Stable ion beam transport through periodic electrostatic structures: linear and non-linear effects

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1 Available online at hsics rocedia rocedia 001 (008) (008) roceedings of the Seventh International Conference on Charged article Otics Stable ion beam transort through eriodic electrostatic structures: linear and non-linear effects Anatoli Verentchikov, Aleander Berdnikov, Mikhail avor * Institute for Analtical Instrumentation, Rizhskij r. 6, St. etersburg, Russia Received Elsevier 9 Jul use 008; onl: received Received indate revised here; form revised 9 Jul date here; 008; acceted date 9 Jul here 008 Abstract The subject of the aer is investigation of stable charged article beam confinement in eriodic electrostatic fields. General conditions of linear stabilit are derived. It is shown that there are similarities between beam confinement in eriodic electrostatic fields and in radiofreuenc uadruolar fields: zone structure of stabilit conditions and a ossibilit to describe the charged article confinement in terms of a seudootential. Shown is the eistence of non-linear resonances. 008 Elsevier B.V. Oen access under CC B-C-D license. ACS: ; 41.0.Cv Kewords: Electron and ion otics; eriodic electrostatic fields; Stabilit; seudootential; on-linear resonances. 1. Introduction The roert of eriodic electromagnetic fields to kee a charged article beam stabl confined is routinel used in charge article accelerators and storage rings where ions or electrons make millions of turns [1]. In such devices where article energies are ver high and the beam hase sace is relativel low, stabilit condition is well described in the linear aroimation []. owadas the interest in stable confinement of charged article beams arises also in low energ devices such as multi-reflection time-of-flight analzers [3-6] and electrostatic tras. Here the hase sace volumes occuied b article beams are much larger (as comared to the sizes of the electrode configurations) and functionalit of the devices can be rovided if the stabilit is retained in resence of non-linear aberrations and eternal erturbations (for eamle, arasitic magnetic fields). Otical conditions for a stable charged article beam transort through a seuence of eriodic mirror-smmetric electromagnetic cells in linear aroimation fields with two lanes of smmetr can be found in Ref. []. In Section of the resent aer the linear model is etended to the general case without such smmetr. Sections 3 and 4 draw attention to a certain analog between the stabilit of charged article motion in dnamic radiofreuenc fields and in electrostatic fields. In Sections 5 we discuss some eculiar features of non-linear ion beam confinement in eriodic electrostatic sstems. These features can be useful for roer design of ion otical devices * Corresonding author. Tel.: address: avor@mail.rcom.ru doi: /j.hro

2 88 A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) utilizing eriodic electrostatic fields. Section 6 demonstrates that the linearl stable motion remains stable in resence of essential non-linear effects and eternal erturbations. reliminar analsis of eamles of non-linear stabilit was given in Refs. [7,8].. Linear model eriodic electrostatic transort sstem consists of a seuence of a large number of identical cells. Such cells can consist of electrostatic lenses like shown in Fig. 1, or alternativel can comrise ion mirrors [3,6] or sector fields [4]. The trajector at the inut of the cell is characterized b lateral coordinates, and momenta,. ote that in such reresentation the conclusions made below remain valid for magnetic sstems as well if ewtonian momenta are substituted b Hamiltonian momenta: = + ( e c) A, = + ( e c) A, where A = ( A, A, Az ) is the magnetic vector otential. Sstem cell d Fig. 1. eriodic sstem consisting of D Einzel lenses. In the articular eamle, the eriod is d = 80 mm with lens electrode length 15 mm, inter-electrode ga 3 mm and electrode otential 430 V. Shown are euiotential lines, samle ion trajectories and direction of focusing and defocusing forces. In linear aroimation the transformation of the inut arameters into outut arameters erformed b a cell is described b the transfer matri = = T. (1) If the sstem ossesses two lanes of smmetr, in linear aroimation the movement is searated into indeendent movements along O and O: = = TO, = = T. () O When the article asses through cells, the total transformation is given b the -th ower of individual matrices: = T, = T, = T. (3) O O Hence, the stabilit of eriodical movement in linear aroimation is rovided if the absolute values of the eigenvalues of the matrices T, T, T O O do not eceed 1, otherwise at some initial conditions the coordinates of trajectories with certain initial conditions grow infinitel.

