Effect of ion-neutral collision mechanism on the trapped-ion equation of motion: a new mass spectral line shape for high-mass trapped ions

ELSEVI E R

and Ion Processes

International Journal of Mass Spectrometryand Ion Processes 167/168 (1997) 185-193

Effect of ion-neutral collision mechanism on the trapped-ion equation of motion: a new mass spectral line shape for high-mass trapped ions S h e n h e n g Guan a, G u o - Z h o n g Li b, Alan G. Marshall a'* aDepartment of Chemistry, Florida State University, Tallahassee, FL 32310 USA bCenter for Interdisciplinary Magnetic Resonance, National High Magnetic Field Laboratory, Florida State University, 1800 E. Paul Dirac Drive, Tallahassee, FL 32310 USA

Received 6 December 1996; accepted 11 March 1997

Abstract

The decay amplitude envelope of an ICR time-domain signal determines its corresponding Fourier transform mass spectral line shape. The commonly accepted FT-ICR frequency-domain unapodized Lorentzian spectral line shape originates from the Langevin ion-neutral collision model, in which an ion is treated as a point charge that induces an electric dipole moment in a neutral collision partner. The Langevin model provides a good description of reactions of low-energy collisions of low-mass positive ions with neutrals. However, the Langevin model is inappropriate for collisions of high-mass gas-phase biopolymer ions with low-mass neutrals. Here, we examine ion trajectories for both Langevin and hard-sphere ion-neutral collision models. For the Langevin model, collision frequency is independent of ion speed, leading to a linear differential equation of ion motion with a frictional damping term linearly proportional to ion velocity. For the hard-sphere model, collision frequency is proportional to ion speed and the frictional damping term is proportional to the square of ion velocity. We show that the resulting (non-linear) equation of ion motion leads to a non-exponential time-domain ICR signal whose amplitude envelope has the form, l/(l + tSt), in which t5 is a constant. Dispersion-vs-absorption(DISPA) line shape analysis reveals that the 'hard-sphere' spectral line shape resembles that of overlaid narrow and broad Lorentzians. We discuss several important implications of the new 'hard-sphere' line shape for ICR spectral analysis, ICR signal processing, collision-based ion activation, and ion axialization. Finally, in the hard-sphere limit, a non-linear frictional damping term will also apply to ions in a Paul trap. © 1997 Elsevier Science B.V. Keywords: FT-ICR; FTICR; FT-MS; FYMS; Paul trap; Quadrupole; Ion trap; Langevin; Hard-sphere; Relaxation; Line shape

1. Introduction In Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometry (also known as FT-MS) [1-6], cyclotron motion of ions can be detected for a prolonged period, leading to potentially ultrahigh mass resolution, mass * Correspondingauthor.

resolving power, and mass accuracy [7-10]. Mass resolution, Am, in FT-ICR is usually defined as the magnitude-mode peak full width at half-maximum peak height; mass resolving power at mass, m, is then defined as m / z ~ [11]. In most applications, collisional damping of ion cyclotron motion principally limits the duration of a detectable ICR time-domain signal. Until now, the commonly assumed collision

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mechanism has been the Langevin model, which treats an ion as a point charge and a neutral as an electric dipole induced by the point charge [12,13]. According to the Langevin model, ionneutral collision cross-section is inversely proportional to ion speed; ion-neutral collision frequency is thus independent of ion speed and the frictional force term in the ion cyclotron equation of motion is linear in ion velocity [14,15]. (Strictly speaking, ion speed should be defined relative to neutral speed. However, during FT-ICR detection, ion speed is significantly higher than that for neutrals and the average neutral speed may be taken as zero.) Moreover, the Langevin model predicts an exponentially damped detected ICR signal, whose corresponding frequency spectrum has a Lorentzian shape [15-17]. Non-Lorentzian line shape in continuous power absorption ICR spectroscopy has been previously treated rigorously and elegantly by Viehland et al. [18,19]. The Langevin model is not only theoretically elegant, leading to a linear equation of ion motion and yielding analytic expressions for both the time-domain signal and its frequency domain spectrum, but also has been used to interpret experimental results for many ion-molecule reactions as well [20]. For reactions of lowmass ions with low-mass neutrals, modeling an ion as a point charge and a neutral as an induced dipole is appropriate. However, analytical and biological applications of FT-ICR MS continue to progress toward ions of ever-higher mass. Here, we examine ion cyclotron motion subject to a hard-sphere ion-neutral collision model, which we consider to be more realistic than the Langevin model [21] for high-mass ions colliding with low-mass neutrals. In the hard-sphere model, the collision cross-section is independent of ion velocity, but the collision frequency is proportional to ion velocity, leading to a (nonlinear) equation of ion cyclotron motion with a frictional damping term proportional to the square of ion velocity (see below). The resulting timedomain ICR signal amplitude is proportional to

