Note 46: Delayed Extraction and Laser Desorption: Time-lag Focusing and Beyond

INTRODUCTION

Recent experiments have shown that the proper use of delayed ion extraction can dramatically improve the mass resolution of laser desorption time-of-flight mass spectrometry (LD-TOFMS) [1-4]. This poster explores two algorithms for the identification of optimal instrumental parameters in these experiments. The first, described in Wiley and McLaren's 1955 paper, is an analytical approach that has been in use for over 40 years [5]. The second, space-velocity correlation focusing, is a recently published computational method that depends on numerical optimization techniques to determine instrumental parameters [6]. Both approaches are used to calculate conditions in which the same physical phenomenon results in improved mass resolution. Wiley and McLaren's calculations were described for a system in which ionization occurred in the gas phase. We now consider the validity of the assumptions made in each approach when they are applied to surface desorption.

Time-lag Focusing

In their landmark 1955 paper [5], Wiley and McLaren described two focusing techniques for time-of-flight mass spectrometry, space and time-lag focusing. Both of these methods were applicable to ions generated in the gas phase. Space focusing was used to minimize peak broadening due to the spatial distribution of ions at the beginning of the time-of-flight measurement. Instrument operating conditions were identified by taking the derivative of an equation for an ion's time-of-flight with respect to the initial position. This derivative was then set equal to zero in order to minimize the spatial dependence of flight time (T). (dT/dX₀=0, where X₀ is the average position at the start of the time measurement.) By supplying a value for X_0, one can calculate the best ratio for the fields in the two acceleration regions of the source. The desired drift energy is then used to establish acceleration voltages in the instrument. An important assumption in this method is that dT/dX₀ does not vary significantly over the range of the spatial distribution of the sample.

Time-lag focusing employs a time delay between the formation of ions and their (time delayed) extraction from the source to reduce the impact of the analyte's initial velocity distribution on the detected peak width. During the delay time, (t) ions move to new positions as a function of their nascent velocities. When the extraction potential is applied, ions with velocities towards the detector will have moved to regions of lower potential energies and those traveling away from the detector will find themselves at a greater potential. Under the proper conditions, the change in potential energy can compensate for the differences in initial velocity. Wiley and McLaren point out that during the time (t) ions will move a distance v₀(t) where v₀ is the initial velocity along the coordinate of interest. This will result in a flight time change of approximately (dT/dX₀)v₀t. Without a delay, the flight time difference between ions of velocity v₀ and 0 is approximately "one-half turn-around time" or mv₀/qE₁. Where m is the mass, q is the charge, and E₁ is the electric field after the extraction voltage is applied. (In their version of this equation, Wiley and McLaren included a factor of 1.02 for unit conversion). Although not explained in the 1955 paper, this equation is derived by taking the derivative dT/dv₀ and assuming the average initial velocity equals zero. Focusing conditions are found when the change in flight time due to the delay compensates for the differences in flight time expected from the various kinetic energies, i.e. (dT/dX₀)v₀t= -dT/dv₀. This is expressed as:

(Reference 5 Equation 13)

in Wiley and McLaren's paper and rearranged to

(Reference 5 Equation 13a)

in order to calculate the proper time delay t.

There are a number of assumptions made in Wiley and McLaren's derivation of this equation. dT/dX₀ and dT/dv₀ are assumed to be relatively constant over the ranges of interest. dT/dX₀ not only needs to be uniform over the volume in which ions are produced but also over the distances represented by v₀t. The derivation also assumes that the average value of v₀ be equal to zero. This was done in the calculation of the derivative dT/dv₀ = mv₀/qE₁ and results in an equation that is independent of v₀. Finally, one must choose a value of X₀ in order to calculate dT/dX₀. Thus a single X₀ is assumed to represent the positions of all of the ions.

The recommended implementation of time-lag focusing involves the selection of space-focusing conditions followed by the variation of a parameter such that dT/dX₀ becomes negative. This violates the space focusing criteria and Wiley and McLaren therefore stress that space focusing and time-lag focusing are mutually exclusive. "The t and dT/dX₀ combination yielding the best overall resolution is a compromise between two extremes." If t is very small then dT/dX₀ must be large and space resolution becomes the limiting factor. If t is too large, ions travel over a range where dT/dX₀ is not sufficiently constant. Consequently, a value of t cannot be calculated under conditions where space focusing has been used to determine the grid potentials. No criteria are given for the selection of which parameter to vary in order to make the derivative negative. Optimum conditions must therefore be found using a certain amount of trial and error.

