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Improved resolution for MALDI-TOF mass spectrometers: a mathematical study
International Journal of Mass Spectrometry and Ion Processes 164 (1997) 19–34
Improved resolution for MALDI-TOF mass spectrometers: a mathematical study Jochen Franzen Bruker–Franzen Analytik GmbH, Fahrenheitstr. 4, D-28359 Bremen, Germany Received 15 October 1996; accepted 28 March 1997
Abstract In time-of-flight (TOF) mass spectrometry with matrix-assisted laser desorption and ionization (MALDI), the delayed acceleration of ions (sometimes called ‘‘time-lag focusing’’, ‘‘pulsed ion extraction’’ or ‘‘delayed extraction’’) has proven to be a major breakthrough with respect to high mass resolution. This theoretical study presents some focusing modes which improve either the resolution at one mass (spot focusing) or the resolution for all masses simultaneously (wide-range focusing), compared with the normal delayed acceleration mode. The study considers: (1) the case of strong correlation between ion location in space and ion speed at the onset of the accelerating field, observed for low mass MALDI ions, and (2) the case of additional spatial distributions disturbing the space–velocity correlation at higher masses. Improved focusing is achieved by acceleration fields varying in time. q 1997 Elsevier Science B.V. Keywords: MALDI-TOF mass spectrometer; Space–velocity correlation
1. Introduction 1.1. Success and drawbacks of delayed acceleration for MALDI-TOF Time-of-flight (TOF) mass spectrometers are outstanding tools for the investigation of high molecular weight substances, limited in principle only by the high-mass behavior of the presently available ion detectors. For ions with a given distribution of initial kinetic energies independent of mass in a linear time-of-flight mass spectrometer, the resolution should be mass independent. The advent of the matrix-assisted laser desorption and ionization (MALDI) process for ion generation [1] has caused widespread applications of TOF mass spectrometry for bio-organic
substances of medium and high molecular weight. However, the resolution for MALDI ions drops down rapidly towards higher masses, in contrast to theoretical expectations. Introduction of the delayed acceleration mode [2–5] in MALDI-TOF mass spectrometry has resulted in improved mass resolution up to R < 20 000 in the mass range of 5000 u to 10000 u for MALDI ions in the reflecting mode of research grade spectrometers. In linear MALDI-TOF mass spectrometers, the mass resolving power is somewhat less but nevertheless remarkably good. For higher masses beyond 10 000 u, the resolution, R, obtained in practice still drops rapidly to values of a few hundreds or even a few tens only. The highly praised delayed acceleration mode also has its disadvantage: it focuses ions of a
0168-1176/97/$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved PII S 0 16 8- 1 17 6 (9 7 )0 0 04 9 -9
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single mass only with optimum resolution, especially in the linear mode. In many fields of application, there is an urgent need for a method which focuses ions of all masses simultaneously. 1.2. Objective of this study This mathematical study investigates possible procedures for resolution improvement in timeof-flight spectrometry by the application of time-varying acceleration fields (‘‘dynamic acceleration’’) under different prepositions using a simple spreadsheet program. It is the objective to show that a ‘‘spot focusing’’ with superhigh resolution can be obtained in a very narrow mass range, adjustable to any mass in the spectrum, as long as there exists a strict space– velocity correlation at the onset of the acceleration. The resolution of this spot focusing drops rapidly towards neighbouring masses. On the other hand, it will be shown that it is possible to achieve a ‘‘wide-range’’ focusing for all masses simultaneously. The resolution in the latter case is somewhat less than that of delayed acceleration in its respective optimum for the mass under observation but allows for a sensitive observation of all ions in the spectrum simultaneously without any sample-specific adjustments. For higher ion masses, MALDI does not truly show an undisturbed space–velocity correlation. The resulting superimposed space distribution of ions destroys any high resolution of the spot focusing mode. In this case, however, the dynamic acceleration in the wide-range focusing mode turns out to give, theoretically, additional phase-space focusing resulting in the prediction of resolution values for high masses in the range of 10 000 u to 100 000 u which were not achievable hitherto. 1.3. The MALDI process as a basis for the success of delayed acceleration The explosion-like expansion of a high-density
plume created from matrix molecules in the laser focus on the sample support plate during the laser flash of the MALDI process accelerates all molecules and ions to about the same average velocity and velocity distribution [6]. Other investigations show that larger molecules seem to have lower average velocities [7,8]. With application of the delayed acceleration method, the neutral molecules and the ions will expand together in a field-free region up to the onset of an accelerating field. Ions may be generated already during the laser shot or at a later time in the expanding plume by reactions of the analyte substance molecules with matrix ions. The onset of the acceleration field, caused by sudden generation of a potential difference between the sample support plate and an intermediate electrode, starts the acceleration of the ions, regardless of whether the ions are generated during the laser flash or at a later time within the expanding plume. Thus, the time distribution of ion generation no longer plays a roˆle. In the classical mode of delayed acceleration, the acceleration field, after being switched on, is held constant during time. Depending on delay time and field strength, a small range of masses in the resulting spectrum shows greatly improved resolution by this method. This improved resolution enables us to conclude that the expansion process shows a very regular characteristic, at least for low masses up to about 10 000 u. 1.4. Space–velocity correlation for MALDI ions The improved resolution of the delayed acceleration can be explained by the existence of a very good correlation between the spatial distance of the ions from the sample support plate, and the velocity of the ions (prior to acceleration) vertical to the surface of the sample support plate and axial to the spectrometer. Fig. 1 shows an example for an ideal space– velocity correlation of seven ions with start velocities from 300 to 900 m s −1, measured vertically to the surface of the sample support plate.
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masses up to about 10 000 u, but the ideal correlation is no longer strictly valid in the higher mass range, and causes resolutions much worse than those which can be expected theoretically. 1.5. Time variations of the acceleration field
Fig. 1. Ideal space–velocity correlation for MALDI ions used in the first part of this study.
After a delay time of 100 ns, the ions have gained distances from the sample support plate of exactly 30 to 90 mm, respectively. Any lateral velocity of the ions is of no meaning here because it influences the divergence of the ion beam only, but by no means the time focusing properties. This space–velocity correlation was first made the basis for a theoretical study of focusing conditions by Colby and Reilly [9]. The authors even investigated the influence of an electric field of the form E(t) = E0 (1 − e − t=tr ), to study the influence of ‘‘slow’’ electronic switching. They found that a small shift value for the delay time could be used to correct for the electronic rise time. No other influences on focusing properties were investigated. The first part of this mathematical study will be based on this ideal hypothetical correlation model; flight times for the seven ions shown in Fig. 1 are calculated and investigated with respect to focusing conditions. Experimental data on the quality of the space– velocity correlation in MALDI processes are not yet available. It seems however, that a rather excellent correlation can be found for ions with
It is the basic idea of the present mathematical study to enhance the resolution by decreasing the speed of the ions with extreme start velocities compared to that of ions of medium start speed. This can be done by a time variation of the accelerating field. Ions of high start velocity stay shortest in this region, ions of low start energy stay longest. If the acceleration drops with time, the integrated energy picked up in this acceleration region corrects the effect of different flight times. An increased resolution can be achieved for a single ion mass only by this decreasing voltage. In contrast, if we increase the accelerating voltage in the first acceleration region, we decrease the resolution compared to that of the delayed acceleration mode, but we can expand the range of best resolution over the full spectrum. Fig. 2 shows the types of alterations of the voltage after the onset of the acceleration investigated in this study. The exponential functions offer additional parameters for the adjustment of maximum resolution. The normal delayed acceleration mode used
Fig. 2. The investigated time functions of the acceleration voltage in the first acceleration region.
