Effects of the source gap on transmission efficiency of a quadrupole mass spectrometer

Rationale Recent trends towards miniature and portable quadrupole mass spectrometry (QMS) entail challenges in instrumental sensitivity, which is influenced by 3D fringe field effects on ion transmission in the Quadrupole Mass Filter (QMF). The relationship of these effects with the gap from the ion source to the QMF entrance (source gap) is significant and little explored. We examine transmission characteristics experimentally and use the results to test the predictive accuracy of a recently developed 3D QMF simulation model. The model is then applied to directly investigate optimal transmission m/z ranges across multiple source gaps. Methods A portable single filter quadrupole mass spectrometer is used to analyse transmission characteristics across a range of common gases. We use an experimental approach originally proposed by Ehlert, enhanced with a novel method for absolute calibration of the transmission curve. Custom QMF simulation software employs the boundary element method (BEM) to compute accurate 3D electric fields. This is used to study the effects of the source gap on transmission efficiency. Results Experimental findings confirm a centrally peaked transmission curve; simulations correctly predict the optimal transmission location (in m/z) and percentage, and extend the experimental trend. We compare several methods for determining fringe field length, demonstrating how the size of the physical source gap influences both the length and the intensity of the fringe field at the QMF entrance. A complex relationship with ion transmission is revealed in which different source gaps promote optimal transmission at differing m/z ranges. Conclusions The presented results map the relationship between the source gap and transmission efficiency for the given instrument, using a simulation method transferrable to other setups. This is of importance to miniature and portable quadrupole mass spectrometers design for specific applications, for the first time enabling the source gap to be tailored for optimal transmission in the desired mass range.

three co-ordinate directions (E x , E y , E z ) to be independent of z position, with E z = 0. 13 Such ideal analysis predicts that as long as stable ions experience sufficient radiofrequency (RF) cycles in the QMF, transmission efficiency does not depend upon m/z. 14 In practice, fields are 3D, having non-zero E z , and all field components subject to change with z position because of finite length electrodes and earthed end plates coupling the ion source and detector to the QMF ( Figure 1A). The resultant fringing electric fields at the entrance and exit of the QMF (see Figure 1B) have a mass-dependent influence on transmission efficiency for a given ion energy and frequency. Due to the fringe field influence on position, velocity and relative field phase, there can be effective defocusing, resulting in loss of signal, 15 or focusing, 16 with a corresponding effect on instrument sensitivity for different m/z. QMF fringe fields are complex because 3D field distribution causes coupling of ion motion in the x, y and z directions. Comparison of the two equipotential plots of Figure 1B illustrates a single example of coupled ion motion: the initial direction of the force exerted by the z component of the field on positive ions (towards or away from the ion source) at a given phase angle is dependent on proximity to the y electrode.

