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Transport of an ion beam through an octopole guide operating in the R.F.-only mode
95
International Journal of Mass Spectrometry and Ion Processes, 93 (1989) 95-105 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
TRANSPORT OF AN ION BEAM THROUGH OPERATING IN THE R.F.-ONLY MODE
PAOLO TOSI, GIORGIO FONTANA, Dipartimento
STEFANO
AN OCTOPOLE
GUIDE
LONGANO and DAVIDE BASS1
di Fisica, Universitci degIi Studi di Trento, I 38050 Povo, Trento (Italy)
(First received 6 June 1988; in final form 20 March 1989)
ABSTRACT Transport of an ion beam through an octopole guide operating in the r.f.-only mode has been investigated both experimentally and by ion trajectory calculations. The octopole transmission band has been characterized by two limit values of the r.f.-potential. The dependence of these limits on the characteristic energy E (e = iw2rnf 2Rg, where m is ion mass, f is frequency and R, is octopole free radius) has been studied considering different configurations of the ion guide.
INTRODUCTION
The electric octopole operating in the r.f.-only mode has been shown to be a powerful tool for guiding ion beams in the energy range from subthermal energies up to few hundred eV. This technique was originally developed by Teloy and Gerlich in 1974 [l], and after that work octopoles found applications in various fields. Octopole-guided beams have been used for measuring integral cross-sections of several ion/molecule reactions as a function of the relative collision energy [2,3]. Reactions of neutrals with state-selected ions have been studied by (a) octopoles in combination with photoionization methods [4,5] and (b) selected ion flow drift tube (SIFDT) experiments where selection of the ion state was obtained by means of specific reactions which occur in an octopole guide [6]. Recently, octopoles have been used for selective ionization of neutrals by charge-exchange reactions. This method has found interesting applications for the detection of pollutants in air at ppm concentrations [7]. Although octopole guides have been successfully used in several experiments, wider application of these devices has been limited so far by the lack of a systematic description of their transmission properties. How transmission depends on ion beam parameters (mass, longitudinal and transverse energies) and electrical and geometrical conditions is not understood in 0168-1176/89/$03.50
0 1989 Elsevier Science Publishers B.V.
96
detail. This is not surprising because the equations of motion are not linear with respect to the positional coordinates_ Moreover, it is not possible to separate the motion into two orthogonal independent components as is currently done for the special case of electric quadrupole mass selectors. The equations of motion cannot be solved by exact analytical methods [8]. An approximate analytical model describing octopole transmission properties has been developed on the basis of the effective potential approximation [1,2,9]. If the frequency of the r.f.-field is sufficiently high, this model describes the ion motion in the r-f.-potential by means of a static effective potential. Recently, Hagg and Szabo followed an alternative approach [8,10-121. They carried out a numerical simulation of the ion motion in multipolar r.f.-fields. Their results do not cover in detail different operating conditions of octopole guides. Nevertheless, they clearly show that octopoles have superior performances with respect to low-order systems for guiding ion beams. The present work has been stimulated by the measurement of octopole transmission properties in a crossed ion beam-molecular beam apparatus. We developed a classical trajectory calculation program which was used to simulate experimental data. The same program has been used to investigate octopole properties over a wide range of operating conditions. The outline of the paper is the following: first, we recall the main properties of the ion motion in an octopole; we then describe the experimental set-up and present experimental data. Finally, trajectory calculations are presented and discussed.
MOTION
OF IONS THROUGH
AN OCTOPOLE GUIDE
The position of an ion in the octopole field can be described by means of a system of cylindrical coordinates R, 0 and Z, where Z is the position measured along the symmetry axis, and R and 8 represent the position of the ion in the transverse plane. The electric potential experienced by the ion is independent from Z and is given by 4
where R, is the free-radius of the octopole, t is the time, f is the frequency and V, is the peak amplitude of the r.f.-potential applied to each pole. The effective potential approximation (EPA) is based on the classical approach of Landau and Lifshitz [9]. Following this model [1,2] the ion motion is separated in two independent terms: a fast one superimposed on a
97
slow one. Under suitable conditions the slow component of the ion motion may be calculated considering an effective potential: 6
where 4 is the ion charge and E is the characteristic E =
is2mf
energy defined by [2]
(3)
'Ri
The limit of validity of the EPA model has been discussed by Gerlich [2] by means of an R-dependent stability parameter V)given by 2 ?J =
o.75qvo
$ i
/& 0
i
He suggested that proper operation of octopole guides is obtained if the following conditions are satisfied: (1) the turning radius R, must be lower than 0.8 R, so that ions do not hit electrodes because of fast oscillations and (2) 77-C0.3 along the path of the ion. In the simple case of ions entering the octopole along the axis, R, depends on their initial transverse kinetic energy Ej and can be calculated from the effective potential (Eq. 2) using the conservation of energy:
