Theory of high-resolution mass spectrometry achieved via resonance

Resonance Ejection from the Paul Trap: A Theoretical Treatment Incorporating a Weak ... Resonance Ejection Ion Trap Mass Spectrometry and Nonlinear Fi...
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Anal. Chem. 1992, 64, 1434-1439

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Theory of High-Resolution Mass Spectrometry Achieved via Resonance Ejection in the Quadrupole Ion Trap Douglas E. Goeringer,' William B. Whitten, J. Michael Ramsey, Scott A. McLuckey, and Gary L. Glish Analytical Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -6365

Beglnning wHh the pseudopotential well approxlmatlonfor Ion motlon in the quadrupole ion trap, analytical expressions are derived for the Ion resonance ejectlon line shape. Incorporation of terms for Ion-molecule collislons and linear variation with tlme of the resonance excltatlon frequency enables ilne shapes to be calculated and displayed as a function of scan rate and relaxationtime. Derlvatlon of the relatlonshlpbetween mass and frequency llne widths allows the mass resoiutlon to be evaluated as a function of fundamental Ion trap operatlng parameters. The results account for the synergistic effect on resolutlonwhen resonanceejection Is combined with retarded scan rates and lndlcate that for any non-zero scan rate there is an optlmum coiilslonai damping value corresponding to a resolutlonmaximum. The effects of the experlmentaivarlables masskharge, neutral pressure, and scan rates on mass resolutlon and accuracy are also evaluated in terms of the model and illustrated In convenlent plots.

INTRODUCTION The use of the quadrupole ion trap for electrodynamic containment of charged particles was introduced by Paul and co-workers.1.2 Fischerz and Rettinghaus3 developed mass spectrometers based on the device which functioned by detecting the secular motion of trapped ions at specific massto-charge ratios. Dawson and Whetten4 later built an instrument which generated mass spectra by periodically pulsing the stored ions out through an end-cap to an electron multiplier. The evolution of the quadrupole ion trap as a modern analytical mass spectrometer can be attributed to a number of more recent performance advances. Stafford, Kelley, and co-workers5developed the mass-selective instability technique enabling mass spectra to conveniently be obtained by linearly ramping the amplitude of the radio frequency (rf)voltage applied to the ring electrode. Various rf/dc methods have been demonstrated for selectively isolating ions of specific mass-to-charge rati0s.69~The application of supplemental signals to the end-cap electrodes has further enhanced the flexibilityof the instrument. Ions are kinetically excited when the frequency of the supplemental signal coincides with one of their secular oscillation frequencies; collision-induced dissociation (CID) can result from such energetically-enhanced collisions with the He bath gas.8

Sequential application of mass-selectiveion isolation and CID enable tandem-in-time mass spectrometry to be accomplished in the ion trap.+10 Use of higher excitation amplitudes results in the ions being ejected from the ion trap, a process known as resonance ejection." Judicious selection of the range of excitation frequencies thus enables mass-selective ion manipulation to be performed over a wide mass range.12-14 In addition, the application of resonance ejection in conjunction with the massselective instability technique, known as axial modulation, enhances mass resolution in the device.15 Selection of appropriate resonance ejection conditions also enables the masslcharge range of the device to be extended from the nominal value of 650 to over 50 OO0.12J6J7However, the resonance ejection technique is accompanied by degradation of the mass resolution at a fixed ramp rate of the ring electrode rf signal due to the direct scaling of mass-to-charge scan rate with the mass range extension factor. Thus, despite such performance enhancements, mass spectrometers based on the quadrupole ion trap have not been viewed as high mass resolution instruments. However, recent research results have shown that high mass resolution can be achieved without sacrificing the masslcharge range by slowing the mass scan rate. Kaiser et a1.18 initially demonstrated the feasibility of the technique using a commercial ion trap mass spectrometer (ITMS) modified to retard the rf scan rate. A similar apparatus was used by Schwartz and co-workers19 to achieve a resolution of 33 000 at mle 502. We have also obtained mass resolution in excess of 45 000 at mle 502 by scanning the resonance ejection frequency.20 More recent results indicate that a mass resolution greater than 1000 000 can be

