Theoretical mass and energy upper limits for thermal ions in Fourier transform ion cyclotron resonance mass spectrometry

International Journal of Mass Spectrometry and Zon Processes, 120 (1992) 193-205

193

Elsevier Science Publishers B.V., Amsterdam

Theoretical mass and energy upper limits for thermal ions in Fourier transform ion cyclotron resonance mass spectrometry Michael A. May, Peter B. Grosshans and Alan G. Marshall’ Departmentof Chemistry. The Ohio State University, I20 West 18th Avenue, Columbus, OH 43210 (USA)

(First received27 January 1992;in final form

22

June 1992)

ABSTRACT Analytic expressions for the probability of radial or axial loss of thermal ions from a hyperbolic ion trap are developed, as a function of trap dimensions, trapping potential, magnetic field strength, ion mass-to-charge ratio, and ion Boltzmann temperature. These equations are readily evaluated to yield ion cyclotron resonance (ICR) upper mass and energy limits above which a massdependent fraction of thermal ions will be lost prior to any ICR excitation event. Non-interacting ions are assumed to & formed initially along the trap z axis with a Maxwell-Boltzmann velocity distribution, and trapped by a quadrupolar electrostatic potential. Under these approximations, the probability of radial ion loss increases strongly and monotonically with ion mass-to-charge ratio and trapping potential, whereas the probability of axial loss is independent of ion mass and inversely related to trapping potential. Both axial and radial ion loss increase with increasing ion thermal energy. For radial ejection (which dominates at high mass-to-charge ratios), an ion “stability” diagram shows boundaries of a constant fraction of ions trapped as a function of ion mass-to-charge ratio and trapping voltage for a given magnetic field strength. Since the electrostatic potential in all common ICR ion traps is approximately quadrupolar near the trap center, the present results can be extended semi-quantitatively to those ICR trap geometries (e.g., cubic, tetragonal, cylindrical, etc.). Elongated or screened traps should significantly extend the upper mass-to-charge ratio limit. Keywork

FT-ICR MS; ion trap.

INTRODUCTION Fourier transform ion cyclotron resonance mass spectrometry (FTICR MS) is a well established technique whereby ions are spatially trapped by

static magnetic and electric fields. The features and advantages of the technique have been repeatedly reviewed [l-14]. Upon radio-frequency (r.f.) electric field resonant excitation at the ion cyclotron orbital frequency, trapped ions absorb energy, increase their cyclotron radii, and the resulting Correspondence to: A.G. Marshall, Department of Chemistry, The Ohio State University, 120 West 18th Avenue, Columbus, OH 43210, USA. ’ Also a member of the Department of Biochemistry.

016%1176/92/$05.00

0 1992 Elsevier Science Publishers B.V. All rights reserved.

194

M.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

spatially coherent orbiting ion packet may be detected by virtue of the differential charge it induces on the trap detection plates [ 15,161. The critical mass, merit, for an ion trapped by a quadrupolar electrostatic potential is the largest mass singly-charged ion with a stable ion cyclotron orbit [17-201, as discussed below. Following their formation in (or injection into) an FT-ICR ion trap, ions of a given mass-to-charge ratio form a packet with a distribution of individual ion kinetic and potential energies. It is obviously important to know the fraction of ions which is permanently lost by contact with the various trap electrodes both before and after r.f. excitation. It is especially important to know how ion loss depends on ion mass-to-charge ratio. Here, we derive expressions for the probability of thermal ion loss along both the radial (x,y) and axial (z) directions prior to electric field excitation, where the magnetic field is taken to point along the z axis. Owing to the analytical simplicity of the quadrupolar potential [l&21] we derive these expressions for the specific case of a hyperbolic (Penning) trap. Our approximations are: (a) the (preionized) neutrals have a Maxwell-Boltzmann velocity distribution; (b) the ionization event deposits no additional translational energy to the ion packet; (c) particle/particle interactions are negligible. Because a quadrupolar potential is maintained near the center of other ICR trap geometries (e.g. cubic and cylindrical) [16], the results obtained here can be extended to these other ion trap geometries. Limitations and extensions of our approach are discussed. THEORY