(5) where
Q=$
(6)
The condition R, -K 0.8 R,, rewritten in terms of Q, becomes Q > Qr, with l/2
i
Q,=5.52
i
2
1
(7)
QI_is proportional to the minimum value of V, which is necessary to confine ions. Measurements of QL can be used for evaluating Ei via Eq. 7. However, we note that this estimate critically depends on the evaluation of R,. For example, if we consider R, -c 0.9 R, instead of R, < 0.8 R,, the calculated E: is two times greater. The condition v(R) -e 0.3 must be satisfied in particular for R = R,. Using Eqs. 4-6 we can rewrite it in terms of Q as Q -C Qu with Qn = 0.008;
(8) t
98
Equations 7 and 8 define two limit curves crossing in the point (Ed, Q,) with &a= 78.1 EI and Q, = 0.6. For E > e0 the condition QL < Q < Qu defines the EPA domain in the E-Q plane. The extension of Eqs. 7 and 8 to the more general case of an ion created at a distance Ri from the octopole axis, for example by photoionization or ion/molecule reactions, is not straightforward. If Ri is lower than 0.8 R,, Eq. 7 becomes l/2
8E;
QI_= i
40.86-(R,/RJ) 1
(9)
The extension of Eq. 8 is more critical. In particular, it is not clear whether the condition 71-C0.3 [2] is strictly valid only for ions with Ri= 0,or has a general validity. In the latter case we obtain
If Ri is sufficiently small (say Ri -=c R,/2), Eqs. 7 and 9 give the same result within a few per cent. On the contrary, a major difference is found when Eq. 10 is used for calculating Qu. In fact, Qu does not increase linearly with E as in the case of Ri= 0,but at large E (E x=-~a) reaches a constant value given by
A more complicated situation occurs when an ion beam is injected into the octopole through a finite-size collimator. In this case, ions can assume all values of R ifrom 0 up to R,,where R, is the collimator radius. An estimate of Qn for which all ions in the beam satisfy the stability conditions, at large E, can be obtained by Eq. 11 with Ri= R,.Of course, ions entering the octopole close to the axis will satisfy the stability condition also for values of
Q’Qw
An alternative approach for estimating the performance of octopole guides was followed by Hagg and Szabo [8,10-121. They calculated ion trajectories in an octopole field by direct numerical integration of exact equations of motion [8,12]. In ref. 8 the ion initial transverse velocity was neglected and several stability diagrams were presented for different values of initial conditions. Hagg and Szabo used in their calculations a system parameter q4 which contains all the terms related to the operating conditions of the octopole. Taking into account that the definition of the r.f.-potential amplitude used in [8] differs from our Eq. 1 by a factor 2 it is
99
easy to verify that q4 = Q. The results of [8] show that the ion motion is stable for values of Q lower than about 50. A more realistic condition was considered in [12], where the distribution of El and the effect of the input collimator were taken into account. Several calculations were carried out. It is not possible to summarize these results by means of a simple parameter, but certain qualitative considerations may be drawn. The limit of Q = 50 does not guarantee 100% transmission even at relatively high masses (see fig. 7c of [12]). Moreover, the effect of the finite size of the entrance hole is certainly important: at E = 38 eV and Q = 7 the transmission of the octopole decreases to 95% when the radius of the entrance hole is chosen equal to 0.3 R,. Unfortunately, calculations reported in [12] have been limited to a few specific cases and their results cannot be extended to more general operating conditions of octopole guides. EXPERIMENTAL
APPARATUS
AND RESULTS
The experimental apparatus is schematically shown in Fig. 1. The whole system is contained into a differentially pumped vacuum chamber with a base pressure lower than lo-’ mbar. Ions are produced by means of an electron bombardment source S and the ion beam is extracted by means of a system of electrostatic lenses Ll. The ion energy is 9 eV. The ion beam is mass selected by means of a first quadrupole Ql which operates at the frequency of 2 MHz with a mass resolution of about 100 [13]. Ions enter the octopole 0 passing through two collimation holes (lenses L2): the radius of the holes R, is 0.2 cm. The octopole is 8 cm long and its free-radius R, is 0.42 cm. The hyperbolic poles are approximated, as usual, by cylindrical rods. The ion beam may be crossed at the center of the octopole by a supersonic molecular beam, not used in this experiment. Ions are then extracted by means of the lens system L3 (radius 0.22 cm) and injected into a second quadrupole 42. Here the mass distribution of the beam may be analyzed to study ion/molecule reactions which may occur in the octopole
L, v-1
I
Ql
Lz
II-II-II=
0
L3
Q2 A
IL-J II~l1~i1~
Fig. 1. Schematic view of the experimental apparatus: S, ion source; Ll, L2, L3, electrostatic lenses; Ql, 42, quadrupoles; 0, octopole; D, detector; Pl, P2, vacuum pumps.
100 120
I 2.5 MHz
I OO
1 10 MHz
I
I
100
200
I
300
v, (Volt)
Fig. 2. Transmission of an At-+ beam through the octopole as a function of the r.f.-amplitude V,, at two different frequencies (2.5 and 10 MHz). Signals (full lines) are independently normalized, assigning the value of 100 to the maximum. Dashed lines represent results of a computer simulation carried out for both frequencies, assuming an MB distribution for the initial transverse energy with (Ei) = 50 meV. The dash-dot line represents calculations for 10 MHz, with (E,‘) =170 meV.