1/(1 + 6t), in which 6 is a constant. Although there is no analytic expression for the magnitude spectrum of such a signal, it can be handled numerically to yield a frequency-domain line shape that is narrower near the center and broader in the wings than a Lorentzian. Dispersion-vsabsorption (DISPA) line shape analysis allows for additional interpretation of the line shape. Finally, we provide several experimental examples to show the validity of the new 'hard-sphere' line shape.

2. Two models for ion cyclotron motional damping FT-ICR experiments are conducted in a Penning ion trap: namely, an approximately axial quadrupolar electrostatic three-dimensional trapping potential in the presence of a strong spatially uniform axial static magnetic field. The electric field in a Penning trap has been treated systematically elsewhere [22]. For present purposes, however, we may neglect the perturbations due to the electrostatic trapping potential, provided that ions are initially compressed into a tight packet at the confer of the trap (e.g. by internal ionization and/or quadrupolar axialization [23]). The classical motion of an ion of mass, m, and charge, q, moving at velocity, v ( = x / + y j + zk), in a magnetic field, B, with ion-neutral collisions represented by a frictional damping force, is described by the modified Lorentz equation mf~ = q(v × B) - m~ vcv

(1)

in which ~ = M / ( M + m ) and M is the mass of a neutral (atom or molecule). ~'c is the ionneutral collision frequency. (~Pc is known as the 'reduced' collision frequency, and can be thought of as the collision frequency an ion would have if it lost all of its non-random lab.frame velocity on each collision [14].) The above equation may be simplified by dividing by m and taking the dot product with v on both

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sides (2)

~.v = q ( v x B).v - ,l~cV.V m

or d(v2)- 2~/ucv2 (3) dt in which v2 =v.v = Ivl 2. We are now in position to study the effects of collisional damping mechanisms on ion motion. 2.1. The ion-induced dipole (Langevin) collision model

The Langevin model treats an ion as a point charge and a neutral molecule as an electric dipole induced by the ion. The classical interaction potential between an ion separated by distance, r, from its induced electric dipole is (in SI units) q2°~ V ( r ) - 47re°r4

(4)

in which t~ is the molecular polarizability in m 3, and e0 is the vacuum dielectric constant. The collision cross-section, tr(v), for the Langevin system may be obtained as [13] _ O(V)lLangevin

q (ot(m+M)~l/21 2eok" mM J v

(5)

and the Langevin collision frequency, (~C)Langevin q ( o l ( m + M ) ' ~ 1/2

(/)C)Langevin= vNtr(V)]Langevin = 2--~0~,

~

J

(6) is therefore independent of ion speed, v. N is the density of neutrals. Eq. (3) is readily solved to yield the commonly accepted exponentially damped ion velocity v(t) = v(0)exp( - r/(PC)Langevint ) = v0e-t/r

(7)

in which v(0) and v(t) are the ion velocity at time zero and time, t, and 1-= 1/(r/(~'C)Langevin ) is the collisional damping lifetime. Because the

187

detected ion cyclotron signal is proportional to ion cyclotron radius [24-26], which is directly proportional to ion cyclotron speed, the detected signal has the same profile as that of Eq. (7). The frequency-domain magnitude spectrum, M(o~), corresponding to the exponentially damped envelope is therefore a Lorentzian