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hitherto has only two adjustable parameters: The time delay t, and the acceleration voltage U 1 between sample support plate and intermediate electrode. In the spot focusing mode, the voltage decreases exponentially, offering the time constant t 1 of the voltage decay as an additional parameter for adjustments. For the wide-range focusing mode, the voltage has to increase. This adds two parameters to the set of adjustable parameters: the additional acceleration voltage V 1 and the time constant t 1 for the exponential increase of the voltage. Because it turned out that less parameters suffice to achieve excellent results, the delay time t usually was fixed to t = 100 ns for the wide-range focusing mode. Linear variations of the voltage, and parabolic variations can be used, too, with very similar results. Since the exponential variation can be much easier implemented electronically, we restrict this presentation to exponential variations of the kind presented in Fig. 2. 1.6. The linear time-of-flight mass spectrometer under study A very simple linear time-of-flight mass spectrometer arrangement is used as a basis for
Fig. 3. Scheme of the linear time-of-flight mass spectrometer used for this study.
this mathematical study, with the usual intermediate acceleration electrode in the ion source used for delayed acceleration. The initial acceleration voltage after switching was chosen to be identical in all calculations: U 1 + U 2 = 30 kV, where U 1 is the initial acceleration voltage between sample support and intermediate electrode, and U 2 is the constant acceleration voltage between intermediate and ground electrode. The geometrical arrangement is shown in Fig. 3. The linear time-of-flight mass spectrometer exhibited in Fig. 3 is shown with a gridless ion source, using an Einzel lens to focus the slightly diverging ion beam generated by the lateral speed of the ions gained in the expanding plume. The mathematical study, however, does not consider the defocusing and refocusing effects shown in this figure. 1.7. Outline of the excel program The flight times for ions of six different masses, with seven start velocities each, as shown in Fig. 1, are calculated in an excel spreadsheet program. For additional space focusing, three different start locations for each of the seven start velocities for the six masses each were chosen, as outlined in Fig. 12 below. The program calculates first, for each ion velocity, the distance of travel during delay time t. After delay time t, the acceleration begins with field strength U 1/d 1, the voltage U 1 being the start voltage between sample support plate and intermediate electrode, and d 1 the distance between both. The time variations of the acceleration field start with the onset of the acceleration. They are approximated as smoothly as possible by a step function with 20 single steps, defined so that the flight time of the ions in each step becomes about equal: The residual distances (d 1 − v ot) of travel inside the first acceleration region are divided, for each of the ions of different velocity, into 20 segments, the lengths of which increase in
J. Franzen/International Journal of Mass Spectrometry and Ion Processes 164 (1997) 19–34
proportion to 1/210:2/210:3/210:…:20/210 (summing up in total to 210/210). Inside these segments of linearly increasing spatial length the field strength is regarded as constant, and its strength is calculated from the exact time of entry into the segment. The increasing lengths of the segments result in roughly equal flight times of the ions inside the segments, thus approximating the wanted smooth increase of the electric field. Inside each of the segments, the following four physical quantities are calculated in a straightforward manner: (a) the acceleration inside the segment due to the selected time function of the voltage, (b) the time of flight through the segment, calculated from start velocity and acceleration, (c) the total time elapsed at the end of the flight through the segment since the onset of the acceleration, and (d) the final velocity at the end of the segment, calculated from start velocity and velocity gain by acceleration. The same four quantities are calculated once for the second acceleration region of the spectrometer. This region has constant field strength and does not need to be divided into segments. The accelerating voltage in the second acceleration region is calculated to be U 2 = (30 − U 1) kV in all cases reported below. From the resulting velocity at the end of the second acceleration region, the total flight time through the spectrometer’s drift path is calculated. The excel program contains a solver which allows one to automatically vary selected parameters in such a way that a calculated final value becomes a minimum. The sum of squared deviations of the resulting flight times from the flight time value of the ion with average start velocity was chosen to be minimized for each of the ion masses. The minimization was performed either for ions of a single mass only (for the spot focusing), or for all masses simultaneously (for the wide-range focusing) by adding up all the sums of squares. For the time variation of the acceleration field, in most cases exponential functions were used.
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Linear variations of the voltage or parabolic variations proportional to the square root of time turned out to give similar results. Inclusions of flat start plateaus of the acceleration voltage after onset of the acceleration do not change the principal behavior. In the following, only results for the exponential variations are shown. For optimizing the resolution of individual masses in the spot focusing mode, the delay time t was included in the set of parameters to be varied by the solver. For the wide-range focusing mode, the delay time was set to be constant at t = 100 ns in spite of the fact that t = 0 ns gives somewhat better results. As explained above, a delay time is needed to eliminate the influence of the time distribution of ion generation in the expanding plume. Flight time resolution, R, is calculated as flight time over maximum peak width at the foot of the mass signal. Since the mass resolution is exactly half the flight time resolution, but mass resolution is conventionally defined at full width at half height (FWHH), the flight time resolutions, R, given here are roughly identical with mass resolutions according to the conventional definition. The validity of the program, in principle, was (1) checked by programs independently developed in mathcad, and (2) by comparison of the results of linear variations by this method with those of exact integrations. The integration of the exponential variation of the acceleration voltage leads to the Lambert W function which by itself can be calculated by complex approximations only. It should be kept in mind, however, that all values given below are by no means obtained by exact integration. All results are approximations only. 1.8. Maximum theoretical resolution for delayed acceleration The excel program was first used to find out the maximum resolution values for individual ion masses for the above linear mass spectrometer in the delayed acceleration mode. The results are
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Fig. 4. Best theoretical flight-time focusing obtained for the delayed acceleration mode by individually optimizing the resolution for masses 1000 u to 32 000 u. Ordinate: relative deviations of flight times (in ppm) from those of ions with 600 m s −1 start velocity. Abscissa: start velocity of the ions (m s −1). Masses m and resulting flight time resolutions R are indicated in the pictures.
exhibited in Fig. 4, showing the flight time deviations (in ppm from the ions of medium start velocity, ordinate) versus the ion’s start velocities (from 300 to 900 m s −1, abscissa). All resolutions in this case were individually maximized for each ion mass. Maximum achievable resolutions turn out to be inversely proportional to the ion mass for this instrument. The ability to resolve the isotopic pattern ceases at about mass 10 000 u, because here the resolution becomes too low to show two adjacent masses separately. In practical experiments with linear time-of-flight instruments, these maximum theoretical resolutions have not yet been achieved, indicating that the ideal space–velocity correlation may not be valid for real MALDI processes. Experimental resolutions are still lower by about a factor of two at lower masses. Towards higher masses beyond mass 10 000 u, experimental resolutions tend to drop rapidly, indicating bad space–velocity correlation. The
parameters for optimum resolution are presented in Table 1. As can be seen from the figure, the focus is of first order for ions of all masses, indicated by the parabolic form of the deviation curve. Ions with extreme start velocities (either too small or too high) have negative deviations, they both need less time through the mass spectrometer. This means that they are faster than the ions with medium start speed.
