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## Trapping Charged Particles

This chapter describes the technical realization for an experimental setup with the purpose to produce, con ne and investigate Be+ and H+2 ions. In the rst section Paul traps are introduced as a possi-bility to trap charged particles. In the second and third section rst the trapping mechanism is reviewed and subsequently a design for a linear Paul trap presented, which suits the requirements. Thereafter follows a description of the beryllium ion source to load the trap with ions. The fth section presents the the vacuum system. A simple test shows that all the components are working as intended. An overview for the planned H+2 source is given in section six. This part is prelim-inary and the nal design may be slightly modi ed. In the last two sections the imaging system and the experiment control software are presented.

**Paul Traps**

harmonic To con ne particles at a given point rc in space, one needs a restor-potential ing force F. For charged particles an electric eld E, described by a harmonic potential = 0 x2 + y2 + z2 ; (2.1) 2r02 seems to allow a simultaneous restoring force in all three directions. But a con ning electrostatic potential in such a form is not possible, since the Laplace equation = 0 must be respected. This leads to the condition ++=0 (2.2) for the potential in equation (2.1), and two simple solutions for the parameters are: = = 1 and = 2; (2.3) = = 1 and = 0: (2.4) no static 3d Obviously an electrostatic potential is always repulsive in at least one con nment direction. This obstacle can be overcome by using an alternating elec-tric eld between parameters of di erent sign, which leads to a cycling of the attractive force between the corresponding directions. In the time average a small mean force remains, con ning the particles. This RF trapping idea was published by Paul in 1958 [37]. The rst realization of such an RF ion trap is based on eq. 2.3, which leads to the geometry of the hyperbolic trap (see Fig. 2.2 a) for a sketch). The design of this apparatus and a description can be found for example in Paul’s Nobel lecture [38].

For the solution in equation (2.4) an alternating electric eld between the x^ and y^ direction has to be established. This leads to the design of linear trap the mass spectrometer and the linear Paul trap. The linear design has the advantage of a better accessibility of the center of the trap, making

it easier to inject particles or manipulate trapped particles.

di erent The main working principle, namely an RF eld between at least two designs electrodes in di erent directions, remains the same for Paul traps, independent of the geometry. However, di erent tasks lead to a broad few ions variety of geometric realizations. In Fig. 2.2 di erent linear traps are shown. The trap in b) is a so called precision trap from Mainz 2.1. Paul Traps 11

Figure 2.2: Di erent Paul traps. a) Sketch of a hyperbolic Paul trap. The ring electrode is cut open to show the hyperbolic shape of the electrodes. b) A precision trap as it is realized at Mainz university. The eleven ngers are each 0:2 mm wide with a spacing of 0:03 mm. c) A possible design for the precapture trap in the GBAR project. The length of the red RF electrodes is 34:8 mm, and the spacing between the DC electrodes 0:2 mm. The total length of this trap is approxi-mately 15 times the length of the precision trap. d) Two ITO chips, a new technical realization of a linear trap, which is under investigation in Mainz university and is designed to capture single or few ions while being able to manipulate the position of the potential minimum very precisely [39, 40]. The trap shown in c) is designed to con ne a laser cooled ion loud which contains more than > 104 particles. This trap is larger > 104 than the precision trap, since it must accomodate a laser beam to ions address the ion cloud. New materials and manufacturing techniques lead to designs like in d), a litographically created chip of conducting segments, where the transparent part is created with an indium tin oxide (ITO) coating. This allows for a conducting, but transparent segment on the chip [41].

The traps b) – c) are also used as design studies for the GBAR project [42]. While b) presents a possible candidate for the nal trap, where + is planned. The traps c) and d) present the photodetachment of H design studies for the pre-capture trap.

The following two sections review rst the mechanism of a linear Paul trap, and present then, based on the previous considerations, a trap design which is suited for this project.