| Fringe fields and transmission efficiency
Few experimental results have been published that treat the effect of fringe fields on ion transmission. 12,17 Early theoretical and experimental work found that QMF ion transmission efficiency will fall for heavier ions which spend a longer period exposed to certain defocusing forces in the y direction, occurring because the typical operating point lies outside the y-stability limit for large portions of the fringe field region. 12,17 Following this, Ehlert employed an experimental approach based on isotope ratio measurements to plot ion transmission efficiency as a function of m/z. 18 We know of no other comparable attempt. The resulting curve (see Figure 2) showed the expected fall in transmission at higher m/z, 12,17 but transmission unexpectedly peaked near m/z 30 and fell at lower m/z; this fall was believed at the time to be an artefact of the RF detection circuitry of the instrument. Following this, however, an approximate linear model of the fringe field (in which E x and E y ramp linearly from zero to full quadrupole fields on attaining a prescribed displacement from the starting position of the ions, typically r 0 ) 16,19,20 predicted a transmission fall-off at lower m/z using phase space dynamics. It was concluded that QMF mass discrimination effects may be predicted as a function of the number of RF cycles (optimal between 1 and 3 RF cycles) spent by ions of different m/z in the fringe field around the entrance to the QMF. section, whose axes are parallel with, and equally spaced around, the z axis, touching an inscribed circle of radius r 0 . The gap between the source end-plate and the QMF electrodes is referred to as the 'source gap'. (B) xz equipotential plots (3D and 2D representations) for a fringe field region with a small source gap (0.25r 0 ), produced using potential values from the custom simulation model. Representations are at two elevations in y, and potential contours are shown up to an axial displacement of 3r 0 from the ion source exit plate [Color figure can be viewed at wileyonlinelibrary.com] Hennequin and Inglebert investigated this model experimentally. 21 Study of their 50% acceptance ellipses reveals a range of agreement with the predictions of linear approximation model from good to poor, depending on the number of cycles spent by ions in the fringe field (n f ), and whether acceptance in x or y was considered. In these experiments, n f was altered by means of ion m/z and energy. The implementation of the linear approximation model assumed a geometric fringe field length (l f ) equal to r 0 . 16 Independent determination of l f was not attempted; the authors observed that this length should be set on the basis of experimental results. Subsequent workers employed a Liebmann iterative relaxation technique to generate an approximation of the fringing field having an exponential form for variation of E x , E y and E z with z position. 22 A cubic version was developed for phase-space analysis and tested against the experimental findings of Hennequin and Inglebert. 23 In the implementation of the cubic model, l f was set at 1.5r 0 , and n f was varied by means of the ions' axial velocities, with a fixed source gap of 0.125r 0 .
Overall improvement in the linear approximation was marginal, despite arguable improvements in y-acceptance for some axial velocities.
An advantage of the exponential model is the avoidance of discontinuous transition to zero E z at the prescribed distance from the source plate. However, the model implies perpetuation of ideal quadrupole field symmetry (between E x and E y ) in the fringe field, corresponding to a lack of compliance with Laplace's Equation. Further, an inherent difficulty of phase-space analysis is that one must neglect the coupling of ion motion in all three dimensions caused by non-zero E z to avoid complex six-dimensional phase-space calculations. 16 The significant remaining discrepancies with experimental data were therefore tentatively ascribed to perturbations in axial velocity (v z ).

| Motivation
Most previous studies have been principally concerned with testing the soundness of theoretical models rather than investigating the transmission effects of varying either the fringe field length or the source gap. In these cases, the fringe field length in terms of number of RF cycles was altered by means of the ion's initial axial velocity (either directly, or using ion mass or energy). This is not equivalent to altering either the geometric fringe field length or the source gap, since (a) altering axial velocity affects not just the number of cycles in the fringe field but also the number of cycles that ions experience in the QMF itself, and thus the resolution; (b) the fringe field length (in number of RF cycles) changes proportionately with the axial velocity of ions, but disproportionately with the source gap; and (c) the fringe field position with respect to the QMF remains the same (other than being scaled in number of RF cycles) when the axial velocity is altered, while the fringe field distribution alters such that it penetrates further into the QMF, when the physical source gap is reduced.
The source gap is a parameter that may easily be altered at the design stage, and can be used to influence transmission efficiency without recourse to ion energy alterations that may be unavailable or counterproductive. A simulation model has recently been developed which accurately and efficiently reproduces 3D field effects (including fluctuations in axial velocity) and can simulate variations in both the source gap and the fringe field length. 24,25 Remaining disparities between existing models of QMF behaviour and experimental observation, and the lack of direct investigation of source gap effects, suggest the potential benefits of an investigation with such a high-fidelity model.
Generation of an experimental transmission curve would be advantageous in light of the single previous example, 18,21 both to compare QMF transmission characteristics obtained with a different set of instrumental parameters (particularly the transmission reduction at lower m/z originally questioned by Ehlert 18 ) and to assess the accuracy of the simulation model. We also explore a method of fixing the absolute transmission efficiency (not attempted by Ehlert).
Once the accuracy of the simulations is established, these can be extended to predict the effects at a wider range of m/z and to investigate variation of the physical source gap. Any such attempt should explicitly define how the fringe field length is determined (we have recently proposed such a method 25 ), and clearly distinguish between the mass selection implications of the fringe field length, and those of the source gap length. We accurately characterise for the first time the relationship of QMF transmission characteristics to these connected but distinct parameters, and use individual ion trajectories to investigate the more complex and realistic picture which emerges.