region. After the second quadrupole ions are detected either by an electron multiplier or by a Faraday cup (D). The octopole guide is supplied using two home-made r.f.-generators operating at the frequency of 2.5 and 10 MHz respectively. Their amplitude is controlled by an external d-c. voltage and stabilized by means of a suitable feedback system. Care is taken to avoid r.f.-imbalance potentials between different rods caused by the presence of uncontrolled parasitic capacitances: this problem may be particularly serious when high-frequency measurements are carried out. The transmission through the octopole guide has been measured for four masses (4, 14, 28 and 40 u) at both the operating frequencies of the octopole, as a function of the r.f.-potential amplitude V,. We show in Fig. 2 an example of experimental data obtained with an Ar+ beam at the two different frequencies. Signals have been independently normalized assigning a full-scale value of 100 to the top of the signal. To define the operating range of the guide we have measured, for each combination of mass and frequency, two extreme voltages V, and Vu which correspond to an attenuation of the signal of 5% with respect to the maximum. At 10 MHz it was not possible to determine V, for masses higher than 14, because this value was higher than the maximum output of our r.f.-generator (about 300 V). Experimental data are presented in Fig. 3, where qVH/.c and qVJ& are plotted as a function of the characteristic energy E. They show, as expected, the existence of a well-defined transmission window. When V, < V, the
101 10
Q
1
0
0.1
I
I
10
100 8
1000
bV)
Fig. 3. Measured octopole transmission limits (95%) are shown in the e-Q plane, with Q = q&/e and E characteristic energy. Asterisks represent the upper transmission limit V0 = V,, and squares correspond to the low transmission limit V0= V,.
octopole field is too weak and ions are lost because of their initial transverse energy. On the other hand, when the octopole field becomes too high (V, > Vu) ions are lost because their motion is strongly perturbed. This interpretation is confirmed by numerical simulation of the ion motion in our octopole. ION TRAJECTORY
CALCULATIONS
Classical trajectory calculations have been demonstrated to be a powerful tool for studying ion motion in electric and magnetic fields. As stated before, this method has been applied to the specific case of octopoles by Hagg and Szabo [8,10-121. Following a similar approach we have integrated the equation of motion d2c
dv2
- f
cos(g,-(p,)p=o
(12)
where cp= 27rft, cp,, is the initial phase, 5 = (X + 1’Y)/R, with X and Y ion transverse coordinates and 5 the complex conjugate of 5. The integration routine uses the Adams method [14]. We ran groups of 1000 trajectories, with Monte Carlo selection of initial conditions. We assumed that the beam is uniformly distributed on the input collimator and that the initial transverse energy is given by a Maxwell-Boltzmann (MB) distribution. This last assumption is somewhat arbitrary. On the other hand, we do not have a measurement of the actual transverse energy distribution of the ion beam. In our preliminary calculations, octopole parameters and the ion longitudinal energy were the same as the experimental values. A qualitative
102
agreement between experimental data and calculations has been found for an average initial transverse energy (Et) = 50 meV, as shown in Fig. 2. In the same figure we show results of calculations carried out with (El) = 170 meV. As expected, they clearly indicate that an increase of (Ef) reduces the transmission at low r.f.-fields. In the EPA frame this fact is expressed by means of Eq. 9. We have already discussed above the limitations to the practical application of this equation, because of the problem of the evaluation of R,. Moreover, Ervin and Armentrout [15] reported a significant difference between the actual value of (Ef) and the value obtained by an EPA analysis of their experimental data. To investigate this problem we have extended our calculations, for different experimental configurations. We have calculated transmission curves (transmission as a function of V,) for different values of the characteristic energy E and of both initial (R,) and final (R,) collimator radii. In all the calculations the octopole length was set to 8 cm, the ion longitudinal energy was 9 eV and the initial transverse ion energy was assumed to follow an MB distribution with ( Eti) = 50 meV. From each transmission curve we obtain a QL value which corresponds to the low corner (95%) of the transmission curve. The corresponding EPA estimation of Q,_ has been obtained by Eq. 9 with two values of Ri (0.1 R, and R,/2) and EI = 150 meV. The last value has been considered taking into account that for an MB distribution about 95% of ions satisfy the condition 0 < El -C 3( E;‘). Figure 4 shows that, as discussed above, the EPA estimate of Q,_ depends very weakly on R,. Similar behaviour is found for simulation results when the radius of the final collimator R f is equal to R, (squares and triangles in
0.05 *
0
10
100 E
1000
(elf)
Fig. 4. Q,_ calculated as a function of E for different input (R,) and final (R f)collimator radii. Squares: R, = R,/2, R, = R,; triangles:R, = 0.1 R,, R, = R,; full circles: R, = 0.1 R,, R, = R,/2; asterisks: R, = R,/2, R, = R,/2. Solid line: QL calculated by Eq. 9, with Ri = R,/2; dashed line: Q,_ calculated by Eq. 9, with Ri = 0.1 R,.
