Influence of the 22-pole Trap Imperfections on theInteraction of Ions with a Neutral Beam

S. Roucka, P. Jusko, I. Zymak, D. Mulin, R. Plasil, and J. GlosıkCharles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

Abstract. The temperature dependence of the rate coefficient of the associativedetachment reaction (H− + H → H2 + e−) was studied using the Atomic Beam—22-pole Trap Apparatus (AB-22PT). A discrepancy between the data measuredat different combinations of ion and atom temperatures was observed. Thiscontribution discusses various imperfections of the 22-pole trap as possible causesof this effect. Numerical models of these imperfections are compared to theexperimental data.

Introduction

This work is motivated by the recent measurement of the associative detachment reaction (AD)H− + H → H2 + e− performed in our laboratory. The measurement was carried out in the AtomicBeam—22-pole Trap apparatus (AB-22PT), which is described by Borodi et al. [2009]; Plasil et al. [2011];Gerlich et al. [2011] in detail. The results of this experiment are being prepared for publication by Juskoet al. [2011]. A basic description of the experiment is presented below and a schematic drawing of theexperiment is shown in figure 1. During the measurement a cloud of H− ions is stored in the cryogenic

Figure 1. Schematic drawing of the AB-22PT experiment configuration. Dimensions are in millimeters.

22-pole radiofrequency trap, which has variable temperature T22PT in the range 10 K < T22PT < 300 K.The ions interact with a cold effusive beam of atomic hydrogen. The beam flows from the accommodatornozzle, which has also a variable temperature TACC in the range 10 K < TACC < 300 K. The ions arethermalized by buffer gas collisions at a temperature close to the T22PT. The temperature of atoms isclose to the accommodator temperature [Borodi et al., 2009]. The interaction temperature T of the twoensembles with different temperatures is given by the arithmetic average

T =TACC + T22PT

2(1)

in case of equal particle masses.Our experimental procedure for measuring the temperature dependence of the rate constant is

described by Jusko et al. [2011] in detail. Since the produced density of H atoms has a complicateddependence on the accommodator temperature, we always perform a sequence of measurements at afixed accommodator temperature and a variable 22-pole temperature. Sets of measurements performedat different accommodator temperatures are then stitched together by the least squares method. Theresult of this procedure for datasets at TACC = 10 K and TACC = 50 K are presented in figure 2 (leftpanel). A discrepancy in the datasets can be seen at low temperatures. This effect can be explained by aT22PT temperature depence of the overlap between the atomic beam and ion cloud. This hypothesis wassupported by measuring the laser photodetachment rate. Multiple measurements of the photodetachmenttemperature dependence were performed with laser pointed into different areas of the trap. Severalmeasurements used a narrow ≈ 1 mm laser beam and other measurements used a wide > 1 cm beam.Under our experimental conditions no temperature dependence of the photodetachment rate should beobserved. However, figure 2 (right panel) shows a significant temperature dependence of the measuredphotodetachment rate. Depending on the precise laser pointing, the photodetachment is decreasing withtemperature at different rate. Only one of the measurements shows an increase of the photodetachmentrate with decreasing temperature. This clearly indicates, that ions are being redistributed by varying the

ROUCKA ET AL.: 22-POLE IMPERFECTIONS

0 20 40 60 80 100 120 140 (K)

0

1

2

3

4

5

6 (1

0cm

s)

==

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0.0

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1.2

Phot

odet

achm

ent r

ate

(arb

. u.)

focused laserdefocused laseroverlap calculated from AD

Figure 2. The discrepancy between rate coefficient measurements performed at different accommodatortemperatures (left panel). Dependence of the photodetachment rate on the 22-pole trap temperature isshown in the right panel. For explanation of the calculated overlap see section Results.

T22PT and are probably concentrating in an effective potential minimum with decreasing temperature.Since the majority of measurements shows a decrease of the photodetachment rate with decreasingtemperature, the effective potential minimum is probably located out of the reach of the neutral or laserbeam.

In the theoretical section below, some basic concepts regarding the interaction of the trapped ionswith a neutral beam are introduced. In the section Results we will analyse and correct the measureddata and provide possible explanations of this effect based on numerical simulations of the ion cloudspatial distribution under various conditions.

The density of the atomic beam was determined by the chemical probing with CO+2 [Borodi et al.,

2009]. The CO+2 has a higher mass than H− and a different RF amplitude was used for the calibration

measurement. Therefore the ion cloud has a different shape and the measured effective H density has tobe corrected by a factor which will be derived in the theoretical section of this article. The measurementwas carried out at T22PT = 30 K and TACC = 50 K.