(8) Louris, J. N.; Cooks, R. G.; Syka, J. E. P.; Kelley, P. E.; Stafford, G. C.; Todd, J. F. J. Anal. Chem. 1987,59, 1677. (9) McLuckey, S. A.; Glish, G. L.; Kelley, P. E. Anal. Chem. 1987,59, 1670. (10)Louris, J. N.; Brodbelt-Lustig, J. S.; Cooks, R. G.; Glish, G. L.; Van Berkel, G. J.; McLuckey, S. A. Int. J . Mass Spectrom. IonProcesses 1990. 94. 15. (11)Fulford, J. E.; Hoa, D.-N.; Hughes, R. J.; March, R. E.; Bonner, R. F.; Wong, G. J. J. Vac. Sci. Technol. 1980, 17, 829. (12) Kaiser, R. E.; Louris, J. N.; Amy, J. W.; Cooks, R. G. Rapid Commun. Mass Spectrom. 1989, 3, 225. (13) McLuckey,S.A.;VanBerkel,G.J.; Glish,G. L. J.Am. Chem.Soc. 1990, 112, 5668. (14) McLuckey, S. A.; Goeringer, D. E.; Glish, G. L. J. Am. SOC.Mass Spectrom. 1991,2, 11. (15) Tucker, D. B.; Hameister, C. H.; Bradshaw, S. C.; Hoekman, D. J.; Weber-Grabau, M. Proceedings of the 36th Annual Conference of Mass Spectrometry and Allied Topics, San Francisco, 1988; p 628. (16) Kaiser, R. E.: Cooks, R. G.: Moss, J.: Hemberger, (1)Paul, W.; Reinhard, H. P.; Von Zahn, U. 2. Phys. 1958, 152, 143. - P. H. RaDid Commun. Mass Spectrom. 1989, 3, 50. (2) Fischer, E. 2. Phys. 1959, 156, 1. (3) Rettinghaus, G. 2. Angew. Phys. 1967,22, 321. (17) VanBerkel, G. J.; Glish, G. L.; McLuckey, S.A. Anal. Chem. 1990, 62, 1284. (4) Dawson, P. H.; Whetten, N. R. J. Vac. Sci. Technol. 1968,5, 11. (18) Kaiser, R. E.; Cooks, R. G.; Stafford, G. C.; Syka, J. E. P.; Hem(5) Stafford, G. C.; Kelley, P. E.; Syka, J. E. P.; Reynolds, W. E.; Todd, J. F. J. Int. J. Mass Spectrom. Ion Processes 1984, 60, 85. berger, P. H. Int. J. Mass Spectrom. Ion Processes 1991, 106, 79. (6) Fulford, J. E.; March, R. E. Int. J . Mass Spectrom. Ion Phys. 1978, (19) Schwartz, J. C.; Syka, J. E. P.; Jardine, I. J . Am. SOC.Mass Spectrom. 1991, 2, 198. 26, 155. (7) Todd,J.F. J.;Bexon, J. J.;Smith,R.D.;Weber-Grabau,M.;Kelley, (20) Goeringer, D. E.; McLuckey, S. A.; Glish, G. L. Proceedings ofthe 39th Annual Conference of Mass Spectrometry and Allied Topics, P. E.; Syka, J. E. P.; Stafford, G. C.; Bradshaw, S. C. 16th Meeting of the Nashville, T N , 1991; p 532. British Mass Spectrometry Society, York, U.K., 1987; p 206. 0003-2700/92/0364-1434$03.00/0

0 1992 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 64, NO. 13, JULY 1, 1992

achieved a t higher mass/charge values.21 The purpose of this paper is to present a theoretical basis for line shape and mass resolution in the quadrupole ion trap when operated in the resonance ejection mode. The pseudopotential well approximation of the Mathieu equation, with a term included to account for the effects of ion-neutral collisions,is employed to analyze the line shape. Examination of the resultant time-dependent differential equation of motion describing the envelope of forced ion oscillations reveals the effect of finite sweep rates on the line shape. The relationship between frequency and mass line width is derived. Finally, an equation predicting asymptotic behavior for the mass resolution as a function of scan rate and collisional damping is obtained.