Radial ion loss Quadrupolar trapping potential

Consider an ion packet formed in a quadrupolar potential produced by an ICR ion trap of hyperbolic geometry (see Fig. 1). Such a trap has three electrodes, a “ring electrode,” and two “end electrodes”. For conventional dipolar excitation and detection, the ring electrode is cut into four quadrants, of which two opposed segments form the excitation electrodes, and the remaining two opposed segments form the detection electrodes. The equations defining the surfaces of these electrodes are [l&22]: Ring electrode: x2 + y* - 2z2 = 4 End electrode: 2 + y* - 2z2 = - 22;

(1)

in which the origin of coordinates is the geometric center of the trap, r. and z, are trap dimensions (r,, is the minimum inner radius of the ring electrode and 22, is the minimum separation of the end electrodes). The direction of a

M.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

195

4-l r

Fig. 1. Schematic perspective (top) and center cross-sectional (bottom) views of a hyperbolic ICR ion trap, with applied magnetic field along the z direction. The two sets of excite/detect electrodes approximate an idealized “ring electrode” (see text).

spatially uniform magnetic field is parallel to the z axis: B = Bk. The electric potential, V(x,y,z), inside the trap is: wri:-(x2+Y2)+2~l

v(xyz)= ,

,

(2)

30 + 22;

in which VTis the difference in potential applied between either end electrode and the ring electrode. The electric field is the negative gradient of that potential: E = - v V(x,y,z)

(3)

and the motion of an ion, of mass m and charge q, in these fields is governed by the Lorentz force equation: F=m$=q(E+vxB)

(4)

in which F is the instantaneous force on an ion moving at velocity, w. The general solution to the differential equations of motion summarized in eqns. (2)-(4) has been given by Byrne and Farago 1231: x(t)+iy(t)=r+exp(-io+t)+r_exp(-io_t) Z(t) =

Z, COS (0,

in which i = m,

t +

0) = Zi COS (0,

Pa) t) + z

sin (0, t)

Vb)

0 is the initial phase’angle for z oscillation, and CO,, co_,

196

M.A. May et aLlInt. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

and CO,are the cyclotron, magnetron, and trapping angular frequencies respectively. The coefficients, r+ and r_, are, in general, mathematically complex (i.e. they describe circular motion in the x-y plane) and their magnitudes are the respective cyclotron and magnetron radii of the ion. Zi and ii are the ion’s initial axial position and velocity respectively. For focusedbeam ionization (e.g. electron ionization) or external ion injection, ions are typically formed (or injected) along the z axis at time, t = 0. Thus, for agiven ion: x(0) = y(0) = 0

(6)

Substitution of eqn. (6) into eqn. (5a) gives Y+ = - r_ (i.e. the magnitudes of the cyclotron and magnetron radii are equal). Let the initial (thermal) velocity of the ion in the x and y directions be ii and pi, respectively. Substituting for r_ in eqn. (5a), differentiating with respect to time, and solving for r+ leads to: r+ =

iAi - 3i

i$ - pi 0, ---

(7)

=UC(l -ET

in which the “unshifted” (i.e. if E = 0) cyclotron frequency, oC, is given by:

wE4B c m

(8)

and the critical mass, mctit:

(9) represents the highest mass ion that can be radially confined under the stated conditions. For radial loss of an ion to occur, the sum of the cyclotron and magnetron radii must equal or exceed r,:

(10) Substitution of eqn. (7) into the above inequality gives (upon rearrangement) the condition for radial loss:

(11) in which ii ~ ((~i_i)* + (~i)2)1’2

(12)

The initial x-y velocity of the ion is necessarily directed radially for ion initially formed on the z axis in a cylindrical coordinate system.