Theory

We only briefly introduce the relevant equations of the theory of motion in oscillatory fields underthe adiabatic conditions. For details refer to Gerlich [1992]. It can be shown that under the adiabaticconditions [Gerlich, 1992] the oscillating electric field described by E(x, t) = E0(r) cos(Ωt+ δ) +∇φs(r)produces an effective potential V ∗ acting on a particle of mass m

V ∗ =q2E2

0

4mΩ2+ qφs . (2)

In case of an ideal linear 2-n pole without DC potential the effective potential can be calculated analyt-ically to give

V ∗ =1

8

(qV0)2

εr2n−2 ; ε =

1

2n2mΩ2r20 ; r = r/r0 , (3)

where V0 is the potential amplitude on the poles and r0 is the inner diameter of the multipole. Theturning radius of a particle with radial component of energy Em trapped in the multipole is then givenby

rm =

(8Emε

(qV0)2

) 12n−2

. (4)

In the atomic beam–ion cloud arrangement used in our experiment, the rate of reaction Ni can beobtained by integrating over the overlap of the atomic beam with the ion cloud

Ni = k(T )

∫r

nH(r)ni(r)dr , (5)

where k(T ) is the reaction rate coefficient, nH is the H atom density and ni is the ion density. We canrewrite this expression in terms of macroscopic variables as

Ni = k(T )NiNH , (6)

ROUCKA ET AL.: 22-POLE IMPERFECTIONS

where Ni is the number of ions in the trap and NH is the effective H atom density defined by

NH =

∫r

nH(r)ni(r)

Nidr . (7)

In our experimental configuration, the ion cloud confined in the 22-polar field has a larger diameter thanthe atomic beam. Due to the flat potential minimum of the 22-pole trap, the cloud can be consideredhomogeneous over the intersection with the atomic beam. The effective H atom density is then propor-tional to the ion density on axis, which is inversely proportional to the cloud cross section area, whichin turn is proportional to the square of the turning radius:

NH ∼ ni(r = 0) ∼ 1

r2m∼(

8Tε

(qV0)2

) −1n−1

. (8)

Since the proportionality is the same for all energies, we have replaced Em with the temperature. Byretaining only the relevant coefficients we obtain a relation for scaling of the effective H atom density inthe 22-pole due to the change of the overlap

NH ∼(Tm

V 20

)− 110

. (9)

This factor can be used to correct the effective H atom density determined by the CO+2 chemical probing.

The RF amplitude used in our experiment was V0 = 25 V for H− and V0 = 50 V for CO+2 . The ratio of

effective densities under these conditions with identical temperatures is NH(H−)/NH(CO+2 ) = 1.271.

The theoretical scaling of the overlap for H− and CO+2 is plotted in figure 3 relative to the overlap

of H− at 150 K. Experimental data in the plot are explained below.

Results

Experimental: We will now discuss a method to extract the scaling of the beam-cloud overlap fromthe observed data. We directly observe the decay rate X(TACC, T22PT) = Ni/Ni = k(T (TACC, T22PT)) ·NH(TACC, T22PT). The interaction temperature T is the average of the reactant temperatures givenby equation (1). We assume, that the dependence of NH on T22PT is caused by some processes inthe 22-pole trap and is not related to the H atom temperature. Therefore it can be separated intoNH(TACC, T22PT) = N(TACC)P (T22PT), where the overlap factor P is defined as unity at the highesttemperature of our measurement. We can now write

X(TACC, T22PT) = K(TACC + T22PT)N(TACC)P (T22PT) , (10)

where K(T ) = k(T/2) for simplicity. From the measured data we now select four measurements per-formed at two different interaction temperatures T using different combinations of T22PT and TACC

X(T ′ACC, T22PT) ; X(T ′ACC, T′22PT) ; X(TACC, T

′′22PT) ; X(TACC, T

′′′22PT) . (11)

The temperatures are related by equations

TACC + T ′′22PT = T ′ACC + T22PT ; TACC + T ′′′22PT = T ′ACC + T ′22PT . (12)

The N factor can be eliminated by dividing the measurements performed at the same TACC.

