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Computer-Controlled Simplex Optimization on a Fourier Transform Ion Cyclotron Resonance Mass Spectrometer John W. Elling,1~2 Leo J. de Koning, F r a n s A. Pinkse, a n d Nico M. M. Nibbering* Institute of Mass Spectrometry, University of Amsterdam, Nieuwe Achtergracht 129, 1018 WS Amsterdam, T h e Netherlands
Henri C. Smit* Laboratory of Analytical Chemistry, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, T h e Netherlands
Computer-controlled formal optimization of the tuning of narrow band Fourier transform ion cyclotron resonance time domain signals has been implemented. The multivariant modlfied slmplex optbnlzatbn method has been taliored to the tuning parameters and the instrument response surface. The narrow band signal has been optlmired on the basis of mass spectrum resolution, line shape, and slgnal to mise ratio. The optlmizatbn performance and reproducibllkyhave been tested over a wide ion mass range and yielded reliable resolution improvements. Computer-controlled optimization of a m / z 1066 Ion signal from a tris(perfluorononyi)-s-triazlne fragment improved the resolution 50-foid, to 1 600 000 In the absorption mode.
INTRODUCTION Fourier transform ion cyclotron resonance (FT-ICR) mass spectrometry has gained considerable popularity since its demonstration by Comisarow and Marshall in 1974 (1-3). The technique and its applications have been extensively reviewed in the literature (4-7). The capabilities of FT-ICR are particularly evident in obtaining very high resolution mass measurements in the narrow band or heterodyne mode. A problem associated with these measurements is the requirement for extremely precise and time-consuming instrument tuning to obtain the maximum resolution possible. The problem arises from the large number of dependent instrument parameters that must be manipulated in the tuning process. Moreover, the parameter settings are highly specific for the ion signal being observed; the instrument should be retuned each time ions with different m / z values are monitored. The tuning problem is made more difficult when the operator has to tune the ion time domain signal-an exponentially damped sum of sinusoids-rather than the more familiar mass spectrum. The FT-ICR mass spectrum is obtained by a Fourier transformation of the time-domain signal. The most successful operators tune by intuition developed by long experience, often using only a subset of the relevant parameters. Even with such experience the day to day tuning results are not always reproducible. Clearly a more efficient method for precise tuning of the FT-ICR heterodyne-mode signal is needed. The purpose of this study is to apply chemometric techniques in the formal optimization of FT-ICR instrument tuning, enabling operators Also affiliated with the Laboratory of Analytical Chemistry, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands. Current address: Department of Chemistry, University of Wisconsin-Madison, Madison, WI 53706.
with less experience to rapidly obtain the maximum resolution possible.
THEORY FT-ICR Tuning. The desirable characteristics of a FTICR frequency-domain spectrum are high signal to noise, S/N, and high resolution of Lorentzian-shaped peaks. However, since tuning of the instrument takes place in the time domain, it is necessary to relate the desirable characteristics of the rims spectrum to characteristics of the observed timedomain signal. The shape of the frequency-domain spectral peak, or line, is determined by the envelope of the ICR time domain signal. A Fourier transformation of an exponentially damped ICR time domain signal yields ideal Lorentzian lines in the frequency spectrum with peaks corresponding to the cyclotron frequencies of the ions generating the signal. If the timedomain ICR signal envelope deviates from the ideal exponential decay, the resulting frequency lines are distorted from the ideal Lorentzian shape. The frequency-domain line resolution is defined in eq 1 as resolution = f/Af5,,% ('1) the frequency assignment, f , of the line divided by the line width at half maximum height, Af509b.f1 The liap width is related to decay time, T , of the exponential damping in the ideal ICR signal by eq 2 (8). From eq 1and 2 it can be seen Af5,,%(magnitude mode) = 3 ' I 2 / ( m )
(2)
that line resolution is inversely proportional to Afm%and so is proportional to the damping constant T . The S/Nin the ICR time-domain signal is directly proportional to the S/Nin the mass spectrum. Thus, increasing the S/Nratio in the time-domain increases the S/N in the frequency domain. From the considerations above, it can be seen that tuning the FT-ICR instrument involves maximizing the time-domain ICR signal T and S / N while preserving the exponentially damped sinusoid appearance. In manual tuning, evaluation of the signal characteristics and the response to parameter changes is done from single ICR signals rapidly collected and displayed in real time. In general, three dependent categories of acquisition parameters are critical to the response of the ICR signal. The first category deals with the electrostatic field in the cell, adjusted by changing the dc voltages applied to the two cell trappings plates, the two cell transmitting plates, and the two cell receiving plates. Improvements in the signal that result from tuning the cell electrostatic field reflect improvements in the efficiency of trapping ions in the cell. The second category involves the excitation of the trapped ions to produce a signal. The power of the ion excitation is controlled by the length and strength of the radio frequency pulse. Correct tuning of these parameters results in maximum excitation of