2. Focusing under ideal space–velocity correlation conditions 2.1. Spot focusing for highest resolution using three control parameters The three variable parameters for the spot focusing mode, delay time t, voltage U 1, and time constant t 1, were varied by the excel solver
Table 1 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV)
211 1.78
300 1.79
427 1.81
610 1.83
874 1.87
1258 1.91
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Fig. 5. Microscopic focusing with individual optimization for six masses. The ions of mass 2000 u are focused with third order, higher masses exhibit focusing of second order. Table 2 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
440 1.69 1.78
447 1.92 2.15
471 2.28 2.26
524 2.76 2.31
610 3.41 2.38
736 4.27 2.52
to search for optimum resolution for each mass individually. Above mass 2000 u, excellent focusing conditions of second order could be found, indicated by deviation curves of third order. Ions of mass 2000 u present even a focus of third order, and ions of mass 1000 u show excellent focusing with a focus of first order.
As can be seen from Fig. 5, the resolution drops, for each doubling of the ion mass, by factors of 3 in the higher mass range. It can be expected that isotopic separation of peaks can be achieved up to mass 50 000 u, if a strict space– velocity correlation can be maintained. In the lower mass range, the resolution seems to have
Fig. 6. Microscopic focusing of second order for all masses shown, achieved by a small hold time (v = 50 ns) of constant voltage before an exponential decrease.
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Table 3 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
432 1.65 1.83
443 1.88 2.18
470 2.23 2.29
524 2.70 2.32
610 3.34 2.38
735 4.19 2.52
a maximum near mass 2000 u. See Table 2 for the parameters for optimum resolution. The focusing singularity can be shifted towards smaller masses if we introduce a small hold time v of constant voltage U 1 before the voltage decreases according to the exponential function. In Fig. 6 below, the hold time amounts to v = 50 ns. For comparison, the optimum parameter values (in addition to v = 50 ns) for this case are listed in Table 3. The values are very similar to those of the case above showing that the onset of the exponential decrease is not very critical. It is somewhat doubtful whether these focus conditions of second and third order really exist or whether they are artifacts created by the kind of approximation. The extremely small flight time differences used for the optimization may cause such artifacts. There is, however, no doubt at all that the decreasing voltage of the spot focusing mode strongly enhances the resolution of single masses. A resolution in excess of millions cannot be
measured experimentally because there is no detector for this kind of resolution, and no presently available transient recorder has the ability to measure the split of a nanosecond required. The line width of mass 2000 u in Fig. 6 amounts to only 0.0012 ns. Even the peak of mass 32 000 u measures only 0.4 ns at its foot needing a transient recorder of at least 20 GHz for the measurement of the peak profile. There is very little chance that we ever will see the extreme resolutions of Figs 6 and 7 in real experiments. Resolutions in the order of 10 000 to 50 000, however, are realistic goals for a spot focusing of the isotopic pattern. 2.2. Spot focusing mode obtained with two control parameters only The exact adjustment of three parameters is experimentally very difficult. Fig. 7 shows the results of a trial to adjust only two parameters. The time constant t 1 was held constant in this experiment (Table 4).
Fig. 7. Resolution for individual ion masses, obtained by variation of two parameters only.
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J. Franzen/International Journal of Mass Spectrometry and Ion Processes 164 (1997) 19–34 Table 4 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
394 1.63 2.5
585 1.69 2.5
509 2.12 2.5
554 2.48 2.5
631 3.23 2.5
731 4.22 2.5
Fig. 8. Microscopic view mode with variation of the delay time t only.
surprisingly good resolution for all masses by control of the delay time t only. Still better results (Fig. 9) give the variation of the start acceleration voltage U 1 only, keeping constant the delay time t and the time constant t 1 (Table 6). This control of the acceleration voltage U 1 only can be easily implemented in the ion source control electronics. No variable time control is required which is somewhat more complicated to achieve. The quality of the resolution is beyond any practical use. More than unit resolution is obtained for the isotopic pattern up to mass 32 000 u. We have to remember that today a resolution above one hundred is still hard to obtain in this mass range.
Handling of the spot focusing mode with the adjustment of two parameters only equals exactly the handling of the delayed acceleration mode where two parameters also have to be controlled. The spot focusing mode, however, gives a drastically better resolution for the mass under investigation. 2.3. Spot focusing mode with single control parameters only The result of the variation of two parameters was too promising not to try to vary only a single parameter. In Fig. 8, the acceleration start voltage U 1 and the time constant t 1 were both fixed (Table 5). The investigation study resulted in a Table 5 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
46 4.0 2.5
77 4.0 2.5
132 4.0 2.5
236 4.0 2.5
432 4.0 2.5
796 4.0 2.5
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Fig. 9. Microscopic view mode with variation of the acceleration voltage U 1 only.
Table 6 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
700 1.42 2.5
700 1.61 2.5
700 1.90 2.5
700 2.34 2.5
700 3.06 2.5
700 4.34 2.5
Fig. 10. Panoramic view with simultaneous focusing for all masses.
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2.4. Wide-range focusing for simultaneous focusing of all ions of the spectrum The spot focusing mode may be nice but more urgently we need a mode which focuses ions of all masses simultaneously, even if top resolution cannot be achieved. Theoretically, this simultaneous resolution for all masses can be obtained by the wide-range focusing mode with increasing voltage according to Fig. 2. The parameters are as follows: delay time t = 100 ns (fixed), voltage U 1 = 1.68 kV, voltage V 1 = 3.51 kV, time constant t 1 = 1.91 ms. The results presented in Fig. 10 for the mass range from 1000 to 32 000 atomic mass units do not look as exciting as the results of the spot focusing. For practical purposes, however, this wide-range focusing shows very promising aspects: a mass spectrometer may be designed needing no sample-specific adjustments at all. The calibration curve for the mass determination will be taken only once for ever, and only be corrected occasionally. Handling the instrument
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becomes very simple, a true push-button apparatus may be built. The question is, however, if these resolutions really can be achieved, or whether the resolutions at higher masses are destroyed by non-ideal correlation between space and velocity. The resolutions of Fig. 10 in the higher mass range are still above any experimental resolutions obtained hitherto.
2.5. Comparison of focusing modes under strict space–velocity correlation Fig. 11 summarizes the results obtained for ideal space–velocity correlation. The mass range of maximum resolution for the spot focusing is extremely narrow, allowing the observation of a single molecular group of ions only. The resolution of the wide-range focusing is, for all masses, about half the resolution which can be achieved in the delayed acceleration mode for individual ion masses.
Fig. 11. Resolution versus mass in mass range 1000 u to 32 000 u for microscopic view (thin lines with high peaks), delayed acceleration (dashed lines), and panoramic view (solid line).
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Fig. 12. Non-ideal space–velocity correlation for further calculations. Two sets of ions with ‘‘wrong distances’’ are shown in addition to the set with ‘‘correct’’ distances at the onset of acceleration.