M0 M(r0) = 1 + ((60- ~0)T) 2

(8)

in which M0 is the maximum peak height and o~0 is the ion cyclotron frequency. Although the Langevin model is obviously oversimplified, experimental rate constants for ion-molecule reactions of low molecular weight ions with light neutrals basically agree reasonably well with the theoretical values. Many modifications, such as the frozen rotor approximation by Dugan and Magee [27] and the average dipole orientation (ADO) idea of Su and Bowers [13], have improved the agreement between predicted and experimental rates. Su and Bowers also included the van der Waals attraction potential ( l / r 6) to demonstrate dependence of reaction rate on collision energy. Their theory has since been improved [28,29]. 2.2. The hard-sphere collision model

The Langevin ion-neutral collision model yields a Lorentzian FT-ICR spectrum which is in excellent agreement with experiment for low-mass ions, such as CH~ at 2 x 10 -5 Torr [17]. However, the Langevin model can be expected to fail for ions of sufficiently high mass, even though the ion cyclotron speed at a given cyclotron radius decreases with mass because of the inverse dependence of ion cyclotron frequency on ion mass. For low-mass ions, the ion/induced dipole interaction causes the Langevin cross-section to be larger than the hard-sphere collision cross-section over the range of ion velocities applicable to the FT-ICR experiment. For cases in which the hard-sphere collision cross-section is larger than the Langevin

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cross-section, as for very high-mass ions, the hard-sphere collision rate applies throughout the course of the FT-ICR transient. In intermediate cases, the hard-sphere collision rate applies at large ion radii of motion (high velocity) whereas the ion/induced-dipole interaction becomes collision rate limiting at low velocities [30]. The simplest model that describes contact between a high-mass ion and a neutral would be the 'hard-sphere' model, for which the ionneutral interaction potential is 0

r~r o

¢o

r > ro

(9)

V(r) =

in which r0 is the sum of the characteristic radii of the ion and the neutral. The collision crosssection, O'Hard Sphere, O'Hard Sphere ----7rr02

(10)

is independent of ion speed, and the reduced collision frequency, (vC)HardSphere, (PC)Hard Sphere = VNaHard Sphere

(11)

is proportional to ion speed. Substituting from Eq. (11) into Eq. (3), we immediately obtain d(v 2) dt -

- 2r/NO'Hard

SphereV3

or

dv

-~ = - ?lNaHard Sphere y2 ------ ~h V2

(12)

in which 6h =r/NaHard Sphere- Although the above equation is non-linear, it is easily solved to yield v(t) - v 0 1 + vorht

(13)

The long-term behavior [t---* ~; v(t)--~ 1/(rht)] indicates that ion speed (and thus the timedomain ICR signal) decays slower than exponentially. Unfortunately, there is no analytical expression for the magnitude spectrum of the so-called ' 1/t'-decay signal of Eq. (13). However, the magnitude spectrum may be computed numerically by FFT.

We have treated two limiting cases. However, in general, the collision cross-section contains both hard-sphere and Langevin components O"= O'Langevin + O'Hard Sphere

( l 4)

and the ion equation of motion is more complicated d(v2) - - 2~Vc v2 + - ~lNaHard Sphere V2 dt

(1 5)

Eq. (15) may be solved numerically.

3. Results and discussion 3.1. Time- and frequency-domain ICR profiles f o r Langevin vs hard-sphere collision models

Fig. 1 shows simulated time-domain ICR signal amplitude envelopes (top) and their corresponding frequency-domain magnitude spectra (bottom) for both Langevin and hard-sphere collision models. The hard-sphere time-domain signal initially decreases faster but eventually decreases more slowly than the uniformly (exponentially) damped signal for a Langevin collision model. As a result, the corresponding hardsphere frequency-domain line shape is narrower near its center but broader far from resonance than the Lorentzian line shape resulting from the Langevin model. 3.2. DISPA line shape analysis

A particularly sensitive and revealing measure for non-Lorentzian spectral line shape is a plot of 'dispersion-vs-absorption' (DISPA) [31,32]. 'Dispersion' and 'absorption' refer to the real and imaginary parts of the complex Fourier transform of a time-domain waveform. As described in a recent review [33], a perfectly Lorentzian spectrum yields a perfectly circular DISPA plot, whose diameter is the absorption-mode (or magnitude-mode) spectral peak height. What makes the DISPA plot useful is that various non-Lorentzian peak shapes produce highly

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S. Guan et al./International Journal of Mass Spectrometry and Ion Processes 167/168 (1997) 185-193

Imaginary 0.5 0.4 0.3 0.2 0.1 0 0 0.2

ICR Voltage Signal 1 ~Time-Domain

Profiles

0.8

0.4

0.2

1 + ~t •

0 0

•

•

,

•

2

.