M.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

197

The probability that eqn. 11 is true is thus the probability that a thermal ion will be lost radially from the ICR trap. For simplicity, we assume that the neutral precursors to the ions have a Maxwell-Boltzmann velocity distribution directly prior to the ionization event, and that ionization imparts no additional energy to the ions. The probability, Pradial,that a thermal ion is radially lost is obtained by integrating the Maxwell-Boltzmann distribution over all radial ion velocities that exceed the right-hand side of eqn. (11): (13) The integral is straightforward,

and the resulting probability of radial loss is:

From eqn. (14), we recognize that the probability for radial loss increases with ion mass, and depends on mtit , r,, co,, and T. In particular, when m reaches merit, the probability of radial loss is unity (i.e. all ions are lost, independent of their initial velocity). When m > mtit, ions do not have a stable cyclotron orbit and the ions simply spiral outward without limit. Zero

trapping

potential

In the absence of any electrostatic field, eqn. (9) shows that the “critical” mass becomes infinite. However, the maximum possible mass-to-charge ratio is still limited by the finite radius of the trap. For ions initially located on the z axis [x(O) = y(O) = 0] with x-y speed, v,.., the “radial” m/q upper limit is given by [24,25]: (15) and the radial upper mass-to-charge ratio limit is determined only by the ion initial velocity and the trap radius, r,,,. Axial

ion loss

Classically, a thermal ion will be lost axially when the sum of its potential and kinetic energies equals/exceeds the electrostatic potential energy, qV,( I$ng = 0), at the end electrodes [26,27]:

05mi~+qV,(~--+fz~) . 1

(4 + 24)

2

qv,

(16)

198

M.A. May et aLlInt. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

At the moment of ionization, r(t = 0) = 0, for ions formed on the z axis. Solving eqn. 16 for the initial z velocity, ii, yields: 4qv,(zi

1

- z?) I’*

‘ii’a[ m(ri + 22;;

For a Maxwell-Boltzmann initial z position distribution Paxid

=

2

(17) initial z velocity distribution [7] and a uniform at t = 0, the probability of axial ion loss is: (Erexp($$)d4

27ckT

(18)

where the second integral yields the probability for axial loss for a given initial z axis Boltzmann-weighted velocity, and the first integral sums over all possible (equally likely) z axis positions. To simplify the form of the integral in eqn. (18), the variable s G (m/2kT)“*ii is introduced: (19) The second integral in eqn. 19 can be expressed as the well-known complementary error function, erfc(u), of argument, u [28]:

Note that eqn. (20) is mass-independent; thus, the probability of axial loss is independent of mass for the case of a thermal ion prior to r.f. excitation in a hyperbolic ICR trap. Furthermore, eqn. (20) can be evaluated numerically [29] to yield the probability for axial ion loss, Paxial,for a given trapping potential, V,, ion Boltzmann temperature, T, and trap aspect ratio, z,/r,. Bloom and Riggin previously derived the probability for the axial retention of thermal ions in an orthorhombic ICR cell [26,27], but for a different model: ICR single-frequency excite/detect in a “drift” cell, with averaging over ion energy rather than ion velocity. RESULTS AND DISCUSSION

Radial or axial storage efficiencies for thermal ions in a hyperbolic trap Radial ion loss

From eqns. (9) and (14), we can identify strategies for minimizing radial loss of ions: decrease T, decrease V,, increase B, increase q (multiply-charged ions), increase the trap aspect ratio (at constant rO), or increase r, at constant aspect ratio. The effect of varying each of the first three parameters is displayed successively in Figs. 2-4, with all other parameters held constant at

M.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

199

1.0

0.8 .+ z d E P

0.8

t 2

0.4

5 0.2

4

1o-3

-

1o-2

..

.

10-l

1

..I

loo

.-

10’

I lo2

Ion Temperature (eV)

Fig. 2. Plot of the probability of thermal singly-charged ion loss from a hyperbolic ICR ion trap as a function of ion kinetic energy, showing axial, radial, and overall ion loss. The conditions are: trap voltage = l.OV across 2cm distance between the centers of the end caps, applied magnetic field = 3.OT, and ion mass = 100~.