3. Focusing under non-ideal space–velocity correlation conditions 3.1. Disturbance of the space–velocity correlation The resolution obtained experimentally with the method of delayed acceleration in comparison with the above mathematical results indicates that the ideal space–velocity correlation is no longer valid at higher masses. In the expanding MALDI plume, molecules of high mass are only as long accelerated with the same amount as ions of low mass, as there exists an extremely high pressure. With pressures in excess of hundreds of atmospheres, heavy molecules experience the same acceleration by viscous friction as the light molecules. With pressures in the atmospheric range and lower, however, many of the lighter molecules pass the heavier molecules leaving them behind with slower speed. The ideal correlation becomes corrupted. Fig. 12 outlines a primitive model of a
disturbed space–velocity correlation. This distribution of the ions in the ‘‘phase space’’ is used for the mathematical experiments in this section. To arrive at this corrupted correlation, the locations of the ions at the onset of acceleration are modified in two ways: absolute dislocations of 3 mm for the seven ions are superimposed by relative dislocations of 10% of the distances. Such wrongly positioned ions are added with dislocations in both directions, too low distances and too high distances, as shown in Fig. 12. Application of this non-ideal correlation to the spot focusing mode immediately reveals that this mode cannot accept any space distribution. The resolution immediately drops down to values of no interest. As will be shown below, ions focused in the normal delayed acceleration mode also suffer a bad resolution. 3.2. Wide-range focusing with additional spatial focusing The resolution for ions of all masses simultaneously was maximized by minimizing the sum of squared deviations of the flight times of all dislocated and non-dislocated ions from the respective flight times of ions with medium start velocity and correct location. The result is presented in Fig. 13 for ions in a large mass range from 5000 to 50 000 atomic mass units. In comparison to the results of Fig. 10, the flight time resolutions do not look that spectacular. Nonetheless the resolutions for high masses above 20 000 atomic mass units are by far better than any resolution obtained hitherto experimentally. Experiments have to be started to find out whether the indicated space focusing really takes place. Parameters for the results in Fig. 13: t = 100 ns, U 1 = 0,90 kV, V 1 = 3.16 kV, and t 1 = 1.24 ms. The parameters given here are not very critical. All parameters can be varied by a few percent without changing the resulting resolution. An instrument designed to operate in this mode will be very stable with respect to resolution. It is
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Fig. 13. Panoramic dynamic focusing in the mass range 5000 u to 50 000 u with resolutions above R = 1000.
easily adjustable for optimum performance. It nevertheless requires stable electronics to maintain a stable calibration curve. Resolution turns out to be even better by about 30% than those in Fig. 13 for the unfavorable case of disappearing delay (t = 0).
3.3. Comparison with the delayed acceleration mode The result of the wide-range focusing mode has to be compared with results of the normal delayed acceleration mode to really recognize
Fig. 14. Delayed acceleration mode under non-ideal correlation, individually optimized for the masses shown.
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Table 7 Parameters for optimum resolution Mass
5000 u
14 000 u
23 000 u
32 000 u
41 000 u
50 000 u
Time delay t (ns) Voltage U 1 (kV)
73 4.15
97 5.32
119 5.97
140 6.42
159 6.78
177 7.07
the space-focusing effect. As can be seen from Fig. 14, the resolutions optimized for individual masses are roughly about half as good as the values obtained in Fig. 13 for all masses simultaneously by the wide-range focusing mode. The parameters resulting from optimization are presented in Table 7. The improvement of the wide-range focusing mode over normal delayed acceleration can be seen from Fig. 15 in an overview diagram. In contrast to results obtained for the ideal space– velocity correlation, the wide-range focusing mode is much superior if the space–velocity correlation is corrupted by the MALDI process.
Fig. 16 presents, in the upper row, three portions of a spectrum (single laser shot) obtained by delayed acceleration. A linear mass spectrometer with 1.3 m flight path was used. The focus was optimized at mass 1048 u, the ions at masses 172 u and 379 u therefore do not show best resolution. In the bottom row, the new wide-range focusing mode was applied. By varying the voltages U 1 and V 1, the ions of masses 172 u and 1048 u were focused simultaneously, exhibiting optimum results also for ions of mass 379 u.
3.4. First experimental results
TOF mass spectrometry, in principle, is well suited for the investigation of high mass ions. The mass independent resolution, theoretically to be expected for ions with equal energy distributions, does not show for MALDI ions, indicating non-equal energy distributions. The success of the delayed acceleration mode in the lower mass range points to the validity of
The first experimental results were obtained after the reviewer of this article greatly regretted that no attempts were made to include experimental checks of the theoretical predictions. The author, therefore, takes the opportunity to include the first experimental measurement here.
4. Conclusions
Fig. 15. Comparison of resolutions of focusing modes with space focusing.
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Fig. 16. First experimental results for the wide-range focusing mode in a linear TOR. Upper row: three ion groups from a spectrum obtained by delayed acceleration, optimized in resolution for the ion group at mass 1046 u. Lower row: the same three groups of ions in a spectrum obtained with wide-range focusing.
velocity distributions with ideal space–velocity correlation. Towards higher masses in the spectrum, however, this correlation seems to become more and more corrupted, and better focusing methods have to be searched for. This study investigated some better focusing procedures for ideal and non-ideal space–velocity correlations. In the case of an ideal correlation, the ‘‘spot focusing’’ mode has shown theoretical resolution values of a quality beyond any practical value. Neither the MALDI process in its present form, nor the available transient recorders or the resolution of today’s detectors will allow one to make use of the superhigh resolution resulting from
this mode approaching the performance of ICR Fourier transform instruments. The resolution is restricted to very narrow mass ranges but can be adjusted to any mass. In this context it is an academic question whether time focusing of second or third order really exists or whether this is an artifact caused by the approximation method used. An independent indication for the (theoretical) existence of focusing of second order in linear time-of-flight instruments can be found in the literature, not mentioned by the authors but visible from one of the figures. There is, however, no doubt at all that the spot focusing mode yields high resolution. The mode
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can be used to look for the isotopic pattern of interest, e. g. to see losses of water or ammonia by irregularities of the isotopic pattern. The spot focusing may further be used to investigate and to improve the MALDI process. It can be expected that improved MALDI processes may be developed capable to enable the acquisition of partial spectra with high resolution. Not that spectacular, but of highly practical value is the ‘‘wide-range focusing’’ mode. This mode works well with ideal space–velocity correlations, even if the resolution is smaller than that of the optimum delayed acceleration mode for a selected ion mass. For non-ideal space–velocity correlations, however, the wide-range focusing mode outperforms by far any adjustment possibility of the delayed acceleration mode. The wide-range focusing mode has the ability of additional phase-space focusing. A first experiment demonstrates that this theoretical capability applies to reality in MALDI-TOF mass spectrometry. This wide-range focusing mode enables us to design mass spectrometers for use in the high mass field needing no sample-specific adjustments at all. Adjustment of parameters is highly uncritical. The calibration curve stays valid unless there are some electronic instabilities. Resolution is much better than that achieved hitherto. A push-button instrument is in reach. Such an instrument may be equipped with the additional feature of a spot focusing mode for an ion signal observed in the wide-range focusing. The simplicity of adjustment by a single parameter (or by two parameters coupled together) will ease the use of such an instrument. Acknowledgements The author would like to thank Dr. Arnim
Holle and Dr. Claus Koster for many fruitful discussions and critical remarks. The latter has written the MATHCAD programs confirming, in principle, the results presented here. Mr. Du¨rer at the Institute of Informatics of the University Bremen has performed the comparison of the linear variations of the acceleration voltage with exact integration values; he also found the Lambert W function as an analytical solution for the integration of the exponential variations. Dr. Arnim Holle obtained the first experimental corroboration for the wide-range focusing shown in Fig. 16.
