Charge Capacity Limitations of Radio Frequency Ion Guides in Their

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Anal. Chem. 2000, 72, 970-978

Charge Capacity Limitations of Radio Frequency Ion Guides in Their Use for Improved Ion Accumulation and Trapping in Mass Spectrometry Aleksey V. Tolmachev, Harold R. Udseth, and Richard D. Smith*

Environmental and Molecular Sciences Laboratory, Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352

The use of radio frequency (rf) ion guides as “linear” twodimensional ion traps and ion guides for ion storage and accumulation, respectively, is becoming increasingly important for realizing improved sensitivity in mass spectrometry. Analytical relationships describing the ion accumulation operation mode of rf ion guides are reported. Comparisons are made between the rf quadrupole ion guide, higher-order rf multipoles and rf stacked ring ion guides, in terms of the charge capacity limitations due to the instability of ions, rf focusing efficiency limits, and effects due to rf ion heating (i.e., collisional activation due to rf oscillations of ions). Analytical relations for the stored charge quantity are derived in the low ion energy approximation, which is shown to be reasonable for the systems considered. The ion density spatial distribution is derived, an exponential form of which proved to provide a good approximation for high-order rf multipoles and stacked ring rf ion guides. The limit on the stored charge dependence upon rf is shown to be directly related to the thermal dissociation thresholds for the ions being studied; the limitation is weaker for higher-order multipoles and stacked ring ion guides. These results suggest that rf quadrupoles provide an optimum configuration when accumulation of a moderate ion density is sufficient (below 109 elementary charges/m). Alternatively, accumulation of an appreciable density for more fragile species, such as noncovalent complexes, may be realized using higherorder multipoles and stacked ring ion guides. External ion accumulation was first introduced in time-of-flight mass spectrometry (TOF MS) as a way of coupling a continuous ion source by accumulating and storing ions in a three-dimensional “quadrupole” ion trap and then “pulsing” the trapped species to the TOF mass analyzer.1,2 Ion accumulation has also been demonstrated using a two-dimensional (2D) ion guide. Such electrodynamic radio frequency (rf) ion guides are now commonly used to accumulate ions prior to transfer to the trapped ion cell used for measurements in FTICR mass spectrometers.3,4 This is (1) Qian, M. G.; Lubman, D. M. Anal. Chem. 1995, 67, 2870-7. (2) Wilhelm, U.; Weickhardt, C.; Grotemeyer, J. Rapid Commun. Mass Spectrom. 1996, 10, 473-477. (3) Senko, M. W.; Hendrickson, C. L.; Emmett, M. R.; Shi, S. D. H.; Marshall, A. G. J. Am. Soc. Mass Spectrom. 1997, 8, 970-976.

970 Analytical Chemistry, Vol. 72, No. 5, March 1, 2000

feasible since an rf ion guide can be used for both the transmission and the accumulation of ions (i.e., also operate as a 2D ion trap). This approach has been attractive since it allows, in principle, one to improve an overall ion utilization up to unit efficiency and to potentially simultaneously preselect ions prior to transfer and mass analysis. Attributes relevant to optimizing performance (e.g., sensitivity) for an ideal 2D ion trap include having a sufficient charge capacity, rf excitation of the stored ions insufficient for undesired activation or fragmentation, the capability for fast ion extraction, and good ion optical qualities of the extracted ion beam. Design of such a device involves taking into account complex issues including the motion of ions in rf multipole fields, fringing field considerations, space charge, and effects due to ion-neutral collisions that properly account for rf excitation. In the present work, we consider the issue of stored charge quantities for rf multipoles and rf stacked ring ion guides. The treatment is primarily based upon approximate analytical approaches enabling facile estimation of key parameters. The limitations on the stored charge capacity are considered due to rf confinement properties and due to rf heating of ions, and the value of a dc potential required to trap the stored ions is also estimated. The spatial density distribution of the stored ions is estimated and compared for important rf ion guide configurations. We also describe and apply a field balance approach for the estimation of space charge distributions and the spatial density distribution of ions in rf multipoles and rf stacked ring guides. The range of applicability of the low-temperature approximation is analyzed, and the stored charge amount is estimated for which the thermal broadening of the radial ion distribution becomes significant. METHODS The rf ion guide configurations considered here are shown schematically in Figure 1. Figure 1a shows an rf quadrupole ion guide, as an example of the rf multipole, having N ) 2 pairs of rods. The input and exit apertures have dc offset Vdc relative to the multipole rods, necessary to trap ions inside. The left cross section view shows a schematic of the rf supply to the multipole rods. It also illustrates the definition of the inscribed radius of a multipole F. The rf voltage notation Vrf used throughout the paper (4) Sannes-Lowery, K.; Griffey, R. H.; Kruppa, G. H.; Speir, J. P.; Hofstadler, S. A. Rapid Commun. Mass Spectrom. 1998, 12, 1957-1961. 10.1021/ac990729u CCC: $19.00