103
Fig. 4). They can be interpolated by a line which is nearly parallel to the EPA curve. Two main conclusions can be drawn: (1) the octopole transmission at low r.f.-fields, i.e. in the left corner of the transmission window, is slightly affected by the size of the input collimator (at least up to R,/2), and (2) the minimum trapping potential V,_ scales as a function of E as predicted by the EPA model. This means that the effective potential approximation gives a good qualitative description of the trapping effect. The quantitative discrepancy between the EPA line and simulation data is a consequence of the particular and somewhat arbitrary choice R, = 0.8 R,. In Fig. 4 we show also simulation results obtained considering a final collimator radius R, = R,/2. This condition corresponds to an experimental configuration where collection of ions at the end of the octopole is limited by the acceptance radius of the next stage, typically a mass analyzer. Calculations show that 95% transmission can still be obtained applying a higher potential (full circles and asterisks in Fig. 4) than in the case of free output (squares and triangles). It is interesting to observe that the effect of the final collimator is to shift-up the QL line without modifying its behaviour as a function of E. In this case, however, an analysis of Q,_ by means of Eq. 9 is meaningless and indeed gives an estimate of (El) much higher than the actual value. If the octopole is used as a reaction cell, the determination of QL is complicated by the simultaneous presence of ions with different masses and transverse energies. To achieve a complete confinement, V, must satisfy the condition V, > Vi_ for all the ions. An obvious question is: what is the maximum acceptable value of V,? In the frame of EPA, Eq. 10 defines an upper limit for V, which at the same time ensures both full transmission and trajectory stability, i.e. transmission through an infinite-length octopole. On the other hand, our experimental data seem to indicate that at low E the whole ion beam is transmitted also for values of Q higher than the upper limit calculated by Eq. 11 (Q = 1.6 for our experimental configuration). By trajectory calculations we can estimate Qn in the same way as was previously done for Q,_. Results are shown in Fig. 5, where we plot calculated values of Q,_ (asterisks) and Qu (triangles), assuming R, = 0.5 R,, R, = R, and (Ei) = 50 meV. In the same figure we show for comparison the corresponding EPA domain. From these results we can conclude that the calculated transmission window in the E-Q plane is larger than the EPA one. We may wonder whether the above results are general or are related to the limited number of r.f.-cycles experienced by ions in the octopole, i.e. are only valid for “sufficiently short” guides. Figure 6 shows results of a simulation carried out for N+ at 2.5 MHz (E = 19.7 ev) assuming an MB transverse energy distribution with (El) = 50 meV. Calculations have been
,
I
10
100
E
. 1000
(eV)
Fig. 5. Calculated transmission window in the E-Q plane assuming Ri = R,/2, R f= R o and (Ei) = 50 meV. The octopole length is 8 cm and the ion longitudinal kinetic energy is 9 eV. Asterisks represent calculated value of Qr, triangles calculated values of Qn. Full circles represent maximum values of Q so that the average ion transverse kinetic energy perturbation is less than 10%. The shaded area corresponds to the EPA domain calculated by Eqs. 9 and 10, with Ej =lSO meV and Ri = R,.
carried out for two lengths of the octopole (8 and 130 cm) and with a longitudinal ion energy of 9 eV. In these conditions 95% of ions fulfil the limit set by Eq. 10 if the voltage V, is less than about 17 V. It is clear from Fig. 6 that ion transmission close to 100% is obtained also for values of V, well over the above limit. When the octopole length is increased by a factor 16 or, equivalently, the length is unchanged but the ion kinetic energy is reduced by a factor 256 (to about 35 mev), a small reduction of ion transmission is observed only for V, > 40 V. For lower values no significant
‘0
40 I
80 I v,
120 I
160 1
200
(Volt)
Fig. 6. Computed transmission for N+ at 2.5 MHz, (El) = 50 meV, input collimator radius R, = 0.1 R,, output collimator radius R, = R,. Squares: 8 cm long octopole; asterisks: 130 cm long octopole.
105
differences are observed. We can then conclude that in real, finite octopoles, also at very low ion energies, we can obtain 100% transmission with operating parameters outside the EPA limits. However, we emphasize that we have discussed only transmission and so far we have not considered the effects of the r.f.-field on the transverse kinetic energy. By trajectory calculations it is possible to evaluate the average transverse kinetic energy of the ion beam (E:) at the end of the octopole. In general, we observe that for a given E, (E:) increases with Q. As an example, the line interpolating full circles in Fig. 5 represents the upper limit of Q which at the same time guarantees full transmission and avoids an energy perturbation greater than 10%. In other words, if the octopole operating point falls over this line the ion beam is transmitted but its average transverse kinetic energy is significantly increased. To our knowledge a general analysis of this problem has not yet been published. Work is in progress in our laboratory. REFERENCES 1 E. Teloy and D. Gerlich, Chem. Phys., 4 (1974) 417. 2 D. Gerlich, in D.C. Lorentz, W.E. Meyerhof and J.R. Peterson (Eds.), Electronic and Atomic Collisions, Elsevier, Amsterdam, 1986, p. 541. 3 P.B. Armentrout, in P. Ausloos and S.G. Lias (Eds.), Structure/Reactivity and Thermochemistry of Ions, NATO AS1 Series, D. Reidel, Dordrecht, 1987, p. 97. 4 S.L. Anderson, F.A. Houle, D. Gerlich and Y.T. Lee, J. Chem. Phys., 77 (1982) 748. 5 J.D. Shao and C.Y. Ng, J. Chem. Phys., 84 (1986) 4317. 6 H. Villinger, J.F. Futrell, A. Saxer, R. Richter and W. Lindinger, J. Chem. Phys., 80 (1984) 2543. 7 W. Lindinger, private communication. 8 C. Hagg and I. Szabo, Int. J. Mass Spectrom. Ion Processes, 73 (1986) 277. 9 L.D. Landau and E.M. Lifshitz, Mechanics, 3rd edn, Oxford University Press, New York, 1976, p. 93. 10 I. Szabo, Int. J. Mass Spectrom. Ion Processes, 73 (1986) 197. 11 C. HIgg and I. Szabo, Int. J. Mass Spectrom. Ion Processes, 73 (1986) 237. 12 C. Hagg and I. Szabo, Int. J. Mass Spectrom. Ion Processes, 73 (1986) 295. 13 Mass filters are supplied by Ionentechnik, Innsbruck, Austria; quadrupole power supplies are QM130 by RIAL, Parma, Italy. 14 LSODE package, Lawrence Livermore Radiation Laboratory. 15 K.M. Ervin and P.B. Armentrout, J. Chem. Phys., 83 (1985) 166.
