We have experimentally investigated the behavior of an electron cloud confined in a Penning trap at weak superimposed magnetic fields. Exciting the motional frequencies of the electrons by an external drive field we found the axial mode split into two components which were identified as center-of-mass and individual electron oscillations. When the trapping potential was varied, rapid electron loss appeared at numerous values of the applied voltage. They are determined by the relation n z ω z + n m ω m =ω c . ω z ,ω m ,ω c are the axial, magnetron, and cyclotron frequency of the trapped electrons, respectively. The reason for this loss is attributed to higher order contributions to the ideal quadrupole trapping potential

Ananthamurthy, S.

Angelescu, C.

Biswas, D.

Paasche, P.

Valenzuela, T.

Werth, G.

ePrints@Bangalore University

UNCLASSIFIEDDefense Technical Information CenterCompilation Part NoticeADP012533TITLE: Collective Motional Resonances and Instabilities of an ElectronCloud Stored in a Penning TrapDISTRIBUTION: Approved for public release, distribution unlimitedThis paper is part of the following report:TITLE: Non-Neutral Plasma Physics 4. Workshop on Non-Neutral Plasmas[2001] Held in San Diego, California on 30 July-2 August 2001To order the complete compilation report, use: ADA404831The component part is provided here to allow users access to individually authored sectionsf proceedings, annals, symposia, etc. However, the component should be considered within-he context of the overall compilation report and not as a stand-alone technical report.The following component part numbers comprise the compilation report:ADP012489 thru ADP012577UNCLASSIFIEDCollective Motional Resonances and Instabilitiesof an Electron Cloud Stored in a Penning TrapD. BISWAS1, P. PAASCHE' , T. VALENZUELA', C. ANGELESCUl,S. ANANTHAMURTHY 2 , G. WERTH''Johannes Gutenberg University, Institutafier Physik, D-55099 Mainz, Germany2Bangalore University, Department of Physics, 560056, Bangalore, IndiaAbstract. We have experimentally investigated the behaviour of an electron cloud confined in aPenning trap at weak superimposed magnetic fields. Exciting the motional frequencies of theelectrons by an external drive field we found the axial mode split into two components whichwere identified as center-of-mass and individual electron oscillations. When the trappingpotential was varied, rapid electron loss appeared at numerous values of the applied voltage.They are determined by the relation nto, + n,,o,, = q.0)o, 0 m, qý are the axial, magnetron, andcyclotron frequency of the trapped electrons, respectively. The reason for this loss is attributed tohigher order contributions to the ideal quadrupole trapping potential.IntroductionThe behaviour of an electron cloud confined in Penning traps has been extensivelyinvestigated experimentally [1] and theoretically [2] in the general context of thecharacterization of non-neutral plasmas. In particular the mode structure of themotional oscillations serves to derive properties of the plasma such as density, rotationfrequency or shape. The experiments that investigate these properties were performedunder conditions, where the temperature of the plasma was low (4K) and theconfinement took place far away from the stability limit of the Penning trap. This limitoccurs when to, 2 = 2(oz.,to, = (elfi)B being the free electron cyclotron frequencyand (t, =(2eV /ind2)" 2 the axial oscillation frequency, in a trap of characteristicdimension d. B is the magnetic field strength and V, the voltage applied to the trap. Inour experiment we confine electrons at weak magnetic fields, approaching the limit ofstability. Furthermore the electron temperature is high in the absence of coolingmechanisms.CP606, Non-Neuitral Plasma Physics IV, edited by F. Anderegg et al.