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The stored waveform inverse Fourier transform (SWIFT) waveform may be tailored for selective ejection or excitation of ions of arbitrary mass-to-charg...
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A d . Chem. 1993, 65, 1288-1294

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Stored Waveform Inverse Fourier Transform Axial Excitation/ Ejection for Quadrupole Ion Trap Mass Spectrometry Shenheng Guan and Alan G. Marshall’*+ Department of Chemistry, 120 West 18th Avenue, The Ohio State University, Columbus, Ohio 43210

A general method for high-resolution ion excitation, ejection,andisolation is developed from linear response theory for resonant dipolar excitation of the axial z-oscillatory motion of ions in a quadrupole (Paul) ion trap operated in rf-only mode. For a spatially uniform dipolar excitation field, the ion z-oscillation amplitude is directly proportional to the amplitude of the Fourier component of the excitation at the axial oscillation frequency of that ion. Thus, one may specify an arbitrary z-motionfrequency-domainspectrum by applying a time-domain stored waveform obtained from the inverse Fourier transform of the corresponding frequency-domainexcitation spectrum. The stored waveform inverse Fourier transform (SWIFT) waveform may be tailored for selective ejection or excitationof ions of arbitrary mass-to-chargeratio ranges. The method includes all other possible excitation/ejection waveforms (e.g., single frequency, frequency sweep) as special cases. The effect of collisional damping during excitation is included in the analytical solution of the ion response. INTRODUCTION The history, principles, and applications of the quadrupolar Paul ion trap have recently been reviewed.’-3 Most of the current analytical and ion/molecule chemistry applications for the Paul ion trap have been modeled after prior similar Fourier transform ion cyclotron resonance mass spectrometry (FT/ICR/MS) techniques and applications, which have also been the subject of many recent reviews.*-18 In both trap

* Author to whom correspondence may be addressed. Also a member of the Department of Biochemistry. (1)March, R. E.; Hughes, R. J. Quadrupole Storage Mass Spectrometry; Wiley: New York, 1989;pp 471. (2) Cooks, R. G.; Kaiser, R. E., Jr. Acc. Chem. Res. 1990,23,213-219. (3)Todd, J. F. J. Mass Spectrom. Reu. 1991,10, 3-52. (4)Freiser, B. S.Chemtracts: Anal. Phys. Chem. 1989,1 , 65-109. (5)Ghaderi, S.Ceram. Trans. 1989,5,73-86. (6)Gord, J. R.; Freiser, B. S. Anal. Chem. Acta 1989,225,11-24. (7) Sharpe, P.; Richardson, D. E. Coord. Chem. Reu. 1989,93,59-85. (8) Wanczek, K.-P. Znt. J. Mass Spectrom. Zon Processes 1989,95, 1-38. (9) Wilkins, C. L.; Chowdhury, A. K.; Nuwaysir, L. M.; Coates, M. L. Mass Spectrom. Reu. 1989,8,67-92. (10)Freiser, B. S. In Bonding Energetics in Organometallic Compounds; Marks, T. J., Ed.; ACS Symposium American Chemical Society: Washington, DC, 1990;pp 55-69. (11) Laude, D. A., Jr.; Hogan, J. D. T M , Tech. Mess. 1990,57,155-159. (12)Lasers in Mass Spectrometry; Lubman, D. M., Ed.; Oxford University Press: New York, 1990. (13)Nibbering, N. M. M. Acc. Chem. Res. 1990,23,279-285. (14)Campana, J. E. In Proc. SPZE Applied Spectroscopy in Material Science; 1991;pp 138-149. (15)Marshall, A. G.; Grosshans, P. B. Anal. Chem. 1991,63,215A+

229A. ~~. ~

(16)Nuwaysir,L.M.; Wilkins,C. L.Proc.SPZEAppl.Spectrosc.Mater. Sci. 1991,112-123. 0003-2700/93/0365-1288$04.00/0