3 A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) The coefficients of transformation matrices satisf the smlectic conditions secific for Hamiltonian sstems [9]. In general case these conditions are = 0, = 0, = 0, = 0, = 1, = 1, which in case of fields with two lanes of smmetr reduce to = 1, (5) = 1. Consider first the case of smmetric fields. Due to smlectic conditions the eigenvalues of the characteristic iϕ olnomial of the individual matri are e + iϕ and e : λ = λ λ = λ = ( + ) λ + ( ) ( + ) λ + 1 ( λ cosϕ i sinϕ)( λ cosϕ + i sinϕ) = λ cosϕ λ + 1. The linear stabilit imlies the following condition for the coefficients of the transformation matri: + + (7) In this case there eists a value ϕ such that + = cosϕ. Hence we can reresent the matri coefficients in the form = cosϕ + M sinϕ, = cosϕ M sinϕ, where M is a constant. Due to the smlectic condition we have ( ) sin = M ϕ and thus = L M sinϕ, = ( 1 L) M sinϕ where L is another constant. Finall, we come to the reresentation cosϕ + M sinϕ L M sinϕ T =, (8) O 1 M sinϕ cosϕ M sinϕ L and it can be easil shown that ( T ) O cos ϕ + M sin ϕ = 1 M sin ϕ L L M sin ϕ cos ϕ M sin ϕ which makes clear that the trajector coordinates remain restricted after assing through arbitrar number of cells. Moreover, the hase sace oint (, ) alwas belongs to an invariant hase sace ellise described arametricall like (4) (6) (9) 1 = 0 M 0M sinτ + 0 cosτ L (10) = cosτ + 0 ( 0M 0L M ) sinτ

4 90 A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) Similar relations are valid for ais O with roer selection of constants K, W and θ : cosθ + W sinθ K W sinθ cos θ + W sin θ = 1 TO = 1 W sinθ cosθ W sinθ W sin θ K K 1 = 0 W 0W sinτ + 0 cosτ K. = cosτ + W K W sin T O, ( ) 0 ( 0 0 ) τ K W sin θ, cos θ W sin θ ote that though formall limiting cases + = + and + = reresent stable transformation, in realit this stabilit is destroed b small variations of the otical sstem. The general case of four-dimensional transformation can be considered similarl. It can be shown that due to smlectic conditions the characteristic olnomial has a smmetric form with + λ A = + + +, B = + λ λ λ = λ Aλ + Bλ Aλ + 1 (11) For linearl stable case the eigenvalues are reresented as the form + iϕ iϕ + iθ iθ ( λ e )( λ e )( λ e )( λ e ) 4 = λ = iϕ e +, iϕ e, iθ e +, 3 ( cosϕ + cosθ ) λ + ( + 4cosϕ cosθ ) λ ( cosϕ + cosθ ) λ + 1 and the stabilit conditions are eressed as: iθ e, the characteristic olnomial has 4 A 4, B 6, B ( A +8) 4, B A, B + A (1) The corresonding stabilit diagram in lane of arameters A and B is shown in Fig.. Fig.. Stabilit diagram corresonding to conditions of E. (1) in the lane of arameters A and B.

5 3. Stabilit zones of charge article motion A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) Consider for simlicit a mirror-smmetric cell, in which case M = 0 in E. (8). The stabilit condition then takes the form 1 < / < 1 (13) with / = cos for some. article beam motion through a seuence of such cells is stable in a certain set of intervals of charged article energies which can be called energ stabilit zones [10]. Indeed, for a ver high article energ a cell erforms onl a weak focusing action so that cos 1. With decreasing the energ of motion the focusing strength of the cell increases. Focusing of initiall arallel beam into the oint at the eit of the cell corresonds to cos = 0. With further increase of the energ cos becomes negative, and finall at cos = 1 the motion becomes instable. However, with further decreasing of the energ an intermediate focus aears inside the cell and in some interval of energies the stabilit condition (13) becomes valid again. Analogousl aear the consecutive stabilit zones. An eamle of the stabilit zones of ion motion in a simle electrostatic tra is given in Fig. 3. ote that stabilit zones with large number become more and more narrow. For this reason, in ractice onl the first and second stabilit zones are used. U = U 0 U = 0 U = U 0 U 0 [KV] rd stabilit zone.40 nd stabilit zone U = 0 (a) Ion trajectories (b) Energ [KeV] 1 st stabilit zone / / (c) Energ [KeV] Energ [KeV] Fig. 3. (a) Simle ion tra as a eriodic electrostatic sstem; (b) energ ranges of stable ion motion as function of tra electrode otential, and (c) deendence of the coefficient cos on the ion energ.