I. Time-Dependent Equation of Motion. Ion motion in the quadrupole ion trap can be described by the Mathieu equation22-25 (eq 1)in which the subscript u refers to motion

d2u + (a, - 29,

cos 25)u = 0

dt2

nt

5 =2

(1)

in either the radial (r) or axial ( z ) direction. In the normal operating mode the end-caps are grounded and an rf voltage is applied to the ring electrode, for which case the reduced parameters q, and a, are defined as q, = -4ev/mr,2n2

qr = 2ev/mr,2n2

a, = - 8 e ~ / m r , 2 ~ ~ a, = 4eu/mr,2n2

(la)

The terms are V, amplitude (0-pk) of the fundamental rf voltage; U,dc voltage; 52, fundamental rf radial frequency; m, ion mass; e, ion charge; ro, radius of the ring electrode; and t, time. The trajectory space for ions can be plotted in terms of qu and a,. For a given ion to simultaneously maintain a stable trajectory in both r a n d z directions, the values of its reduced parameters must lie within overlappingr and z regions corresponding to stable solutions of the Mathieu equation. The technique of resonance ejection uses a supplementary (ac) voltage applied to the end-cap electrodes. For the situation in which the dc voltage is zero and the supplementary voltage is V,, the z motion may be written as a forced Mathieu equation

dt

- 29, cos (2& =

4e V ,

motion at the secular frequency. When such an approximation is made, eq 2 can then be rewritten as eq 3.

It is useful for our purposes to convert eq 3 to the timedependent form shown in eq 4, in which wo is the secular oscillation frequency for ionic mass-to-charge mle. Near resonance, i.e. when w, wo, such ions absorb power from the

dt2

THEORY

exp(i2~) v'5mroa2

(2)

with w, indicating the radial frequency of the supplementary signal. The pseudopotential well approximation, originally developed by Wuerker et alSz6and Major and Dehmelt,27 assumes the ionic z motion for qr < 0.4 to be composed of a fundamental rf ripple superimposed on simple harmonic (21) Williams, J. D.; Cox, K.; Morand, K. L.; Cooks, R. G.; Julian, R. K., Jr.; Kaiser, R. E. Proceedings of the 39th Annual Conference of Mass Spectrometry and Allied Topics, Nashville, TN, 1991; p 1481. (22) Dawson, P. H. Quadrupole Mass Spectrometry and Its Applications: Elsevier: Amsterdam, 1976. (23) Lawson, G.;Todd, J. F. J.; Bonner, R. F. Dyn. Mass Spectrom. 1975,4, 39. (24) March, R. E.; Hughes, R. J. In Quadrupole Storage Mass Spectrometry;Winefordner,J. D., Kolthoff, I. M., Eds.;ChemicalAnalysis Series 102; John Wiley & Sons: New York, 1989. (25) McLachlan, N . W. Theory and Applications of Mathieu Functions; Clarendon: Oxford, U.K., 1947. (26) Wuerker, R.F.; Shelton, H.; Langmuir, R. V. J.Appl. Phys. 1959, 30, 342. (27) Major, F. G.;Dehmelt, H. G. Phys. Reo. 1968, 170, 91.

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+ w:z

FS

= - exp(iw,t) m

wo =

n

"5

(4)

supplementary signal and increase the amplitude of their z trajectories. During resonance ejection, which is employed for masdcharge range extension and resolution enhancement, the maximum ion excursions exceed the z axis dimensions of the ion trap. The effects of ion-neutral collisions on ion motion can be introduced into eq 4 by including a damping factor, c(dz/dt), where c is the reduced collision frequency (see below); the modified equation can then be summed over all ions to obtain d2z, + c 2 dz + w,,?z, = Fa exp(iw,t)

(5) dt2 dt where zc is the center of the trapped-ion charge. Major and Dehmelt27 have noted the analogy between the oscillating electric-dipole moment of the trapped ion cloud and the magnetic moment in NMR spectrometry. Following their treatment, z, motion can be described by z, = Z,(t) exp(io,t)

where Z, is the amplitude of the slowly varying envelope of rapid z, oscillations. Inserting the appropriate expressions fordzJdt and dzzJdt2intoeq 5 gives eq 6 describingZ,motion.