X(T ′ACC, T22PT)

X(T ′ACC, T′22PT)

=K(T ′ACC + T22PT)

K(T ′ACC + T ′22PT)

P (T22PT)

P (T ′22PT)(13)

X(TACC, T′′22PT)

X(TACC, T ′′′22PT)=K(TACC + T ′′22PT)

K(TACC + T ′′′22PT)

P (T ′′22PT)

P (T ′′′22PT)(14)

After calculating the ratio of equations (13) and (14) the unknown rate coefficient K can be eliminatedwith use of relations (12)

P (T ′′′22PT) = P (T ′′22PT)P (T ′22PT)

P (T22PT)

X(T ′ACC, T22PT)

X(T ′ACC, T′22PT)

X(TACC, T′′′22PT)

X(TACC, T ′′22PT). (15)

ROUCKA ET AL.: 22-POLE IMPERFECTIONS

The above equation is a recursive relation for determining P . Since no discrepancy in the measured datais apparent at temperatures above 50 K, we assume that P is equal to unity above 50 K. This assumptionprovides us with a sufficient boundary condition for solving the equation (15). This technique was appliedon the experimental data from figure 2 (left panel). The results are presented in figure 3. The obtainedoverlap factor was then used to correct the rate constant measurements for publication by Jusko et al.[2011].

These quantitative results clearly support the qualitative analysis performed in the introduction. Atlow temperatures, the ions should be concentrated in the potential minimimum on axis and the overlapshould increase. However, the observations indicate, that the potential minimum is located outside ofatomic H beam.

An additional measurement was performed at T22PT = 70 K by measuring the associative detach-ment rate as a function of RF amplitude in the trap. According to equation (2), by operating the trap ata low RF amplitude V0 = 7.5 V we were able to achieve the same effective potential for H− as for CO+

2

at V0 = 50 V. The correction factor was thus determined experimentally by calculating the ratio of ob-served reaction rates with a result NH(H−)/NH(CO+

2 ) = 1.37±0.08. The error estimate was determinedpurely from the goodness of fit. The observed value differs slightly from the theoretical one. However,the difference is well below 10 %. Due to the discrepancy of the H− overlap scaling from theory (figure2) we expect a slightly different correction factor at 30 K where the chemical probing was performed. Aconservative error estimate of the calibration procedure 20 % was used in our error analysis.

0 20 40 60 80 100 120 140 160 [K]

0.0

0.2

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0.8

1.0

1.2

1.4

rela

tive

over

lap

+

observed overlap scalingsimulation: ideal traptheory: /

Figure 3. The theoretical beam-cloud overlap scaling in comparison with numerically and experimen-tally obtained results. The solid line corresponding to the numerical model is indistinguishable from thetheoretical dashed line. The experimentally observed overlap scaling of H− ion cloud differs significantlyfrom the theory at low temperatures. The ratio of CO+

2 and H− overlaps is, however, close to thetheoretical value.

Simulations: The observed effect of decrease in the overlap with decreasing temperature can beexplained by various imperfections of the 22-pole trap. We have set up several simplified models ofpossible scenarios. A two dimensional model of an infinitely long 22-pole trap is used in all studied cases.The effective potential is calculated from equation (2). The DC potential φs and the RF potential φ0 arecalculated separately by solving the Poisson equation with Dirichlet boundary condition on the rods andwith a zero Dirichlet boundary condition on the distant domain boundary outside of trap. The Poissonequation was solved by means of the finite element method. The mesh was generated by gmsh [Geuzaineand Remacle, 2009]. The DOLFIN software from the FEniCS project [Logg and Wells, 2010] was usedfor solving the Poisson equation and all subsequent data processing. Once the effective potential wascalculated, the thermal ion distribution given by the Boltzmann distribution was obtained from

ni(r) = exp

(−V

∗(r)

kBT

)/∫trap

exp

(−V

∗(r′)

kBT

)dr′ . (16)

Finally, the overlap was obtained by integrating the fraction of ions in the beam, which has diameterapproximately 5 mm. The whole procedure was repeated with parameters of CO+

2 and H− for severaltemperatures in order to obtain the temperature dependence of the overlap. In our plots, the overlap isnormalized to the value corresponding to H− trapped in the ideal trap at 150 K under our experimentalconditions. The results calculated for the ideal trap are plotted in figure 3. The simulated curves areindistinguishable from the theoretical curves, which validates the assumptions of the theoretical modeland the correctness of our numerical model.

ROUCKA ET AL.: 22-POLE IMPERFECTIONS

The first case studied by our model is the possibility of mechanical deformation of the 22-pole. Weinvestigate a possible offset between the rod assemblies of opposite polarities. This effect was observedby Hlavenka et al. [2009] and investigated using numerical simulation by Otto et al. [2009]. Our modelreplicates the results of Otto et al. [2009]. By simulating high enough offset, it is possible to mimic thebehavior observed in our experiment, since ions start hiding in the ten potential minima outside of thebeam at low temperatures. See figure 4 for a plot of the calculated H− density distribution at 30 Kand the overlap temperature dependence. However, a very high deformation of 0.24 mm is needed toexplain the observed behavior. In our 22-pole, the rods are fixed on both sides of the trap and such adeformation seems to be an improbable explanation of the observed effects.