| Method
Measurements were made using a commercial single filter portable Ehlert's method 18 was followed to produce a transmission efficiency curve for this instrument (enhanced by a method to determine absolute transmission efficiency). The method compares measured and known isotopic abundances to establish transmission efficiency ratios τ s1 =τ s2 between pairs of isotopes of several elements, where S 1 and S 2 refer to isotopic species 1 and 2 of the same element S. The approach assumes that the ion source behaviour will be the same for both isotopes. Equation 1 then allows comparison of the ratio in detected ion current (I s2 =I s2 Þ with the known isotopic ratio (A s1 /A s2 ) to determine the transmission efficiency ratio of the QMF for each pair of isotopes: where G S1 and G S2 are the gains of the detector used for the respective species. Provided that the same detector gain is used for two isotopes of the same element, then in Faraday mode G S1 /G S2 = 1, while in multiplier mode, it has been shown 26 that: It is assumed that the transmission curve can be approximated by an equation whose logarithm takes the form of Equation 3, if a suitable order can be chosen: With the exception of C 0 , all polynomial coefficients (C 1 , C 2 , etc.) are found through least squares curve fitting of Equation 4 (which follows from Equation 3) for all isotopic pairs: The choice of C 0 then sets the scale of the transmission curve generated by substituting C 1 to C 7 back into Equation 3 and raising it to an exponent of 10. Ehlert 18 chose C 0 arbitrarily to give τ = 100 at m/z = 0. In our case, RF-only transmission data were compared to determine τ empirically as a percentage of the total ion current entering the QMF (Appendix A).

| Experimental transmission curve
For the new transmission curve ( Figure 3A), neon, argon, krypton and xenon were introduced into the quadrupole mass spectrometer inlet  to elapse between measurements taken from different gases; in these cases, the instrument was allowed to reach its base pressure before introducing the subsequent gas. Immediately before introducing each gas, residual gases in the chamber (where apparently present in significant quantity) at the m/z of the target analytes were measured for possible subtraction from the target signal. However, even when present, the residual gas component was found to be negligible in its effect. Table 1 provides the ion transmission ratios (τ S1 /τ S2 ) of isotopic pairs of different species calculated from known abundance ratios (A S2 /A S1 ) 28 We investigated lower order polynomials to order 3 (standard error 0.012; R 2 = 0.922.) Above order 3 there was no effect on the m/z position of peak transmission, while for order 3, the optimal transmission position was shifted slightly to m/z 85. The effect on peak shape was more noticeable: reducing the polynomial order produced a broader, flatter transmission peak, with (below order 6) elimination of the slight hump on the low m/z side. Overall, the minor impact appeared to support the robustness of the generated curve.

| Absolute transmission efficiency
To convert the relative transmission curve predicted by Equation 5 into absolute transmission efficiency (τ absolute ) as shown in Figure 3A (RF-only mode). The latter may be taken to represent the input current, I 1i , of component i at the source aperture. That is: Obtaining the ratio I 2i /I 1i is not a trivial exercise. Attempting this directly by choosing the local peak in the RF-only spectrum representing I 1i was found to be a complex process subject to human error and the precise calibration of the instrument's mass scale.
Reliability is further reduced because the determination of I 1i for a single isotope introduces significant additional analytical error. The proportions of doubly ionised peaks increased the complexities. A more feasible approach was therefore formulated (fully described in Appendix A) using the reasonable assumption that the maximum RF-only ion current measured during introduction of a single element represents the total ion current at the source (I 1 ) of that element. Using relative transmissions already available from Equation 5, we applied correcting factors to the measured ion currents for all N components.
These were then summed and divided by the peak RF-only value to obtain an estimate of X i for the component under consideration (for example, X 1 ): Substitution of X i into Equation 5 for τ, the calculated C 0 is identical regardless of which component of the given element is used, but will, in practice, vary by element due to analytical error. Comparison

