Anal. Chem. 1996, 68, 4257-4263
Resonance Ejection from the Paul Trap: A Theoretical Treatment Incorporating a Weak Octapole Field Alexander A. Makarov
HD Technologies Ltd., 95-98 Atlas House, Simonsway, Manchester M22 5PP, U.K.
In response to the growing experimental evidence of the importance of nonlinear phenomena in ion trap operation, a new theoretical model of ion ejection is developed. The pseudopotential well approximation for forced ion oscillations in an ion trap under the conditions of ionmolecule collisions is modified to include octapole perturbations on the quadrupole field. Ion ejection is investigated using the first-order Mitropol’skii asymptotic method for both infinitesimal and finite scan rates. It is shown that the combined action of collisional damping and nonlinearity distorts the resonance curve in such a way that “quenching” of oscillations takes place. As a result, with appropriate excitation and direction of scanning, the amplitude increases as if no damping exists! The main characteristics of the jump are derived as functions of scan rate and used for analytical estimation of mass resolution, mass peak width, and excitation voltage. Satisfactory agreement between calculated and experimental peak widths is demonstrated for the range of scanning rates in excess of 6 orders of magnitude. Although Paul’s ion trap1 forms the heart of thousands of analytical mass spectrometers around the world, and almost 15 years have passed since its commercial introduction, some important aspects of its operation are still not characterized adequately. The most well-known is the observed but unexplained ultrahigh mass resolution in the resonant excitation mode at low scanning rates. Here and below, ultrahigh mass resolution is used to indicate mass resolutions R which exceed the relative accuracy of the ion trap electrode geometry, R > 104-105. First reported in the early 1990s by several experimental groups,2-7 this phenomenon has not been given an adequate quantitative explanation within the framework of the existing theory for ion traps. To explain the observed high mass resolution, the generally used pseudopotential well model8,9 was extended by Goeringer and co(1) Paul, W.; Reinhard, H. P.; Von Zahn, U. Z. Phys. 1958, 152, 143. (2) Kaiser, R. E.; Cooks, R. G.; Stafford G. C.; Syka, J. E. P.; Hemberger, P. H. Int. J. Mass Spectrom. Ion Processes 1991, 106, 79. (3) Schwartz, J. C.; Syka, J. E. P.; Jardine, I. J. Am. Soc. Mass Spectrom. 1991, 2, 198. (4) Goeringer, D. E.; McLuckey, S. A.; Glish, G. L. Proceedings of the 39th ASMS Conference of Mass Spectrometry and Allied Topics, Nashville, TN, May 1924, 1991; p 532. (5) March, R. E., Londry, F. A.; Wells, G. J. Proceedings of the 41st ASMS Conference of Mass Spectrometry and Allied Topics, San Francisco, CA, May 31-June 4, 1993; p 790. (6) Londry, F. A.; Wells, G. J.; March, R. E. Rapid Commun. Mass Spectrom. 1993, 7, 43. (7) Williams, J. D.; Cox, K.; Morand, K. L.; Cooks, R. G.; Julian, R. K.; Kaiser, R. E. Proceedings of the 39th Annual Conference of Mass Spectrometry and Allied Topics, Nashville, TN, May 19-24, 1991; p 1481. S0003-2700(96)00653-1 CCC: $12.00
© 1996 American Chemical Society
workers10 by incorporating a sinusoidal external driving force and viscous damping due to collisions with gas. Within their model, the ion resonance ejection line shape was derived as a function of scan rate and collisional relaxation time. Later, this theory was extended by Arnold et al.11 to account for the amplitude dispersion at ejection. Although these newer theories predict correctly the increase of mass resolution with decreasing scan rate, they still underestimate the experimental results by a factor of >10! This is rather unusual as, for mass spectrometers, experimental mass resolution is normally much lower than the theoretically predicted one. In the light of existing knowledge of ion trap operation, in general, and the resonance excitation mode, in particular, the sought theoretical model should not only provide reasonable quantitative estimations of mass resolution but also address some important puzzles of resonance ejection. For example, why have instruments with the stretched geometry been found to provide higher mass resolution than ideally hyperbolic ones?12 What are the factors producing the observed strong anisotropy of mass resolution for different directions