References [1] M. Karas, F. Hillenkamp, Anal. Chem., 60 (1988) 2299–2301. K. Tanaka et al., Rapid Commun. Mass Spectrom. 2 (1988) 151. [2] W. C. Wiley, I. H. McLaren, Time-of-flight mass spectrometer with improved resolution, Rev. Sci. Instrum. 26 (1955) 1150. [3] R. S. Brown, J. J. Lennon, Mass resolution improvement by incorporation of pulsed ion extraction in a matrix-assisted laser desorption/ionization linear time-of-flight mass spectrometer, Anal. Chem. 67 (1995) 1998. [4] R. M. Whittal, L. Li, High-resolution matrix-assisted laser desorption/ionization in a linear time-of-flight mass spectrometer, Anal. Chem. 67 (1995) 1950. [5] P. Juhasz et al., Applications of delayed extraction matrixassisted laser desorption ionization time-of-flight mass spectrpmetry to oligonucleotide analysis, Anal. Chem. 68 (1996) 941. [6] R. C. Beavis, B. T. Chait, Chem. Phys. Lett. 181 (1992) 479. [7] L. M. Russon, R. M. Whittal, L. Li, Measurement of MALDI initial ion velocities, Proceedings of the 44th ASMS Conference on Mass Spectrometry and Allied Topics, Portland, OR, May 12–16, 1996, p. 731. [8] P. Juhasz, M. L. Vestal, S. A. Martin, Novel method for the measurement of the initial velocity of ions generated by MALDI, Proceedings of the 44th ASMS Conference on Mass Spectrometry and Allied Topics, Portland, OR, May 12–16, 1996, p. 730. [9] S. M. Colby, J. P. Reilly, Space–velocity correlation focusing, Anal. Chem. 68 (1996) 1419–1428.
Improved resolution for MALDI-TOF mass spectrometers: a mathematical study Jochen Franzen Bruker–Franzen Analytik GmbH, Fahrenheitstr. 4, D-28359 Bremen, Germany Received 15 October 1996; accepted 28 March 1997
Abstract In time-of-flight (TOF) mass spectrometry with matrix-assisted laser desorption and ionization (MALDI), the delayed acceleration of ions (sometimes called ‘‘time-lag focusing’’, ‘‘pulsed ion extraction’’ or ‘‘delayed extraction’’) has proven to be a major breakthrough with respect to high mass resolution. This theoretical study presents some focusing modes which improve either the resolution at one mass (spot focusing) or the resolution for all masses simultaneously (wide-range focusing), compared with the normal delayed acceleration mode. The study considers: (1) the case of strong correlation between ion location in space and ion speed at the onset of the accelerating field, observed for low mass MALDI ions, and (2) the case of additional spatial distributions disturbing the space–velocity correlation at higher masses. Improved focusing is achieved by acceleration fields varying in time. q 1997 Elsevier Science B.V. Keywords: MALDI-TOF mass spectrometer; Space–velocity correlation
1. Introduction 1.1. Success and drawbacks of delayed acceleration for MALDI-TOF Time-of-flight (TOF) mass spectrometers are outstanding tools for the investigation of high molecular weight substances, limited in principle only by the high-mass behavior of the presently available ion detectors. For ions with a given distribution of initial kinetic energies independent of mass in a linear time-of-flight mass spectrometer, the resolution should be mass independent. The advent of the matrix-assisted laser desorption and ionization (MALDI) process for ion generation [1] has caused widespread applications of TOF mass spectrometry for bio-organic
substances of medium and high molecular weight. However, the resolution for MALDI ions drops down rapidly towards higher masses, in contrast to theoretical expectations. Introduction of the delayed acceleration mode [2–5] in MALDI-TOF mass spectrometry has resulted in improved mass resolution up to R < 20 000 in the mass range of 5000 u to 10000 u for MALDI ions in the reflecting mode of research grade spectrometers. In linear MALDI-TOF mass spectrometers, the mass resolving power is somewhat less but nevertheless remarkably good. For higher masses beyond 10 000 u, the resolution, R, obtained in practice still drops rapidly to values of a few hundreds or even a few tens only. The highly praised delayed acceleration mode also has its disadvantage: it focuses ions of a
0168-1176/97/$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved PII S 0 16 8- 1 17 6 (9 7 )0 0 04 9 -9
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single mass only with optimum resolution, especially in the linear mode. In many fields of application, there is an urgent need for a method which focuses ions of all masses simultaneously. 1.2. Objective of this study This mathematical study investigates possible procedures for resolution improvement in timeof-flight spectrometry by the application of time-varying acceleration fields (‘‘dynamic acceleration’’) under different prepositions using a simple spreadsheet program. It is the objective to show that a ‘‘spot focusing’’ with superhigh resolution can be obtained in a very narrow mass range, adjustable to any mass in the spectrum, as long as there exists a strict space– velocity correlation at the onset of the acceleration. The resolution of this spot focusing drops rapidly towards neighbouring masses. On the other hand, it will be shown that it is possible to achieve a ‘‘wide-range’’ focusing for all masses simultaneously. The resolution in the latter case is somewhat less than that of delayed acceleration in its respective optimum for the mass under observation but allows for a sensitive observation of all ions in the spectrum simultaneously without any sample-specific adjustments. For higher ion masses, MALDI does not truly show an undisturbed space–velocity correlation. The resulting superimposed space distribution of ions destroys any high resolution of the spot focusing mode. In this case, however, the dynamic acceleration in the wide-range focusing mode turns out to give, theoretically, additional phase-space focusing resulting in the prediction of resolution values for high masses in the range of 10 000 u to 100 000 u which were not achievable hitherto. 1.3. The MALDI process as a basis for the success of delayed acceleration The explosion-like expansion of a high-density
plume created from matrix molecules in the laser focus on the sample support plate during the laser flash of the MALDI process accelerates all molecules and ions to about the same average velocity and velocity distribution [6]. Other investigations show that larger molecules seem to have lower average velocities [7,8]. With application of the delayed acceleration method, the neutral molecules and the ions will expand together in a field-free region up to the onset of an accelerating field. Ions may be generated already during the laser shot or at a later time in the expanding plume by reactions of the analyte substance molecules with matrix ions. The onset of the acceleration field, caused by sudden generation of a potential difference between the sample support plate and an intermediate electrode, starts the acceleration of the ions, regardless of whether the ions are generated during the laser flash or at a later time within the expanding plume. Thus, the time distribution of ion generation no longer plays a roˆle. In the classical mode of delayed acceleration, the acceleration field, after being switched on, is held constant during time. Depending on delay time and field strength, a small range of masses in the resulting spectrum shows greatly improved resolution by this method. This improved resolution enables us to conclude that the expansion process shows a very regular characteristic, at least for low masses up to about 10 000 u. 1.4. Space–velocity correlation for MALDI ions The improved resolution of the delayed acceleration can be explained by the existence of a very good correlation between the spatial distance of the ions from the sample support plate, and the velocity of the ions (prior to acceleration) vertical to the surface of the sample support plate and axial to the spectrometer. Fig. 1 shows an example for an ideal space– velocity correlation of seven ions with start velocities from 300 to 900 m s −1, measured vertically to the surface of the sample support plate.
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masses up to about 10 000 u, but the ideal correlation is no longer strictly valid in the higher mass range, and causes resolutions much worse than those which can be expected theoretically. 1.5. Time variations of the acceleration field
Fig. 1. Ideal space–velocity correlation for MALDI ions used in the first part of this study.
After a delay time of 100 ns, the ions have gained distances from the sample support plate of exactly 30 to 90 mm, respectively. Any lateral velocity of the ions is of no meaning here because it influences the divergence of the ion beam only, but by no means the time focusing properties. This space–velocity correlation was first made the basis for a theoretical study of focusing conditions by Colby and Reilly [9]. The authors even investigated the influence of an electric field of the form E(t) = E0 (1 − e − t=tr ), to study the influence of ‘‘slow’’ electronic switching. They found that a small shift value for the delay time could be used to correct for the electronic rise time. No other influences on focusing properties were investigated. The first part of this mathematical study will be based on this ideal hypothetical correlation model; flight times for the seven ions shown in Fig. 1 are calculated and investigated with respect to focusing conditions. Experimental data on the quality of the space– velocity correlation in MALDI processes are not yet available. It seems however, that a rather excellent correlation can be found for ions with
It is the basic idea of the present mathematical study to enhance the resolution by decreasing the speed of the ions with extreme start velocities compared to that of ions of medium start speed. This can be done by a time variation of the accelerating field. Ions of high start velocity stay shortest in this region, ions of low start energy stay longest. If the acceleration drops with time, the integrated energy picked up in this acceleration region corrects the effect of different flight times. An increased resolution can be achieved for a single ion mass only by this decreasing voltage. In contrast, if we increase the accelerating voltage in the first acceleration region, we decrease the resolution compared to that of the delayed acceleration mode, but we can expand the range of best resolution over the full spectrum. Fig. 2 shows the types of alterations of the voltage after the onset of the acceleration investigated in this study. The exponential functions offer additional parameters for the adjustment of maximum resolution. The normal delayed acceleration mode used
Fig. 2. The investigated time functions of the acceleration voltage in the first acceleration region.