realistic experimental values: T = 300 K (i.e. ion translational energy of O.O26eV), V, = 1 V, B = 3T, q = 1.602 x 10-19C, z. = 1 cm, and r. = flcm. For m/z = 100 thermal ions (m in u, z in multiples of the elementary charge), Figs. 2-4 show that radial loss is negligible (c 5%) for ion translational energy <40eV(at V,= lVandB=3T);orfor V-<434V(forT=300K andB=3T);orforB>O.O6T(atT=300Kand V,=lV). In other words, radial ion loss is negligible for low-mass ions under 1.0

Radial -

lo5

lo-2 10-l

100

10’

lo2

lo3

Trap Voltage (V)

Fig. 3. Plot of the probability of thermal singly-charged ion loss from a hyperbolic ICR ion trap as a function of trap voltage (across 2 cm distance between end cap centers) showing axial and radial ion loss. Ion temperature = 300 K, applied magnetic field = 3.OT, and ion mass = 100 u.

200

M.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

1

Radial

0.0 0.00

0.02

0.04

0.06

Static Magnetic Field

0.06

0.10

(Tesla)

Fig. 4. Plot of the probability of thermal singly-charged ion loss from a hyperbolic ICR ion trap as a function of applied magnetic field strength showing axial, radial, and overall loss. Ion temperature = 300 K, trap voltage = l.OV across 2cm, and ion mass = 100~.

virtually all typical experimental conditions. However, Fig. 5 shows that radial loss of room-temperature thermal ions remains negligible (at VT= 1 V and B = 3 T) until just below the critical mass [17], at which radial loss increases very steeply. Figure 6 offers one type of “ion stability” diagram, in which each curve represents the highest mass-to-charge ratio at which a given fraction (lOO%, IO%, and 1%) of thermal ions is radially lost from the hyperbolic trap, as a function of trapping potential, for each of three magnetic field strengths (3, 7, and 14T). The m/z value at which 100% of ions are radially lost (i.e. 1

l.O-

0

1QOQQ

2QOOO

30000

4QQQQ

6OkO

m/z

Fig. 5. Plot of the probability of thermal singly-charged ion loss from a hyperbolic ICR ion trap as a function of ion mass, showing axial, radial, and overall loss. Ion temperature = 300 K, trap voltage = l.OV, and applied magnetic field = 3.OT.

h4.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

201

10”

10’

10”

105

lo4

lo3 lo’*

10”

loo

lo’

Trap Voltage (V)

Fig. 6. Radial mass-to-charge ratio upper limits for thermal ions in a hyperbolic ICR ion trap, showing the m/z values at which 100% (uppermost curve in each set), 10% (middle curve in each set), and 1% (lowermost curve in each set) of the ions are radially lost as a function of trap voltage (measured across 2cm distance between end cap centers). Each set of curves corresponds to one of three different magnetic field strengths (3, 7, and 14T).

topmost curve in each set in Fig. 6) is simply the critical mass initially defined by Ledford et al. [ 171,for that trapping potential and magnetic field strength. Axial ion loss

Inspection of eqn. (20) suggests methods to minimize axial loss: decrease T, increase V,, increase q, or increase the trap aspect ratio (z,/r,). Plots of axial loss vs. ion translational energy, trap voltage, and magnetic field strength are also shown in Figs. 2-4. For example, Fig. 2 shows that axial loss is negligible (< 5%) for thermal ions at translational energy < 0.07 eV (at V, = 1 V across 2cm) or for room-temperature thermal ions at V, > 0.5 V across 2cm. Finally, Fig. 5 illustrates that axial loss of thermal ions is mass-independent; rather, axial loss depends only upon translational energy z component. Experimentally, axial loss of resonantly excited ions is known to be massdependent if the electric r.f. excitation field contains a z component, as for cubic [30,3 11, cylindrical [32], tetragonal, and non-infinitely extended hyperbolic traps [33-391 or if the trap voltage changes suddenly [40]. The z ejection may be effectively eliminated by “shimming” the r.f. electric field to near-uniformity, by the addition of “guard. rings” [39,41,42] or trap electrode segmentation [43], or by use of a capacitively-coupled three-section elongated trap [44].