-w,2zc + aiw,[

21+ [3 1+ .(

iw,z,

+

[%I) +

If 2, is assumed to be a slowly varying function of time, the usual approximations wo = we, 00 + w, = 2w0, wo >> c near resonance are made, and the definition & = w, - wo is used, then eq 6 is simplified to (7)

which describes the time-dependent 2, motion. T is the relaxation time for decay of the amplitude of resonantlyexcited Z, motion due to ion-molecule collisions. 11. Determination of Frequency Line Width. Motion of trapped ions in the axial direction can be detected by resonant power absorption,2 frequency-tuned circuita,3 and image current measurements.28 Alternatively, the supplementary excitation signal can be increased until the maximum z axis ion excursions begin to exceed the axial dimensions of the trap; the ejected ion current can then be measured with an electron multiplier.29 Because the secular frequency, wo, (28) Syka, J. E. P.; Fies, W. J. Proceedings of the 35th Annual Conference of Mass Spectrometry and Allied Topics, Denver, CO, 1987; p 767. (29) Dawson, P.H.; Whettan, N. R. J. Vac. Sci. Technol. 1968,5,11.

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ANALYTICAL CHEMISTRY, VOL. 64, NO. 13, JULY 1, 1992

is a function of the mass/charge ratio (see below), the mass spectrum for trapped ions can be obtained by scanning Aw and monitoring the response, Z,(t). The frequency spectrum and line shape for &(A,) can be obtained from &(t) if A,(t) is known. Equation 7 is the basic form obtained by Jacobsohn and Wangsness30for the sweep rate dependent line shape of the vector amplitude of the magnetic moment in NMR spectrometry, as noted by Major and Dehmelt.27 We proceed in a similar manner in the following development. Assuming c to be independent of ion velocity (see below), the complex solution of eq 7 is obtained.

I

0

If A, is swept linearly with time, A J t ) = at, and the sweep rate dAwidt= a rad s - ~ . It is convenient to introduce the dimensionless variable, x = A t , so that eq 8 becomes ZJX) =

Flgure 1. Plot of normalized lZ,l vs ijY7 for d a s values of 0.4, 2, 4, and 8 indicated by solid, dotted, dashed, and dash-dot curves, ~ be converted to mass units from Am = respectively. A w can

W V ' ~ / ( Q W I~ , 7 .

/

The general aspects of the 2, line shape are more evident after eq 9 is integrated by parts to yield

30 a

-Bexp

lo2

10' 1O D

1

i__p A

10.'

1 oo

1o2

10' &r

Equation 10 indicates that the line shape is the superposition of time-independent and time-dependent terms. The first term is a Lorentzian function dependent on the instantaneous value of A,(t). The exponential term represents a damped oscillation; its phase and amplitude are contained in the coefficient B, which is a function of the explicit time dependence of Aw. The quantity B is a constant if there exists a time, tl = x1/&, beyond which a / ( l / s+ iAw)2is negligible.30 Inthelowscanratelimit,&7> 1. In this case the integral over x' can be approximated by a Fresnel integral,27,30and eq 9 then becomes Z,(X)

E

4; (---

-iF, -

exp

-x

ixJ

+

...

2mw0 &7 This expression is essentially the frequency transform of the impulse response function for the damped oscillator. The peak amplitude is inversely proportion to 6since ions (30)Jacobsohn, B. A,; Wangsness, R. K., Phys. Reu. 1948, 73, 942.

Normalized line width plotted as a function of ATfor selected scan rates and amounts of damping. The solld curve was calculated from eq 13. The line width In mass unlts, Am, can be obtained by multiplying the displayed values by (2V5m)/(qz!27). Figure 2.

experience fewer cycles of excitation during passage through the resonance line as the scan rate increases. The line width, Awo = aAt = &Ax, in this limit can be written Awo (rad s-l) = U T

(12)