6 9 A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) The zone structure of motion stabilit allows one to design band-ass filters based on eriodic arras of electrostatic fields. An eamle of such a band-ass filter is a twisted uadruole of Ref. [11]. Zone structure of stable charged article motion in electrostatic eriodic sstems is similar to such structure in dnamic radiofreuenc (RF) uadruoles. However, in the latter sstems the role of the energ las the value 1/ = mω r /(4eV), where is a so-called stabilit arameter, m and e are article mass and charge, is the circular freuenc of RF field, V is its amlitude and r is the uadruole aerture radius. Thus in RF uadruoles stabilit zones corresond to certain article mass ranges whereas in eriodic electrostatic sstems the corresond to certain energ ranges. 4. Concet of seudootential in eriodic electrostatic sstems Consider a charge article motion through a eriodic electrostatic lens arra of Fig. 1 in the first stabilit zone at high energies. Then cos = 1 d/f is close to 1, where d is half a length of the cell as shown in Fig. 1 and f is the focal length of one lens. In this case ϕ d / f and after assing through n cells the -coordinate of the article is n = cos ( n d / f ) + a df sin( n d / f ) (14) where a is the tangent of the angle of the trajector with resect to the otic ais. Ignoring a detailed behavior of the article trajector inside the individual cells, we see that this trajector is close to a harmonic one: ( z) coskz + a / k sinkz (15) with the eriod of z = π / k, where k = 1/ df. The harmonic te of the secular (large scale) motion is illustrated in Fig. 4. Fast oscillations Secular motion Fig. 4. Trajector of the charged article with the kinetic energ K = 1 KeV in the eriodic arra of lenses of Fig. 1. The total length of the drawing corresonds to 100 cells. It is seen that the article eeriences small oscillations at the background of the harmonic secular motion. Such a te of the harmonic motion would have observed if the article moved in a field of the uadratic seudootential U ( ) = U 0 / r0 where U 0 is the otential at = r 0. The gradient g = de( ) / d of the seudootential field strength is related to the arameter k b the relation k = ge/(k ), where K is the kinetic energ of the article. Thus the seudootential field strength gradient g = K /(edf ) deends on the kinetic energ of the charged article. ote that the concet of the seudootential of the eriodic electrostatic field is similar to such concet in RF uadruolar fields [1]. In the latter case the amlitude of the seudootential is U 0 = V / 4 with being the stabilit arameter of the RF field and V its amlitude. Thus, in RF field the seudootential deends on the article mass unlike in the case of electrostatic eriodic sstems where the seudootential deends on article energ.

7 5. on-linear stabilit of charged article motion A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) Most interesting, however, are non-linear roerties of eriodic electrostatic sstems. Motion of ions close to the otic ais of the sstem can be considered as linear. With increasing the distance from this ais, the ion motion becomes essentiall non-linear. Though aberrations must have disersed the ion beam, recise numerical simulation shows that surrisingl this beam still remains confined in the vicinit of otic ais if the condition (7) or (1) of linear stabilit is satisfied. This fact has a strict mathematical background derived from the theor of non-linear dnamics of Hamiltonian sstems [13-15] and can be relied on in an eriodic sstem. The detailed analsis of the alication of this theor is outside the scoe of this aer. However, this mathematical theor allows making imortant conclusions concerning the stabilit of non-linear motion in eriodic otical sstems: Linear stable ion motion leads to non-linear stabilit which means that even ion trajectories ver different from linear ones remain stable that is, the stabilit derived from the linear criterion is conserved in resence of essential erturbation of the linear motion. In articular, if the linear stabilit criterion is fulfilled, ion motion remains also stable in resence of relativel strong eternal erturbations b arasitic electromagnetic fields, in case these erturbations retain the Hamiltonian structure of the fields (that is, for eamle, are not stochastic). The non-linear stabilit in an infinitel long eriodic sstem is valid u to infinitel long beam ath length or, in other words, u to infinite time of motion. The stabilit condition can be broken b certain resonance conditions. For eamle, in the ion tra of Fig. 3 the resonances occur when the ratio of freuenc of eriodic motion along the ais of the tra to the freuenc of erendicular secular motion is a rimitive rational fraction like 1:4 or :3. Unfortunatel, although the general theor [13-15] in rincile enables to estimate the range where the nonlinear stabilit is conserved, the deriving of ractical criterion from this comlicated mathematics is difficult and remains subject for the future investigation. 6. Simulation eamles of non-linear stabilit As a first eamle we consider ion motion in a eriodic seuence of D electrostatic lenses as shown in Fig. 1. This motion is essentiall linear onl for ver small values of the comonent K r of the kinetic energ erendicular to the lens ais. The linear stabilit is demonstrated in Fig. 5 and corresonds to almost erfect eriodic focusing of trajectories starting from a oint at the otic ais. Such focusing occurs when the K r < 0.03 ev. For larger values of K r about 0.1 ev geometric aberrations are well noticeable alread after several first cells of the sstem; for eamle, after assing through 8 cells the eriodic focusing is comletel ruined. However, Fig. 5 shows that the amlitude of ion motion remains unchanged after assing through 100 cells. The stable character of article motion retains further on indeendentl of the number of the sstem cells. A similar effect is observed in resence of strong erturbation factors such as an eternal magnetic field. In Fig. 6 shown is stable ion beam motion in resence of the magnetic field with the flu densit B = 300 G. At such field flu densit the ions of the mass 100 amu would be thrown awa from the channel to the lens electrodes alread in the second cell, if the eriodic electrostatic focusing was absent. In resence of this focusing the ions are stabl confined within the channel in site of the fact that the behavior of their trajectories is essentiall non-linear. ote that surel the non-linear stabilit becomes weaker when charged article energies are close to the boundaries of the stabilit diagram. In case of a mirror smmetric cell with two lanes of smmetr the best degree of stabilit is reached when cos = cos = 0.