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the trapped ions without causing collisions with the cell walls. In this category, the frequency of the excitation pulse is critical to the uniformity of the excitation energy over the monitored bandwidth. The final category is the strength of the signal, adjusted by controlling the number of ions generated. In the case of electron impact ionization, the number of ions can be manipulated by varying the electron beam pulse length or the current through the filament or both. Simplex Method. For the convenience of the reader and clarity of the discussion later, the simplex optimization procedure is briefly reviewed. The simplex method was originally suggested by Spendley et al. (9) and applied by Morgan and Deming (10).It is suited to an FT-ICR system of dependent parameters in that several parameters are changed simultaneously in a “hill climbing” approach to the problem. In a system of n parameters, simplex optimization first defines a geometric figure in n dimensions (the simplex) possessing (n 1)vertices, where each dimension represents an adjustable parameter and each vertex is a set of parameter values for an experimental evaluation. Initially, (n 1)experiments are required to determine the response of the system to each vertex in the first simplex. In this case the response is defined by characteristics of the ICR signal, as discussed below. At the conclusion of the first (n 1)experiments the responses are ranked, identifying the vertices that caused the best (B), worst (W), and next worst (NW) responses. By reflecting the worst response vertex through a point, P, located on the opposite face of the simplex, its reflection, R, is generated. The response to R is tested, and if it is better than the response to W, W is replaced with R and a new simplex formed. Repeating the procedure of experimental evaluation, ranking, reflection, and logical selection, the simplex progresses toward an optimum in that parameter space. Many suggestions for modification of this basic simplex technique have been made, each aiming to improve the speed and reliability of the optimization. Several modifications were incorporated into the automatic tuning procedure: a modified simplex algorithm allows the simplex to converge more rapidly toward an optimum by expansion or contraction along the line of reflection if R is better than B or if R is worse than W, respectively (11).The simplex may also converge faster if the point P on the hyperface of the simplex is weighted toward the vertices with the best responses (12). In experimental optimizations, precautions must be taken to prevent simplex oscillation between two poor vertices by replacing the NW vertex with R if the worst vertex is the previous R (13). Further precautions must be taken if noise is present in the response. The simplex can be prevented from getting stuck on an unusually high response by testing every vertex retained in (n 1) successive simplexes (9). Response Criterion. Any optimization technique requires a criterion to define the response of the system that is to be maximized or minimized. Equation 3 has been empirically developed to numerically mimic the operator’s visual evaluation of the FT-ICR time-domain signal; maximizing the response of eq 3 maximizes the desirable frequency characteristics of the FT-ICR signal discussed above. In eq 3, A is
+
+
+
+
RESPONSE = A r ( 1 - F)
(3)
the initial amplitude of the ICR signal, T is the relaxation time of the exponential damping in the ICR signal, and F is a factor determined by the deviation of the observed signal from the ideal exponential decay envelope. The initial amplitude, A , is calculated as shown in eq 4, by summing the absolute values of the first 50 points of the sampled time-domain signal, S(k). (4)
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The relaxation time of the ion signal is calculated for use in eq 2 with a ratio method proposed by McLachlan (14). McLachlan’s method is selected for its professed stability in the presence of noise. The fit factor, F, is calculated as shown in eq 5. Equation F = 5(1.0 - CC) (5) 5 was developed empirically. In eq 5, the weighting factor 5 was chosen to yield an F equal to 1for a very bad exponential fit and CC is the correlation coefficient of a weighted leastsquares fit to the ICR signal. The least-squares fit is carried out on the natural log of 20 points (one point is the absolute value sum of 100 time-domain points) evenly sampled over the first 3 T of the ICR signal. The fit of a straight line to the natural log of these points is weighted with an exponential function of decay time equal to the signal T . The CC reflects how well the straight line, determined by least squares, fits the 20 points.