types, ions are confined by the joint effect of an approximately quadrupolar electrostatic dc potential and a second force provided by either a quadrupolar rf electric potential (Paul trap) or a static magnetic field (Penning or ICR trap). For the Paul ion trap, ion motions may be analyzed into two “normal modes”: axial z-oscillation between the end caps of the ion trap and radial oscillation toward and away from the symmetry axis passing through the centers of the two end caps. In ICR, ions move in three normal modes: axial oscillation as in the Paul trap and two circular “magnetron” and “cyclotron” orbital motions in a plane perpendicular to the magnetic field direction (z-axis). Parallels between the principles of operation of the two kinds of traps will be reported in detail elsewhere. In this paper, we demonstrate the strong correspondence between dipolar excitation of ion cyclotron orbital motion in the ICR ion trap and dipolar excitation of axial z-oscillation in the Paul trap. New capabilities, experiments, and applications for z-excitation and z-ejection in the Paul trap are proposed and discussed. In a quadrupolar Paul ion trap, the high-frequency, highamplitude rf voltage applied between the ring and end-cap electrodes forms a pseudopotential in the z-direction that can be approximated as a quadrupolar electrostatic potential19 and ion motion in such a potential can be modeled as a harmonic oscillator when the corresponding qp-value (see Theory) is small. The pseudopotential model treats ion z-motion as a superposition of motion a t the fundamental rf frequency and a harmonic oscillation at the “secular” frequency of the normal-mode z-oscillation. If the rf “ripple” component of the ion z-motion is neglected, the remaining ion z-motion may be described by a linear differential equation equivalent to a one-dimensional harmonic oscillation in the z-direction. An important tool for ion trap applications is resonant excitation of axial z-motion, generated by a dipolar excitation field (i.e., two excitation voltage signals differing by 180° in phase, applied to the two end-cap electrodes) at the ion “secular”or “natural” z-oscillation frequency corresponding to a given q,-value (see Theory). Resonant excitation of z-motion may be considered as an action that “burns a hole” in the ion stability diagram. By varying either the quadrupolar rf frequency or amplitude, one can effectively move the “hole” along the q,-axis in the stability diagram. A mass spectrum may be obtained by scanning (a) the quadrupolar rf frequency, (b) the quadrupolar rf amplitude, or (c) the dipolar z-excitation frequency to eject ions axially (i.e., increase the ion z-displacement until it exceeds the distance from the center of the trap to either end cap) for external ion detection. Dipolar resonant z-excitation for selective ejection of ions was first used by March and co-workers20s21 for study of ion(17)Marshall, A. G.; Schweikhard, L. Int. J. Moss Spectrom. Ion Processes 1992,1181119, 37-70. (18)Schweikhard, L.;Alber, G. M.; Marshall, A. G. Phys. Scr. 1992 4fi. .., ~,9a-(in2. - - - - - -. (19)Major, F. G.; Dehmelt, H. G. Phys. Reu. 1968,270,91-107. (20)Armitage,M. A.; Fulford, J. E.; Duong, N.-H.; Hughes, R. J.;March, R. E. Can. J . Chem. 1979,57,2108-2113. 0 1993 American Chemlcal Soclety

ANALYTICAL CHEMISTRY, VOL. 65, NO. 9, MAY 1, 1993

molecule reactions. By use of resonant axial ejection, SchwartzZ2and 0 t h e r s ~ ~ have 3 ~ ~extended the mass range of ion trap mass spectrometry from m/z650 to 45 O00; slowing the scan rate has been shown to increase mass resolvingpower (but not necessarilymaas accuracy)to 100 O00 at m / z 1350.24 Broad-band ion isolation and ejection by use of resonant dipolar z-excitation has been demonstrated by McLuckey et al.25 Goeringer et al.26 recently introduced a theory for resonant ejection in which the effects of various instrumental parameters on mass resolving power are analyzed for frequency-swept dipolar z-excitation; much of the experimental and conceptual framework for that work had previously been developed for dipolar frequency-sweep excitation of ion cyclotron orbital motion more than a decade earlier.27bz8 Finally, Julian et al.% have reported the use of stored waveform inverse Fourier transform (SWIFT) e~citation30.3~ in place of the previously conventional frequency-sweep dipolar z-excitation. The equations of ion cyclotron motion in the presence of a uniform dipolar excitation rf electric field are linear, and the response of the cyclotron motion is linear.32,33The linear response is the theoretical foundation of the SWIFT excitation method introduced by Marshall and co-workers.30,31,34-37In the SWIFT method, a desired excitation magnitude spectrum is specified, and its corresponding time-domain waveform is then synthesized by inverse Fourier transformation. The demonstrated advantages of the SWIFT method over all the previous excitation methods include the following: uniform excitation over specified mlz range(@,high excitation-ejection selectivity?’ extension of FTIICR detection dynamic and convenience in use. In the Paul ion trap, the equation of z-motion of ions subjected to dipolar z-excitation is linear under the approximations of a uniform excitation field and pure quadratic pseudopotential. In this paper, we begin by solving analytically the equation for forced ion z-motion; we then develop a linear response theory for dipolar z-excitation in the absence of collisional damping during the excitation event and prove that the postexcitation ion z-oscillation amplitude is proportional to the magnitude spectrum (at the ion secular

-

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frequency) of the time-domain excitation voltage signal and that the postexcitation phase of the ion z-displacement is described by the phase spectrum of the time-domain z-excitation voltage signal. We then present a general algorithm for generation of an excitation waveform which excites ions to any desired z-amplitude over any number of arbitrary mlz ranges, so that ions may be selectively ejected or isolated. The present formalism closely follows that previously developed for excitation of ion cyclotron orbital motion, except that the ion axial motion in the Paul trap is linear, whereas the ion cyclotron orbital motion is circular. Therefore, the SWIFT technique developed previously for ICR cyclotron excitation may be easily adapted to excitation of z-motion of ions in RF ion traps, for high-resolution z-excitation and z-ejection.