8 94 A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) Electrostatic lens arra Cell number Linear stabilit for araial ions on-linear stabilit far from otic ais Fig. 5. Stabilit of trajectories of ion with the kinetic energ K = 100 ev in eriodic sstem of Fig. 1, in first (1 8) and last (93 100) cells B = 300 G Fig. 6. Stabilit of trajectories of ions with the mass 100 amu in the sstem of Fig. 5 in resence of eternal magnetic field (with field flu densit B = 300 G) directed erendicularl to the lane of the drawing. The behavior of the article trajector in a eriodic sstem can be described b a hase-sace ortrait, that is a seuence of oints characterizing this trajector after assing through each individual cell at the hase sace lane {, a}. In the linear aroimation these oints are located at an ellise and the hase advance is the same after each cell. In resence of aberrations these oints remain at some uasi-ellitic closed curve, but the hase advance is different in different cells. In Fig. 7,a the just mentioned oints are shown for article motion in the eriodic arra of lenses of Fig. 1. In the linear aroimation (for articles starting under the angle 1 degree with resect to the otic ais) with the article energ K = 100 ev the hase advance is 90 degrees. With a larger starting angle the hase advance deends on this angle and is no more constant. A ver interesting feature of the non-linear motion is the resence of non-linear resonances characterized b some favorite hase advances. In Fig. 7 this case is realized at the starting angle of 11 degrees with the hase advance of = /7. on-linear resonances aear when two conditions are fulfilled simultaneousl. First, the motion is essentiall non-linear so that there is an energ echange between the motions along the otic ais and in the erendicular direction. Second, the eriod of the cell and the eriod of the secular motion in the erendicular direction are roortional or form a simle fractional ratio. The resonance is characterized b the following feature:

9 A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) in site of the fact that the individual hase advances are different from /7, the oints at the hase-sace lot do not fill the ellise-like curve uniforml and randoml, as for starting angles of 7 or 15 degrees in Fig. 7, but tend to kee in the vicinit of the hases n/7 for most numbers n. 15 on-linear transformation for ions far from otic ais on-linear Hamiltonian resonances K = 100 ev α 0 = 1 α 0 = 7 α 0 = α 0 = 15 α [ ] 5 0 (a) [ [mm] ] Linear transformation ( = 90º) for araial ions 0 = 10.5º 0 = 10.9º z (b) 0 = 11.3º Fig. 7. (a) Sets of oints characterizing the trajector of a charged article in the eriodic lens arra of Fig. 1, after assing through consecutive cells of the arra. The article starts from the otic ais under some angle a with resect to this ais. The oints corresonding to the adjacent cells are connected b dotted lines. (b) Ion trajectories corresonding to starting angles around 11 degrees after assing through 100 cells (the ste of initial angles is 0.05 degrees).