of scanning?13 Why is this anisotropy reversed when an ion trap of compressed geometry is used as opposed to the stretched one?14 Also, why are ultrahigh mass resolution experiments not very sensitive to variations of gas pressure2-7 in ion traps? Although the previous analysis of the experimental results (for example, Doppler effect analogy applied by Williams and coworkers13 and Eiden and co-workers15) already clearly revealed the importance of nonlinearity for increasing mass resolution, no analytical model yet exists, probably due to the complexity of the system. Instead, a number of numerical approaches have been developed.16-24 Numerical simulation is very useful for tracing so-called “black holes” in the stability diagram or predicting (8) Wuerker, R. F.; Shelton, H.; Langmuir R. V. J. Appl. Phys. 1959, 3, 342. (9) Major, F. G.; Dehmelt, H. G. Phys. Rev. 1968, 170, 91. (10) Goeringer, D. E.; Whitten, W. B.; Ramsey, J. M.; McLuckey, S. A.; Glish, G. L. Anal. Chem. 1992, 64, 1434. (11) Arnold, N. S.; Hars, G.; Meuzelaar, H. L. C. J. Am. Soc. Mass Spectrom. 1994, 5, 676. (12) Louris, J. N.; Stafford, G. C., Jr.; Syka, J. E. P.; Taylor, D. Proceedings of the 40th ASMS Conference of Mass Spectrometry and Allied Topics, Washington, DC, May 31-June 5, 1992; p 1003. (13) Williams, J. D.; Cox, K. A.; Cooks, R. G.; McLuckey, S. A.; Hart, K. J.; Goeringer, D. E. Anal. Chem. 1994, 66, 725. (14) Wang, M. Proceedings of the 43rd Annual Conference of Mass Spectrometry and Allied topics, Atlanta, GA, May 21-26, 1995; p 1121. (15) Eiden, G. C.; Barinaga, C. J.; Koppenaal, D. W. J. Am. Soc. Mass Spectrom. In press. (16) Wang, Y.; Franzen, J. Int. J. Mass Spectrom. Ion Processes 1992, 112, 167. (17) Wang, Y.; Franzen, J.; Wanczek, K. P. Int. J. Mass Spectrom. Ion Processes 1993, 124, 125. (18) Wang, Y.; Franzen, J. Int. J. Mass Spectrom. Ion Processes 1994, 132, 155. (19) Franzen, J. Int. J. Mass Spectrom. Ion Processes 1993, 125, 165.
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nonlinear resonances. So far, it has not been used to study relations between experimental variables in the resonant excitation mode, presumably due to the large number of these variables. In this work, it will be shown that, even in the age of computerization, analytical approaches still have surprising power to unravel such relations. The pseudopotential well model10,11 is further developed with the clear intention to avoid any new variables in the model as much as possible. The only new term introduced is the nonlinearity of an ion trap, taken to be octapole distortion of the trapping field and, to a lesser extent, hexapole distortion of the excitation field. The introduction of nonlinearity changes dramatically the physical picture of ion confinement. Analysis of this picture is based essentially on the results of the theory of vibrations and, perhaps unexpectedly, relates the phenomenon of ultrahigh mass resolution to the quenching of oscillations in radio frequency (rf) resonators or the instability of crankshafts.25 These results are applied to ion trap mass spectrometry in pursuit of simple but informative theories. Satisfactory agreement with experimental data is demonstrated for the first time. Implications for improving analytical parameters are discussed.
inside the ion trap is modulated with the angular frequency Ω and may be represented as
Φ(r,z,t) ) A2
V 2 0
r0
(20) Franzen, J. Int. J. Mass Spectrom. Ion Processes 1994, 130, 15. (21) March, R. E.; McMahon, A. W.; Londry, F. A.; Alfred, R. L.; Todd, J. F. J.; Vedel, F. Int. J. Mass Spectrom. Ion Processes 1989, 95, 119. (22) Londry, F. A.; Alfred, R. L.; March, R. E. J. Am. Soc. Mass Spectrom. 1993, 4, 687. (23) Julian, R. K., Jr.; Cooks, R. G. Int. J. Mass Spectrom. Ion Processes 1993, 123, 85. (24) Julian, R. K., Jr.; Nappi, M.; Weil, C.; Cooks, R. G. J. Am. Soc. Mass Spectrom. 1995, 6, 57. (25) (a) Mitropol’skii, Y. A. Problems of the Asymptotic Theory of Nonstationary Vibrations; Israel Program for Scientific Translations: Jerusalem, 1965. (b) Mitropol’skii, Y. A. Problems of the Asymptotic Theory of Nonstationary Vibrations; Daniel Davey: New York, 1965. (26) Beaty, E. C. Phys. Rev. A 1986, 33, 3645. (27) March, R. E.; Hughes R. J. Quadrupole Storage Mass Spectrometry; WileyInterscience: New York, 1989. (28) Landau, L. D.; Lifshitz, E. M. Mechanics; Pergamon Press: Oxford, UK, 1969.