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hitherto has only two adjustable parameters: The time delay t, and the acceleration voltage U 1 between sample support plate and intermediate electrode. In the spot focusing mode, the voltage decreases exponentially, offering the time constant t 1 of the voltage decay as an additional parameter for adjustments. For the wide-range focusing mode, the voltage has to increase. This adds two parameters to the set of adjustable parameters: the additional acceleration voltage V 1 and the time constant t 1 for the exponential increase of the voltage. Because it turned out that less parameters suffice to achieve excellent results, the delay time t usually was fixed to t = 100 ns for the wide-range focusing mode. Linear variations of the voltage, and parabolic variations can be used, too, with very similar results. Since the exponential variation can be much easier implemented electronically, we restrict this presentation to exponential variations of the kind presented in Fig. 2. 1.6. The linear time-of-flight mass spectrometer under study A very simple linear time-of-flight mass spectrometer arrangement is used as a basis for
Fig. 3. Scheme of the linear time-of-flight mass spectrometer used for this study.
this mathematical study, with the usual intermediate acceleration electrode in the ion source used for delayed acceleration. The initial acceleration voltage after switching was chosen to be identical in all calculations: U 1 + U 2 = 30 kV, where U 1 is the initial acceleration voltage between sample support and intermediate electrode, and U 2 is the constant acceleration voltage between intermediate and ground electrode. The geometrical arrangement is shown in Fig. 3. The linear time-of-flight mass spectrometer exhibited in Fig. 3 is shown with a gridless ion source, using an Einzel lens to focus the slightly diverging ion beam generated by the lateral speed of the ions gained in the expanding plume. The mathematical study, however, does not consider the defocusing and refocusing effects shown in this figure. 1.7. Outline of the excel program The flight times for ions of six different masses, with seven start velocities each, as shown in Fig. 1, are calculated in an excel spreadsheet program. For additional space focusing, three different start locations for each of the seven start velocities for the six masses each were chosen, as outlined in Fig. 12 below. The program calculates first, for each ion velocity, the distance of travel during delay time t. After delay time t, the acceleration begins with field strength U 1/d 1, the voltage U 1 being the start voltage between sample support plate and intermediate electrode, and d 1 the distance between both. The time variations of the acceleration field start with the onset of the acceleration. They are approximated as smoothly as possible by a step function with 20 single steps, defined so that the flight time of the ions in each step becomes about equal: The residual distances (d 1 − v ot) of travel inside the first acceleration region are divided, for each of the ions of different velocity, into 20 segments, the lengths of which increase in
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proportion to 1/210:2/210:3/210:…:20/210 (summing up in total to 210/210). Inside these segments of linearly increasing spatial length the field strength is regarded as constant, and its strength is calculated from the exact time of entry into the segment. The increasing lengths of the segments result in roughly equal flight times of the ions inside the segments, thus approximating the wanted smooth increase of the electric field. Inside each of the segments, the following four physical quantities are calculated in a straightforward manner: (a) the acceleration inside the segment due to the selected time function of the voltage, (b) the time of flight through the segment, calculated from start velocity and acceleration, (c) the total time elapsed at the end of the flight through the segment since the onset of the acceleration, and (d) the final velocity at the end of the segment, calculated from start velocity and velocity gain by acceleration. The same four quantities are calculated once for the second acceleration region of the spectrometer. This region has constant field strength and does not need to be divided into segments. The accelerating voltage in the second acceleration region is calculated to be U 2 = (30 − U 1) kV in all cases reported below. From the resulting velocity at the end of the second acceleration region, the total flight time through the spectrometer’s drift path is calculated. The excel program contains a solver which allows one to automatically vary selected parameters in such a way that a calculated final value becomes a minimum. The sum of squared deviations of the resulting flight times from the flight time value of the ion with average start velocity was chosen to be minimized for each of the ion masses. The minimization was performed either for ions of a single mass only (for the spot focusing), or for all masses simultaneously (for the wide-range focusing) by adding up all the sums of squares. For the time variation of the acceleration field, in most cases exponential functions were used.
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Linear variations of the voltage or parabolic variations proportional to the square root of time turned out to give similar results. Inclusions of flat start plateaus of the acceleration voltage after onset of the acceleration do not change the principal behavior. In the following, only results for the exponential variations are shown. For optimizing the resolution of individual masses in the spot focusing mode, the delay time t was included in the set of parameters to be varied by the solver. For the wide-range focusing mode, the delay time was set to be constant at t = 100 ns in spite of the fact that t = 0 ns gives somewhat better results. As explained above, a delay time is needed to eliminate the influence of the time distribution of ion generation in the expanding plume. Flight time resolution, R, is calculated as flight time over maximum peak width at the foot of the mass signal. Since the mass resolution is exactly half the flight time resolution, but mass resolution is conventionally defined at full width at half height (FWHH), the flight time resolutions, R, given here are roughly identical with mass resolutions according to the conventional definition. The validity of the program, in principle, was (1) checked by programs independently developed in mathcad, and (2) by comparison of the results of linear variations by this method with those of exact integrations. The integration of the exponential variation of the acceleration voltage leads to the Lambert W function which by itself can be calculated by complex approximations only. It should be kept in mind, however, that all values given below are by no means obtained by exact integration. All results are approximations only. 1.8. Maximum theoretical resolution for delayed acceleration The excel program was first used to find out the maximum resolution values for individual ion masses for the above linear mass spectrometer in the delayed acceleration mode. The results are
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Fig. 4. Best theoretical flight-time focusing obtained for the delayed acceleration mode by individually optimizing the resolution for masses 1000 u to 32 000 u. Ordinate: relative deviations of flight times (in ppm) from those of ions with 600 m s −1 start velocity. Abscissa: start velocity of the ions (m s −1). Masses m and resulting flight time resolutions R are indicated in the pictures.
exhibited in Fig. 4, showing the flight time deviations (in ppm from the ions of medium start velocity, ordinate) versus the ion’s start velocities (from 300 to 900 m s −1, abscissa). All resolutions in this case were individually maximized for each ion mass. Maximum achievable resolutions turn out to be inversely proportional to the ion mass for this instrument. The ability to resolve the isotopic pattern ceases at about mass 10 000 u, because here the resolution becomes too low to show two adjacent masses separately. In practical experiments with linear time-of-flight instruments, these maximum theoretical resolutions have not yet been achieved, indicating that the ideal space–velocity correlation may not be valid for real MALDI processes. Experimental resolutions are still lower by about a factor of two at lower masses. Towards higher masses beyond mass 10 000 u, experimental resolutions tend to drop rapidly, indicating bad space–velocity correlation. The
parameters for optimum resolution are presented in Table 1. As can be seen from the figure, the focus is of first order for ions of all masses, indicated by the parabolic form of the deviation curve. Ions with extreme start velocities (either too small or too high) have negative deviations, they both need less time through the mass spectrometer. This means that they are faster than the ions with medium start speed.