© 2000 American Chemical Society Published on Web 01/11/2000

Figure 1. Schematic diagram of the rf ion guides. (a) Multipole ion guide. The case of a quadrupole, N ) 2, is shown. The dc voltage layout is shown on the left isometric view: Vdc offset is applied to the input and exit orifices; the rods are dc grounded. The rf connection scheme is depicted on the cross section right view. (b) stacked ring ion guide; the exit part is not shown. The r and z axes show the cylindrical coordinate system used; dSR is the distance between ring electrodes; F is the radius of the ring electrode aperture.

corresponds to the amplitude of the rf voltage, or half of the maximum potential difference between the neighboring rods, also called peaf-to-peak voltage Vp-p ) 2Vrf. Figure 1b shows a schematic of the stacked ring ion guide. The exit part, which is symmetrical to the front part, is not shown. The radial coordinate r and axial z are shown. The dimensions used are spacing between rings dSR and the radius of the rings’ apertures F. An estimate of the charge capacity of the 2D rf ion guide may be obtained on the basis of the equilibrium condition between the space charge repulsion field ESC and the effective rf focusing field E*: ESC ) -E*. This relationship gives the equilibrium condition for the case of negligible radial kinetic energy of ions and zero temperature of a buffer gas, T f 0. This relationship is convenient due to its simplicity; we later consider the effect of real buffer gas temperatures. For a cylindrically symmetric ion guide configuration

ESC(Rq) ) QL/2π0Rq

(1)

Here QL is the charge per unit length of the ion guide, Rq is the ion cloud radius, and 0 is the dielectric permittivity constant. At equilibrium, the space charge field is compensated by effective rf focusing field E*, defined as the gradient of the effective potential V*: 5,6

E* ) -∇V*; V* ) qErf2/4Mω2

and e is the elementary charge. These relations are general and may be applied for any rf ion guide (or ion trap) configuration, provided that adiabatic conditions apply. We assume further that the buffer gas pressure is low enough so that its influence on the effective rf potential is negligible; this is justified in the pressure range P < 1 Torr and rf frequency f ∼ 1 MHz, corresponding to the present study.7 We first consider the case of cylindrical symmetry with the space charge density qn and rf field independent of z (n, number density of ions). We may introduce a partial linear charge density ql(r) as a charge per unit length of the ion cloud occupying radial interval from 0 to an arbitrary radial position inside the ion cloud, r < Rq. Thus, the total linear charge density QL ) ql(Rq). The field balance requirement is also valid inside the ion cloud. Due to symmetry, the space charge field is defined only by the partial linear charge for a given radial position; the outer charges result in the zero total field. Thus, the field balance inside the ion cloud is

ESC(r) ) q1(r)/2π0r ) -E*(r), r e R

and we arrive at the following relation between the charge density qn(r) and the rf effective potential:

qn(r) )

(5) Dehmelt, H. G. Adv. At. Mol. Phys. 1967, 3, 53-72. (6) Gerlich, D. In State Selected and State-to State Ion-Molecule Reaction Dynamics. Part 1. Experiment; Ng C. Y., Baer M., Eds.; Wiley: New York, 1992; Vol. LXXXII, pp 1-176.