© 2002 American Institute of Physics 0-7354-0050-4/02/$19.00360ExperimentWe used a trap with hyperbolic electrodes of characteristic dimension d = 2 cm. Themagnetic field along the z-axis was produced by two Helmholtz coils of 27 cm meandiameter, driven by currents up to 25 A producing a field up to 0.01 T. The trap hasbeen used previously in laser spectroscopic experiments and had holes of several mmdiameter drilled into the ring electrode. These holes together with possiblemisalignments, caused higher order contributions to the trapping potential, which inthe ideal case is a quadrupole potential.The trap was loaded with electrons from a tungsten filament placed several cm belowone endcap electrode in pulses of several ms length. After a preset time which couldbe varied between 10 ms and virtually infinity, detection took place by a tuned circuitconsisting of an inductance and the trap electrodes as capacitance. The circuit wasweakly excited at its resonance frequency o, (10 MHz). When the positive trapvoltage, applied to the ring electrode, is ramped down, the electrons axial oscillationfrequency o), changes. At a certain voltage wo. coincides with co,. Then the electronsabsorb energy from the circuit, which consequently is damped. The r.f. amplitudeacross the circuit is rectified and the corresponding signal is proportional to the storedelectron number. The total number is estimated from observed space charge shifts toabout 105. The method of detecting the stored electron cloud is in principlenondestructive. During the detection process, however, the electrons gain energy fromthe tank circuit. In addition they are ejected from the trap as consequence of higherorder components in the trapping potential as shown below. This results in a totalstorage time of only a few second. Therefore we decided to ramp the trapping voltagedown to negative voltages. This ensures elimination of all electrons from the trap andeach creation-detection cycle starts with a completely empty trap.Axial oscillationWhen we apply an additional r.f. field to one of the electrodes or to the electron gunusing it as antenna and sweep the frequency of this field, the electrons become excitedwhenever it coincides with one of the motional modes. Some of the electrons leave thetrap and the motional resonances appear as minima in the detection signal. Fig. 1*shows such a spectrum which contains the fundamental oscillation frequencies 0'(magnetron), (o, (axial), and w,' (perturbed cyclotron) as well as some linearcombinations or multiple of these frequencies. The number of observed combinationsdepends on the amplitude of the applied r.f. filed. A high resolution scan of O(z or 20),shows, that these frequencies have two components: A broad asymmetric minimumaccompanied on the high frequency side by a narrow more symmetric minimum(Fig.2). Changing the stored ion number shows that the position of the broad minimumis shifted to higher frequency with decreasing electron number while the sharp361minimum does not change its position. When extrapolating both frequencies tovanishing electron number they coincide (Fig. 3). From this we conclude that the broadminimum corresponds to the excitation of the individual electrons in the cloud whichexperience the space charge of the surrounding cloud, while the sharp minimum is thecenter-of-mass mode of the whole ion cloud, which, of course, is independent of spacecharge. This observation is similar to the behavior of a molecular ion cloud stored in aPaul trap [3]. Similarly the appearance of the coherent oscillation of the ion cloud canbe explained as a parametric instability of the axial motion under the influence of theadditional drive field. As outlined in [3] the excitation of the coherent mode requires aminimum threshold excitation amplitude when the ion motion is damped. As maindamping mechanism we consider collisions of the in cloud with rest gas molecules,although our background pressure is of the order of 10-9 mbar. This assumption isjustified by the fact that the threshold amplitude in our experiments is about 0.2 Vwhereas in the corresponding experiment on molecular ions it was of the order of 10V. This change in order of magnitude reflects the difference in collisional cross sectionof electrons or molecular ions with background molecules.o) '-2 (o - (03n 0 2"o (0 '-0) 0 -)'+0=0)2500 1 Il15 15U 2- 1000< 5000 20 40 60 80 100 120 140 160Excitation frequency [MHz]FIGURE 1. Motional spectra of a stored electron cloud in a Penning trap for different amplitudes of ar.f. drive field. Some of the identified resonances are labeled, the others are axial and magnetronsidebands to these frequencies.362Non-collective Collective(individual) (cloud centre of mass1600' 1400O 1200S10000S800o 600S40020067 68 69 70 71 72 73 74 75 76 77Excitation frequency (MHz)FIGURE 2. High resolution scan of the frequency 2w, showing the collective (individual) and non-collective (center-of-mass) components of this mode80 V . 