International Journal of Mass Spectrometry and Ion Processes, 93 (1989) 95-105 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
TRANSPORT OF AN ION BEAM THROUGH OPERATING IN THE R.F.-ONLY MODE
PAOLO TOSI, GIORGIO FONTANA, Dipartimento
STEFANO
AN OCTOPOLE
GUIDE
LONGANO and DAVIDE BASS1
di Fisica, Universitci degIi Studi di Trento, I 38050 Povo, Trento (Italy)
(First received 6 June 1988; in final form 20 March 1989)
ABSTRACT Transport of an ion beam through an octopole guide operating in the r.f.-only mode has been investigated both experimentally and by ion trajectory calculations. The octopole transmission band has been characterized by two limit values of the r.f.-potential. The dependence of these limits on the characteristic energy E (e = iw2rnf 2Rg, where m is ion mass, f is frequency and R, is octopole free radius) has been studied considering different configurations of the ion guide.
INTRODUCTION
The electric octopole operating in the r.f.-only mode has been shown to be a powerful tool for guiding ion beams in the energy range from subthermal energies up to few hundred eV. This technique was originally developed by Teloy and Gerlich in 1974 [l], and after that work octopoles found applications in various fields. Octopole-guided beams have been used for measuring integral cross-sections of several ion/molecule reactions as a function of the relative collision energy [2,3]. Reactions of neutrals with state-selected ions have been studied by (a) octopoles in combination with photoionization methods [4,5] and (b) selected ion flow drift tube (SIFDT) experiments where selection of the ion state was obtained by means of specific reactions which occur in an octopole guide [6]. Recently, octopoles have been used for selective ionization of neutrals by charge-exchange reactions. This method has found interesting applications for the detection of pollutants in air at ppm concentrations [7]. Although octopole guides have been successfully used in several experiments, wider application of these devices has been limited so far by the lack of a systematic description of their transmission properties. How transmission depends on ion beam parameters (mass, longitudinal and transverse energies) and electrical and geometrical conditions is not understood in 0168-1176/89/$03.50
0 1989 Elsevier Science Publishers B.V.
96
detail. This is not surprising because the equations of motion are not linear with respect to the positional coordinates_ Moreover, it is not possible to separate the motion into two orthogonal independent components as is currently done for the special case of electric quadrupole mass selectors. The equations of motion cannot be solved by exact analytical methods [8]. An approximate analytical model describing octopole transmission properties has been developed on the basis of the effective potential approximation [1,2,9]. If the frequency of the r.f.-field is sufficiently high, this model describes the ion motion in the r-f.-potential by means of a static effective potential. Recently, Hagg and Szabo followed an alternative approach [8,10-121. They carried out a numerical simulation of the ion motion in multipolar r.f.-fields. Their results do not cover in detail different operating conditions of octopole guides. Nevertheless, they clearly show that octopoles have superior performances with respect to low-order systems for guiding ion beams. The present work has been stimulated by the measurement of octopole transmission properties in a crossed ion beam-molecular beam apparatus. We developed a classical trajectory calculation program which was used to simulate experimental data. The same program has been used to investigate octopole properties over a wide range of operating conditions. The outline of the paper is the following: first, we recall the main properties of the ion motion in an octopole; we then describe the experimental set-up and present experimental data. Finally, trajectory calculations are presented and discussed.
MOTION
OF IONS THROUGH
AN OCTOPOLE GUIDE
The position of an ion in the octopole field can be described by means of a system of cylindrical coordinates R, 0 and Z, where Z is the position measured along the symmetry axis, and R and 8 represent the position of the ion in the transverse plane. The electric potential experienced by the ion is independent from Z and is given by 4
where R, is the free-radius of the octopole, t is the time, f is the frequency and V, is the peak amplitude of the r.f.-potential applied to each pole. The effective potential approximation (EPA) is based on the classical approach of Landau and Lifshitz [9]. Following this model [1,2] the ion motion is separated in two independent terms: a fast one superimposed on a
97
slow one. Under suitable conditions the slow component of the ion motion may be calculated considering an effective potential: 6
where 4 is the ion charge and E is the characteristic E =
is2mf
energy defined by [2]
(3)
'Ri
The limit of validity of the EPA model has been discussed by Gerlich [2] by means of an R-dependent stability parameter V)given by 2 ?J =
o.75qvo
$ i
/& 0
i
He suggested that proper operation of octopole guides is obtained if the following conditions are satisfied: (1) the turning radius R, must be lower than 0.8 R, so that ions do not hit electrodes because of fast oscillations and (2) 77-C0.3 along the path of the ion. In the simple case of ions entering the octopole along the axis, R, depends on their initial transverse kinetic energy Ej and can be calculated from the effective potential (Eq. 2) using the conservation of energy:
(5) where
Q=$
(6)
The condition R, -K 0.8 R,, rewritten in terms of Q, becomes Q > Qr, with l/2
i
Q,=5.52
i
2
1
(7)
QI_is proportional to the minimum value of V, which is necessary to confine ions. Measurements of QL can be used for evaluating Ei via Eq. 7. However, we note that this estimate critically depends on the evaluation of R,. For example, if we consider R, -c 0.9 R, instead of R, < 0.8 R,, the calculated E: is two times greater. The condition v(R) -e 0.3 must be satisfied in particular for R = R,. Using Eqs. 4-6 we can rewrite it in terms of Q as Q -C Qu with Qn = 0.008;
(8) t
98
Equations 7 and 8 define two limit curves crossing in the point (Ed, Q,) with &a= 78.1 EI and Q, = 0.6. For E > e0 the condition QL < Q < Qu defines the EPA domain in the E-Q plane. The extension of Eqs. 7 and 8 to the more general case of an ion created at a distance Ri from the octopole axis, for example by photoionization or ion/molecule reactions, is not straightforward. If Ri is lower than 0.8 R,, Eq. 7 becomes l/2
8E;
QI_= i
40.86-(R,/RJ) 1
(9)
The extension of Eq. 8 is more critical. In particular, it is not clear whether the condition 71-C0.3 [2] is strictly valid only for ions with Ri= 0,or has a general validity. In the latter case we obtain
If Ri is sufficiently small (say Ri -=c R,/2), Eqs. 7 and 9 give the same result within a few per cent. On the contrary, a major difference is found when Eq. 10 is used for calculating Qu. In fact, Qu does not increase linearly with E as in the case of Ri= 0,but at large E (E x=-~a) reaches a constant value given by
A more complicated situation occurs when an ion beam is injected into the octopole through a finite-size collimator. In this case, ions can assume all values of R ifrom 0 up to R,,where R, is the collimator radius. An estimate of Qn for which all ions in the beam satisfy the stability conditions, at large E, can be obtained by Eq. 11 with Ri= R,.Of course, ions entering the octopole close to the axis will satisfy the stability condition also for values of
Q’Qw
An alternative approach for estimating the performance of octopole guides was followed by Hagg and Szabo [8,10-121. They calculated ion trajectories in an octopole field by direct numerical integration of exact equations of motion [8,12]. In ref. 8 the ion initial transverse velocity was neglected and several stability diagrams were presented for different values of initial conditions. Hagg and Szabo used in their calculations a system parameter q4 which contains all the terms related to the operating conditions of the octopole. Taking into account that the definition of the r.f.-potential amplitude used in [8] differs from our Eq. 1 by a factor 2 it is
99
easy to verify that q4 = Q. The results of [8] show that the ion motion is stable for values of Q lower than about 50. A more realistic condition was considered in [12], where the distribution of El and the effect of the input collimator were taken into account. Several calculations were carried out. It is not possible to summarize these results by means of a simple parameter, but certain qualitative considerations may be drawn. The limit of Q = 50 does not guarantee 100% transmission even at relatively high masses (see fig. 7c of [12]). Moreover, the effect of the finite size of the entrance hole is certainly important: at E = 38 eV and Q = 7 the transmission of the octopole decreases to 95% when the radius of the entrance hole is chosen equal to 0.3 R,. Unfortunately, calculations reported in [12] have been limited to a few specific cases and their results cannot be extended to more general operating conditions of octopole guides. EXPERIMENTAL
APPARATUS
AND RESULTS
The experimental apparatus is schematically shown in Fig. 1. The whole system is contained into a differentially pumped vacuum chamber with a base pressure lower than lo-’ mbar. Ions are produced by means of an electron bombardment source S and the ion beam is extracted by means of a system of electrostatic lenses Ll. The ion energy is 9 eV. The ion beam is mass selected by means of a first quadrupole Ql which operates at the frequency of 2 MHz with a mass resolution of about 100 [13]. Ions enter the octopole 0 passing through two collimation holes (lenses L2): the radius of the holes R, is 0.2 cm. The octopole is 8 cm long and its free-radius R, is 0.42 cm. The hyperbolic poles are approximated, as usual, by cylindrical rods. The ion beam may be crossed at the center of the octopole by a supersonic molecular beam, not used in this experiment. Ions are then extracted by means of the lens system L3 (radius 0.22 cm) and injected into a second quadrupole 42. Here the mass distribution of the beam may be analyzed to study ion/molecule reactions which may occur in the octopole
L, v-1
I
Ql
Lz
II-II-II=
0
L3
Q2 A
IL-J II~l1~i1~
Fig. 1. Schematic view of the experimental apparatus: S, ion source; Ll, L2, L3, electrostatic lenses; Ql, 42, quadrupoles; 0, octopole; D, detector; Pl, P2, vacuum pumps.
100 120
I 2.5 MHz
I OO
1 10 MHz
I
I
100
200
I
300
v, (Volt)
Fig. 2. Transmission of an At-+ beam through the octopole as a function of the r.f.-amplitude V,, at two different frequencies (2.5 and 10 MHz). Signals (full lines) are independently normalized, assigning the value of 100 to the maximum. Dashed lines represent results of a computer simulation carried out for both frequencies, assuming an MB distribution for the initial transverse energy with (Ei) = 50 meV. The dash-dot line represents calculations for 10 MHz, with (E,‘) =170 meV.