•

,

•

•

4

•

r

•

•

•

6

,

•

•

•

8

,

-

•

10

*

,

•

12

Tlme (ms)

Magnitude Spectrum

i FT of

1_

]1/

1_

, . . . f . . . * . . . i - - . T . . * l . . . ,

v0 - 5 0 0 H z

v0

v0 + 5 0 0 H z

Fig. 1. Time domain profiles (top) and frequency-domain magnitude-mode positive-ion spectra (bottom) of ion cyclotron signals simulated from Langevin and hard-sphere ion-neutral collision mechanisms. The commonly accepted Langevin model predicts an exponentially damped time-domain amplitude and a Lorentzian frequency-domain peak shape. The hard-sphere model gives rise to an initially rapid and slower long-term decrease in time-domain amplitude, and a magnitude spectral peak with a sharp central component and very broad base.

characteristic deviations from the DISPA 'magic' circle• For example, superposition of two or more Lorentzians of different frequency displaces a DISPA curve outside the reference circle whose diameter is the absorption-mode spectral peak height, whereas superposition of two or more Lorentzians of different width displaces the DISPA curve inside the reference circle [34]• Imperfect phasing of the FFT data rotates the DISPA circle about the origin [35], and so on. DISPA plots for Langevin (Lorentzian spectral line shape) and hard-sphere collision models are shown in Fig. 2. As expected, the numerically computed FFT of an exponentially damped

Lanpvln Model

0.4 0.6 Real

0.8

1.0

Fig. 2. Dispersion-vs-absorption (DISPA) plot of imaginary vs real parts of a complex FFT of simulated ICR time-domain signals for each of two collision models. The numerically computed Lorentzian data for the Langevin model fall exactly on a perfect circle (thin curve), whereas the hard-sphere model data deviate far inside the circle (much like a superposition of a narrow and a broad Lorentzian--see text).

sinusoidal time-domain signal produces DISPA data lying on the reference circle. On the other hand, the DISPA curve for the hard-sphere line shape is displaced markedly inside the reference circle, much as would be expected for a superposition of narrow and broad Lorentzians.

3.3. Experimental evidence for hard-sphere FT-ICR mass spectral line shape High-resolution MALDI/FT-ICR experiments were carded out on an Finnigan FTMS 2000 instrument equipped with a dual-trap. The detailed instrumental configuration and procedures have been reported previously [36]. Briefly, ions generated by a UV laser pulse from a N2 laser, were cooled by collisions with Ar gas and then axialized by azimuthal quadrupolar excitation before being transferred to the analyzer trap for detection• The residual Ar pressure in the analyzer was kept constant at 0.5 × 10 -8 Torr. A low trapping potential (0.5 V) limited the number of trapped ions so as to minimize frequency drift and peak shift. Ions were excited by broadband chirp dipolar excitation to a coherent cyclotron radius of ~0.65 cm and detected in heterodyne mode. All data were acquired until signals decayed to the noise level, and zero-filled once before FPT. No apodization was applied•

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Fig. 3 (top) shows the pseudomolecular ion region of a MALDI FT-ICR magnitude-mode positive-ion mass spectrum of angiotensin I. High mass resolution allows for complete separation between adjacent isotopic peaks. Further mass scale expansion (Fig. 3, bottom) clearly confirms the characteristic sharp central peak and very broad base expected for the hard-sphere line shape. When matched to the same peak width at half-maximum peak height, the magnitudemode hard-sphere line shape of Fig. 1 is essentially congruent to the peak in Fig. 3 (bottom). Figs. 4 and 5 show similarly plotted MALDI FT-ICR magnitude-mode negative-ion mass spectra for a fullerene, C70, and a trinucleotide, (pdC)3. The mass scale-expanded segments of both spectra again clearly show the characteristic hard-sphere line shape•