4.S79 Collective7 Non-collective>,780' 7776 0750 1 2 3 4 5 6 7 8 9 10Relative amount of electrons (a.u.)FIGURE 3. Space charge shift of the collective and non-collective resonances. Both frequenciescoincide at vanishing electron numberInstabilities In Electron ConfinementWhen we varied the trapping potential we observed that it was impossible to detectstored electrons at values which are within less than about half of the maximumvoltage allowed for stable confinement. This voltage follows from the stability363criterion for Penning traps , " = 2w02 : V,,,a = (e/4m)d 2B2. Also at distinct lowervalues of V minima in the detected ion number were observed. The strength of theseminima as well as their number increases with the storage time. When the storage timeexceeds several seconds, storage of electrons is virtually impossible. Fig. 4 illustratesthe observation for a given strength of the magnetic field.When we plot our observations for different magnetic fields, keeping the storage timeconstant and calibrating the trapping voltage in percentages of V,,,a,, we find that theinstabilities occur at the same positions. These positions are given by a simple relationbetween the motional frequencies of the electrons:nh (q +l,1 (O`,, = (O`, (1)nz,nm are integers.This relation can be understood in analogy to similar observations in a Paul trap[4,5]: When the trap potential contains multipole components in addition to the idealquadrupole part it can be described by a series expansion:0I(r,O, (p) = V C,, (r/d)2 n PE, (cos0), (2),,=owhere P2, are the Legendre polynomials and C2, are constants. It has been shown forPaul traps [6] that for a potential with nonvanishing coefficients C2,, the ion motionbecomes unstable for operating conditions which fulfill the relation"nrW + n.w. = Q (3)where a) and a. are the radial and axial oscillation frequencies of the ion motion andS2 the driving frequency of the trapping potential. When we consider an electronorbiting in a constant magnetic field of a Penning trap with the cyclotron frequency (0,the constant inhomogeneous electric trapping field acts as a time varying electric fieldin the rest frame of the electron. Thus the cyclotron frequency replaces the drivingfrequency in eqn. (3). The axial frequency remains unchanged and the radial frequencyis replaced by the magnetron frequency yielding eqn. (1). Fig. 4 shows that theobserved instabilities can be well assigned to those operating points as predicted byeqn. (1).ConclusionsWe have observed the simultaneous appearance of individual (non-collective) andcenter-of-mass (collective) resonances in the excitation of the axial oscillation of astored electron cloud in a Penning trap. At operating point determined by a simplerelation between the axial, magnetron and cyclotron oscillation frequencies of theelectrons, the confinement becomes unstable. This is attributed to the existence ofhigher order components in the electric potential of the trap.AcknowledgementsOur experiments were supported by the Deutsche Forschungsgemeinschaft. S.A.acknowledges support from the DST-DAAD Indo-German exchange Program3642Coz+ 2cOm-cOcI 2 m+ C==%2000 11000 +2COw=)I I . .. ..c0 I : * [ IE ~z2i. =10 20 30 40 50 60 70 80 90 1002000Do I .i;;..r... .,.... v ... ..  ..,L........ ,......... W .... ..10 20 30 40 50 60 70 80 90 10010 20 30 40 50 60 70 80 90 1002000 .100010 20 30 40 50 60 70 80 90 100Ring voltage (% of Vmax)FIGURE 4. Observed instabilities of electron confinement and assignement topredictions following eqn (1)ReferencesI. Weimer, C.S., Bollinbger.J.J., Moore,F.L., and Wineland, D.J, Phys. Rexv. A 49, 3842-3852 (1994)2. Dubin, D.H., Phys. Rev. Lett. 66, 2076-2079 (1991)3. Alheit, R. et at., Phys. Rev A 56, 4023-4031 (1997)4. Aiheit, R., Henning,C, Morgenstern, R, Vedel, F. and Werth, G., Appl. Phys.B 61, 277-283 (1995)5. Alheit,R., Kleineidam, S., Vedel, F., Vedel,M., and Werth, G.,Int. J. Mass Spectr. Ion Proc. 154,155-169 (1996)6. Wang, Y., Franzen J., and Wanczek K.P., Int. J. Mass Spectr. Ion Proc. 124, 125-144 (1993)365

Collective motional resonances and instabilities of an electron cloud stored in a Penning trap

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Collective motional resonances and instabilities of an electron cloud stored in a Penning trap

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