THEORY Equation of z-Motion of Ions in an rf Quadrupole Ion Trap. In a Paul or Quistor ion trap, axial motion of an ion of mass, m, and charge, q , is described by the Mathieu equation’

d2z/dgZ - 2q, cos 2gz = 0

(la)

[ = at12

(1b) (IC)

in which Vrf and SZ are the rf voltage amplitude (applied between the end caps and the ring electrode) and angular frequency, respectively, and PO is the radius of the ring electrode at the z = 0 midplane of the trap. Typically (but not necessarily38), r-02 = 2202, in which 220 is the separation between the two end-cap electrodes. The Mothieu equation has been studied in detail by McLachlan.39 Analytical solutions of the equation are not available, and its behavior is understood largely on the basis of a “stability” diagram,’ showing the boundary values of dc and rf voltage at which ion trajectories are stable within the trap. As discussed in (21) Fulford, J. E.; Hoa, D.-N.; Hughes, R. E.; March, R. E.; Bonner, R. F.; Wong, G. J. J . Vac. Sci. Technol. 1980, 17, 829-835. more detail below, we limit our present treatment to rf-only (22) Schwartz, J. C.; Syka, J . E. P.; Jardine, I. J. Am. Soc. Mass operation (i.e., dc trap potential is zero). Major and Dehmelt Spectrom. 1991, 2, 198-204. developed a method called the pseudopotential well approach (23) Kaiser, R. E., Jr.; Cooks, R. G.; Stafford, G. C., Jr.; Syka, J . E. P.; Hemberger, P. H. Int. J. Mass Spectrom. Ion Processes 1991,106, 79that neglects the high-frequency oscillating terms in the 115. (24) Williams,J.D.;Cox,K.A.;Cooks,R. G.;Kaiser,R.E.,Jr.;Schwartz, secular frequency 0scillation.1~When q2 is small (e.g., qr < 0.41,they showed that eq l a may be simplified to J. C. Rapid Commun. Muss Spectrom. 1991, 5, 327-329. ~~

~

~

(25) McLuckey, S. A.; Goeringer, D. E.; Glish, G. L. J.Am. Soc. Mass Spectrom. 1991,2, 11-21. (26) Goeringer, D. E.; Whitten, W. B.; Ramsey, J. M.; McLuckey, S. A,; Glish, G. L. Anal. Chem. 1992, 64, 1434-1439. (27) Comisarow, M. B.; Marshall, A. G. Chem. Phys. Lett. 1974, 26, 489-490. (28) Marshall, A. G.; Roe, D. C. J. Chem. Phys. 1980, 73, 1581-1590. (29) Julian, R. K., Jr.; Cox, K.; Cooks, R. G. In Proceedings, 40th American Society of Mass Spectrometry, Annual Conference on Mass Spectrometry & Allied Topics; Washington, D.C., 1992; pp 943-944. (30) Marshall, A. G.;Wang, T.-C. L.; Ricca, T. L. J.Am. Chem. Soc. 1985, 107, 7893-7897. (31)Marshall, A. G.; Wang, T.-C. L.; Chen, L.; Ricca, T. L. In Fourier Transform Mass Spectrometry; M. V. Buchanan, Ed.; ACS Symposium Series 359; American Chemical Society: Washington, DC, 1987; pp 2133. (32) Guan, S. J. Am. Soc. Mass Spectrom. 1991, 2, 483-486. (33) Grosshans, P. B.; Marshall, A. G. Anal. Chem. 1991, 63, 20572061. (34) Wang, T.-C. L.; Ricca, T. L.; Marshall, A. G. Anal. Chem. 1986, 58, 2935-2938. (35) Chen,L.; Wang,T.C. L.;Ricca,T. L.; Marshal1,A. G. Anal. Chem. 1987,59, 449-454. (36) Chen, L.; Marshall, A. G. Rapid Commun. Mass Spectrom. 1987, 1, 39-42. (37) Chen, L.; Marshall, A. G. Int. J. Mass Spectrom. Ion Processes 1987, 79, 115-125.

d2zldt2+ wZ2z = 0

(2) Equation 2 describes simple harmonic oscillation along the z-axis, at an ’axial” oscillation (secular or natural) frequency, w,

= qrQ/2(2) ‘1’ = ( 2 ) 1 / 2 q ~ r f / m r o 2 ~

(3)

Note that the axial secular frequency, wr, is inversely proportional to the ion mass-to-charge ratio. An axial excitation voltage waveform, f V z ( t ) / 2applied , between the end-cap electrodes generates a dipolar excitation field inside the trap. Near the center of the trap, the excitation field can be considered to be spatially uniform. The effect of ionneutral collisions may be represented by adding to eq 1 a collisional damping force, -72, proportional to the ion z-velocity: (38) Knight, R. D. Int. J. Mass Spectrom. Ion Processes 1983, 51, 127-131. (39) McLachlan, N. W. Theory and Applications of Mathieu Functions; Clarendon: Oxford, England, 1947.

