig. 4.6] that the sharp resonance stays at the same place but the broad, asymmetric one shifts to higher frequencies with decreasing number of electrons. This frequency shift is explained [Yu89] by the inuence of space charge on the electrons motion. Each electron sees not only the trapping potential from the electrodes but also the perturbation due to the other electrons in the trap. The eective potential for each electron is smaller that the one supplied by the electrodes. Since we are looking to 2! z / p V and the screening is / N 2 (where we denote by N the number of electrons in the trap) the frequency should shift linearly with the number of electrons in the trap. This dependency of the screening arise from assuming that the cloud is an uniformly charged sphere [Yu89]: The frequency shift in the axial motion, is given by:! 0 z =! z s 1,! p 2 3! 2 2 with! p = q2 n " o m (4.1)

21 4.3 Collective and individual oscillation 20 Figure 4.5: Electron number as a function of the cooling potential. Solid line: expected electron number for a Boltzman distribution of 16:6eV mean kinetic energy. The trap's for a potential depth was 45eV n is the charge density in the trap. Based on this equation we have calculated [insert in Fig.4.6] the frequency shift. For a cloud size of 5mm and an amount of10 7 electrons which is the expected order of magnitude, we can reproduce the measured shift. These space charge eects, however, do not aect the motion of the cloud as a whole because the center of mass, and of charge, of the cloud is moving near the center of the trap independently of the number of electrons and the cloud only sees the potential delivered by the electrodes. Thus we expect no shift of this frequency with changing electron number, as experimentally observed Asymmetry An other feature that should be explained is that the broad resonance arising from the non-collective motion of the individual electron is very asymmetric. The electron cloud is so big that some electrons move far away from the center of the trap. There the potential is no longer harmonic and dierent electrons will oscillate at dierent frequency. We simulated [Fig. 4.4]the resonance assuming an octupole component of the trapping potential of strength C 4 = 10,3 (Normalized to the strength of the quadrupole potential C 2 = 1) [Gud96] and could well reproduce the observed line shape Threshold The collective oscillation appears only if the excitation amplitude is larger than a certain threshold amplitude [Fig. 4.7]. This observation was also made by [Alh97]. The only mechanism that explains this threshold is that the motion is damped. Then a certain excitation amplitude is required to overcome a critical damping. The only reasonable damping mechanism for the oscillation would be collisions with background gas atoms. Even at background pressure of about 10,10 mbar as in our experiment the collision rate may be suciently large if we consider the electron cloud as a simple by particle.

22 4.3 Collective and individual oscillation 21 In a similar experiment on H 2+ ions in a Paul trap [Alh97] at about the same background pressure a threshold amplitude for the exciting r.f. eld of 10mV was found. In our case on electrons we observe values of the order of 0.2mV. This value was obtained by extrapolation of the experimental values of the resonance's width as a function of the excitation amplitude using the equation given in the box [Fig. 4.7]. The fact that the threshold amplitude for electrons is more than one order of magnitude smaller then in the case of H 2+ under similar pressure conditions reects the diferen cross sections for electrons and molecular ions in collisions with neutral atoms. Figure 4.6: Space charge eect in the non-collective and collective oscillations Figure 4.7: Threshold behavior of the collective oscillation. The tted function is y = y 0 +A(1,e V=V thr ), which gives a threshold amplitude of V thr =0:180:04mV

23 Chapter 5 Discussion and Outlook In this work we have reported about a series of measurements held in order to understand the behavior of trapped charged particles. We have observed that the motion in a real, non-ideal, Penning trap is the composition of several oscillations. These oscillation can be decomposed in two component, one due to the incoherent motion of each electron in the trap and one due to the coherent motion of the cloud as a whole. This collective/non-collective behavior has been proved on the 2! z oscillation but there are some indications that this behavior can be a general feature of the motion in a Penning trap. This is an investigation that can be performed in the future. This behavior may have application in mass separation processes because of the sharpness of the collective oscillation. It allows easier selection of one species q from a multi-component cloud. Also it is of importance that the frequency q 2! z /, does not depend on the number of particles in m the cloud and it reduces the uncertainty when the oscillations frequencies of ions are used to calibrate the electric or magnetic eld in a trap.