| Simulated transmission curve
To test the predictions of our simulation model, particularly in terms of the maximum transmission efficiency and its location on the m/z scale, we simulated ion transmission as a function of m/z for the experimental instrument, with and without fringe field effects. This also allows investigation of ion transmission at very low m/z, which is difficult to measure experimentally assuming typical ion energies and, as discussed in the previous section, is not possible to predict confidently from experimental data at higher m/z using a curve-fitting equation.
To minimise statistical variations while still allowing time-efficient computation, 20,000 ion trajectories were simulated at each mass step of 0.02 m/z units. The results described here are for a precisely centered source plate aperture, emitting a uniformly distributed beam of ions having identical energy (5 eV), random distribution of initial RF phase angle and travelling parallel to the system axis. This allowed us to determine the effects of varying some of these parameters, the results of which are available in Figure S1 (supporting information).  Figure 3A, including a very similar peak transmission position (albeit slightly flatter and broader) and a similar m/z range of overall optimal transmission. The absolute percentage of transmission is moderately higher than that calculated experimentally; this could be symptomatic of imperfect setting of simulation parameters (e.g. spread in initial ion energy and angle) or experimental imperfections (e.g. isotopic fractionation effects or a slightly off-axis source plate aperture). 30,31 Similar factors could be responsible for the slight differences in curve shape. Furthermore, as discussed in section 2, Ehlert's approach 18 was followed in this work, which used a single gauge calibration for all gases. Preliminary investigations suggest that the differences in shape would be reduced by calibrating the gauge separately to each gas. Overall, the level of agreement in terms of order of magnitude suggests merit in the RF-only comparison method of fixing absolute transmission and supports the fundamental accuracy of the simulation method. Comparison between the 2D and 3D simulated curves confirms that the transmission peak is purely due to the effect of 3D fringe fields.
While the broad trends of previous models are borne out, 12,16 with a pattern of transmission reductions at both lower and higher m/z, the BEM also predicts clear fluctuations (non-smooth) in ion transmission which are not a result of statistical variations as discussed in our previous work. 24,25 Furthermore, the simulation model indicates a sharp rise in transmission below m/z 4 which was difficult to investigate experimentally. This is to be expected in the case of ions travelling parallel to the system axis since, as m/z tends to zero, their initial velocity must tend to infinity, guaranteeing their arrival at the detector. The trend is reduced, but still apparent, for a small initial half angle between the ion beam and the system axis, as seen in Figure S1 (supporting information). For the specific quadrupole mass spectrometer dimensions and RF deployed in this study, experimental and simulation evidence agree that optimal ion transmission occurs when the QMF is operated at a range from about m/z 60 to 90.

| Fringe field length (l f )
Many earlier works have assumed for convenience a fringe field length (l f ) of fixed distance equal to r 0 . 16,20 While the fringe field length is impossible to determine since it never completely disappears even in the region between the QMF electrodes, there will be a distance into this region where its effects become negligible. We have previously    increases; the optimal position shows greatest mobility for source gaps between 0.5r 0 and r 0 . Increasing the source gap beyond 1.5r 0 significantly compromises the transmission but with little change to the optimal m/z range. It is also noticeable that the spread of m/z experiencing optimal transmission reduces as the source gap length is increased. There is little appreciable difference in the transmission peak height (i.e. optimal transmission efficiency) for source gaps ≤r 0 .

| Source gap effect and fringe field length
The optimum value increases markedly for a source gaps of 1.5r 0 and reduces again for a 2r 0 gap.
High transmission with a long source gap will be reduced if ions are introduced with an angle spread ( Figure S3, supporting information), but the trend and peak position remain the same. The effects predicted in Figure 4A are also similar for a QMF with hyperbolic electrodes ( Figure S4, supporting information). In general, the height of the transmission curve tends to reduce slightly for source gaps increasing from 0.25r 0 to r 0 . The peak of the transmission curve will also shift by changing ion energies but the trend in transmission curve height and shape differs from the source gap change. Figure 4B shows the ion transmission as a function of the approximate number of RF cycles spent by ions in the fringe field, N f (r 0 units). N f may be calculated by: , 19 where l f is the length of the fringe field, f is the frequency of the RF voltage, m is the mass of the ion, ϵ z is the accelerating voltage of the ion source in  The results presented in Figure 4B show that peak transmission for all source gaps occurs in fringe field dwell times between~0.6r 0 and 1.3r 0 RF cycles, in accordance with the previous analysis. 16,19,25 While this approximate general relationship between N f and transmission efficiency is clear, the specific range of N f for optimal transmission is dependent on the individual source gap. Usually increasing the source gap promotes optimal transmission for ions experiencing smaller number of RF cycles in the fringe field. The effect of the source gap on the optimal transmission position with respect to N f (highlighted by red dots) is most significant between gaps of 0.25r 0 and 1.5r 0 .
Smaller source gaps are clearly beneficial for optimal transmission at higher m/z, and vice versa. To illustrate a potential application of these findings, the simulated spectra in Figure 5 show that a smaller source gap of~0.25r 0 is beneficial for transmission efficiency of stable ions with m/z 120 (for the given QMF settings), while, for a low-mass ion such as m/z 12, a larger source gap of~1.5r 0 is beneficial. It is clear Smaller source gaps promote optimal transmission at higher m/z without the need for increased ion energies. It is possible to predict in detail the optimum source gap length for transmission at a given range of m/z when designing and manufacturing an instrument for specific application. These conclusions are particularly significant for portable and miniature instruments, where sensitivity is a key concern.