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)]
(
f 4 r2 3 + z - 3z2r2 + r4 2 r2 8 0
(1)
where r,z are cylindrical coordinates, t is time, r0 is the radius of the ion trap, V0(0-p) is the peak amplitude of the trapping voltage, and f ) A4/A2. For certainty, let us assume here and below that f > 0. Expression 1 may be transformed using the ideas of the pseudopotential well approximation.8 This approximation assumes that the ion motion is composed of a small ripple at the angular frequency Ω, superimposed on a simple harmonic motion at much lower frequency ω0. Using classical mechanics,28 it may be shown that, in the rf-only ion trap, the motion of the ions with mass-tocharge ratio m/e along “smoothed” trajectories is equivalent to the motion in the pseudopotential:
∫
1 〈| ∇Φ dt|2〉 ) 2m/e t 3 e A2 2 2 r2 4f 4 3 2 2 V0 z + + 2 z - z r - r4 4 r 4 16 mΩ2 r02 0
Ueff(r,z) ) THEORY 1. Equation of Motion. As was shown by Wang and Franzen,18 the potential distribution in any experimental ion trap may be represented as a sum of an ideal quadrupole field and weak, higher multipole fields introduced by small deviations from the ideal electrode structure. Here and below, weights of multipole components of ith power in potential distribution are represented by dimensionless parameters Ai (i ) 1, 2, 3, 4, ...). In the typical symmetric electrode configurations, only even multipoles (octapole, dodecapole, etc.) differ from zero. Though the asymmetry of injection and detection holes makes this statement not entirely true, arising odd multipoles are small and will not be considered in the first-order theory below. Calculation of parameters A2, A4, A6 for symmetrical ion trap geometries has been performed by Wang and Franzen.18 When excitation voltage is applied across end-caps, odd multipoles (dipole, hexapole, etc.) become really important. In this case, the data of Beaty26 could be used (note the difference in defining the multipoles between this paper and that of Wang and Franzen18). In a first-order approximation, adopted in this paper, only the octapole component is superimposed on the quadrupole field. In the rf-only mode of trap operation,27 the potential distribution
[
cos(Ωt) z2 -
( )[
)]
(
(2)
where 〈...〉 means averaging on the period 2π/Ω, |...| means the modulus of the vector, and ∇Φ is the gradient of the potential. In the resonance excitation mode, an ac voltage of amplitude Vs(0-p) is applied to the each end-cap electrode 180° out of phase, producing an additional excitation field at frequencies close to that of the secular motion. The equation for three-dimensional ion motion in the pseudopotential may be written as
m
d2b r + e∇Ueff ) -e∇Uexc dt2
(3)
where br is the ion radius vector,
Uexc(r,z,t) ) -A1Vs sin(ωt)
[
(
)]
g 3 32 z + z - rz r0 r 3 2 0
(4)
and g ) 2A3/A1. The effects of ion-neutral collisions on ion motion are to be introduced into eq 3 in the same manner as was done by Goeringer et al.10 After substituting eq 1 into eqs 2 and 3 and neglecting terms of fourth and higher powers on z and r, the resulting equation for axial motion becomes
(
)
d2z dz 3 + C + ω02z + γω02 z3 - r2z ) 2 dt 8 dt
[ ( )]
Fs sin(ωt) 1 +
where
3g 2 r2 z 2 r02
(5)
ω0 )
q zΩ
(6a)
2x2
Fs )
e A1Vs m r0
(6b)
qz )
e 4A2V m r 2Ω2 0
(6c)
8f r02
(6d)
γ)
and ω0 is the secular frequency of ion oscillations in the pseudopotential well, qz is the parameter of the Mathieu equation (modified by A2 from the usual definition27), Fs is acceleration caused by excitation, and γ is a parameter of nonlinearity. C is the reduced collision frequency and represents the rate of nonreactive momentum relaxation at near-thermal energies in neutral bath gas:10, 22