region. After the second quadrupole ions are detected either by an electron multiplier or by a Faraday cup (D). The octopole guide is supplied using two home-made r.f.-generators operating at the frequency of 2.5 and 10 MHz respectively. Their amplitude is controlled by an external d-c. voltage and stabilized by means of a suitable feedback system. Care is taken to avoid r.f.-imbalance potentials between different rods caused by the presence of uncontrolled parasitic capacitances: this problem may be particularly serious when high-frequency measurements are carried out. The transmission through the octopole guide has been measured for four masses (4, 14, 28 and 40 u) at both the operating frequencies of the octopole, as a function of the r.f.-potential amplitude V,. We show in Fig. 2 an example of experimental data obtained with an Ar+ beam at the two different frequencies. Signals have been independently normalized assigning a full-scale value of 100 to the top of the signal. To define the operating range of the guide we have measured, for each combination of mass and frequency, two extreme voltages V, and Vu which correspond to an attenuation of the signal of 5% with respect to the maximum. At 10 MHz it was not possible to determine V, for masses higher than 14, because this value was higher than the maximum output of our r.f.-generator (about 300 V). Experimental data are presented in Fig. 3, where qVH/.c and qVJ& are plotted as a function of the characteristic energy E. They show, as expected, the existence of a well-defined transmission window. When V, < V, the
101 10
Q
1
0
0.1
I
I
10
100 8
1000
bV)
Fig. 3. Measured octopole transmission limits (95%) are shown in the e-Q plane, with Q = q&/e and E characteristic energy. Asterisks represent the upper transmission limit V0 = V,, and squares correspond to the low transmission limit V0= V,.
octopole field is too weak and ions are lost because of their initial transverse energy. On the other hand, when the octopole field becomes too high (V, > Vu) ions are lost because their motion is strongly perturbed. This interpretation is confirmed by numerical simulation of the ion motion in our octopole. ION TRAJECTORY
CALCULATIONS
Classical trajectory calculations have been demonstrated to be a powerful tool for studying ion motion in electric and magnetic fields. As stated before, this method has been applied to the specific case of octopoles by Hagg and Szabo [8,10-121. Following a similar approach we have integrated the equation of motion d2c
dv2
- f
cos(g,-(p,)p=o
(12)
where cp= 27rft, cp,, is the initial phase, 5 = (X + 1’Y)/R, with X and Y ion transverse coordinates and 5 the complex conjugate of 5. The integration routine uses the Adams method [14]. We ran groups of 1000 trajectories, with Monte Carlo selection of initial conditions. We assumed that the beam is uniformly distributed on the input collimator and that the initial transverse energy is given by a Maxwell-Boltzmann (MB) distribution. This last assumption is somewhat arbitrary. On the other hand, we do not have a measurement of the actual transverse energy distribution of the ion beam. In our preliminary calculations, octopole parameters and the ion longitudinal energy were the same as the experimental values. A qualitative
102
agreement between experimental data and calculations has been found for an average initial transverse energy (Et) = 50 meV, as shown in Fig. 2. In the same figure we show results of calculations carried out with (El) = 170 meV. As expected, they clearly indicate that an increase of (Ef) reduces the transmission at low r.f.-fields. In the EPA frame this fact is expressed by means of Eq. 9. We have already discussed above the limitations to the practical application of this equation, because of the problem of the evaluation of R,. Moreover, Ervin and Armentrout [15] reported a significant difference between the actual value of (Ef) and the value obtained by an EPA analysis of their experimental data. To investigate this problem we have extended our calculations, for different experimental configurations. We have calculated transmission curves (transmission as a function of V,) for different values of the characteristic energy E and of both initial (R,) and final (R,) collimator radii. In all the calculations the octopole length was set to 8 cm, the ion longitudinal energy was 9 eV and the initial transverse ion energy was assumed to follow an MB distribution with ( Eti) = 50 meV. From each transmission curve we obtain a QL value which corresponds to the low corner (95%) of the transmission curve. The corresponding EPA estimation of Q,_ has been obtained by Eq. 9 with two values of Ri (0.1 R, and R,/2) and EI = 150 meV. The last value has been considered taking into account that for an MB distribution about 95% of ions satisfy the condition 0 < El -C 3( E;‘). Figure 4 shows that, as discussed above, the EPA estimate of Q,_ depends very weakly on R,. Similar behaviour is found for simulation results when the radius of the final collimator R f is equal to R, (squares and triangles in
0.05 *
0
10
100 E
1000
(elf)
Fig. 4. Q,_ calculated as a function of E for different input (R,) and final (R f)collimator radii. Squares: R, = R,/2, R, = R,; triangles:R, = 0.1 R,, R, = R,; full circles: R, = 0.1 R,, R, = R,/2; asterisks: R, = R,/2, R, = R,/2. Solid line: QL calculated by Eq. 9, with Ri = R,/2; dashed line: Q,_ calculated by Eq. 9, with Ri = 0.1 R,.