EXPERIMENTAL SECTION A description of the CMS-47 Bruker FT-ICR instrument used in these experiments and a general operating procedure has been published previously (15). The automatic tuning software was developed on the Bruker dedicated ASPECT 3000 minicomputer in the PASCAL language supplied with the computer. The low mass sample used was background water. High m/z ion signals were obtained from fragments resulting from electron bombardment of tris(perfluoronony1)-s-triazine introduced to the cell by direct insertion probe. Tris(perfluoronony1)-s-triazine was purchased from PCR Research Chemicals, Inc., Gainesville, FL. Ions with the same nominal mass but different elemental composition were generated by electron bombardment of gaseous n-butyl n-propyl ketone introduced to the cell through a leakvalve. RESULTS AND DISCUSSION Simplex Implementation. The Bruker ICR operating system allows PASCAL object code to be invoked from command files. One command file is used to run the PASCAL optimization procedure. The simplex procedure consists of two parts: The setup program allows the user to choose the parameters to be included in the simplex, define the initial step size and boundaries of these parameters, create the start simplex, and define the stop criterion. The run program evaluates a signal resulting from a vertex, ranks the response, makes the logical selections, and determines the next vertex to be evaluated. In the command file created for automatic tuning, the setup program is executed first. The menu-driven setup program can be used to list and modify the tuning parameters to be included in the simplex. On the Bruker CMS-47 the 11 parameters discussed above (and any other parameter under computer control) may be included in the simplex optimization procedure. However, in routine automatic tuning it is convenient to limit the set of parameters to be optimized. Because only the relative adjustments of the plate voltages determine the ion trapping efficiency, one of the cell receiving plates is held at a constant 0.200 V. The remaining five plate voltages are included in the simplex. The excitation pulse strength is also not included in routine optimizations because it is not a continuous variable in the region of interest. The strength is usually held at 115 V, for all optimization experiments and the excitation power controlled by varying the length of the pulse. Neither parameter controlling the signal strength is included in routine simplex optimizations since initial experiments have indicated that the upper limit of ion generation caused unstable and irreproducible ICR signals, making simplex optimization impossible. The length of the ionization pulse
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1 4
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5 50
5 00 LOG
6 00
6 50
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I RESONANCE FREOUENCY 1
Flgure 2. Criticality of three tuning parameters expressed as initlal step
size (percent change in the parameter value from the optimum that led to a 70% drop in the response) versus the log of the ion resonance frequency and versus the ion m l z : (0)receiving plate voltage: (*) trapping plate voltage; (+) excitation pulse length.
i
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)*
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I O E V I I T I O N FROM OPTIMUM
Figure 1. Response surface cross sections created by varying one receiving plate voltage (0),one trapping plate voltage ('), and the excitation pulse length (+) around the optJmwn values for three signals: (a) H20+' ion (mlz 18) signal from background water resonating with a frequency of ca. 4001 kHz; (b) C,,F2,N2+ triazine fragment ion (mlz 571) signal resonating at ca. 128 kHz; (c)C2proN,+ triazine fragment ion ( m l z 1066) resonating at ca. 67 kHz. The ceii pressure was in the order of 3 X lo-' mbar for all the cross-section experiments.