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ANALYTICAL CHEMISTRY, VOL. 65, NO. 9, MAY 1, 1993

Then, eq 7 becomes tl = clq/2zom

(4b)

in which c1 is a geometric factor for the hyperbolic trap (ro2 = 2z0*)typically used in rf and Penning ion traps. A numerical evaluation by a relaxation method yields c1- O A 4 O Equation 4 is a linear differential equation which may be solved in several ways. First, one may solvethe simpler problem of the response to a &function ('impulse") excitation and then take the convolution of that response with an arbitrary (timedomain) excitation waveform.41 Alternatively, one may solve eq 4 directly by Laplace transform methods (see below). Solution of the z-Motion Equation. Let the initial 2-position and z-velocity of the ion be z(0)and i ( O ) , respectively; the Laplace transform of eq 4a then becomes42 S2Z(S)- sz(0) - i ( 0 ) + y[sZ(s) - Z(0)l - (w2)2Z(s)= qV,(s)

e-iw~'L~oteiw2'rV,'*(~) d7) + z ( 0 ) e-?'' cos(w,")

(i(o)/w,')e-f'' sin(w,'t) (9) Response to z-Excitation. It is convenient to specify an excitation period from -T It I0, so that the observation of the ion response takes place from t = 0 onward. The initial conditions [z(O), i(0)l thereby transform to [ z ( - T ) , i(-T)], and eq 9 becomes

e-iw~'L~Teiu,'rVz'*(r) d r ) + e-7'T[z(-T)(cos(w,'T) (y'/w,')

V,(s)

L(V,(t))

F(s) L ( f ( t ) )= ~ o m e -fs(tt ) d t

sin(o,'T))

+ (Z(-T)/w:)

+

sin(w,'T)I e-?'' cos(w,'t)

+

e-f'T[z(-T)((yf/w,') cos(w,'T) - sin(w,'T)) + ( i ( - ~ ) / w , ~COS(W,~T)I ) e-ft sin (w,'t) (10)

(5a) in which Z(S) = L(z(t))

+

(5b)

If the ions have zero initial z-velocity, t(-T) = 0, and zero initial z-displacement, ~ ( - 7 ' )= 0, eq 10 simplifies to

(5c)

z ( w , t ) = ~e~Y"/2iw(eiwtF1(~,t) - e-'"'F1*(w,t)),

Z(s) is the Laplace transform (denoted by the symbol L)of the z-position, z ( t ) , and VJs) is the Laplace transform of the excitation waveform, V,(t). Solving eq 5a for Z(s),we obtain

t > -T ( l l a ) in which w,' has been replaced by w for simplicity in notation, and

and

Fl*(w,t) is the complex conjugate of F,(w,t). If the excitation

has nonzero amplitude only during the period [-T,O], then Fl(w,t>O) = Fl(w,O) is the Fourier transform of V,'(t) (= V,(t)

eft) because the integral limits in eq l l b may be extended to f-.

Applying the inverse Laplace transform to eq 6, we obtain the desired expression for the time-dependent ion z-position, z(t):

M ( o ) and @(u) are the magnitude spectrum and phase spectrum of the product of the dipolar z-excitation voltage waveform, V,(t), and the damping factor, e?,'. We obtain the z-oscillationresponse to that excitation during the observation period, t > 0

= (q/w)M(w)e-?"sin + ut], t > o (12) Because the sinusoidal function in eq 12 determines phase but does not affect amplitude, the postexcitation time-domain amplitude of z-oscillation at excitation frequency, w, is therefore proportional to the magnitude spectrum at that frequency, M(w),of the product of the voltage waveform and the collisional factor, exp(y't): z(w,t)

z(0)[sl es" - s2 eS>']+ [ Z ( O )

+ yz(0)I [e"" - es2"1](7)

In order to simpiify the algebra, we define

z-amplitudeb) = ( t l / w ) M ( w ) (13) Interpretation of eq 13 is much simplified in the limit that the excitation event is much shorter than the collisional relaxation exponential damping time constant, l / y (Le., no collisionsduring the excitation event), so that the z-oscillation (40) Gabrielse, G. Phys. Reu. A. 1984,29, 462. (41) Marshall, A. G.; Verdun, F. R. Fourier Transforms in NMR, Optical, and Mass Spectrometry: A User's Handbook; Elsevier: Amsterdam, 1990. (42) Ross,S. L. Differential Equations, 3rd ed.; John Wiley & Sons: New York. 1984.