2. Focusing under ideal space–velocity correlation conditions 2.1. Spot focusing for highest resolution using three control parameters The three variable parameters for the spot focusing mode, delay time t, voltage U 1, and time constant t 1, were varied by the excel solver
Table 1 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV)
211 1.78
300 1.79
427 1.81
610 1.83
874 1.87
1258 1.91
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Fig. 5. Microscopic focusing with individual optimization for six masses. The ions of mass 2000 u are focused with third order, higher masses exhibit focusing of second order. Table 2 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
440 1.69 1.78
447 1.92 2.15
471 2.28 2.26
524 2.76 2.31
610 3.41 2.38
736 4.27 2.52
to search for optimum resolution for each mass individually. Above mass 2000 u, excellent focusing conditions of second order could be found, indicated by deviation curves of third order. Ions of mass 2000 u present even a focus of third order, and ions of mass 1000 u show excellent focusing with a focus of first order.
As can be seen from Fig. 5, the resolution drops, for each doubling of the ion mass, by factors of 3 in the higher mass range. It can be expected that isotopic separation of peaks can be achieved up to mass 50 000 u, if a strict space– velocity correlation can be maintained. In the lower mass range, the resolution seems to have
Fig. 6. Microscopic focusing of second order for all masses shown, achieved by a small hold time (v = 50 ns) of constant voltage before an exponential decrease.
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Table 3 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
432 1.65 1.83
443 1.88 2.18
470 2.23 2.29
524 2.70 2.32
610 3.34 2.38
735 4.19 2.52
a maximum near mass 2000 u. See Table 2 for the parameters for optimum resolution. The focusing singularity can be shifted towards smaller masses if we introduce a small hold time v of constant voltage U 1 before the voltage decreases according to the exponential function. In Fig. 6 below, the hold time amounts to v = 50 ns. For comparison, the optimum parameter values (in addition to v = 50 ns) for this case are listed in Table 3. The values are very similar to those of the case above showing that the onset of the exponential decrease is not very critical. It is somewhat doubtful whether these focus conditions of second and third order really exist or whether they are artifacts created by the kind of approximation. The extremely small flight time differences used for the optimization may cause such artifacts. There is, however, no doubt at all that the decreasing voltage of the spot focusing mode strongly enhances the resolution of single masses. A resolution in excess of millions cannot be
measured experimentally because there is no detector for this kind of resolution, and no presently available transient recorder has the ability to measure the split of a nanosecond required. The line width of mass 2000 u in Fig. 6 amounts to only 0.0012 ns. Even the peak of mass 32 000 u measures only 0.4 ns at its foot needing a transient recorder of at least 20 GHz for the measurement of the peak profile. There is very little chance that we ever will see the extreme resolutions of Figs 6 and 7 in real experiments. Resolutions in the order of 10 000 to 50 000, however, are realistic goals for a spot focusing of the isotopic pattern. 2.2. Spot focusing mode obtained with two control parameters only The exact adjustment of three parameters is experimentally very difficult. Fig. 7 shows the results of a trial to adjust only two parameters. The time constant t 1 was held constant in this experiment (Table 4).
Fig. 7. Resolution for individual ion masses, obtained by variation of two parameters only.
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J. Franzen/International Journal of Mass Spectrometry and Ion Processes 164 (1997) 19–34 Table 4 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
394 1.63 2.5
585 1.69 2.5
509 2.12 2.5
554 2.48 2.5
631 3.23 2.5
731 4.22 2.5
Fig. 8. Microscopic view mode with variation of the delay time t only.
surprisingly good resolution for all masses by control of the delay time t only. Still better results (Fig. 9) give the variation of the start acceleration voltage U 1 only, keeping constant the delay time t and the time constant t 1 (Table 6). This control of the acceleration voltage U 1 only can be easily implemented in the ion source control electronics. No variable time control is required which is somewhat more complicated to achieve. The quality of the resolution is beyond any practical use. More than unit resolution is obtained for the isotopic pattern up to mass 32 000 u. We have to remember that today a resolution above one hundred is still hard to obtain in this mass range.
Handling of the spot focusing mode with the adjustment of two parameters only equals exactly the handling of the delayed acceleration mode where two parameters also have to be controlled. The spot focusing mode, however, gives a drastically better resolution for the mass under investigation. 2.3. Spot focusing mode with single control parameters only The result of the variation of two parameters was too promising not to try to vary only a single parameter. In Fig. 8, the acceleration start voltage U 1 and the time constant t 1 were both fixed (Table 5). The investigation study resulted in a Table 5 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
46 4.0 2.5
77 4.0 2.5
132 4.0 2.5
236 4.0 2.5
432 4.0 2.5
796 4.0 2.5
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Fig. 9. Microscopic view mode with variation of the acceleration voltage U 1 only.
Table 6 Parameters for optimum resolution Mass
1000 u
2000 u
4000 u
8000 u
16 000 u
32 000 u
Time delay t (ns) Voltage U 1 (kV) Time constant t 1 (ms)
700 1.42 2.5
700 1.61 2.5
700 1.90 2.5
700 2.34 2.5
700 3.06 2.5
700 4.34 2.5
Fig. 10. Panoramic view with simultaneous focusing for all masses.
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2.4. Wide-range focusing for simultaneous focusing of all ions of the spectrum The spot focusing mode may be nice but more urgently we need a mode which focuses ions of all masses simultaneously, even if top resolution cannot be achieved. Theoretically, this simultaneous resolution for all masses can be obtained by the wide-range focusing mode with increasing voltage according to Fig. 2. The parameters are as follows: delay time t = 100 ns (fixed), voltage U 1 = 1.68 kV, voltage V 1 = 3.51 kV, time constant t 1 = 1.91 ms. The results presented in Fig. 10 for the mass range from 1000 to 32 000 atomic mass units do not look as exciting as the results of the spot focusing. For practical purposes, however, this wide-range focusing shows very promising aspects: a mass spectrometer may be designed needing no sample-specific adjustments at all. The calibration curve for the mass determination will be taken only once for ever, and only be corrected occasionally. Handling the instrument
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becomes very simple, a true push-button apparatus may be built. The question is, however, if these resolutions really can be achieved, or whether the resolutions at higher masses are destroyed by non-ideal correlation between space and velocity. The resolutions of Fig. 10 in the higher mass range are still above any experimental resolutions obtained hitherto.
2.5. Comparison of focusing modes under strict space–velocity correlation Fig. 11 summarizes the results obtained for ideal space–velocity correlation. The mass range of maximum resolution for the spot focusing is extremely narrow, allowing the observation of a single molecular group of ions only. The resolution of the wide-range focusing is, for all masses, about half the resolution which can be achieved in the delayed acceleration mode for individual ion masses.
Fig. 11. Resolution versus mass in mass range 1000 u to 32 000 u for microscopic view (thin lines with high peaks), delayed acceleration (dashed lines), and panoramic view (solid line).
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Fig. 12. Non-ideal space–velocity correlation for further calculations. Two sets of ions with ‘‘wrong distances’’ are shown in addition to the set with ‘‘correct’’ distances at the onset of acceleration.