**A linear Paul Trap**

RF and DC In eq. 2.4 an alternating potential between electrodes in x^ and y^ di-potentials rection can establish con nement in the x^y^ plane. To con ne particles along the z^ direction additional electrodes are needed. Fig. 2.3 shows an example of a linear Paul trap with its RF electrodes, the center electrodes and the endcaps.

Figure 2.3: Sketch of a linear Paul trap. Electrodes 1 and 8 are the RF electrodes. Electrodes 3 and 6 are the center electrodes. The remaining ones are the endcap electrodes.

While the center electrodes are supplied with an RF voltage VRF = U0 + V0 cos [ t], the endcap electrodes are supplied with a constant voltage U1. The complete potential can be written as:

(x; y; z; t) = RF (x; y; t) + endc: (x; y; z) (2.5)

In the following two subsections rst the x^y^-con nement and the con-ditions for a stable RF trapping are derived, then the con nement along the z^-axis with a static DC potential are treated in more de-tail.

**Trapping in the x^y^-plane**

the As a rst step in understanding the trapping dynamics in the x^y^-quadrupole plane a two dimensional quadrupole is treated. Real linear Paul traps usually do not use hyperbolic electrodes, but rather cylindric ones. In a second step a geometric correction factor will be introduced, to account for this di erence.

Figure 2.4: Cross section of hyperbolic electrodes with the inner radius r0. Also shown is the potential for two di erent times t = 0 and t = ; the attractive potential is cycling between the electrodes.

The hyperbolic electrodes shown in Fig. 2.4 establish a two dimen- the sional quadrupole eld in the x^y^ plane. With 0 = U0 + V0 cos[ t] the potential two dimensional potential is

RF (x; y; t) = (U0 + V0 cos [ t]) x2 y2 + r02 : (2.6) 2r02

The electric eld is given by E = , so the equations of motion eq. of for an ion of mass m and charge q can be written as: motion x q (U0 + V0 cos [ t]) x = 0; (2.7) mr2 0 y + q (U0 + V0 cos [ t]) y = 0

Figure 2.5: Lissajous like trajectories of two charged particles with mass 9 amu and charge 1 e. The initial conditions xi; xi are the same in both cases, where the red arrow represents the direction of the initial velocity. Left: Stable trajectories with = 2 13 :37 MHz, V0 = 518 V and U0 = 1 V, and therefore (jaxj ; jqxj) = (4:96 10 ; 0:13).

Right: Unstable trajectories with = 32 13:37 MHz, V0 = 200 V and U0 = 4 V, and (jaxj ; jqxj) = (1:98 10 ; 0:050).

Mathieu the equations of motion can be transformed into a Mathieu equation equation for both directions u = x; y: d2u + (au + 2qu cos [2 ]) u = 0: (2.12) unstable In Fig. 2.5 numerical solutions for di erent initial conditions and RF and stable voltage parameters are shown. These solutions can be divided into solutions two di erent categories: unstable and stable solutions. For unstable solutions the trajectory diverges (and the particles collide with the electrodes), while for stable solutions the trajectory is restricted to a nite area in space. The general solution to the Mathieu equation can be written as [37]: u ( ) = A e c2n e2in + B e c2n e 2in ; (2.13) where A and B are constants depending on the initial conditions, and the stability parameter and the coe cients c2n depend only on the parameters au and qu. As the name would suggest the stability parameter = l + im determines if the solution u ( ) describes a unstable or stable solution: RF trap. Left: Only the overlapping regions lead to stable solutions.

Right: The rst stable region. l 6= 0: In that case either e or e diverges and therefore also the solution u ( ) diverges. This unstable solution leads to the loss of the particle. l = 0 and m 2 R: The particle has a periodic amplitude in the x^y^ plane and the solution is stable. The solutions with m 2 Z describe the transition between stable and unstable solutions.