ANALYTICAL CHEMISTRY, VOL. 65, NO. 9, MAY 1, 1993

postexcitation amplitude is nonzero only for w = w, (in the collisionless limit) and M(w) is the magnitude spectrum of the excitation voltage waveform alone. Note that 7 is proportional to qlm (eq 4b) and that the ion secular z-oscillation frequency, w, is also approximately proportional to q/m (eq 3). We may combined eqs 3,4b, and 13 to yield the final desired result: z-amplitude(w) = &(w)

(14a)

in which

= C1P02n/(2)”2vrf (14b) is the proportionality constant. Ww) is the initial phase of the ion z-oscillation position immediately following dipolar z-excitation (eq 12). When collisions during the excitation period cannot be neglected, the z-oscillation frequency, w2‘, is no longer equal to the undamped natural frequency, wz (see eq 8b). However, in the (usual) limit that the ion mass is much greater than the mass of the neutral (typically He), the collisional damping constant, y,is inversely proportional to mass-to-charge ratio. Therefore, the z-oscillation frequency, wI‘, is also inversely proportional to m/q. K is still independent of mass-to-charge ratio, although ita value deviates from the undamped value in eq 14b. In the presence of a collision gas, the frequency spectrum of the z-oscillation response is described by a Lorentzian line shape, K

which has a finite spectral peak width, 2y’, compared to zero spectral peak width in the absence of collisional relaxation. Frequency selectivity is thus inherently limited by collisional damping. M(w) is the magnitude spectrum of the product of the excitation voltage waveform and the collisional factor, exp(y’t) (see eqs 8c and 8d). Thus, the effect of collisions is that, in order to attain a given z-oscillation amplitude at the end of the excitation period, higher excitation power is needed to overcomethe collisional energy loss during excitation. The implications of eq 14 will now be discussed.

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FT/ICR/MS30,31,34-3’,44-46 also applies to ion z-excitation1 ejection in an rf-only ion trap. An excitation waveform which achieves an arbitrary ion z-amplitude response profile may be synthesized by inverse Fourier transformation of an arbitrary frequency-domain excitation magnitude spectrum corresponding to the desired mlz ranges over which excitation or ejection is to be generated (see below). The excitationphase spectrum determines only the initial postexcitation phase of the ion z-oscillation amplitude response, which in most cases is unessential. Therefore, the excitation-phase spectrum may be specified so as to reduce dynamic range in the time-domain excitation waveform.35~44~45,47 A general algorithm for generating an optimally uniform amplitude time-domain excitation waveform from a specified excitation magnitude spectrum has previously been developed.32,44t45 The excitation power of the waveform may be distributed uniformly over the excitation period, thereby reducing the time-domain voltage dynamic range.32,35,44+45 In addition, the Gibbs oscillations41normally found in the final magnitude spectrum as a result of truncation of the timedomain waveform may be reduced by interpolation (smoothing) of the sharp boundaries between regions of different excitation magnitude in the excitation magnitude spectrum46 or by apodization of the time-domain SWIFT excitation waveform.47 The SWIFT waveform generation algorithm starts by dividing the mass-to-charge ratio excitation range into bands of different excitation magnitude. Interpolation smoothing is then applied a t each excitation magnitude boundary; the degree of interpolation is limited by the discrete mass resolution required a t the boundary. For example, mass resolution a t the extremes of the mlq range may be less critical than mass resolution near a mass-selectivewindow elsewhere in the mlq range of interest. The required SWIFT excitation frequency spectrum is then generated from the specified excitation magnitude spectrum according to the relation between ion axial z-oscillation frequency and ion mass-tocharge ratio in an rf-only Paul trap (eq 3, with w, replaced by 27rv,):

(16)

RESULTS AND DISCUSSION Algorithm for Generating SWIFT z-Excitation Waveforms. The most obvious and most important feature of eq 14 is that the postexcitation z-oscillation amplitude response to an arbitrary time-domain dipolar excitation waveform is proportional to the forward Fourier transform of the excitation waveform itself. Stated another way, the ion response is independent of ion axial oscillation frequency and thus is independent of ion mass-to-charge ratio, mlq. The extent to which an ion of a given mlq is excited depends only on how much excitation magnitude is applied at that frequency. In this respect, excitation of axial oscillatory motion in an rfonly ion trap is the same as excitation of ion cyclotron orbital motion (which is also frequency-independent and thus independent of mlq) in an ICR ion trap.28.43 Thus, computationally intensive ion trajectory simulations are not required in order to determine the postexcitation ion z-oscillation amplitude. For example, the z-oscillation amplitude as a function of frequency in response to single-frequency excitation is a sinc function,41as in FT/ICR/MS. As another example, the response to a frequency-sweep (‘chirp”) excitation is a Fresnel function,4l as in FT/ICR/MS.26 Most generally, eq 13implies that stored waveform inverse Fourier transform excitation previously developed for (43) Comisarow, M. B. J. Chem. Phys. 1978, 69, 4097-4104.