1 d 1 dq1 dV* ) 0 r , r e Rq 2πr dr r dr dr

(

)

(4)

Equation 4 allows one to estimate the ion density in any rf ion guide for which the cylindrical symmetry and z-independent behavior may be assumed. However, the approximation of low ion energy leads to an underestimation of the maximal ion cloud radius Rq (see below). An alternative approach has been reported, where the ion density is derived using a self-consistent Poisson’s equation and the Gibbs distribution; eq 4 also corresponds to the low-temperature limit, T f 0. 8 RESULTS AND DISCUSSION Charge Capacity of the rf Multipoles. For the case of a multipole having N pairs of rods, the absolute value of the rf field intensity is

Erf ) (NVrf/F)(r/F)N-1

(5)

where F is the aperture radius and Vrf is the amplitude of the rf voltage applied to the rods. Thus, we have

(2)

where Erf is the amplitude of the local rf electric field, the angular frequency is ω ) 2πf, M is the ion mass, q ) ke, the ion’s charge,

(3)

V* )

( )()

q NVrf M 2Fω

2

r F

2(N-1)

(6)

and (7) Tolmachev, A. V.; Chernushevich, I. V.; Dodonov, A. F.; Standing, K. G. Nucl. Instrum. Methods Phys. Res. B 1997, 124, 112-119. (8) Guo-Zhong, Li; Jarrell, J. A. Proceedings of the 45th ASMS Conference on Mass Spectrometry and Allied Topics, Orlando, FL, 1998; p 491.

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E* ) -2(N - 1)

( )()

(N - 1)q NVrf V* )r ω 2MF3

2

r F

2N-3

(7)

Using field balance eq 3, we obtain the low-temperature estimate of the charge per unit length of a multipole:

( )( )

q NVrf q1(r) ) π0(N - 1) M Fω

2

N ) 2 (quadrupole): A2 ) 4qVrf/Mω2F2 ) qM (13)

2(N-1)

r F

(8)

In this low-temperature approximation, we assume that the position of an ion is completely defined by the balance of the space charge field and effective rf potential force; the distortions caused by ion kinetic energy and thermal ion-bath gas interactions are neglected. Equation 8 may be used to derive the ion density distribution as follows:

qn(r) )

the maximum useful stored charge. However, in practical cases, the stability limit may occur for lower AN(r) values, in the range ∼0.3-1. The stability condition (12) holds for any radial position of stored ions. In the case of a 2D quadrupole, the condition is independent of the radial coordinate:

( )( )

1 dq1(r) q NVrf ) 0(N - 1)2 2πr dr M F2ω

2

r F

2(N-2)

(9)

Here qM is the Mathieu parameter, which defines stability conditions in an rf-only quadrupole as qM < 0.9. 9 This is the so-called low-mass cutoff, which is very sharp for rf-only quadrupoles, due to independence of the stability criterion on r. In general, for any N the rf multipole parameters should be adjusted so as to provide stability conditions taking into account the radial ion distribution and mass-to-charge ratio(s) of the stored ions. In practice, AN of eq 12 should be 0.3.6 We use the unit limit (12), since our aim is to estimate 972

Analytical Chemistry, Vol. 72, No. 5, March 1, 2000

Here

This equation gives an upper limit to the amount of charge stored in an rf multipole. The limit may be approached only if the stability parameter AN(r) (12) is adjusted so that it approaches 1 at the maximal radial position, AN(F) ) 1. Note that AN(r) ∼ 1 implies that rf oscillations of ions develop an amplitude comparable to a local scale of variations of the rf field amplitude Erf(r). Thus, an ion cloud boundary extends beyond a maximum average ion radius by the amplitude of the fast oscillations. Taking this into account, one may derive a more accurate limit for the stored charge, which must be less than QMAX (15). The relation (15) is independent of frequency, multipole radius F, and ion mass and charge; only the multipole order N and rf voltage Vrf are significant. We conclude that the radial size of the multipole ion guide does not influence its stored charge capacity as long as the same rf voltage Vrf may be applied without exceeding the stability requirement. Comparing the instability offset in quadrupoles and higherorder multipoles, we see that in the quadrupole case the trap can always be filled up to the limiting capacity when the ion cloud radius approaches F, unless the combination of F, Vrf, ω, and M/q is such that A2 > ∼1, in which case the trap fails to trap any ions. In contrast, for a multipole with N > 2, the trap will normally fill up to the ion cloud radius at which the stability parameter approaches ∼1, and no ions will be trapped at radii beyond this limit. Finally, we estimate a maximum value for stored charge (15). For the case of a quadrupole, N ) 2, Vrf ) 1000 V: QMAX ∼ 3 × 10-8 C/m ∼ 2 × 1011 e/m. Note that eq 15 assumes ideal (9) Dawson, P. H., Ed. Quadrupole Mass Spectrometry and Its Applications; NorthHolland: Amsterdam, 1976.