The overall 2,line shape, given by the solution of eq 9, can be obtained by numerical integration or from the analytical expressions of Jacobsohn and W a n g s n e ~ s .Figure ~ ~ 1 shows the results of numerical integration of eq 9 for selected values of the dimensionless quantity A T . The ordinate, lZcl,and abscissa, Aw,have been normalized to ( F , r ) / mand T , respectively. As expected from the general form of eq 10, the damped oscillations disappear and the line shape becomes Lorentzian for &T values less than -1/2. Additional insight can be gained by plotting individual h 0 7 values as a function of AT.Figure 2 is such a plot obtained from a series of resonance line shapes (similar to Figure 1) corresponding to selected a and T values. Rather than attempting an analytical evaluation of eq 9 for the overall line width, an inspection of Figure 2 indicates AWOTis approximately equal to the steadystate limit of 2 f i (eq 11)for %"& 5 while for &T 1 -4 the value of AUOT= (AT)' (eq 12). Therefore, we empirically approximate the overall line width as the sum of the steady-state and h i g h - A s expressions (eq 13), which is Awo (rad s-l) *

+ UT = d

c

2a +C

(13)

plotted as the solid curve in Figure 2. The continuous curve

ANALYTICAL CHEMISTRY, VOL. 64, NO. 13, JULY 1, 1992

fits the individual points well, indicating that eq 13 is a good approximation for line widths obtained via solution of the time-dependent differential eq 7. Figure 2 indicates that the line width asymptotically approaches a damping-dependent minimum value, ( 2 l h ) / ~as, a decreases. Furthermore, for large 6 7 values the oscillator does not respond as rapidly as the supplementary frequency is changing, hence the apparent line width is given by the product of the scan rate, a, and the natural response time, 7 , of the oscillator. 111. Relation between Mass Resolution and Frequency Line Width. Mass resolution and line width in the ion trap can be related by proceeding from the fundamental expression m = fieV/wortR Differentiating eq 14 gives dm=(*)

qz C 0.4

(14)

dVWdtQ

wo,R

The first term of eq 15 indicates that a small change in the amplitude of the fundamental rf voltage is proportional to a minor deviation in mass. The second and third terms indicate that any small increment in the secular frequency or the fundamental frequency is also linearly related to a small mass increment; the minus sign indicates that wo and R are inversely related to m. Consequently, mass analysis via resonance ejection can be achieved by ramping the rf voltage level, by varying the frequency of the supplementary excitation, or by sweeping the fundamental rf frequency (while fixing the remaining variables). In order to relate the line width in frequency units to that in mass units, we set dV and dQ to zero in eq 15 and divide by eq 14. Making the additional assumption that dwo = AWOand dm = Am yields wo/Awo = m/ Am

(16) This indicates that the mass scale resolution is numerically equal to the frequency scale resolution for qz C 0.4. Substituting the expressions for wo and @, into eq 16 produces eq 17, the mass resolution in terms of the frequency line width

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is equal to zo, or r O / h Thus, the optimum resolution would be achieved when lZclat a given scan rate exactly equalled zo and would correspond to a critical value of the excitation voltage, V,. Note that since the peak value of IZ,I decreases with increasing scan rate, damping, and masslcharge value, as seen in Figure 1, the critical excitation voltage required will be proportional to those parameters. However, under critical excitation voltage conditions, the magnitude of the ion signal would be low. Since the equation of motion (eq 5 ) is linear, IZcI is proportional to the excitation voltage. Consequently, an increased signal level would be attainable at the expense of resolution by operating somewhat above the critical value for V,. Schwarz et al.19have experimentally confirmed that the amplitude of the excitation voltage is critical in obtaining high mass resolution and is dependent on masslcharge value and scan speed. It is also of interest to examine the effects of the instrumental parameters q, and R on mass resolution. The resonance ejection q, value for an ion of given masslcharge ratio can be increased by elevating the (fixed) amplitude of the fundamental rf voltage (duringfrequency-sweepresonance ejection) or by increasing the (fixed) frequency of the supplementary excitation signal (during rf-ramp resonance ejection). Equation 17 indicates that the mass resolution at constant AWO,Le., fixed scan rate and damping, can be improved by independently increasing q, or R. In such cases the enhanced resolution results from the increase in wo with qr or R (eqs 3 and 4). Schwarz et al.,l9 Syka et al.,31 and we20 have experimentally observed this effect. For a given mass/ charge ratio, the corresponding maximum q, value can be determined from eq l a by substitution of the maximum attainable rf amplitude and the specific values for the parameters ro and R. It is interesting to note a potential additional benefit associated with operation at q, > 0.4. The approximation wo = (q,Q)/(2.\/2) becomes inaccurate for q, values greater than 0.4 because the dependence of 8, on q, becomes nonlinear. A plot of wo vs q, determined from the more complicated successive approximation method24 indicates that wo = (q,)y