radial coordinates. Therefore, the maximum of mass resolution may be achieved. However, this balance is likely to be strongly dependent on the experimental parameters and requires extensive numerical modeling. This factor may account for the reported sensitivity of mass spectra to the starting conditions of the mass scan.31 In this paper, it will be assumed that experimental parameters are already optimized to provide maximum mass resolution. As will be shown below, the main features of resonance excitation in nonlinear ion traps start to show up at fairly large amplitudes of axial oscillations15 after this stabilization already occurred and r/z , 1. At this stage, the role of r- and z-coupling becomes small relative to the z3 term and does not affect the general trend of excitation.25 Mathematically, it means that axial motion becomes effectively decoupled from radial motion and may be described in the first approximation by a dimensionless equation:
d2x dx + x + x3 ) E sin(θ(τ)) + 2η 2 dτ dτ
(8)
where
x
mn p e C≈ m + mn kT 20
m + mn R mmn
(7)
τ ) ω0t )
qzΩ
(9a)
t
2x2
x ) xγz )
2x2f z r0
(9b)
y ) xγr )
2x2f r r0
(9c)
where mn is the mass of the neutral, R is the polarizability of the neutral (in F‚m2), 0 is the absolute permittivity of a vacuum (0 ) 8.854 × 10-12 F/m), k is the Boltzmann constant, and p and T are the pressure and temperature of the collision gas, respectively. Expression 7 may be regarded as a very rough estimation only because it is not expected to be valid for higher qz or higher ion energies characteristic for resonant excitation. However, it will be shown below that, within the proposed model, this uncertainty in C does not influence significantly the final results. Equation 5 contains a nonlinear term on r which results in coupling of the radial and axial motions of the ions. The frequency of radial oscillations is about one-half that of axial oscillations, which may give rise to an internal nonlinear resonance.29 The simplest mechanical system which could be described by similar equations is a pair of pendulums suspended from one filament.30 Though, in the general case, the motion of this system is extremely complex, the initial phase difference between pendulums may be chosen in such a way that nonlinearity drives pendulums out of resonance with each other, and only one pendulum is strongly excited, while the other is preserved in lowexcited steady state.30 In the case of the ion trap, it means that experimental conditions could be chosen in such a way that amplitude of axial oscillations will increase while radial amplitude is stabilized at some steady-state level. This saturation level is small relative to r0, and, much more importantly, it is expected to be practically independent of the initial conditions due to the presence of collisional damping and nonlinear excitation (which automatically restores the saturation value as soon as some energy is lost in collisions). It means that the spread of averaged r2 in eq 5 is much smaller than the initial radial spread of the beam; therefore, the resulting frequency shifts for different ions could be orders of magnitude lower than the relative spread of initial
where the terms are τ, dimensionless time; x, dimensionless axial coordinate; y, dimensionless radial coordinate; E, the dimensionless excitation amplitude; η, the dimensionless coefficient of viscous damping; and ν(τ), the dimensionless excitation frequency. It could be shown that, for most ion traps, η , E, x2 , 1, and therefore asymptotic methods of the theory of vibrations may be applied when ν(τ) is close to 1 and moderate excitation is used (so that E , 1, E , x). In a typical experiment, the rf voltage V0 is scanned to provide scanning of ω0 while the excitation frequency remains constant. However, in eqs 8 and 9 above, the dimensionless oscillation frequency is always 1 while the dimensionless excitation frequency ν is scanned. Therefore, the forward scan of V0 is equivalent to the reverse scan of ν(τ). When the secular oscillation frequency ω0 is scanned at a rate aω (rad/s2) and aω/ ω02 , 1,
(29) Nayfeh, A. H.; Mook, D. T. Nonlinear oscillations; Wiley: New York, 1979. (30) Ovchinnikov, A. A.; Erikhman, N. S. Sov. Phys. Uspekhi 1982, 25, 738.