The magnitude spectrum of the excitation voltage signal is obtained by dividing the z-oscillation amplitude by the (constant)factor K (see eq 14b). The correspondingexcitationphase spectrum may then be calculated by the Guan method.32~~~845 Inverse Fourier transformation of the excitation spectrum (magnitude and phase) then yields the desired time-domain excitation waveform. Example. Suppose that we desire to excite ion z-oscillations with the spectral magnitude shown in Figure l a (collisionless limit): singly charged ions between ml Im I m2are to be ejected (Le., excited to a z-oscillation amplitude larger than the distance from the trap center to either end cap); singly charged ions between m3 Im I m4 are to be excited (but to z-oscillation amplitude less than that for ejection); and singly charged ions of any other mass are not to be excited at all. The specified excitation magnitude spectrum is first interpolatively smoothed46(Figure lb), in order to reduce the corresponding time-domain amplitude near the start and end of the excitation period (and thus produce more uniform excitation magnitude). The excitation magnitude boundaries at ml and m4 may be more heavily (44) Guan, S. J. Chem. Phys. 1989, 91, 775-171. (45) Guan, S.; McIver, R.T., Jr. J. Chem. Phys. 1990,92,5841-5846. (46) Guan, S. J. Chem. Phys. 1990, 93, 8442-8445. (47) Goodman, S.; Hanna, A., U S . Patent No. 4,945,234, issued 31 July, 1990.

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ANALYTICAL CHEMISTRY, VOL. 65, NO. 9, MAY 1, 1993

z-amplitude

z-amplitude

z-am plit

I I I

0

z-am plit

I I I I

I

iw I I

Construction of a SWIFT timedomain z-excitation waveform for mass-selectlve axial excitatlon or ejection of ions in an rf-only quadrupole ion trap. (a)z-Osclliatlon excitation magnitude spectrum in the mlq domain. Singly charged ions of ml < m < m2 are to be < m < m4 are to be ejected from the trap; ions in the range, z-exclted below the ejection threshold; and ions of other mass are unaffected. (b) Smoothed z-osclliatlon excitation magnitude mlq spectrum. (c) Frequencydomain excitation magnitude spectrum correspondingto the mlgdomain spectrum of (b). (d) Phase spectrum associated with (c), deslgned to distribute the timedomain excitation waveform amplitude uniformly over the excitation period. (e) Timedomaln excitatlon vobge signal synthesized by inverse Fourier transformation of the magnitude and phase spectra of (c) and (d). Flgur 1.

smoothed because high mass selectivity is not usually as important at the extremes of the detected mass range, whereas less smoothing is applied a t the m2and m3boundaries where mass selectivity is more important. The smoothed excitation magnitude mass spectrum is then converted by eqs 14a and 16 to give the corresponding frequency-domain voltage magnitude spectrum of the excitation waveform, M(u)(Figure IC). Because ion mlq and ion z-oscillation frequency are reciprocally related in the rf-only Paul trap Cjust as ion mlq and ion cyclotron orbital frequency in an ICR ion trap), the excitation magnitude boundaries a t ml < m2 < m3 < m4 in the mlq domain correspond to u1> u2 > u3 > u4 in the frequency domain. The excitation-phase spectrum (Figure Id) may be calculated from eq 3 of ref 45. Finally, inverse Fourier transformation of the excitation spectrum defined by Figure lc,d gives the desired time-domain excitation SWIFT waveform shown in Figure le. Note that the amplitude envelope of the time-domain SWIFT waveform is relatively uniform throughout the waveform, due to the phase-encodement of Figure Id. Also note that the nearzero amplitude at the beginning and end to the time-domain waveform ensures near-elimination of Gibbs oscillations in the excitation magnitude spectrum. Time-Domain Excitation and Response Profiles. (A) Frequency-Sweep Excitation. It is of interest to examine the real-time behavior of the ion response (eq 11)to dipolar axial z-excitation. Figure 2 shows the ion z-oscillation amplitude response to a frequency-sweep excitation (collisionless limit). Figure 2a shows the time evolution of the excitation voltage, for a linearly time-varying (see Figure 2b) frequency sweep from a maximum frequency, urnax, to zero frequency. The z-oscillation amplitude response for an ion