3. Focusing under non-ideal space–velocity correlation conditions 3.1. Disturbance of the space–velocity correlation The resolution obtained experimentally with the method of delayed acceleration in comparison with the above mathematical results indicates that the ideal space–velocity correlation is no longer valid at higher masses. In the expanding MALDI plume, molecules of high mass are only as long accelerated with the same amount as ions of low mass, as there exists an extremely high pressure. With pressures in excess of hundreds of atmospheres, heavy molecules experience the same acceleration by viscous friction as the light molecules. With pressures in the atmospheric range and lower, however, many of the lighter molecules pass the heavier molecules leaving them behind with slower speed. The ideal correlation becomes corrupted. Fig. 12 outlines a primitive model of a
disturbed space–velocity correlation. This distribution of the ions in the ‘‘phase space’’ is used for the mathematical experiments in this section. To arrive at this corrupted correlation, the locations of the ions at the onset of acceleration are modified in two ways: absolute dislocations of 3 mm for the seven ions are superimposed by relative dislocations of 10% of the distances. Such wrongly positioned ions are added with dislocations in both directions, too low distances and too high distances, as shown in Fig. 12. Application of this non-ideal correlation to the spot focusing mode immediately reveals that this mode cannot accept any space distribution. The resolution immediately drops down to values of no interest. As will be shown below, ions focused in the normal delayed acceleration mode also suffer a bad resolution. 3.2. Wide-range focusing with additional spatial focusing The resolution for ions of all masses simultaneously was maximized by minimizing the sum of squared deviations of the flight times of all dislocated and non-dislocated ions from the respective flight times of ions with medium start velocity and correct location. The result is presented in Fig. 13 for ions in a large mass range from 5000 to 50 000 atomic mass units. In comparison to the results of Fig. 10, the flight time resolutions do not look that spectacular. Nonetheless the resolutions for high masses above 20 000 atomic mass units are by far better than any resolution obtained hitherto experimentally. Experiments have to be started to find out whether the indicated space focusing really takes place. Parameters for the results in Fig. 13: t = 100 ns, U 1 = 0,90 kV, V 1 = 3.16 kV, and t 1 = 1.24 ms. The parameters given here are not very critical. All parameters can be varied by a few percent without changing the resulting resolution. An instrument designed to operate in this mode will be very stable with respect to resolution. It is
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Fig. 13. Panoramic dynamic focusing in the mass range 5000 u to 50 000 u with resolutions above R = 1000.
easily adjustable for optimum performance. It nevertheless requires stable electronics to maintain a stable calibration curve. Resolution turns out to be even better by about 30% than those in Fig. 13 for the unfavorable case of disappearing delay (t = 0).
3.3. Comparison with the delayed acceleration mode The result of the wide-range focusing mode has to be compared with results of the normal delayed acceleration mode to really recognize
Fig. 14. Delayed acceleration mode under non-ideal correlation, individually optimized for the masses shown.
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Table 7 Parameters for optimum resolution Mass
5000 u
14 000 u
23 000 u
32 000 u
41 000 u
50 000 u
Time delay t (ns) Voltage U 1 (kV)
73 4.15
97 5.32
119 5.97
140 6.42
159 6.78
177 7.07
the space-focusing effect. As can be seen from Fig. 14, the resolutions optimized for individual masses are roughly about half as good as the values obtained in Fig. 13 for all masses simultaneously by the wide-range focusing mode. The parameters resulting from optimization are presented in Table 7. The improvement of the wide-range focusing mode over normal delayed acceleration can be seen from Fig. 15 in an overview diagram. In contrast to results obtained for the ideal space– velocity correlation, the wide-range focusing mode is much superior if the space–velocity correlation is corrupted by the MALDI process.
Fig. 16 presents, in the upper row, three portions of a spectrum (single laser shot) obtained by delayed acceleration. A linear mass spectrometer with 1.3 m flight path was used. The focus was optimized at mass 1048 u, the ions at masses 172 u and 379 u therefore do not show best resolution. In the bottom row, the new wide-range focusing mode was applied. By varying the voltages U 1 and V 1, the ions of masses 172 u and 1048 u were focused simultaneously, exhibiting optimum results also for ions of mass 379 u.
3.4. First experimental results
TOF mass spectrometry, in principle, is well suited for the investigation of high mass ions. The mass independent resolution, theoretically to be expected for ions with equal energy distributions, does not show for MALDI ions, indicating non-equal energy distributions. The success of the delayed acceleration mode in the lower mass range points to the validity of
The first experimental results were obtained after the reviewer of this article greatly regretted that no attempts were made to include experimental checks of the theoretical predictions. The author, therefore, takes the opportunity to include the first experimental measurement here.
4. Conclusions
Fig. 15. Comparison of resolutions of focusing modes with space focusing.
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Fig. 16. First experimental results for the wide-range focusing mode in a linear TOR. Upper row: three ion groups from a spectrum obtained by delayed acceleration, optimized in resolution for the ion group at mass 1046 u. Lower row: the same three groups of ions in a spectrum obtained with wide-range focusing.
velocity distributions with ideal space–velocity correlation. Towards higher masses in the spectrum, however, this correlation seems to become more and more corrupted, and better focusing methods have to be searched for. This study investigated some better focusing procedures for ideal and non-ideal space–velocity correlations. In the case of an ideal correlation, the ‘‘spot focusing’’ mode has shown theoretical resolution values of a quality beyond any practical value. Neither the MALDI process in its present form, nor the available transient recorders or the resolution of today’s detectors will allow one to make use of the superhigh resolution resulting from
this mode approaching the performance of ICR Fourier transform instruments. The resolution is restricted to very narrow mass ranges but can be adjusted to any mass. In this context it is an academic question whether time focusing of second or third order really exists or whether this is an artifact caused by the approximation method used. An independent indication for the (theoretical) existence of focusing of second order in linear time-of-flight instruments can be found in the literature, not mentioned by the authors but visible from one of the figures. There is, however, no doubt at all that the spot focusing mode yields high resolution. The mode
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can be used to look for the isotopic pattern of interest, e. g. to see losses of water or ammonia by irregularities of the isotopic pattern. The spot focusing may further be used to investigate and to improve the MALDI process. It can be expected that improved MALDI processes may be developed capable to enable the acquisition of partial spectra with high resolution. Not that spectacular, but of highly practical value is the ‘‘wide-range focusing’’ mode. This mode works well with ideal space–velocity correlations, even if the resolution is smaller than that of the optimum delayed acceleration mode for a selected ion mass. For non-ideal space–velocity correlations, however, the wide-range focusing mode outperforms by far any adjustment possibility of the delayed acceleration mode. The wide-range focusing mode has the ability of additional phase-space focusing. A first experiment demonstrates that this theoretical capability applies to reality in MALDI-TOF mass spectrometry. This wide-range focusing mode enables us to design mass spectrometers for use in the high mass field needing no sample-specific adjustments at all. Adjustment of parameters is highly uncritical. The calibration curve stays valid unless there are some electronic instabilities. Resolution is much better than that achieved hitherto. A push-button instrument is in reach. Such an instrument may be equipped with the additional feature of a spot focusing mode for an ion signal observed in the wide-range focusing. The simplicity of adjustment by a single parameter (or by two parameters coupled together) will ease the use of such an instrument. Acknowledgements The author would like to thank Dr. Arnim
Holle and Dr. Claus Koster for many fruitful discussions and critical remarks. The latter has written the MATHCAD programs confirming, in principle, the results presented here. Mr. Du¨rer at the Institute of Informatics of the University Bremen has performed the comparison of the linear variations of the acceleration voltage with exact integration values; he also found the Lambert W function as an analytical solution for the integration of the exponential variations. Dr. Arnim Holle obtained the first experimental corroboration for the wide-range focusing shown in Fig. 16.
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