is typically set by the user to the minimum length necessary to observe a signal in one experiment and, together with the filament current, held constant during optimization. Before entering the automatic tuning command f i e and the setup program, the user is required to have a rough signal stored to disk. The start position of the simplex is defined by the initial values of the tuning parameters of this rough signal. These parameter values make up the first vertex in the start simplex. The initial step sizes of each parameter, used to create the other start simplex vertices in the setup program from the user-supplied start position, are critical to the speed and
reproducibility of the optimization. Default values for the step sizes have been found empirically as follows: after an optimum had been reached, a cross section of the response surface in each direction was made by varying one parameter stepwise around its optimum, collecting a signal at each step, and calculating the response of the signal with eq 3. Parts a-c of Figure 1 illustrate the response surface cross sections of some of the tuning parameters made from optimized signals from ions with m/z 18,571, and 1066, respectively. From the cross sections for each parameter, the step size used to set up the first simplex was determined to be a value for which the response drops to 30% of the maximum. As expected, the initial step sizes for all the tuning parameters are dependent on the cyclotron frequency of the ion. In Figure 2, the dependence of some parameter's step size on the ion m / z and the related ion cyclotron frequency is shown, illustrating the increasing criticality of the tuning parameters when monitoring signals from ions with b h e r m/z values. Equations describing the lines in Figure 2 and similar lines for the remaining tuning parameters are used by the setup program to determine default step sizes for each parameter as a function of ion cyclotron frequency when creating the start simplex. The boundaries of the n-dimensional space in which the simplex moves are also defiied in the setup program. Bounds of the dc offset voltages are the limits of the instrument-from 0 to 10.0 V. The excitation pulse length simplex variable is allowed to range from 20.0 to 200.0 ps. The excitation pulse frequency can move in a range half the detection bandwidth on either side of the starting frequency. In the setup routine, one of several stop criteria can be chosen by the user to define when an optimum has been reached. The simplex can be terminated when the average T of the simplex response signals will yield a line resolution (eq 1 and 2) greater than a minimum specified by the user. Alternately, the simplex can be terminated when the standard deviation of the simplex responses is at a level that would be caused by the noise influence on the signal alone. Use of the latter stop criterion allows the simplex to proceed until there is no more directional information available from response changes to parameter changes, when the simplex has reached an optimum. Following successful completion of the setup program, the automatic tuning command file enters a loop that carries out the stepwise simplex optimization. In the loop the tuning parameters corresponding to the vertex being tested are read from the disk, the interface is initialized, and a signal is
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1
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1065.0
+ 6.0
1067.0
1068,'O
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'
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'
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Flgure 3. Optimization of the signal from C,y+ and C3H50+ions (m/ z 57) which were generated by electron bombardment of C,H&OC3H5 (cell pressure CB. 4 X 10" mbar). (a) The magnitude mode frequency spectrum resulting from a poorly tuned signal, which was sampled for 450 ms. The lines have a resolution of 80 000 and the splitting is an artifact of poor tuning. (b) The frequency spectrum with 640000 line resolution obtained from the signal after optimization. The optimized signal was sampled for 1440 ms.
generated and stored back to the disk. Any series of ion generation, selection, detection options can be used per experiment and any number of experiments can be accumulated to produce the signal to be optimized. In the loop the run program determines the response of the signal in the disk file via eq 3 and executes the simplex procedure. At the conclusion of the run program and the loop, the parameters of the next vertex to be evaluated are passed to the ICR signal generation commands via the same disk file. During the process of optimization, the tuning software adjusts the acquisition period to be 5.2 times the largest T of the simplex responses. This ratio of T to acquisition period yields a minimum error in McLachlan's method of T calculation (14). Optimizations. The automatic tuning procedure has been tested on the signal from background water ions with m / z 18. Five simplexes were initiated a t different starting positions and terminated in accordance with the second stop criterion discussed above. The improvement in resolution upon optimization ranged from 2- to 11-fold, depending of course, on the resolution of the starting signal. Optimum excitation frequency has been investigated for a heterodyne-mode signal arising from cyclotron resonance of two ions with the same nominal mass but different elemental compositions. Figure 3 illustrates an example of starting and optimized frequency spectra. The tuning of the excitation and electrostatic field has not only improved the resolution and line shape but also slightly altered the resonance frequency of the lines and the relative peak amplitudes. The latter occurs as a result of the adjustment of the excitation pulse frequency-in the center of the frequency ranges shown. In Figure 3, the excitation pulse frequency has been optimized to a value midway between the two cyclotron resonance frequencies, so distributing the excitation energy uniformly over the monitored ions. The full advantage of the computer-controlled tuning procedure is illustrated by Figure 4, which shows a dramatic
Figure 4. Optimizationof the signal from C&&+ ions with m / z 1066 generated by electron bombardment of trls(perf1wrononyib.s-triazine. (cell pressure ca.2 x lo4 mbar). (a) Magnitude mode mass spectrum with 22000 line resolution resulting from the 20-fold accumulated starting signal. The tlmedomaln signal was acquired for 500 ms. (b) The mass spectrum with 1 600 000 absorption-mode line resolution obtained after optimization of the starting signal. Each of the 70 accumulated signals was sampled for 28 s.