0.24 mm

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tive

over

lap

+

observed overlap scalingsimulation: ideal trapsimulation: 0.24 mm offset

Figure 4. Influence of an offset between the two rod assemblies. The H− density distribution at 30 K(left panel). The color palette is normalized to unity in the maximum of this plot and is valid for allsubsequent plots. The cloud-beam overlap as a function of temperature (right panel).

The other investigated case is the possibility of complete grounding of one of the rods. A moreprobable scenario would be only a partial disconnecting from the RF combined with charging up of therod. The effect of charging or patch fields is investigated separately in the next paragraph though. Seefigure 5 for a plot of the calculated overlap temperature dependence and H− density distribution at 30 K.Interestingly, the grounding of one of the rods has virtually no effect on the overlap. Therefore it cannotexplain the observed behavior.

0 20 40 60 80 100 120 140 160 [K]

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0.8

1.0

1.2

1.4

rela

tive

over

lap

+

observed overlap scalingsimulation: ideal trapsimulation: grounded electrode

Figure 5. Influence of grounding of one electrode. The H− density distribution at 30 K (left panel).The cloud-beam overlap as a function of temperature (right panel).

Finally, the influence of patch fields was investigated. The patch fields are caused by inhomogeneitiesof the polycrystalline metal and by surface coating with thin layers of impurities. They can occur onvarious scales with various intensities. For our study a case of a large patch with a small ±10 mVpotential was chosen, because its effect resembles more closely the observed behavior. The magnitudeand scale of the simulated patch is in agreement with the measurements of patch effects on metalsurfaces [Darling et al., 1992; Rossi and Opat, 1992]. See figure 6 for a plot of the calculated ion densitydistribution at 30 K and overlap temperature dependence. The effect of both positive and negative patchwas investigated. As expected, the effect of attractive potential distortion is more pronounced, since theions are accumulated in the deeper potential minimum outside of the beam at low temperatures.

Although a quantitative agreement with the measured data is not achieved, the magnitude of the

ROUCKA ET AL.: 22-POLE IMPERFECTIONS

+10mV DC -10mV DC-10mV DC

0 20 40 60 80 100 120 140 160 [K]

0.0

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1.0

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rela

tive

over

lap

+

observed overlap scalingsimulation: ideal trapsimulation: -0.01 V patchsimulation: +0.01 V patch

Figure 6. Influence of positive and negative patch fields. The H− density distribution at 30 K (leftpanel). The cloud-beam overlap as a function of temperature (right panel).

effect is well reproduced by the simulation with realistic input parameters. As the patch effects cannotbe expected to be translationally symmetric, a fully 3D model would be needed to account properly forthe patch distribution and the end-electrode effects.

Conclusion

A discrepancy in the data measured using the 22-pole trap was observed. By analysis of the mea-sured data a correction factor was obtained, which was then used to correct the final results. Severalpossible explanations of the observed effect were proposed. All of the explanations were based on pos-sible imperfections of the 22-pole trap. From the computational simulations of these imperfections weconclude, that the most probable of our explanations is the presence of DC patch fields in the trap. Theratio of overlaps between CO+

2 and H−, which is important for our H density calibration procedure, doesnot differ from the theoretical value by more than 20 % in our models at 30 K. We therefore use 20 %as the error estimate for the H density calibration.

A possible solution to the problem of patch fields is cleaning of the trap described by Gerlich [1992].Alternatively, the effect on H beam calibration can be eliminated by trapping the CO+

2 in the sameeffective potential as a reference measurement with H−.

Acknowledgments. This work is a part of the research plan MSM 0021620834 and grant OC10046financed by the Ministry of Education of the Czech Republic and was partly supported by GACR (202/07/0495,202/08/H057, 205/09/1183, 202/09/0642), by GAUK 25709, GAUK 406011, GAUK 388811 and by COST ActionCM0805 (The Chemical Cosmos). We wish to thank prof. D. Gerlich for providing us with his know-how of iontrapping technique in valuable discussions. The AB-22PT instrument has been developed in Chemnitz in thegroup of prof. D. Gerlich with contributions from S. Schlemmer, G. Borodi, and A. Luca and the help from manyother people. Since 2010 the instrument is operated in the Faculty of Mathematics and Physics of the CharlesUniversity in Prague. We thank the Chemnitz University of Technology and the DFG for lending us the 22-poletrap instrument.

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