M.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

202

Overall (radial plus axial) storage efficiency for thermal ions in a hyperbolic trap

For ions in a quadrupolar potential, the solutions to the equations of motion separately involve an r coordinate motion (eqn. (5a)) and a z coordinate motion (eqn. (5b)); for this case, axial and radial loss are independent (in the absence of particle/particle interactions). The net probability that an ion will be lost, either radially or axially, is the sum of the probabilities of radial and axial loss (&al + Pradial),minus the probability of loss in both directions (&al x I’mdial). In Fig. 2, the overall ion loss curve is virtually indistinguishable from the axial-only loss curve for ion translational energy up to x 40 eV (at V, = 1 V across 2 cm, B = 3 T, m/z = 100). Figure 3 shows that for room-temperature ions (T = 300 K), axial loss predominates at low trap voltage (< 1 V), whereas radial loss shows a sudden and overwhelming onset at x 430 V across 2 cm (for m/z = 100, B = 3 T). Figure 4 shows radial (and overall) ion loss of room-temperature ions (T = 300 K) is significant only at very small magnetic field strength (B < O.O6T, at V, = 1 V across 2cm, m/z = 100). Finally, Fig. 5 shows that radial (and overall) loss of roomtemperature ions (T = 300 K) become significant only for m/z 2 35 000 (at V, = 1 V across 2 cm, B = 3 T). Storage efficiency for thermal ions in ICR traps of non-hyperbolic shape

Other common ICR trap geometries include tetragonal (of which cubic is a special case) or cylindrical configuration. A quadrupolar potential, V(x,y,z), exists near the center of each of these traps: V(x,y,z) = b

[

y-

-$(x2 + y2 -

2z2) 1

(21)

in which 01and y are constants determined by the trap geometry and a is a characteristic trap dimension [16]. Thus, the equations developed here for the hyperbolic trap are directly applicable near the geometric center of a tetragonal or cylindrical ICR trap. For example, comparison of eqn. (21) with eqns. (2) and (9) yields an expression for the critical mass for a tetragonal trap of dimensions a x a x c, merit= qa2B2/8aVT, in which a is a trap geometry factor [45]. Next, if r,, = a/2 in eqn. (14), an expression for radial ejection from a tetragonal ICR trap is obtained:

eadial

=

exp

The probability for axial ion loss from a tetragonal ICR cell is obtained by an

M.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

203

analysis similar to that for the hyperbolic cell, with the axial trap limit now given by z = f c/2: 1dz. C

(23)

Alternatively, for a cylindrical trap, the same eqn. (21) describes the quadrupolar potential near the center of that ICR trap when the corresponding cylindrical trap parameters are substituted for a, y, and a [46]. For a cylindrical trap of diameter, d, and length, c, eqn. (22) gives the radial loss probability, provided that a is replaced by d. Likewise, the axial ejection probability for the cylindrical trap is given by eqn. (23), in which ccylindfi& replaces ctetragonal and Ycylindrical

rep1aces

hetragonal -

Limitations of the present theory In the Theory section, we assumed that the pre-ionized neutrals have a well-defined Boltzmann temperature. Of course, for any FT-ICR experiment, it is necessary that the initial distribution of magnetron and cyclotron orbital radii be small compared to the trap radius, in order that subsequent r.f. electric field excitation produce a phase-coherent ion packet [47]. Moreover, for axial loss, we suppose that ions are distributed uniformly along the z axis (i.e. r = 0), a good approximation just after ions are formed by electron ionization. (The quite different FT-ICR MS m/q upper limit corresponding to the initial ion velocity distribution resulting from matrix-assisted laser desorption has been described elsewhere [25].) It is also worth noting that ions initially off-axis can be axialized by a combination of quadrupolar electric field resonant excitation in the presence of ion/molecule collisions [48,49]. In any case, the position and velocity distributions of the ions evolve in time owing to the trapping potential, ion/neutral collisions, ion/ion interactions, and (especially) r.f. electric field excitation. For example, Rempel et al. have derived “pre-excitation” expressions for the FT-ICR ion distribution as a function of the ion z axis amplitude [38], and Chen et al. [50] have analyzed the detected ICR signal strength as a function of the initial ion z distribution. Thus, experimental FT-ICR upper mass and energy limits will be lower than those calculated here. Extension of upper mass-to-charge ratio limit As seen from eqn. (9) (critical mass) and eqn. (14) (upper mass limit for finite-size trap), the radial upper mass-to-charge ratio limit may be increased by reducing the trapping potential or by increasing the aspect ratio. For