improvement of the resolution upon optimization of a signal ion. The absorption mode from the m / z 1066 C22F40Na+ resolution of 1600 000 in Figure 4b compares favorably with a recently reported high resolution of 60 000 for detection of a m/z 5922 poly(propy1ene)glycol 4000 fragment ion obtained with a FT-ICR instrument at 7 T (16). The improvement as a result of this low-pressure optimization can be seen in Figure 4. Figure 4a shows the mass spectrum resulting from a 20-fold accumulated starting signal. Accumulation was necessary as no signal was visible in a single experiment. Figure 4b is the mass spectrum obtained from the signal associated with the 76th simplex vertex. The carbon-13 isotope peak intensity in Figure 4b is 26.1% at m / z 1067-close to the expected intensity of 25.8%. The resolution improvement achieved would be difficult, if not impossible, to obtain manually since manual tuning requires generation of enough ions for a visible signal in one experiment. Signals of such strength usually have inferior resolution to weaker signals. However, it is very difficult to manually tune an accumulation of weak signals; the effect of small parameter changes must be evaluated between signals that can take more than 5 min to be generated. Therefore, the ultrahigh resolution demonstrated is partially a result of being able to tune an accumulation of very weak signals. The reproducibility of the optimum position found by the simplex procedure has been investigated with each of the signals discussed above. For several independent optimizations of the same ion signal, the simplex reliably converged to the same optimum position from different start positions when the starting simplex was created with the step sizes described above. Widely divergent starting positions or initial simplexes created with default step sizes much larger than those describe above sometimes converged to local optimum positions. This is not anticipated to be a problem; in routine use of the automatic tuning procedure it is unlikely that any operator will define a start position as unusual as those tested nor step sizes larger than the default size. The relationship between the S / N ratio of the time-domain signal and the reproducibility of the calculated response value
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1
10 0
I
-
__
0
SIGNAL / N O I S E
Flgure 5. Reproducibility of the calculated signal response (eq 3) versus the time-domain SIN (eq 6). The reproducibility is measured as the percent standard deviation from the average A (+), 7 (0),and response (*) of 10 signals independently recorded at each SIN ratio.
(eq 3) has been investigated. The S / N ratio of the time-domain signal has been calculated with TD
S / N = [ ( 2 S ( k ) ) - Ni-TD]/(Ni.TD) k=l
(6)
In eq 6 the absolute value sum of all TD time-domain points of the sampled signal minus the noise in the points (the experimentally estimated standard deviation of the noise, N , times TD) is divided by the same noise contribution. Time-domain signals of variable S / N ratios were generated by varying the number of accumulations of identical experiments with regard to ion generation and detection. Figure 5 illustrates the percent standard deviation of the A's, and responses of time-domain signals as a function of the S / N ratio. In order for the simplex to converge to a small optimum position, it has been found that the response value to the simplex signals from eq 3 must be reproducible with a standard deviation of 10% or less. From Figure 5 it can be seen that such a reproducibility corresponds to a S / N ratio of at least 1.0 in the time-domain signal being optimized. The progress of one simplex optimization procedure is illustrated in Figure 6. The optimization process can be seen to proceed in two stages: an initial stage where the response increases rapidly and a second stage where the simplex is near the optimum and the response becomes nearly constant. In all cases studied, the simplex achieved 90% of the eventual optimal response in less than 50 simplex evaluations. A weighted centroid (12) has recently been implemented, reducing the number of