(31) Londry, F. A.; March, R. E. Int. J. Mass Spectrom. Ion Processes 1995, 144, 87.
E)
η)
Fs
xγ )
2
ω0
e A1Vs 16x2f m q 2Ω2r 2 z
(9d)
0
x
x2 mn p e 1 C ≈ 2 ω0 qzΩ m + mn kT 20
m + mn mmn
R
dθ ω(τ) ) ≡ ν(τ) dτ ω0
Analytical Chemistry, Vol. 68, No. 23, December 1, 1996
(9e)
(9f)
4259
ω0(t) ) ω0i + aωt
(10)
ν(τ) ) νi - bτ
(11)
This implies that
where dimensionless scan rate is given by
b)
aω 2
)
ω0
8 aω qz Ω2
(12)
2
and bτ , 1. Index i denotes the initial value at t ) 0. Alternatively, the dimensionless scan rate b may be expressed in more familiar terms of mass scan rate am (Thomsons/s32) using the relation
am aω ) ω0 M/Z Thomsons32
with M/Z ion specific mass in through absolute mass m and charge e as
M/Z ≡
(13)
being expressed
m/e m0/e0
(14)
where m0 is atomic mass unit (m0 ) 1.66054 × 10-27 kg), e0 is elementary charge (e0 ) 1.602 × 10-19 C). Then,
b)
am 1 2x2 am ) M/Z ω0 qzΩ M/Z
(15)
2. Asymptotic Theory of Nonlinear Ion Trap. Detailed analysis of eq 8 by asymptotic methods of the theory of vibrations was first performed by Mitropol’skii.25 The solution of eq 9 in a first approximation is given by
x ) a cos(θ(τ) + ψ)
(16)
where dimensionless amplitude a and phase ψ should be determined from the following differential equations (variables as defined above):
da E ) -ηa cos ψ dτ 1 + ν(τ) 3 E dψ ) 1 - ν(τ) + a2 + sin ψ dτ 8 (1 + ν(τ))a
}
(17)
Equations 17 are best analyzed in two stages.25 First, the stationary mode of the nonlinear system is considered. Subsequently, this static picture is animated, i.e., attention will be focused on the transition through the resonance. 3. The Stationary Mode. By letting all transition rates in eq 17 tend to zero, i.e., da/dτ f 0, dψ/dτ f 0, b f 0, the dependence of amplitude a on the excitation frequency ν can be obtained as shown in Figure 1, curve MABDCN (the resonance (32) Cooks, R. G.; Rockwood, A. L. Rapid Commun. Mass Spectrom. 1991, 5, 93.
4260
Figure 1. Resonance curve for the nonlinear oscillator in the stationary mode. The dimensionless amplitude of oscillations in the axial direction is plotted against the dimensionless frequency of the excitation. The curve MABDCN shows qualitatively the change in the calculated maximum axial excursions of the ion cloud as the excitation frequency is changed, either from low to high frequency or vice versa. Real systems are unable to follow the curve at B (increasing frequency) or at D (decreasing frequency), so that “jumps” occur at D f A and B f C.
Analytical Chemistry, Vol. 68, No. 23, December 1, 1996
curve). The peculiar shape of the resonance curve is reminiscent of an ocean wave breaking over, in clear contrast with the symmetric shape of the analogous curve for a linear system.11,25 This “turning-over” is caused solely by introduction of the nonlinearity into eq 8. If the damping in the system is sufficiently low, typical one-to-one correspondence of frequency and amplitude disappears in the frequency range from νD to νC. Instead, for each frequency one could count there three values of amplitude. This is just the place where new physics starts. Indeed, in the hypothetical case of infinitely slowly increasing frequency from zero (point M in Figure 1), the amplitude of oscillations gradually increases until it reaches point B. As the frequency is increased beyond B, it cannot follow the curve from B to D in Figure 1. Instead, a sudden quenching of the oscillations occurs, and amplitude falls from aB down to that in point C.25 Further increase of ν results in transition from C toward N. If the frequency ν is, alternatively, decreasing (following the curve from N toward D), then the amplitude gradually increases until point D is reached. Again, as the frequency is bound to decrease, the curve cannot be followed from D to B. Instead, the system jumps to point A, i.e., the amplitude increases without any further decrease in frequency, as if there is no friction or nonlinearity. Therefore, damping induced by collisions with bath gas does not have an effect on the mass resolution of an ion trap. As will be shown below, the main “art” in obtaining ultrahigh mass resolution in the resonance excitation mode is clever choice of E and η in such a way that ions are ejected to the detector (i.e., condition a > adet is satisfied at some moment), while the difference in νD for different ions of the same mass is kept as low as possible. If ions are not ejected, the further decrease of the frequency will result in the gradual decrease of the amplitude along the branch AM of the resonance curve. Analysis of eq 17 in the stationary mode also gives important estimates for some points on the resonance curve:
aB ≈ E/2η
(the amplitude in point B)
(18)
Figure 2. Nonstationary resonance curve in the vicinity of the resonance for different scan rates, E ) 1.1 × 10-3, η ) 3 × 10-5 (corresponding to excitation voltage Vs ≈ 5 V and buffer gas pressure p ≈ 10-3 Torr for the parameters cited in the practical example). Dashed line represents a part of the curve MABDCN from Figure 1. For M/Z ) 614 Th, b ) 5 × 10-6 corresponds approximately to the scan rate of commercial instruments.