The python package scipy.special contains special functions, amongst stability other things also the functions for the characteristic value of the even diagram and odd Mathieu function: mathieu a(m,q) and mathieu b(m,q). In fig. 2.6 the functions for m = 0; 1; 2 are plotted, which enclose the stable regions for the x^ and y^ direction. To con ne a charged particle the trajectory must be stable in x^ and y^ direction, therefore only the overlapping regions in Fig. 2.6 are of interest. Experimentally the region with a smaller qu and therefore also a smaller voltage is easier to access, for which reason linear Paul traps are usually operated in the rst stable region.

The trajectory of a charged particle in a time constant eld and a Adiabatic rapidly oscillating eld can be separated into two components: a Approximasmooth secular motion S(t) and a fast oscillating micromotion M(S; t), tion following the secular motion [43, 44].

In the following an one-dimensional approach outlines this separation.

The potential can be expressed by a time independent part S (x) and an oscillating part M (x; t), so that one can write for the resulting forces: F (x) = q d S (x) (2.14) dx f(x; t) = q d M (x; t) = f0(x) cos [ (t t0)] (2.15) dx

If now M S and M_ S_, then M can be treated as perturbation of S and one can expand the equation of motion: m S (t) + m M (S; t) = F (S) + f (S; t) + M @x (F (x) + f (x; t))jS + O(M2) (2.16)

Since the smooth and the oscillating part must be separately equal one can write for the oscillating terms: m M (S; t) = f (S; t)

Using eq. (2.15) this leads to an expression for M: ( ; t) = f0(S) cos [ (t t )]

Averaging eq. (2.16) over one period of the micromotion one obtains the secular motion, as shown in Fig. 2.7 on the right side. The fast oscillating terms vanish, so that one can write mS = F (S) + T T M @xf(x; t)jS dt: (2.19)

Using the expression for the force from eq. (2.15) and for the secular motion from eq. (2.19) one can write: mS = F (S) 2m 2 f0(x) @f0(x)jS ; (2.20) which can be rewritten mS = @x q S(x) + q2 (@x M (x))2 : (2.21)

One can now identify the pseudopotential energy: pseudo potential (x) = q2 (@ (x))2 (2.22)

This expression shows that the pseudopotential is con ning in the x^ and y^ direction. Finally, the complete secular potential energy in the x^y^-plane is q2V 2 qU0 sec(x; y) = 0 (x2 + y2) + (x2 y2): (2.25) 4m 2r04 2r02

It can be written in terms of secular oscillation frequencies as secular frequencies sec(x; y) = 1 m !2(x2 + y2) + 1 m !2 (x2 y2) = 1 m e !2 + !02 x2 + 1 m 0 !2 !02 y2; (2.26)

These frequencies can also be expressed in terms of the parameters au, qu and 2 qu2 2 2 au 2

!e = and !0 = ; 8 4 so that one can write: !x=y = 2 q2 au : (2.27) 2 u potential The potential depth can be estimated using eq. 2.26 as the e ective depth potential at the electrode location: sec;depth = q2V02 qU0 : (2.28) 4m 2r2 2 cylindrical Mathieu’s di erential equation assumes hyperbolically shaped elec-electrodes trodes as boundary conditions. These are di cult to produce, and in 1958 Paul, Reinhard and von Zahn replaced the hyperbolic electrodes successfully by electrodes with a circular cross section with re=r0 = 1:16 [37].

With the increasing computer power it was shown in 1971, that a ratio of re=r0 = 1:1468 leads to an even better approximation of the quadrupole eld [45]. Nowadays it is easy to verify these results. The potential distribution in a quadrupole with circular cross section can be expressed as a series [45]: (x; y; t) = (U0 1 Cn r 2(2n+1) cos [2 (2n + 1) ] + 1 : + V0 cos [ t]) n=0 r0 2 X

Figure 2.8: Cross section of cylindrical electrodes for a linear Paul trap. The characteristic parameters are R, the distance between the trap center and the electrodes, and the radius for the cylindric elec-trodes, re.