3.4. Absence of prior evidence for hard-sphere line shape Given the marked difference shown here between the spectral line shapes resulting from Langevin and hard-sphere collision models, and given that at sufficiently high ion velocity, the ~100 80 (M*H)i

Anglotensin I

I:1 29! O

1296

[ 1297

"

t "

1298m/z 1299

A

1300

....

J 60!

M-

|00! j4o I

20~ 01,

.

i, .)oo

. . . . . . 1296.75 1296.80 1296.85 1296,90 1296,95 m/z

Fig, 3. Matrix-assisted laser desorption/ionization (MALDI) FTICR positive-ion mass spectrum of the small peptide, angioteinsin 1. Top: quasi-molecular ion, (M + H) ÷, region. Bottom: mass scale-expansion of the monoisotopic species. Mass resolving power, mlAmso~ -~ 200000, in which Ams0~ is the full width at half-maximum magnitude-mode peak height. The base of the peak is clearly visible out to -- 10 line widths, as expected for hard-sphere line shape.

1 L 840

..,

C7o

841 m/z

.. 842

843

100

140 i 2°i 839.96

840.00111 m/z

840.04

Fig. 4. MALDI FT-ICR negative-ion mass spectrum of a fullerene, C 7o. Top: molecular ion, M-, region. Bottom: mass scale-expansion of the monoisotopic species. The characteristic hard-sphere line shape is evident.

hard-sphere collision rate should significantly exceed the Langevin collision rate, it is logical to wonder why the hard-sphere line shape has until now not been reported experimentally. For one thing, early detailed analysis of FT-ICR line shape was conducted at a relatively low magnetic field of - 1 Tesla [17], so that the Langevin model holds even at observable (~1 cm) ion cyclotron orbital post-excitation radius. Second, FT-ICR peak shape for high-mass ions has not been investigated quantitatively. Third and most important, FT-ICR instruments typically employ { 100!80i 6~

[M_H]i

°882

I 240~ oi I;: 0

~100~

.3

884'

Trinucleotlde, [dpC] 3

~s

m/z

.d

~t

.8

00

| 6o4 ! 401

~_ 2 o i o L ....

884.10

_

. . . . . . . .

884,14

m/z

884.18

Fig. 5. Negative-ion MALDI FT-ICR mass spectrum of the trinucleotide, (pdC)3. Top: quasi-molecular ion, (M - H)-, region. Bottom: mass scale-expansion of the monoisotopic species. The characteristic hard-sphere line shape is again evident.

S. Guan et al./lnternational Journal of Mass Spectrometry and Ion Processes 167/168 (1997) 185-193

time-domain 'windowing' (i.e. multiply the timedomain signal by a specific weighting function), and typical weight functions (Blackman-Harris, sine-wave, etc. [11]) suppress the early part of the time-domain data and thus suppress the 'broad' component of the FFT hard-sphere line shape. That is experimentally appropriate if the goal is to flatten the spectral baseline, but is not appropriate if one seeks to extract ion-neutral collision frequency and cross-section from the spectral line shape.

4. Implications of ion cyclotron motional damping models 4.1. ICR spectral line shape

The Lorentzian is the universally assumed unapodized line shape in FT-ICR mass spectrometry [ 17]. Various peak-finding algorithms are based on the Lorentzian shape or its apodized forms. For example, the center of a magnitudemode spectral peak resulting from FFT of a truncated exponentially damped time-domain noiseless ICR signal can be located exactly by fitting the three highest-magnitude spectral data to a magnitude-mode Lorentzian [37]. Alternatively (and more commonly), a Lorentzian (Eq. (8)) can be approximated as a parabola near the peak center M° =M0(1 - r2(co- coo)2 M(w) = 1 + ((co- CO0)T) 2 + 0(4))