multipolar fields up to the maximum radius F, which is not the case in practical devices. Most significantly, for quadrupoles, the maximum ion cloud radius will be somewhat less than F. The stability limits (12) and (15) also contribute to overestimations, as discussed above. Thus, one might expect to trap a maximum 1, to avoid potential wells between ring Analytical Chemistry, Vol. 72, No. 5, March 1, 2000

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electrodes. Such a ring guide corresponds in terms of charge capacity to a multipole having 2N > 6 rods. Using the same rf voltage as above for a quadrupole, Vrf ) 1000 V, we estimate a higher capacity by at least a factor of 3/2, QMAX > 3 × 1011 e/m. This upper estimate corresponds to the case of low-energy ions at zero temperature buffer gas and assumes the ion cloud fills the whole ion guide aperture 2F. To approach the maximum capacity, it is necessary to adjust rf parameters so that

ASR(F) ) 2qVrf/Mω2δ2 ∼ 1

Charge Capacity Limit due to rf Heating of Ions. Ions interacting with an rf field perform oscillations with corresponding frequency. In adiabatic approximation, when the amplitude of the oscillations is smaller than a characteristic scale of variations of the rf field amplitude distribution, the average kinetic energy Krf of the rf motion is exactly equal to the effective potential V* (2).5,6

Krf ) V*

(26)

The kinetic energy (in volts per unit charge) is assumed to be

Krf ) 〈Mu2/2q〉

(27)

u being the velocity of rf-induced ion motion. Collisions of a fastmoving ion with buffer gas molecules may lead to its fragmentation, as commonly used in mass spectrometry based upon 3D ion traps10-13 and, more recently, using 2D rf multipole ion guides.14-16 The requirement for storing of ions that are sensitive to the collisional-induced dissociation creates an additional limitation on the number of ions that may be stored in an rf 2D trap/ion guide. To get an upper estimation of the rf heating, we assume that the time of storing ions in the rf guide is long enough for the internal ion temperature (Tion) to reach an equilibrium corresponding to the average collision energy as defined in eq 27. A similar approach was previously used to interpret experimental results on selective fragmentation of ions in an rf collisional quadrupole.14 This approach assumes that the temperature Tion corresponds to a thermal bath where the average collision energy in the center of mass system is the same as in the real situation. Thus: (10) March, R. E.; Huges, R. J. Quadrupole Storage Mass Spectrometry; John Wiley: New York, 1991. (11) Goeringer, D. E.; McLuckey, S. A. J. Chem. Phys. 1996, 104 (6), 22142221. (12) Goeringer, D. E.; McLuckey, S. A. Rapid Commun. Mass Spectrom. 1996, 10, 328-334. (13) Asano, K. G.; Goeringer, D. E.; McLuckey, S. A. Int. J. Mass Spectrom. 1998, 179. (14) Dodonov, A.; Kozlovsky, V.; Loboda, A.; Raznikov, V.; Sulimenkov, I.; Tolmachev, A.; Kraft, A.; Wollnik, H. Rapid Commun. Mass Spectrom. 1997, 11, 1649-1656. (15) Campbell, J. M.; Collings, B. A.; Douglas, D. J. Proceedings of the 45th ASMS Conference on Mass Spectrometry and Allied Topics, Orlando, FL, 1998; p 40. (16) Loboda, A. V.; Krutchinsky, A. N.; Spicer, V.; Ens, W.; Standing, K. G. Proceedings of the 45th ASMS Conference on Mass Spectrometry and Allied Topics, Orlando, FL, 1998; p 37.

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Tion ) T + (m/M)(qKrf/Cg)

(28)

where T, m, and Cg are the temperature, molecular mass, and molecular heat capacity of the buffer gas, respectively, for the case where M . m. Using eq 26, the rf heating constraint may be formulated as

5 kT m qKrf m V* < < T or M Cg M 2 q

(29)

where Cg ) 5/2k; k is the Boltzmann constant. Considering the CID limit on the effective potential value V* above which the storage of intact ions is unlikely

V* < VCID )

5 kT M 2 q m

(30)