t

-

0 t Flguro 2. Dipolar z-excitation voltage (a)and excitation frequency (b), and ion z-oscillation amplitude response (b, c) for ions whose axial oscillation frequency is 2u,,/3 (b) or u,,/3 (c)vs time. Frequency is swept from u,, to zero. The rightmost limit of each response profile (b, c) represents the final z-osclliation amplitude at the end of the excitation period, in the limit of no collisionsduring excitation. Note that the ion z-oscillatlon amplitude represents the maximum (not the instantaneous)ion zdisplacement (see text). whose natural z-oscillation frequency is u,,,/3 or 2um,,/3 is shown in Figure 2c,d. Three aspects of Figure 2c,d deserve comment. First, the greatest change in ion axial z-oscillation amplitude during excitation occurs when the excitation frequency exactly matches the ion natural (secular) axial z-oscillation frequency (dotted vertical lines in the figure). Second, the rightmost point in each curve represents the final postexcitation axial z-oscillation amplitude in the absence of collisions: if that response amplitude exceeds the distance from the center of the trap to either end cap, then those ions will be ejected from the trap. Third, the instantaneous z-oscillation amplitude itself oscillates as the frequency sweep approaches and passes through "resonance" (dotted lines in the figure), as previously noted for frequency-sweep excitation in ICR.28 However (unlike the ICR case), one cannot simply infer that an ion whose instantaneous z-amplitude is greater than the distance between the center of the trap and the end cap will be ejected, because the ion z-displacement may not be maximal at the instant of maximum z-oscillation z-amplitude. In the absence of collisions, ejection is assured only if the final postexcitation z-amplitude exceeds the minimum ejection distance. The 'wiggles" in the ion z-oscillation amplitude response may be reduced in size by slowing the frequency-sweep rate; however (as in the prior ICR case), the sweep rate should still be kept high enough that essentially no collisions occur during the sweep. (B) SWIFT Excitation. A more general case is shown in Figure 3 (collisionless limit), which shows the z-oscillation amplitude responses of ions whose z-oscillation natural frequencies are shown as U b , u,, and U d in Figure ICto the dipolar SWIFT z-excitation waveform of Figure l e (shown again in Figure 3a for reference). As for frequency-sweep excitation, several aspects deserve special mention. First, the rightmost point in each curve again represents the final postexcitation axial z-oscillation amplitude in the absence of

ANALYTICAL CHEMISTRY, VOL. 65, NO. 9,MAY 1, 1993

I

0

t 'I,"" ' I '

z-amplitude

~ ( w , t =) l m i 2

-t c)

0 -t

z-amplitude

(17)

1 - smr12 [M(0)I2cos2[Ww) + u t ] (18) 2 The time-averaged kinetic energy is obtained by averaging , z-oscillation: eq 18 over one period, 2 ~ 1 0of

f

z-amplitude

the ion translational energy for collision-induced reactions or fragmentation. It is thus of interest to use eq 11to predict the average axial kinetic energy gained by ions during the excitation process. The postexcitation ion z-velocity is obtained from the time derivative of eq 11: i ( 0 , t )= rlM(0) cos[aqw) + ut], t > 0 The kinetic energy then takes the form,

'I'

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i 0 Flgure 3. Dipolar z-excltation SWIFT voltage vs time (a) and ion z-osclllatlon amplitude response vs time (b-d) for ions whose axial osclllation frequency is vbr v,, or vd in Figure IC, respectively. The rightmost ilmlt of each response profile (b-d) represents the final zosclilatlon amplltude at the end of the excitation period, in the limit of no collisionsdurlng excltation. Note that the Ion zoscillatii amplitude represents the maximum (not the instantaneous) Ion zdisplacement (see text).

collisions: if that response amplitude exceeds the distance from the center of the trap to either end cap, then those ions will be ejected from the trap. Second, the z-amplitude postexcitation response is clearly twice as large (rightmost point of Figure 3d compared to rightmost point of Figure 3b) for an ion subjected to twice the excitation magnitude (compare the excitation levels at Yb and Vd in Figure IC).Also the ion for which zero z-excitation amplitude was specified (v, in Figure IC)shows a net zero postexcitation z-oscillation amplitude (rightmost point in Figure 3c). Third, because the rate of increase of phase decreases with frequency in the excitation-phasespectrum of Figure Id, the ion of high natural z-oscillation frequency (Yd in Figure IC)is excited (Figure 3d) before the ion (Figure 3b) of lower natural z-oscillation frequency (Yb in Figure 1c)-for comparison, phase increases quadratically with frequency in the excitation-phasespectrum (not shown) of a linear frequency sweep from high to low frequency. Fourth, wiggles in the ion z-oscillation response amplitude are observed. The amplitude of the wiggles may be reduced by increasing d20(w)ldw2(up to an upper limit of d@(w)/do5 T per frequency-domain data point35) in the excitation-phase spectrum. The duration of the wiggles may be reduced by increasing the number of SWIFT excitation frequency-domain data points, resulting in a longer timedomain SWIFT waveform. It is worth noting that even an ion which receives zero net excitation at the end of the excitation period (v, in Figure IC,and the rightmost point of Figure 3c) may be driven to a significant z-oscillation amplitude during the excitation process. Finally, as in Figure 2c,d, the instantaneous z-oscillation amplitude in Figure 3b-d does not necessarily reflect the instantaneous z-oscillation displacement. Ion Axial Kinetic Energy. One purpose for selective excitation of z-oscillation is for ejection of ions of specific mlq ratios. A complementary purpose is for selective excitation of ions of one or more mlq ratios in order to increase

Therefore, the time-averagedpostexcitation ion kinetic energy is proportional to the power spectrum, P(w) = [M(w)]2,of the product of the time-domain excitation waveform and exp(y't).