(16)

in which 0(4) represents terms of fourth or higher powers of (co - coo). By fitting a few data points near the peak center to a parabola, one can obtain the central frequency, COo,peak height, M0, and the time-domain exponential damping time constant, T. One of the problems in analysis of the l/t decaying time-domain signal is that there is no analytical expression for its magnitude spectrum. Fitting of data points with the model function is

191

thus difficult and may well be a non-linear problem. On the other hand, because of the slow (compared with exponential) long-term decay of the time-domain signal, one may apodize the signal with, e.g., a Gaussian. In that case, the magnitude spectrum will be approximately Gaussian and the peak position and height can be easily extracted. For unapodized time-domain data, frequencydomain absorption-mode amplitude and magnitude (obtained by the linear FT operation) are each still proportional to number of ions [38]. Thus, either peak height or peak area may be used for quantitative analysis. However, the peak base is broadened in the hard-sphere line shape, making it difficult to deconvolve contributions from neighboring peaks. 4.2. Axialization and ion remeasurement

All theoretical treatments of axialization are based on the Langevin collision model, for which the equations of ion motion subject to azimuthal two-dimensional quadrupolar excitation may be solved analytically [23,39]. Unfortunately, no analytical solution for axialization is possible in the hard-sphere collision limit, due to the presence of a non-linear collisional damping term. However, if we consider that the effect of collisions is merely to reduce ion kinetic energy, then the qualitative effects of axialization based on magnetron-to-cyclotron motion conversion remain valid. Our preliminary simulations show little change in the mass-selectivity of axialization. For an ion prepared with an initial cyclotron radius r0 and zero magnetron radius, the damping of cyclotron motion depends strongly on the collision damping mechanism. From the relation between cyclotron radius r and ion speed v (r = v/w, co is cyclotron angular frequency), we have r(t) = roe-t/~ (Langevin model) r(t) -

r0 (Hard sphere model) 1 +wro6ht

(17) (18)

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192

For example, for an axialization experiment conducted in a dual Penning trap, one may estimate the period required for the ion cyclotron radius to decrease to the radius of the conductance limit (re)

tr(t)=rc(Langevin)= r I n ( r°']

(19)

k, r c J

tr(t) rc(Hard Sphere) = wrorh \ rc :

/

'pseudopotential' limit [43,44], (4eV)/(mr2~ 2) < 0.4, in which r0 is the radial distance from the center of the trap to the ring electrode surface, V is the amplitude of the RF voltage applied to the ring electrode, and f~ is the RF frequency, the equation of z-motion reduces to simple harmonic oscillation, and the time- and frequency-domain responses reduce to those obtained here.

(20)

Acknowledgements Note that for the Langevin model, relaxation rate (the reciprocal of the lifetime, r) is proportional to m/q for m much larger than M. In contrast, since ~0 ~ q/m, the hard-sphere relaxation rate varies inversely with mass-to-charge ratio, so that ions of high mass-to-charge ratio relax more slowly back to the center of the trap. Thus, for high-mass ions, higher collision gas pressure may be needed for efficient ion remeasurement [36,40]. Finally, the Langevin relaxation rate is independent of magnetic field, but (since c0 B), the hard-sphere relaxation rate increases directly with increasing magnetic field.

5. Conclusion and future directions The ion trajectory model based on the hardsphere damping mechanism provides a better representation of the motion of high-mass ions routinely encountered in MALDI and electrospray FT-ICR MS experiments. Since the hardsphere unapodized FT-ICR mass spectral line shape is not characterized by a single relaxation parameter, new criteria for mass resolution and mass resolving power may be warranted. Finally, the non-linear velocity-damping term discussed here for ions in a Penning trap will be equally relevant for ions confined in a Paul trap [41]. In that case, the usual Mathieu equation for ion motion [42] becomes a non-linear differential equation without an analytical solution. Thus, not even the time-domain ion trajectory is available analytically. However, in the usual

This work was supported by grants from NSF (CHE-93-22824), the NSF National High Field FT-ICR Mass Spectrometry Facility (CHE-9413008), Florida State University, and the National High Magnetic Field Laboratory in Tallahassee.

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