It should be noted that the VCID definition (30) is necessarily arbitrary, and it does not take into account the threshold energy for the ion fragmentation. A lower limit than (30) would probably apply for labile noncovalent complexes of biomolecules (such as produced by electrospray ionization17). However, where eqs 29 and 30 apply, the rf input to ion temperature is not appreciable and should be appropriate for ion storage. A more detailed description of the ion dissociation due to the rf heating must take into account a dissociation threshold energy, the ion trapping time, a number density of a bath gas, and all other relevant parameters. The simplified approach using a characteristic effective potential VCID (30) gives only a rough estimate of the dissociation offset but is useful here for comparison of different rf trap configurations. Now we can estimate the amount of stored charge corresponding to the effective potential satisfying eq 30. For rf multipoles, from eqs 6 and 8

QL ) 4(N - 1)π0V* < 4(N - 1)π0VCID

(31)

We see that the stored charge limitation from CID energy is defined only by the dissociation threshold VCID. The ion guide parameters, including radial dimensions, rf voltage, and frequency, have no influence. The only way to increase the charge capacity for a given VCID is to increase the number of rods (2N). To estimate the limit (31) for a quadrupole (N ) 2) at M/q ) 1000 Da, we have VCID ∼2 eV and arrive at QL < 2.3 × 10-10 C/m ∼ 1.4 × 109 e/m. The value is much less than the full charge capacity of rf multipole estimated earlier as QMAX ∼ 3 × 10-8 C/m ∼ 2 × 1011 e/m. This means that only a small fraction of a quadrupole aperture can be filled by ions while avoiding rf heating and subsequent fragmentation, in this case Rq/F < 0.1 (if Vrf ) 1000 V and stability term qM approaches 0.9). For the case of an octopole, N ) 4, and the same M/q, according to eq 31, the charge limit is 3 times larger than for a quadrupole: QL 8 rods. Comparing the stacked ring guide to the quadrupole and octopole cases above, we see that the rf fragmentation limit here is higher than for the quadrupole and similar to that for the 2N pole device having N ) πF/dSR + 1. We also expect a more extended ion cloud, as in case of multipoles, i.e., the ion density radial distribution shallow at the axis, with a major fraction of ions concentrated at the r ) Rq position. An advantage of the stacked ring guide is the possibility of creation of any desired axial dc potential profile. This is useful both for application of the end trapping dc and for fast and efficient ion extraction. One additional consequence of these relationships concerns the quality of the ion beam extracted from the 2D ion trap. We may estimate the radial component of the kinetic energy of exiting ions Kr as approximately equal to the average energy of rf oscillations; see eqs 26 and 27. From eq 31, Kr ∼ V* ) QL/4π0(N - 1). We see that higher N results in a reduced radial energy spread for extracted ions. Generally, we conclude that it is possible to increase the stored charge limit due to rf heating by using higher-order multipoles or the stacked ring ion guide. A smaller distance between neighboring electrodes relative to the ion guide aperture provides a larger ion capacity for multipoles and for stacked ring guides. On the other hand, this solution leads to very flat potential well at the axis, so the ion cloud tends to become distributed along the inner cylindrical surface of an ion guide. By contrast, in the rf quadrupole ions are better axialized, which facilitates extraction and subsequent focusing. Thus, the rf quadrupole provides a better solution if the requirements on the stored charge amount

are not severe and/or the collisional stability of ions is not crucial. Radio frequency ion heating becomes less important at higher M/q; see eq 30, VCID ∝ M/q. Charge Capacity in Terms of dc Trapping Potential. The charge capacity per unit length estimated above, is limited by rf focusing and rf heating properties of the ion guides. One must also consider the dc potential Vdc (i.e., the trapping potential) that is necessary at the both ends of the ion guide to trap the stored charge. We consider the case of an rf quadrupole ion guide; a similar approach may be applied for higher-order multipoles and stacked ring guides. We assume here that the linear charge density QL(z) is constant along the axis of a 2D trap, which is true when the length is much greater than the radial dimension. (Numerical simulations of the QL(z) distribution (to be reported elsewhere) show that the density at the ends does not deviate much from its value inside the trap.) As an approximation, we replace the actual configuration by a semi-infinite coaxial cylindrical system of charges. The inner cylinder of radius Rq having a constant charge density qn, corresponds to the ion cloud of the stored charge, and the outer cylinder of radius F corresponds to the ion guide electrodes, linear density dq/dz ) -πRq2qn. The trapping dc potential must create an axial field sufficient to balance the fringing field of the space charge Ef:

Ef )