9 νD ) νA ≈ 1 + aD2 8 2 1/3 aD ≈ E 3 aA ≈ 2aD
( )
(19)
(coordinates of points A and D; for point D, the equation dν/da ) 0 should be applied in addition to eq 17). These estimations are valid for η , E/a and aD2 , 1, which is generally the case for all practical ion traps. 4. Transition through the Resonance. When frequency ν is changed with finite rate, new nonstationary effects appear.25 Using the exact numerical solution of eq 17 for specific b, the resulting nonstationary resonance curve may be derived (Figure 2). New features of this curve are few small oscillations following the quenching and the small shifts of the frequency of transitions D f A and B f C. The slower the transition, the smaller these shifts are and the higher the frequency of oscillations. Also, smaller excitation is required for the same change of the amplitude at lower scan rates. Actual dependencies of transition parameters have been investigated by numerical integration of eq 17 by the Runge-Kutta method for different scan rates, excitation, and damping. The results of integration for dimensionless frequency of transition νD (i.e., frequency at which amplitude aD is achieved), duration of transition S (in units of τ), and amplitude increase λ (ratio of maximum amplitude amax to aD) are represented in Figure 3 and appear to be virtually independent (within a few percent) of E and damping η for η < 0.05E. The latter condition is typically true for ion traps. All dependencies approach saturation levels at b 10-7. Otherwise, the constant value, λ ≈ 2.98, can be used. Comparison with Experiment. Formulas 30-32 may be exemplified for a commercial Finnigan MAT ion trap with parameters Ω/2π ) 1.1 MHz, r0 ) 0.01 m, and z0 ) 0.00783 m. For given r0 and z0, Figure 13 from the paper of Wang and Franzen18 defines f ) A4/A2 ≈ 0.02. Then, for qz ) 0.43, formula 30 gives minimum mass resolution about 2.11(M/Z) at standard scan rate 5555 Th/s, increasing up to 2.7 × 104(M/Z) at scan rate 0.1 Th/s. Figure 4 provides graphical comparison of experimentally measured mass peak widths5 and those theoretically predicted by formula 31. It clearly shows that, at high scan rates, experimental widths are close to or even lower than those predicted by the model, while at lower scan rates they are increasingly worse than could be expected theoretically, i.e., other factors (higher-order aberrations, space charge, deviations from
Figure 4. Variation of mass peak width as a function of a scanning rate for qz ) 0.43 and M/Z ) 131 Th. Experimental data obtained by March et al.5 for perfluorotributylamine are indicated by ×. Solid line indicates the theoretical curve obtained for qz ) 0.43 and other parameters according to the text.
the pseudopotential well approximation, jitter of electronics, etc.) become increasingly important. Though the presented model agrees satisfactorily with experimental results for qz ) 0.43, the highest values of mass resolution in ion trap experiments have been achieved at qz substantially higher than levels of applicability of pseudopotential well approach (typically, qz < 0.4, refs 8 and 9). The irresistible temptation to apply the model beyond the limits of its applicability is partially justified by accounting for ripple at rf frequency in formulas 3032 and relatively good (within