y(qJ > 1

for

qz > 0.4

which can be combined with the definition for q, to give

Proceeding as with eqs 14-16 yields and the fundamental operating parameters. Setting dwo and dR to zero in eq 15 produces a similar result for the voltage scale resolution, indicating that the analysis of resonance ejection for mass resolution enhancement presented here applies to both voltage and supplementary frequency scan methods.19-21 Although the fundamental rf scan method has not been employed in the ion trap, similar reasoning reveals that these results should apply to that technique as well.

RESULTS AND DISCUSSION Effect of Parameters V., qz, and Q. In this model, we have assumed that resonance ejection occurs when the maximum ion kinetic energy in the z direction, E,, exceeds the z depth of the Dehmelt pseudopotential well in which the ion is oscillating, D2.27

The maximum ion velocity in the z direction can be approximated by u,, = w0IZcJ,so that E, = eD, when lZ,l

m/Am = -y(wo/Awo) which indicates that the mass resolution can actually become greater than the frequency resolution by the multiplicative factor y for q, > 0.4. However, at increasingly higher qz the simple harmonic motion model becomes less accurate since the relative contribution of oscillatory motion at the fundamental rf frequency becomes increasinglylarge. This suggests that an optimum q, value may exist which is less than the value of 0.908 corresponding to the boundary of the stability region. In similar experiments with resonance ejection, Louris et al.32 have noted an increase in resolution with 8, and a resolution maximum in the vicinity of p , = 0.73 (q, = 0.83). They attributed the loss in resolution at higher @, to the (31) Syka, J. E.P.; Schwartz, J. C.; Louris, J. N.; Amy, J. W.; Fies, W. J.; Fenske, S.A. Proceedings of the 39th Annual Conference of Mass Spectrometry and Allied Topics, Nashville, TN, 1991; p 544. (32) Louris, J.; Freuler, K.; Kirkish, S.; Schwartz, J.;Stafford, G.; Syka, J.; Taylor, D.; Tucker, D.; Zhou, J. Proceedings of the 39th Annual Conference of Mass Spectrometry and Allied Topics, Nashville, TN, 1991; p 542.

ANALYTICAL CHEMISTRY, VOL. 64, NO. 13, JULY 1, 1992

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10'

10' c

Y

3

a

1on

I 10'

1O D

6. Flguro 3. Normalized peak shift vs AT for different scan rates and amounts of damping. The shlft, Am, In units of mass can be found by multiplying the displayed values by (2&m)l(q2n7). The solid ilne representsthe least-squareslinear regression flt to the indlvidual points.

relatively shallow potential well depth near the edge of the stability diagram. Effect of Damping and Scan Rate on Mass Accuracy and Resolution. It is apparent from Figure 1that the line center shifts as &T increases, implying that such an effect must be considered in order to attain high mass accuracy when such resonance ejection techniques are employed. A graphical analysis for the deviation in line center, (WLC - W O ) ~ or (AWLC)T,as a function of &T is shown in Figure 3. The individual points were obtained from the same series of line shapes used to construct Figure 2; the solid line indicates the least-squares fit to a first-order model. An expression predicting the true line center, WO, can be obtained from the regression coefficients.

The accuracy of the predicted line center could presumably be improved by fitting the points to a higher-order polynomial. The line center shift effect is similar to that seen for the increased line width at high &T, i.e., the supplementary frequency changes faster than the oscillator can respond, so that the apparent line center varies with &T. As noted in the Introduction, experimental results have established that a reduction in scan rate results in mass resolution enhancement in the ion trap. An equation relating mass resolution to the damping and scan rate is readily obtained by substituting eq 13 into eq 17.