∫ ∫ ∞

z)0

2πqnrz dr dz r)0 C0(r2 + z2)3/2 Rq

∫

πRq2qnz dz

∞

C0(F2 + z2)3/2

z)0

(

)

)

Rq 1 πqnRq 2 (34) C0 F Here C0 ) 4π0. The dc field (Edc) created by the trapping potential Vdc is dependent on the particular configuration of the end electrodes. For an approximate estimate we use the following relation:

Edc ≈ Vdc/F

(35)

Assuming the balance of the two field intensities (34) and (35) and inserting QL ) qnπRq2, we derive an estimate for the dc potential as a function of the trapped charge linear density QL:

Vdc )

(

2F 1 Q -1 C0 L Rq

)

(36)

In the case of a quadrupole, we can express relative ion cloud radius as a function of QL and dimensionless stability parameter qM (13), using (11):

() Rq F

2

)

QL qMπ0Vrf

(37)

Thus, the ion cloud cross section relative to the area of the quadrupole aperture is proportional to the ratio of the charge per unit length QL to the rf voltage amplitude Vrf. All other parameters (i.e., M/q, F, ω) are insignificant as long as the qM is kept constant, consistent with the stability requirement, qM < 0.9. Combining Analytical Chemistry, Vol. 72, No. 5, March 1, 2000

975

(36) and (37) we find

Vdc )

(

)

qM Rq Rq 2V 4 rf F F

(38)

This simple relationship allows one to estimate the relative ion cloud radius Rq/F corresponding to the saturation of the accumulated charge. The approach used here provides a rough estimate of the trapping potential needed, but still allows elucidation of the important parameters and estimation of characteristic values. For example, for a quadrupole having linear charge QL ∼ 2 × 10-10 C/m ∼ 109 e/m, estimated due to rf heating (31), eqs 37 and 38 give Rq/F ∼ 0.08 and Vdc ∼ 40 V (for Vrf ) 1000 V and qM ∼ 1). Estimation of the Temperature Effect on the Ion Density Distribution. All the above estimates assumed that thermal interactions of ions with buffer gas molecules are insignificant. A density distribution of ions taking into account temperature has been derived in ref 7 for negligible space charge; the relationship including space charge effects is given in ref 8. Here we estimate the ion density for which the thermal effects start to become significant. As shown above, in the zero kinetic energy-zero temperature approximation, the density distribution of ions stored in an ion guide has an abrupt cutoff at a certain radial position Rq, at which an equilibrium occurs between the space charge electric field and the effective rf focusing field. We designate the density at Rq as n0. We take into account the thermal broadening of the distribution in the first-order approximation, assuming that a certain gradient of the density is produced at the Rq position. This gradient creates a diffusive outward flow of ions, which must be balanced at equilibrium by the external field:

D(dn/dr) ) - µn(ESC - E*)

(39)

Figure 3. Estimation of the thermal broadening ∆r of the distribution density n(r); see text. Low-temperature approximation, solid line; thermal broadening, dashed line.

∆r2 ≈ (kT/q2)(160/n0)

The ∆r gives a characteristic radial scale for the thermal broadening of the density distribution. An alternative approach to the thermal broadening estimation is to use the Debye length, λD ) (0kT/q2n)1/2, i.e., the exponential length of the external field penetration into a plasma. The approach was recently used for an estimation of the low-temperature charge density distribution in the Paul trap.19 The ∆r estimated from eq 43 is larger than λD by a factor of 4, thus giving more constraining conditions for the low-temperature approximation to apply. rf Quadrupole. Consider first the case of the quadrupole, for which zero temperature ion density is independent of radius for r < Rq. Here the zero temperature approximation may be justified when the thermal broadening ∆r is small compared to the ion cloud radius Rq:

D, the diffusion coefficient, and µ, mobility, are related to each other by18

µ ) (q/kT)D

(43)

n0 . 160kT/q2 Rq2

(44)

QL ) π Rq2qn0 . QkT ) 16π0(kT/q)

(45)

(40) thus

Assuming that the broadening of the distribution has a characteristic radial width ∆r and is approximately symmetric relative to Rq position, we estimate

dn/dr ∼ n0/∆r; n(Rq) ∼ n0/2

(41)

The field imbalance ∆E ) ESC - E* is related to the decrease of space charge for the Rq position and is estimated as

∆E ) ∆ESC ) ∆q1/02πr ) qn0∆r/80

(42)

Here we have estimated the decrease of total linear charge ∆ql as ∆ql ) qn0πr∆r/4, from the triangle area to the left at the lowtemperature ion cloud boundary r ) Rq; see Figure 3. Substituting eqs 42 and 40 into 39, we obtain an estimation of ∆r: (18) McDaniel, E. W.; Mason, E. A. The Mobility and Diffusion of Ions in Gases; Willey-Interscience: New York, 1973.