204

M.A. May et al./Int. J. Mass Spectrom. Ion Processes 120 (1992) 193-205

example, the electrostatic radial electric field in a tetragonal trap with aspect ratio of 6 : 1 is reduced by a factor of 10’ at the trap center, compared to that of a cubic trap of the same radius [51]. Alternatively, placement of grounded screens just inside each trap plate can reduce the radial electric field by a factor of x 100 throughout the interior of the trap [52]. Thus, the upper mass-tocharge ratio limit owing to radial ion loss may be increased significantly by relatively simple changes to the trap configuration. ACKNOWLEDGMENTS

The authors thank Xinzhen Xiang, Troy Wood, Ruidan Chen, Mingda Wang, and Lutz Schweikhard for helpful discussions. This work was supported by the U.S.A. National Science Foundation (CHE-90-21058) and The Ohio State University. REFERENCES 1 B. Asamoto and R.C. Dunbar, Analytical Application of Fourier Transform Ion Cyclotron Resonance Mass Spectrometry, VCH, New York, 1991. 2 M.P. Chiarelli and M.L. Gross, in C.S. Creaser and A.M.C. Davies (Eds.), Analytical Applications of Spectroscopy, Royal Society of Chemistry, London, 1988, p.2263. 3 R.B. Cody, Analysis, 16 (1988) 30. 4 B.S. Freiser in Bonding Energies in Organometallic Compounds, Vol. 428, American Chemical Society, Washington, D.C., 1990, p. 55. 5 S. Ghaderi, Ceram. Trans., 5 (1989) 73. 6 C.D. Hanson, E.L. Kerley and D.H. Russell, in J.D. Winefordner (Ed.), Treatise in Analytical Chemistry, Vol. 11, Wiley, New York, 1988, p. 117. 7 D.A. Laude, Jr. and J.D. Hogan, Tech. Messen, 57 (1990) 155. 8 D.M. Lubman (Ed.), Lasers in Mass Spectrometry, Oxford University Press, New York, 1990. 9 A.G. Marshall and P.B. Grosshans, Anal. Chem., 63 (1991) 215. 10 A.G. Marshall and L. Schweikhard, Int. J. Mass Spectrom. Ion Processes, 118/l 19 (1992) 37. 11 N.M.M. Nibbering, Act. Chem. Res., 23 (1990) 279. 12 P. Sharpe and D.E. Richardson, Coord. Chem. Rev., 93 (1989) 59. 13 K.-P. Wanczek, Int. J. Mass Spectrom. Ion Processes, 95 (1989) 1. 14 C.L. Wilkins, A.K. Chowdhury, L.M. Nuwaysir and M.L. Coates, Mass Spectrom. Rev., 8 (1989) 67. 15 M.B. Comisarow, J. Chem. Phys., 69 (1978) 4097. 16 P.B. Grosshans, P.J. Shields and A.G. Marshall, J. Chem. Phys., 94 (1991) 5341. 17 E.B. Ledford, Jr., D.L. Rempel and M.L. Gross, Anal. Chem., 56 (1984) 2744. 18 L.S. Brown and G. Gabrielse, Rev. Mod. Phys., 58 (1986) 233. 19 D.L. Rempel, E.B. Ledford, Jr., S.K. Huang and M.L. Gross, Anal. Chem., 59 (1987) 2527. 20 W.W. Yin, M. Wang, A.G. Marshall and E.B. Ledford, Jr., J. Am. Sot. Mass Spectrom., 3 (1992) 188. 21 R.D. Knight, Int. J. Mass Spectrom. Ion Processes, 51 (1983) 127.