The rst term is the quadrupole term, the second one a 12-pole term, etc. The best approximation to a pure quadrupole potential is an electrode radius such that the 12-pole coe cient C1 disappears, all higher contributions are magnitudes of order smaller.

Transforming eq. (2.29) into cartesian coordinates and using some trigonometric identities one can write for the potential along the x^ axis for U0 = 0 and t = 0: x2 x6 (x; y = 0; t = 0) = V0 C0 + C1 +::: (2.30) r02 r06

Here an e ective radius r0 was introduced, which is expected to be close to R.

An option to simulate the potential which is caused by multiple elec- SIMION trodes is the commercial software SIMION 8.1. This software can be used to de ne the geometry of the electrodes via .gem- les. It then solves the Laplace equation numerically on a mesh, where the resolu-tion of the mesh is 1 point=mm. In order to obtain a better resolution all the following results are obtained with a 10=1 scaled geometry. The result for each independent electrode is saved in a so called .patxt- le format, and then imported into Python 3. Here the potential arrays are rescaled to a newly de ned voltage for the corresponding electrode. The geometry is also rescaled. Thereafter the total potential caused by the i independent electrodes can be summed up: X SIM,DC (x; y; z) = i(x; y; z) (2.31) To determine the values of the coe cients C0 and C1 the geometry of four cylindric electrodes with the parameters R = 3:5 mm, l = 12 mm and re = 2:975 mm; 3:325 mm; ::: was de ned in multiple .gem- les (an example le and an overview of the resulting geometry can be found in the appendix A.1). The mesh consists in each case of 261 261 141 points. The results were imported to python and rescaled such, that each two opposite set of electrodes have the same voltage, the two RF electrodes with 1 V and the other two electrodes with 0 V. Thereafter the total potential SIM,DC along the x^ axis was tted with the rst two terms of eq. 2.30. In g. 2.9 the coe cients C0 and C1 are plotted as a function of the ratio re=R. For a ratio of re=R = 1:15 0:01 the contribution of the 12-pole term vanishes, and the approximation with a pure quadrupole potential is best justi ed. These calculations agree with what can be found here [46], where a more detailed analysis is carried out.

Data points from SIMION for re=R = 4:0=3:5 = 1:1429 and a pure quadrupole t. correction In the case of our trap this ratio is not the ideal value, since re=R = factor 4:0=3:5 = 1:1429, which is slightly smaller. Therefore the 12-pole term does not vanish completely, and one obtains C0 = 0:5055 and C1 = 3:823810 4. In the practical case one is interested in the region in the proximity of the trap center, so that r=r0 1. Therefore C0 (r=r0)2 C1 (r=r0)6 and the quadrupole approximation is valid.

The t with a pure quadrupole term of the form x2=r02 + leads to = 0:500 and to = 0:5055. The discrepancy between the theoretic value th = 1=2 and the value from the t can be explained by the fact that not all higher order terms in eq. 2.29 become zero. By introducing r0 = R (2.32) and demanding x =R2 = x =2r02 rection factor = 0:9945. This correction factor has to be taken into account when deriving the secular frequencies (see section 2.2.3).

#### Trapping in the z^-direction

In the section before it was shown how trapping in the x^y^ plane can be achieved using an RF potential. Con nement along the z^-direction can be accomplished by applying DC potentials at the endcap elec-trodes.

Figure 2.10: Design for a linear Paul trap, consisting of RF and cen-ter electrodes (light grey), and four endcap electrodes (dark red) for con nement along the z^-direction. endcaps In g. 2.10 the endcap electrodes are highlighted. The RF electrodes are not segmented. One can see that with positive endcap voltage the potential is con ning along the z^ direction and decon ning along the x^ and y^ directions. Close to the center of the trap, the potential has a quadrupolar behaviour that is modeled by endc: (x; y; z) = U1 z2 1=2 x2 + y2 + ; (2.33) 2zeff2 where U1 is the voltage applied to the endcap electrodes. To estimate the e ective parameter zeff for the speci c geometry shown in g. 2.10 SIMION 8.1 was used again. The .gem- le with a mesh of 241 241 401 points can be found in the appendix A.2.