It is informative to analyze eq 18 graphically by plotting the normalized mass resolution, (m/Am)(l/[q,W, vs scan rate for several selected T values, as illustrated in Figure 4. It is clear from the figure that an increase in ultimate mass resolution results from decreased damping (lower c, higher T ) but requires lower scan rates to attain. For relatively high scan rates the mass resolution will be inversely related to scan rate and proportional to damping. At some sufficiently low scan rate the mass resolution should become independent of scan rate and inversely proportional to damping. However, for any non-zero scan rate an optimum damping value should exist which corresponds to a resolution maximum. Further evaluation of the dependence of mass resolution on experimental variables, rather than the decay constant T , requires development of an expression relating the reduced frequency c to mass, charge, and pressure. Evaluation of Reduced Collision Frequency, c. In this paper, c represents the rate of ion momentum relaxation due

Scan Rate (rad-s-')

Mass resolution normalized by q2Q as a function of scan rate, with A, varying linearly with time, for several values of T . The scan rate, dmldt, in mass units per second can be determined by multiplying the displayed values by (2&m)l(q n). The pseudopotential approximationsused here are valid for 0 +?q2 < 0.4. The value of il Is determined by the instrumentation,typically 2 4 1.1 X 10') rad Flgure 4.

S-1.

only to collision processes which are nonreactive and does not include ion loss at the electrodes during passage through resonance. M (19) m+M M is the neutral mass and u is the collision frequency c = u(-)

u

= nUpD

(20)

where n is the number density of the neutrals, ur is the relative velocity of the ion and neutral, and UD is the diffusion cross section. For near-thermal ion energies the dominant ionneutral force can be described as originating from a polarization potential, V, = -(e2a,)/(2tl) ( r is the ion-neutral separation and a, is the polarizability of the neutral) surrounding a hard-sphere potential. For particles subject to such attractive r-4 potentials, orbiting collisions are associated with impact parameters smaller than a critical value, b0.33 Scattering subsequent to such orbiting collisions is isotropic. Thus, UD Uorb

where the terms are p , the reduced mass for the ion-neutral pair, and to, the absolute permittivity of vacuum. Contributions from large impact parameter collisions associated with nonspiraling orbits are also assumed to be negligible at low ion-molecule collision energies.34 When eqs 20 and 21 are substituted into eq 19 and the ionic mass is taken to be much greater than the neutral mass (thenormal case as helium is typically the buffer gas), eq 22 is obtained, which indicates c

n/(mle)

m >> M

(22)

that the ion collisional relaxation rate is a velocityindependent function of the neutral pressurelmass-to-charge ratio. In the event that ion-molecule interactions other than orbiting collisions significantly contribute to UD, eq 7 may not easily be solved since c then becomes a function of ion velocity. For example, the collision frequency due to the repulsive part of the potential is velocity dependent and may, in some cases, contribute to c at large values of 2,.However, since B e a ~ c h a mhas p ~ shown ~ minimal errors to be associated (33) Vogt, E.; Wannier, G. H. Phys. Reu. 1964, 95, 1140. (34) McDaniel, E. W. Collision Phenomena in Ionized Gases; Wiley: New York, 1969. (35) Beauchamp, J. L. J. Chem.Phys. 1967, 46,1231.

ANALYTICAL CHEMISTRY, VOL. 64, NO. 13, JULY 1, 1992

with weakly velocity dependent collision terms in ICR theory, in the resulta presented here we expect similar small deviations due to such interactions. Dependence of Mass Resolution on Neutral Pressure and mle. Equation 22 can be used to determine the effects of neutral pressure and ionic masslcharge ratio in different regions of Figure 4. The scan rate dependent portion of each curve in Figure 4 is ml Amq,Q = c 1 4 f i a

Upon substitution for c, eq 23 is obtained, which indicates that for relatively high scan rates, the normalized mass resolution in the ion trap should increase with buffer gas pressure. This point is evidenced by the experimental mlAmq$

a

n/(m/e)a

(23)

observation that resolution in the Finnigan ITMS is enhanced by the addition of helium.36 Equation 23 also predicts that mass resolution at such scan rates should be inversely proportional to the ion masslcharge ratio and scan rate. For a given scan rate, however, a resolution maximum will be reached correspondingto an optimum combination of pressure and masslcharge ratio. Above this optimum, the resolution will decrease with damping and become scan rate independent. The scan rate independent region of each curve in Figure 4 is mlAmq,Q = 1 1 2 f i c