103
Fig. 4). They can be interpolated by a line which is nearly parallel to the EPA curve. Two main conclusions can be drawn: (1) the octopole transmission at low r.f.-fields, i.e. in the left corner of the transmission window, is slightly affected by the size of the input collimator (at least up to R,/2), and (2) the minimum trapping potential V,_ scales as a function of E as predicted by the EPA model. This means that the effective potential approximation gives a good qualitative description of the trapping effect. The quantitative discrepancy between the EPA line and simulation data is a consequence of the particular and somewhat arbitrary choice R, = 0.8 R,. In Fig. 4 we show also simulation results obtained considering a final collimator radius R, = R,/2. This condition corresponds to an experimental configuration where collection of ions at the end of the octopole is limited by the acceptance radius of the next stage, typically a mass analyzer. Calculations show that 95% transmission can still be obtained applying a higher potential (full circles and asterisks in Fig. 4) than in the case of free output (squares and triangles). It is interesting to observe that the effect of the final collimator is to shift-up the QL line without modifying its behaviour as a function of E. In this case, however, an analysis of Q,_ by means of Eq. 9 is meaningless and indeed gives an estimate of (El) much higher than the actual value. If the octopole is used as a reaction cell, the determination of QL is complicated by the simultaneous presence of ions with different masses and transverse energies. To achieve a complete confinement, V, must satisfy the condition V, > Vi_ for all the ions. An obvious question is: what is the maximum acceptable value of V,? In the frame of EPA, Eq. 10 defines an upper limit for V, which at the same time ensures both full transmission and trajectory stability, i.e. transmission through an infinite-length octopole. On the other hand, our experimental data seem to indicate that at low E the whole ion beam is transmitted also for values of Q higher than the upper limit calculated by Eq. 11 (Q = 1.6 for our experimental configuration). By trajectory calculations we can estimate Qn in the same way as was previously done for Q,_. Results are shown in Fig. 5, where we plot calculated values of Q,_ (asterisks) and Qu (triangles), assuming R, = 0.5 R,, R, = R, and (Ei) = 50 meV. In the same figure we show for comparison the corresponding EPA domain. From these results we can conclude that the calculated transmission window in the E-Q plane is larger than the EPA one. We may wonder whether the above results are general or are related to the limited number of r.f.-cycles experienced by ions in the octopole, i.e. are only valid for “sufficiently short” guides. Figure 6 shows results of a simulation carried out for N+ at 2.5 MHz (E = 19.7 ev) assuming an MB transverse energy distribution with (El) = 50 meV. Calculations have been
,
I
10
100
E
. 1000
(eV)
Fig. 5. Calculated transmission window in the E-Q plane assuming Ri = R,/2, R f= R o and (Ei) = 50 meV. The octopole length is 8 cm and the ion longitudinal kinetic energy is 9 eV. Asterisks represent calculated value of Qr, triangles calculated values of Qn. Full circles represent maximum values of Q so that the average ion transverse kinetic energy perturbation is less than 10%. The shaded area corresponds to the EPA domain calculated by Eqs. 9 and 10, with Ej =lSO meV and Ri = R,.
carried out for two lengths of the octopole (8 and 130 cm) and with a longitudinal ion energy of 9 eV. In these conditions 95% of ions fulfil the limit set by Eq. 10 if the voltage V, is less than about 17 V. It is clear from Fig. 6 that ion transmission close to 100% is obtained also for values of V, well over the above limit. When the octopole length is increased by a factor 16 or, equivalently, the length is unchanged but the ion kinetic energy is reduced by a factor 256 (to about 35 mev), a small reduction of ion transmission is observed only for V, > 40 V. For lower values no significant
‘0
40 I
80 I v,
120 I
160 1
200
(Volt)
Fig. 6. Computed transmission for N+ at 2.5 MHz, (El) = 50 meV, input collimator radius R, = 0.1 R,, output collimator radius R, = R,. Squares: 8 cm long octopole; asterisks: 130 cm long octopole.
105
differences are observed. We can then conclude that in real, finite octopoles, also at very low ion energies, we can obtain 100% transmission with operating parameters outside the EPA limits. However, we emphasize that we have discussed only transmission and so far we have not considered the effects of the r.f.-field on the transverse kinetic energy. By trajectory calculations it is possible to evaluate the average transverse kinetic energy of the ion beam (E:) at the end of the octopole. In general, we observe that for a given E, (E:) increases with Q. As an example, the line interpolating full circles in Fig. 5 represents the upper limit of Q which at the same time guarantees full transmission and avoids an energy perturbation greater than 10%. In other words, if the octopole operating point falls over this line the ion beam is transmitted but its average transverse kinetic energy is significantly increased. To our knowledge a general analysis of this problem has not yet been published. Work is in progress in our laboratory. REFERENCES 1 E. Teloy and D. Gerlich, Chem. Phys., 4 (1974) 417. 2 D. Gerlich, in D.C. Lorentz, W.E. Meyerhof and J.R. Peterson (Eds.), Electronic and Atomic Collisions, Elsevier, Amsterdam, 1986, p. 541. 3 P.B. Armentrout, in P. Ausloos and S.G. Lias (Eds.), Structure/Reactivity and Thermochemistry of Ions, NATO AS1 Series, D. Reidel, Dordrecht, 1987, p. 97. 4 S.L. Anderson, F.A. Houle, D. Gerlich and Y.T. Lee, J. Chem. Phys., 77 (1982) 748. 5 J.D. Shao and C.Y. Ng, J. Chem. Phys., 84 (1986) 4317. 6 H. Villinger, J.F. Futrell, A. Saxer, R. Richter and W. Lindinger, J. Chem. Phys., 80 (1984) 2543. 7 W. Lindinger, private communication. 8 C. Hagg and I. Szabo, Int. J. Mass Spectrom. Ion Processes, 73 (1986) 277. 9 L.D. Landau and E.M. Lifshitz, Mechanics, 3rd edn, Oxford University Press, New York, 1976, p. 93. 10 I. Szabo, Int. J. Mass Spectrom. Ion Processes, 73 (1986) 197. 11 C. HIgg and I. Szabo, Int. J. Mass Spectrom. Ion Processes, 73 (1986) 237. 12 C. Hagg and I. Szabo, Int. J. Mass Spectrom. Ion Processes, 73 (1986) 295. 13 Mass filters are supplied by Ionentechnik, Innsbruck, Austria; quadrupole power supplies are QM130 by RIAL, Parma, Italy. 14 LSODE package, Lawrence Livermore Radiation Laboratory. 15 K.M. Ervin and P.B. Armentrout, J. Chem. Phys., 83 (1985) 166.