simplex evaluations necessary tQ realize 90% of the optimum response, thus speeding up the optimization process. The speed of the computer-controlled optimizations of the tuning is dominantly determined by the total acquisition time. The time taken by software execution is negligible. As discussed above, the best FT-ICR performance occurs when the number of ions generated is minimal and the ICR signal is summed over a number of experiments to achieve a satisfactory S/Nratio. However, large numbers of accumulations increase the total acquisition time; a balance between tuning speed and high resolution must be made. For any optimization process, the execution time can be roughly estimated as 50-60 times the time required to accumulate enough experiments for one signal. In the experiments to date, total optimization times have ranged between 20 min for strong signals from low mass ions to longer than 2 h for multiple accumulations of weak signals from high mass ions. As currently implemented, the simplex optimization technique has proven more than satisfactory in meeting the goals TIS,
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Flgure 6. Progress of a simplex optimization of a signal from C ~ F , N , " ions (m / z 1066) generated by electron bombardment of tris(perfluorononyi)-s-triazine (cell pressure ca.6 X lov8mbar). The 7 of the
highest response to a vertex is plotted against the number of vertices evaluated in the optimization process. of computer-controlled tuning of Fourier transform ion cyclotron resolution signals. In our experience, the rate of resolution improvement surpasses that of manual tuning, which is an advantage if for any reason the system under study permits a limited tuning time. Still, the success of automatic tuning is only guaranteed if the scan to scan reproducibility of the ICR signal allows a proper evaluation of the response criterion, implying that the above-described method does not provide the solution for the problem associated with tuning of ICR signals stemming from ions that are generated under irreproducible conditions, such as experienced in laser desorption studies (17). The successful use of the weighted centroid to speed the simplex progress toward an optimum suggests that further modifications of the simplex algorithm (18) may be fruitful. Mathematical modeling of the response surface of the instrument may suggest fitting equations that can be used to more efficiently determine the location of the R vertex.
LITERATURE CITED Comisarow. M. B.; Marshall, A. G. Chem. Phys. Lett. 1974, 25, 282-283. Comisarow, M. 6.; Marshall, A. 0. Chem. Phys. Len. 1974, 26, 489-490. Comisarow, M. B.; Marshall, A. G. Can. J . Chem. 1974, 52, 1997- 1999. Laude, D. A.; Johiman, C. L.; Brown, R. S.; Weil, D. A,; Wilkins, C. L. Mass Spectrom. Rev. 1986. 5, 107-166. Marshall, A. G. Acc. Chem. Res. 1985, 78, 316-322. Cody, R. B.; Kinsinger, J. A.; Ghaderi, S.; Amster, 1. J.; McLafferty, F. W.; Brown, C. E. Anal. Chim. Acta 1985, 178, 43-66. Nibbering, N. M. M. Reel. Trav. Chim. Pays-Bas 1986. 705, 245-253. Marshall, A. G; Comisarow. M. B.; Parisod, G. J . Chem. Phys. 1979, 7 1 , 4434-4443. Spendley, W.; Hext, G. B.; Himsworth, F. R. Technometrics 1962, 4 , 441-461. Morgan, S. L.; Deming, S. N. Anal. Chem. 1974, 46, 1170-1181. Nelder, J . A.; Mead, R. Computer J . 1065, 78, 308-313. BenerMge, D.; Wade, A. P. Talanta 1985, 32, 709-722. King, P. G.; Deming, S. N.; Morgan, S. L. Anal. Lett. 1975, 8 , 369. McLachlan, L. A. J . M g n . Res. 1977, 26, 223-228. De Koning, L. J.; Fokkens, R. H.; Pinkse, F. A,: Nibbering, N. M. M. Int. J . Mass Spectrom. Ion Processes 1987, 77, 95-105. Ijames. C. F.; Wilkins, C. L. J . Am. Chem. SOC. 1988, 170. 2687-2688. McCrery, D. A.; Gross, M. L. Anal. Chim. Acta 1985, 778, 105-116. Van der Wiel, P. F. A. Anal. Chlm. Acta 1980, 722,421-433.
RECEIVED for review August 15, 1988. Accepted November 14, 1988. J. W. Elling gratefully acknowledges the financial support of the International Rotary Foundation and L. J. de Koning, F. A. Pinkse, and N. M. M. Nibbering thank the Netherlands Organization for Scientific Research (SON/ NWO) for continuous support.