Substitution for c yields mlAmq,Q

a

(mle)ln

(24)

Thus, in such a scan rate regime, mass resolution should increase with masslcharge ratio and be inversely related to buffer gas pressure. However, for a given scan rate a damping optimum will exist below which the damping vs scan rate curve becomes scan rate dependent. At that point the resolution dependence on pressure and masslcharge ratio is again described by eq 23. Differentiating eq 13 enables an optimum damping value to be determined for any given scan rate.

The maximum normalized resolution, [ml(Amq$)l,,, for a given scan rate is obtained by substitution of eq 25 into eq 18.

(mlAmq,Q),, = 1/8(31’4)&

(26)

Equation 26 is indicated by the solid line in Figure 4. Thus, the point of tangency with each curve represents an optimum scan rate-damping combination. Although a lower scan rate would result in a slight resolution improvement (at constant damping), the corresponding maximum resolution would be (36) Kelley, P. E.; Stafford, G. C.; Fies, W. J.; Syka, J. E.P.; McIver, R. T.; Hunter, R. L.; Todd, J. F. J.; Bexon, J. J. Proceedings of the 32nd Annual Conference of Mass Spectrometry and Allied Topics, San Antonio, TX, 1984; p 505.

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attained by decreasing the damping to the value calculated from eq 25, i.e., a new point of tangency. While this model adequately explains the observed qualitative dependence of mass resolution on experimental variables and parameters, a comparison with quantitative results indicates that the calculated values appear (within experimental error) to underestimate experimental resolutions by a factor of 3-5. Initial evaluation of the fundamental line shape in terms of resonantly-excited ion motion was necessary because that process is the foundation for resonance ejection. The apparent discrepancy in resolution results from the fact that the full-width at half-maximum of the entire line shape, calculated from the model for resonantly-excited ion motion, was arbitrarily used as an indication of resolution. However, because the optimum resolution would be realized when IZ,I becomes exactly equal to 20, the ejected ion line shape under near-optimum resolution conditions would correspond to only a portion of the calculated line profile near the apex. It is important to realize that even under such conditions the entire ion population can be ejected, provided that enough cycles of the excitation signal are experienced during passage through the apex. Examination of Figure 1 reveals that the full-width at half-maximum for any such truncated apex is some fraction of that for the complete line profile, with the fraction dependent on the ratio Z,/zo. The overall effect is to reduce the observed line width compared with the calculated value, thus increasing the corresponding experimental mass resolution. These results indicate that the pseudopotential well approximation for the quadrupole ion trap, modified for the effects of ion-molecule collisions and linearly-varying resonance excitation frequency, can be used to describe the envelope of z axis motion for stored ions during resonance excitation a t low qz values. The resultant time-dependent differential equation, which is in the form of a driven harmonic oscillator with damping, is useful in elucidating the fundamental factors influencing line shift, line width, and resolution when resonance ejection is used to effect mass discrimination. The model indicates as a first approximation that the shift in line center varies with 6. In addition, at relatively high scan rates the apparent line width is proportional to the scan rate and relaxation time, so that increasing the neutral pressure or reducing the masslcharge ratio or scan rate results in narrower line widths. Furthermore, for sufficiently low scan rates the line width becomes independent of scan rate and proportional to damping, so that mass resolution in such a regime increases with masslcharge ratio and relaxation time. Thus, an optimum value of damping should exist for any non-zero scan rate. The maximum achievable resolution will be inversely proportional to the square root of the scan rate, provided that the optimum combination of scan rate and damping is established. However, the ultimate resolution will likely be limited by practical considerations such as space charge, instrument stability, signallnoise ratio, and ion residence times.

ACKNOWLEDGMENT This research was sponsored by the U.S.Department of Energy, Officeiof Basic Energy Sciences, under Contract DEAC05-840R21400 with Martin MariettaEnergy Systems, Inc. RECEIVED for review November 18, 1991. Accepted March 20, 1992.