10 96 A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) The trajector lots corresonding to initial starting angles close to the resonance condition in the considered case are shown in Fig. 7,b. It is clearl seen that the trajectories in the vicinit of the resonance starting angle are concentrated along one ath. With increasing number of cells, these trajectories will be still concentrated along this ath unlike the non-resonant trajectories whose hases are disersed in the whole hase range (though both resonant and non-resonant trajectories remain stable u to infinite number of cells). ote that the resonance region is uite narrow (about 10.9 ± 0. degrees of starting angles). A similar eamle of a non-linear resonance in a simle electrostatic tra of Fig. 3 is illustrated in Fig. 8. The shown case corresonds to the ratio of the freuencies of the aial (in z-direction) to the erendicular (in - direction) motion of :1. The hase-sace ortrait of the motion consists of the sets of oints concentrated near the hases of 0, 90, 180 and 70 degrees. (t) a z(t) z (a) (b) Fig. 8. (a) A non-linear trajector in the simle electrostatic tra of Fig. 3 and the corresonding lots of aial motion and of the motion in the direction erendicular to the otic ais, showing the ration of motion eriods of :1; (b) the hase-sace ortrait of the non-linear resonance. ote that such simle structure and conditions of resonances eist onl in case the article motion is essentiall two-dimensional. For essentiall 3D motion this structure can no more be described b simle resonance conditions. 7. Conclusion A dnamic effect of stable charged article confinement in the vicinit of otic ais in eriodic electrostatic fields is of imortance for charged article otic sstems with long ath lengths such as multi-reflection time-offlight analzers or electrostatic tras. Stable ion confinement in electrostatic fields is similar to such effect in radiofreuenc uadruole fields. However, characteristics (stabilit zones, secular freuencies etc.) in case of eriodic electrostatic fields are functions of ion energ and not ion mass as in case of a uadruolar radiofreuenc field. The stable confinement arises at an energ higher than some threshold one and becomes weaker at ver high energies. However, at lower energies there also eist narrow stabilit zones. The most relevant feature of the stabilit under consideration is that (in site of growing aberrations) fulfilling a condition of the linear stable ion motion leads to a non-linear stabilit even for trajectories ver different from linear ones. Moreover, if a linear stabilit criterion is fulfilled, the charged article motion remains stable in resence of strong eternal erturbations, that is resists arasitic electromagnetic fields. Though in general ion motion in eriodic sstems is described b uasi-ellitical hase-sace ictures, there eist ecetional cases non-linear Hamiltonian resonances. These resonances are characterized b favorite hases of ions after assing through consecutive cells of the eriodic sstem.

11 A. Verentchikov et al. / hsics rocedia 1 (008) A. Verentchikov et al. / hsics rocedia 00 (008) References [1] E.D. Courant et al., hs. Rev. 88 (195) [] H. Wollnik, Otics of Charged articles, Acad. ress, Orlando, FL (1987). [3] H. Wollnik, A.Casares. Int. J. Mass Sectrom. 7 (003) 17. [4] M. Tooda, D.Okumura, M.Ishihara, I.Katakuse, J. Mass Sectrometr 38 (003) 115. [5] H. Wollnik, A.Casares, D.Radford, M.avor. ucl. Instrum. Meth. A519 (004) 373. [6] A.. Verentchikov, M.I.avor, u.i.hasin, M.A.Gavrik, Technical hsics 75(1) (005) 73. [7] A. Verentchkov, M.avor, Etended Abstract ASMS 003 ( [8] A.. Verenchikov, M.I.avor, autchnoje riborostroenie, 14() (004) 46. [9] M. Berz, IEEE Trans. Electron Dev. ED-35 (1988) 00. [10] A.. Banford, The transort of charged article beams, E. & F.. Soon Ltd., London (1966). [11] H.A. Simonjan, ribor i Technika Ekserimenta, 3 (196) 7. [1] A.V. Tolmachev, H.R.Udseth, R.D.Smith, Anal. Chem. 7 (000) 970. [13] F.M. Arscott, eriodic Differential Euations, Macmillan, ew ork (1964). [14] A. Katok, B.Hasselblatt, Introduction to the Modern Theor of Dnamical Sstems, Cambridge Universit ress (1996). [15]. Glendinning, Stabilit, Instabilit and Chaos: an introduction to the theor of non-linear differential euations, Cambridge Universit ress (1994).