Effect of Collisions on Ion z-Oscillation Response to Arbitrary Excitation. First, we should note that because the mass of the analyte ion of interest is usually much larger than the mass of the neutral buffer gas (usually He),collisions in the quadrupole ion trap are well-modeled by collisional damping, in which each collision reduces the ion velocity by an amount small compared to the ion velocity itself. For example, for an ion of m/z 400,a given collision will reduce the ion momentum by a factor of only 41(4 + 400)= 175,on average. Thus, a large number of collisions may be required to reduce the ion speed by a factor of (say) 2 or more. Therefore, even if ions collide many times during the excitation period, collisional damping of their z-oscillation amplitude may yet be negligible. For example, at 1 mTorr of helium buffer gas pressure, agiven ion will collide on average 30 000 timesis with helium buffer gas. Thus, for an excitation spanning 0-50 kHz in 1 ms, ions are excited to significantly larger z-oscillation amplitude in -0.1 ms (see Figure 2c), during which there are only -3 collisions, each of which reduces the speed of an ion of mlz 400 by only 1% , As seen in Figure 2, ions of different mlz may be excited at different times during the excitation period. If the excitation (e.g., a linear frequency sweep) is designed to produce resonant ejection of the ions, then collisional damping should not significantly affect the observed relative abundances of sequentially ejected ions of different miz values (since ions of a given mlz value are axially ejected almost as soon as they reach the ejection z-oscillation amplitude threshold), except for a frequency shift:

-

(4(0,)2- y2)1'2/2 (8b) However, since both wz and y are (inversely) proportional to mlq, wLI should still be proportional to qlm, and internal mass calibration should be unaffected in the absence of space charge and nonquadrupolar electric field effects. In the usual other case in which excitation is designed to excite (but not eject) ions, then ions excited early in the excitation period will be subject to subsequent collisional damping for a longer period than ions excited late in the excitation period (compare parts c and d of Figure 2). However, since the subsequent period for ion-neutral collisions and/or reactions is usually long compared to the excitation period, we may again neglect the effect on relative abundances of product ions due to collisions during excitation. In summary, although we have included the effect of collisions during the excitation period, we can to a good approximation neglect such effects experimentally,for either ion axial ejection or 'tickling". 0 , '=

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ANALYTICAL CHEMISTRY, VOL. 65, NO. 9, MAY 1, 1993

dc and rf vs rf-Only Mode, We have limited the present treatment to a quadrupole ion trap operating in rf-only mode, because the ion z-oscillation frequency is then inversely proportional to ion mass-to-charge ratio, mlq. As the ratio of dc voltage to rf voltage increases, the z-oscillation frequency approaches a limiting frequency which is inversely proportional to (m/q)l/*(as for the ICR ion trap, which is usually, but not always,48operated with dc-only trapping potential). (Dipolar z-excitation and detection of axially distributed ions in ICR have previouslybeen described49.) For a nonzero ratio of dc to rf voltage, the z-oscillation frequency for an ion in a Paul trap has a nonmalytic (but numerically evaluable) relation to mlq, In any case, the present formalism may still be applied; it is just that a uniform-magnitude excitation (48) Gorshkov, M. V.; Guan, S.;Marshall, A. G.Rapid Commun. Mass Spectrom. 1992, 6,166-172. (49) Schweikhard, L.; Blundschling, M.; Jertz, R.; Kluge, H.-J. Int. J. Mass Spectrom. Ion Processes 1989, 89, R7-Rl2.

segment in the mlq domain may no longer be uniform in magnitude in the corresponding frequency-domainexcitation range. We leave further details for future investigation. Finally, we suggest that a preferred mode of quadrupole ion trap operation may be to cool ions initially a t high buffer gas pressure and then pump down to much lower pressure for subsequent ion excitation and detection, thereby avoiding any problem of unequal excitation of ions of different mlz during the detection period. ACKNOWLEDGMENT This work was supported by grants (to A.G.M.) from NSF (CHE-90-21058),NIH (GM-31683),The Ohio State University, and Amoco Corp.

RECEIVED for review August 5, 1992. Accepted January 27, 1993.