After preparing the results in python 3 the endcap electrodes were scaled to 1 V and all other electrodes to 0 V. The total potential SIM,DC (x; y; z) along the x^, y^ and z^ axis was plotted and can be seen in g. 2.11.

Figure 2.11: Top: SIMION potential along the x^, y^ and z^ axis for the geometry shown in Fig. 2.10. At the endcaps the applied voltage is 1 V, and 0 V elsewhere. Bottom: Adjustment of the parameter zeff depending on the number of points which are used to t the potential.

e ective This result has been tted with the expression in eq. 2.33. The length bottom part of g. 2.11 shows the dependency of zeff for the three axes, when varying the t intervall around x; y; z = 0. One can es-timate the value for the e ective length with zeff 11:7 mm. This is approximately twice the size of the half electrode distance with 6:2 mm, which can be expected since the RF electrodes are not segmented. The potential minimum is consistent in all three dimensions with = SIM,DC (0; 0; 0) = 0:021 V.

**The E ective Potential**

With the investigations of the last two sections eq. (2.5) can be rewrit-ten in order to gain some quantitative insight to the e ective trapping potential. First the RF part can be replaced with the expression for the secular potential in eq. (2.25) with r0 = R, where = 0:9945. In a next step the endcap potential is replaced with eq. (2.33), with ze = 11:7 mm. By sorting the equation by variable we nally obtain an expression for the e ective potential energy

**Table of contents :**

**1 Introduction **

1.1 What and Why – Spectroscopy of H+2 2

1.1.1 The Hydrogen Atom

1.1.2 The Hydrogen Molecular Ion

1.1.3 H+2 Spectroscopy at LKB

1.2 How – Be+ Coulomb Crystals

1.3 Overview of this Work

**2 Trapping Charged Particles **

2.1 Paul Traps

2.2 A linear Paul Trap

2.2.1 Trapping in the ^x^y-plane

2.2.2 Trapping in the ^z-direction

2.2.3 The Eective Potential

2.3 Design and Implementation of a linear Paul Trap

2.3.1 Trap Accessibility

2.3.2 DC Voltage Supply

2.3.3 RF Voltage Supply

2.4 Loading Ions

2.4.1 Beryllium Oven

2.4.2 E-Gun

2.5 The Vacuum System

2.6 The H+2 Source

2.7 Imaging

2.8 Experiment Control via Python 3

**3 Cooling Trapped Ions **

3.1 Doppler Cooling

3.2 Cooling Lasers at 313nm for Beryllium Ions

3.2.1 Master Slave System

3.2.2 Fiber Laser

3.2.3 Bow Tie Cavity

3.3 Other Laser Sources in this Experiment

3.3.1 The REMPI Laser

3.3.2 The Spectroscopy Laser

3.3.3 The Dissociation Laser

**4 Crystallized Ions**

4.1 Coulomb Crystals

4.2 First crystallized Be+ Ions

4.3 Minimizing the Micromotion

4.4 Magnication of the Imaging System

4.5 Three-dimensional Be+ Crystals

4.6 Sympathetic Cooling

4.7 Flat Be+ Crystals and sympathetically cooled H+2 Ions

4.7.1 A two-dimensional Be+ Crystal

4.7.2 A single H+2 ion in a two-dimensional Be+ Crystal

4.7.3 Multiple H+2 Ions in a at Be+ Crystal

4.8 A three-component Crystal

4.9 A large Be+/H+2

Coulomb Crystal

**5 Summary **

Appendices

A .gem-les for SIMION

A.1 .gem-le for the optimal rod size

A.2 .gem-le to estimate zeff

A.3 .gem-le for SIM(x; y; z)

**References **