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Analytical Chemistry, Vol. 72, No. 5, March 1, 2000

Here we introduce QkT, a linear charge density below which the thermal broadening produces a significant effect on the spatial distribution. Again, as in the rf heating limit (31), condition 45 is independent of M/q and the quadrupole parameters. We see that the limit (31) is larger than (45): VCID ∼ 2 V . 4kT/q ∼ 0.1 V. This confirms that a wide range of the stored charge amount below the rf heating limit exists for which the approximation used here is reasonable. The case of low space charge may also be of practical interest. In this case, when QL , QkT, we may use the space density distribution:7 (19) Li, G.-Z.; Guan, S.; Marshall, A. G. J. Am. Soc. Mass Spectrom. 1998, 9, 473-481.

( )

n(r) ∼ exp -

r2 Mω2F4 ; where RkT2 ) kT 2 2 2 RkT q Vrf

(46)

At low space charge conditions, the capacitance of an rf quadrupole Cq in terms of trapping dc potential becomes constant and independent of the stored charge amount:

Cq t

QL C0RkT ) for QL , QkT Vdc 2F - RkT

(47)

Here we have used the relationship (36), in which the ion cloud radius Rq is substituted by RkT, assuming that the Gaussian density distribution (46) may be replaced by the cylindrical ion cloud for the approximate estimation. Thus, we have two ranges for the accumulated charge QL for which the dependence of Vdc on QL is quite different. For low QL, the capacitance Cq is constant. For QL > QkT from eqs 36 and 37

Cq )

C0Rq , 2F - Rq

() Rq F

2

)

4QL for QL G QkT C0qMVrf

(48)

Figure 4 illustrates the resulting dependence of the rf quadrupole capacitance on QL. Higher-Order Multipoles and Stacked Ring Ion Guide. It is convenient here to use the approximate exponential form of the space density distribution (19), which may be applicable in the zero temperature approximation for multipoles having N . π and for rf stacked ring ion guides with aperture radius F . dSR (see subsection, Charge Capacity of the rf Stacked Ring Ion Guide). We can express the requirement of insignificant thermal broadening of the distribution as ∆r , δ/2, based upon eq 43 to arrive at

n0 . 640kT/q2δ2

(49)

Here n0 is the maximum density at cutoff, n0 ) n(Rq). Taking the integral of the exponent of eq 21 and taking into account Rq ≈ F and Rq . δ, we obtain for the total stored charge

F kT QL ≈ πFδqn0 . 64π0 ) QM kT δ q

(50)

Here we have introduced the amount of charge QM kT at which the thermal broadening essentially changes the density distribution relative to the zero temperature exponential form. We see that the QM kT for higher-order multipoles is larger than the similar quantity for the rf quadrupole QkT (45) by a factor of 4F/δ . 1; i.e., temperature has a significant effect on the ion density distribution for relatively higher total linear charge density. Comparing the limit given by eq 50 to the rf heating limit (33) we see that

QCID QM kT

)

5M F ; where QCID ) 4π0 V/CID 32m δ

(51)

Figure 4. Charge capacity of the rf quadrupole ion guide, defined as the ratio of linear charge density to dc trapping potential; see eqs 47 and 48, Vrf ) 1000 V; the stability parameter (13) qM ) 0.05 for the upper solid curve and qM ) 0.5 for the lower dashed curve. The dotted inserts show the intermediate region where both the temperature and space charge must be taken into account. The QkT (45) indicates the characteristic charge density above which the ion cloud radius is defined by space charge rather than by temperature. The QCID is the linear charge density limit due to rf heating; see eq 30. The maximum linear charge density QMAX (15) corresponds to the maximum possible ion cloud radius approaching the quadrupole rods.