M.A. May et aLlInt. J. Mass Spectrom. Ion Processes 120 (1992) 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

49 50

51

52

193-205

205

L. Schweikhard, M. Lindinger and H.-J. Kluge, Rev. Sci. Instrum., 61 (1990) 1055. J. Byrne and P.S. Farago, Proc. Phys. Sot., 86 (1965) 801. A.G. Marshall and F.R. Verdun, Fourier Transforms in NMR, Optical, and Mass Spectrometry: A User’s Handbook, Elsevier, Amsterdam, 1990, p. 460. T.D. Wood, L. Schweikhard and A.G. Marshall, Anal. Chem., 64 (1992) 1461. M. Bloom and M. Riggin, Can. J. Phys., 52 (1974) 436. M. Bloom and M. Riggin, Can. J. Phys., 52 (1974) 456. W.H. Beyer (Ed.), CRC Standard Mathematical Tables, CRC Press, Boca Raton, FL, 1987. M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, U.S. Department of Commerce, U.S. Government Printing Office, Washington, D.C., 1972. M.B. Comisarow, Adv. Mass Spectrom., 8 (1980) 1698. M.B. Comisarow, Int. J. Mass Spectrom. Ion Phys., 37 (1981) 251. M.B. Comisarow and A.G. Marshall, U.S.A. Patent 3,937,955, 14 February, 1976. P. Kofel, M. Allemann, H. Kellerhals and K.-P. Wanczek, Int. J. Mass Spectrom. Ion Processes, 74 (1986) 1. W.J. van der Hart and W.J. van de Guchte, Int. J. Mass Spectrom. Ion Processes, 82 (1988) 17. W.J. van de Guchte and W.J. van der Hart, Int. J. Mass Spectrom. Ion Processes, 95 (1990) 317. M. Alleman, P. Kofel, H. Kellerhals and K.-P. Wanczek, Int. J. Mass Spectrom. Ion Processes, 75 (1987) 47. SK. Huang, D.L. Rempel and M.L. Gross, Int. J. Mass Spectrom. Ion Processes, 72 (1986) 15. D.L. Rempel, S.K. Huang and M.L. Gross, Int. J. Mass Spectrom. Ion Processes, 70 (1986) 163. M. Wang and A.G. Marshall, Anal. Chem., 62 (1990) 515. S.A. Hofstadler and D.A. Laude Jr., Int. J. Mass Spectrom. Ion Processes, 101 (1990) 65. H. Sommer, H.A. Thomas and J.A. Hipple, Phys. Rev., 82 (1951) 697. CD. Hanson, M.E. Castro, E.L. Kerley and D.H. Russell, Anal. Chem., 62 (1990) 520. P. Caravatti and M. Allemann, Org. Mass Spectrom., 26 (1991) 514. SC. Beu and D.A. Laude Jr., Proc. 39th ASMS Conference on Mass Spectrometry and Allied Topics, Nashville, TN, 1991, p. 130. J.B. Jeffries, S.E. Barlow and G.H. Dunn, Int. J. Mass Spectrom. Ion Processes, 54 (1983) 169. P.B. Grosshans and A.G. Marshall, Anal. Chem., 63 (1991) 2057. M. Wang and A.G. Marshall, Int. J. Mass Spectrom. Ion Processes, 100 (1990) 323. L. Schweikhard, S. Becker, G. Bollen, H.-J. Kluge, G. Savard, H. Stoizenberg, U. Wiess and R.B. Moore, Proc. 38th ASMS Conference on Mass Spectrometry and Allied Topics, Tucson, AR, 1990, p. 415. L. Schweikhard, S. Guan and A.G. Marshall, Int. J. Mass Spectrom. Ion Processes, 120 (1992) 71. R. Chen, P.B. Grosshans and A.G. Marshall, unpublished results, 1992. P.B. Grosshans, M. Wang, T.L. Ricca, E.B. Ledford, Jr. and A.G. Marshall, Proc. 36th ASMS Conference on Mass Spectrometry and Allied Topics, San Francisco, CA, 1988, p. 592. M. Wang and A.G. Marshall, Anal. Chem., 61 (1989) 1288.