24 List of Figures 2.1 Penning trap Electric and Magnetic elds in a Penning trap Motion in a Penning trap Simulated trajectory in the x-y plane of a real Penning trap including an octupole term in the potential with coecient C 4 =10,3 for the two cases n =! + =!, =8:7 and n =! + =!, = Vacuum chamber Homogeneity of the magnetic eld Electrodes Detection Scheme Signal sample Microwave feed-through Measurement Cycle Experimental setup Calibration of the Magnetic eld Calibration of the Potential in the ring Full frequency scan Frequency scan for dierent storage potentials. The dots are the centers of the resonances Comparison of the antenna excitation mode to a parametric quadrupolar mode 18

25 LIST OF FIGURES ! z resonance. The simulation of the non-collective resonance (dashed line) assumes an octupole contribution to the trapping potential of strength C 4 =10, Electron number as a function of the cooling potential. Solid line: expected electron number for a Boltzman distribution of 16:6eV mean kinetic energy. The trap's for a potential depth was 45eV Space charge eect in the non-collective and collective oscillations Threshold behavior of the collective oscillation. The tted function is y = y 0 + A(1, e V=V thr ), which gives a threshold amplitude of V thr =0:18 0:04mV 21

26 Bibliography [Alh97] R. Alheit, X. Z. Chu, M. Hoefer, M. Holzki and G. Werth Nonlinear collective oscillations of an ion cloud in a Paul trap Phys. Rev. A 56, 4023(1997) [Bro86] L.S. Brown and G. Gabrielse Geonuim Theory: Physics of a single electron or ion in a Penning trap. Rev. Mod. Phys. 58, 1, pp.233(1986) [Gud96] Th. Gudjons, F.Kurth, P.Seibert and G. Werth Ca + in a Paul trap Proceedings of the Workshop: "Frequency Satandards Based on Laser-Manipulated Atoms and Ions" Feb 12-14,1996, in Schierke, Germany [Hüb97] K. Hübner, H.Klein, Ch.Lichtenberg, G.Marx and G. Werth Instabilities of ion connement in a Penning trap Europhys. Lett. 37 (7), pp (1997) [Kla99] A. Klaas Diplomarbeit Mainz (1999) [Pen36] F.M. Penning Physica (Utrecht) 3, 873(1936) [Pie54] J.R. Pierce Theory and Design of ELECTRON BEAMS Macmillan & CO., London (1954) [Sch95] L. Schweikhard, J. Ziegler, H.Boop and K. Lützenkirchen The trapping condition and a new instability of the ion motion in the ion cyclotron resonance trap. International Journal of Mass Spectrometry and Ion Processes 141 (1995) [Sta94] S. Stahl Diplomarbeit Mainz (1994) [Win75] D. Wineland and H.Dehmelt Line shifts an widths of axial, cyclotron and G-2 resonaces in tailored, stored electron(ion) cloud. International Journal of Mass Spectrometry and Ion Physics 16 (1975) [Yu89] J. Yu, M. Desaintfuscien, and F. Plumelle Ion Density Limitation in a Penning Trap due to the Combined Eect of Asymmetry and Space Charge. Appl. Phys. B 48,51-54 (1989)

27 Acknowledgements I thank Prof. Günter Werth for his kindness and his nice guiding through physics and for giving me the opportunity to work in his wonderful group. I thank Priv. Doz. Wolfgang Quint and Dr. Hartmut Häner for their help and their always interesting remarks. I also thank P.Paasche, who shared my oce-laboratory because without his help this work would not have been possible. The members of the "Arbeit Gruppe Werth" for their sympathy and for making with their happiness and good humor the work easier. I thank all my friends and SEACAVA, Association of valencian students in Germany, for helping me to "survive", so far from home, in Germany. Thank also to my "Mitbewohner" Bea and Mievi for beeing so nice and helping me with my "Deutsch". Special thanks to Victor Olmos because he brought me to Mainz and he made possible that I knew Prof. Werth. Finalmente quiero agradecer a mi familia el apoyo que he recibido durante mis largos años de estudio. Os quiero.