Thus, we find that for sufficiently heavy ions (M . 6m), there is a range of QL below the rf heating limit (33) where the results calculated for zero temperature are indeed reasonable. The consideration of the temperature broadening of the ion density distribution should properly take into account the influence of the ion motion in intense rf and dc fields, which may increase the ions’ diffusion temperature (i.e., an effective temperature that replaces T in eq 40 to account for the average ion kinetic energy above thermal); this will be a subject of a future publication. The estimates of the thermal broadening based upon this diffusional approach are applicable only when the free path of ions is small compared to a characteristic scale of the system under consideration. This may not be the case for operation at lower pressures. Under these conditions, one may estimate the thermal broadening by the distance to which an ion having a given thermal kinetic energy may escape beyond the ion cloud radius Rq when traveling against the action of the effective focusing field excess over the space charge field. The resulting limit is lower than QkT (45) and QM kT (50) by a factor of 16. Thus, an ion having a thermal kinetic energy cannot escape far from the Rq position as a result of single free path movement. CONCLUSIONS The operation of rf ion guides as 2D ion traps has been modeled. The charge capacity limits due to rf focusing and ion stability have been obtained. In the case of rf multipoles, the charge capacity is shown to be independent of rf frequency, multipole radius F, and ion mass and charge; only the multipole order N and rf voltage are significant. Thus, for a given rf voltage, the radial size of the multipole ion guide does not influence its stored charge capacity. For stacked ring guides, we arrive at a similar relationship; instead of N, the ratio of the ring radius to the spacing multiplied by π applies. The spatial density distribution of ions has also been derived; the exponential form provides a reasonable approximation for Analytical Chemistry, Vol. 72, No. 5, March 1, 2000

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higher order rf multipoles and stacked ring rf ion guides. Interestingly, the space charge results in a radial ion density distribution concentrated around the ion cloud outer boundary Rq, with the inner axial region having a much lower density. By contrast, in an rf quadrupole the ion density is constant, independent of radius in the r < Rq interval. The limit of the stored charge set by rf heating and dissociation has also been considered. It is shown that the stored charge density which causes the onset of collisional dissociation is primarily defined by the dissociation energy threshold. The ion guide parameters, including radial dimensions, rf voltage, and frequency, have no significant influence. The only way to increase stored charge for a given dissociation threshold is to increase the number of multipole rods 2N. The dc potential (e.g., applied on the “end caps”) necessary to trap ions inside the rf ion guide and the thermal broadening of the zero temperature spatial density distribution have been estimated. This approximation is justified when the stored charge is sufficiently large so that the ion cloud radius exceeds the thermal radius resulting from a Boltzman distribution. In all cases, the rf quadrupole has a somewhat smaller charge capacity than higher-order rf multipoles and stacked ring rf ion guides. On the other hand, it has a unique property of better confinement charge close to the axis, thus providing potentially higher efficiency for ion extraction. Thus, for a stored charge below 109 charges/m, rf quadrupoles provide an optimal choice for ion “accumulation” (e.g., before a time-of-flight or ICR mass analyzer). Alternatively, extensive accumulation of fragile species, such as noncovalent complexes, may be better realized using higher-order multipoles and stacked ring ion guides. Finally, these results shed light on recent observations related to ion accumulation times and the use of high trapping (end cap)

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potentials.4 It is clear that extended accumulation of ions can result in “overfilling” of the ion trap, causing expansion of the radial dimension of the ion cloud and increasing the collisional excitation of trapped ions. Since heating of the outer part of the ion cloud can be expected to increase the effective temperature of the entire ion cloud, the overall extent of activation and dissociation can be easily affected. While this can be disadvantageous, it can also provide an alternative to other ion excitation schemes. In fact, using a segmented multipole or stacked ring devices, it is feasible to use the axial field strength to compress an ion cloud along the axial dimension, so as to expand the radial dimension. Such an approach would easily and rapidly allow controlled excitation of an ion cloud for obtaining precise levels of activation/dissociation. Experimental studies using this approach may also provide important information on nonequilibrium effects related to ion cloud shape and estimates of ion mixing. It should be noted that these phenomena and their treatment span the continuum between the cases of 2D and 3D ion traps. ACKNOWLEDGMENT Portions of this research were supported by the U.S. Department of Energy, Office of Biological and Environmental Research, and the National Institutes of Health National Center for Research Resources through Grant RR12365. The Pacific Northwest National Laboratory is operated by Battelle Memorial Institute for the U.S. Department of Energy through Contract DE-AC06-76RLO 1830.

Received for review July 2, 1999. Accepted December 2, 1999. AC990729U