The Application of Time-Lag Focusing to Samples Desorbed From a Surface

In most laser desorption techniques (e.g. MALDI), a broad initial velocity distribution is presumed to be a major factor leading to reduced mass resolution. However, it is clear that the time-lag focusing algorithm, as described by Wiley and McLaren, cannot be applied directly to this problem. This is due to the non-zero value for the average initial velocity. Fortunately, it is possible to change their approach and remove the assumption that v₀ equals zero. In this case the equation for the derivative dT/dv₀ is:

where a_x is the acceleration in each region, v₁ and v₂ are the velocities after the first two regions, and L is the length of the drift region. A value for t could now be calculated given values for X₀, v , voltages, and distances. However, it is also necessary to examine the validity of the other assumptions involved in the Wiley and McLaren approach.

The first assumptions discussed above are that the values of dT/dX₀ and dT/dv₀ are uniform over the spatial and velocity distributions of the ions. (Note that the spatial distribution we are concerned with is that found at the time of ion extraction not just that present at the moment of ionization.) The assumptions can be studied by examining plots of flight time verses initial position or initial velocity. Three previously published examples [2] are shown in Figure 1. These plots show data for an ion of m/z=23981 calculated using different values for the potentials of the extraction voltages. In each case the value of t has been optimized. Note that the x-axis could be labeled in either velocity or position at any given value of t, because the two variables are correlated by the simple equation X₀ = v₀t when the sample is desorbed from a surface. Since all ions start in approximately the same place, v₀t may be used to represent position-not just displacement as was done by Wiley and McLaren. The initial velocity range that we must be concerned with has been experimentally determined by a number of workers [7-11] and is approximately 750 +/- 500 m/s. It is clear that dT/dX₀ and dT/dv₀ (the slopes of the curves in Figure 1) vary substantially over this range. They are even found to change sign. The variation is greatest when several minima and maxima occur as in Figure 1C. This is a particular problem, since this third order structure is indicative of the best extraction potentials.

In order to calculate a value for t, using Wiley and McLaren's approach one assumes that there is a single value of X₀ that adequately represents all of the ions. When the sample is desorbed from a surface, one must also select a value for v_0, since dT/dv₀ is no longer independent of the initial velocity. Figure 1C shows how critical the selection of these variables are. For example, the values of dT/dX₀ and dT/dv₀ change substantially between 850 and 950 m/s. It is disconcerting to know that a small change in the chosen value of v₀ will make a dramatic difference in the calculated value of t, particularly when the range of v₀ is known to be large. This suggests that the calculated t will properly focus only a small fraction of the ions. The selection of X₀ presents an additional problem. X₀ should be approximately related to v₀ by the relation X₀ = v₀t. However, a value of t is needed to solve this equation and t cannot be calculated until X₀ is known.

Using the approach of Wiley and McLaren, the implementation of time-lag focusing to desorbed samples would involve the selection of space-focusing conditions and then changing a parameter such that dT/dX₀ becomes negative. However, space focusing conditions cannot be determined until X₀ is known, which in turn requires t. The requirement that dT/dX₀ be negative is also questionable for surface desorption. Figure 1 illustrates several examples where t is optimized, but a positive slope can be found for any probable value of v₀. Figure 1 also shows the importance of optimizing instrumental parameters other than t. Each plot was produced with conditions for which t was optimized but other potentials were differed. However, the range of calculated flight times is clearly much narrower in Figure 1C. Wiley and McLaren provide no criteria for the selection of parameters other than t except to suggest that one start with space focusing conditions.

Figure 1. Calculated Flight Times As a Function Of Initial Velocity (and position) for various instrumental conditions. The delay time (t) has been optimized in all cases.

Space-Velocity Correlation Focusing (SVCF)

SVCF is an alternative method for determining delayed extraction conditions. It is a numerical process in which iterative calculations are used to optimize instrumental parameters [6]. The algorithm is called Space-Velocity Correlation Focusing, because it takes advantage of correlation between the distributions of ion positions and velocities at the time of extraction. Briefly, the approach involves finding a relationship between the two distributions, using the resulting expression to eliminate a variable (X₀ or v₀) from the equation for ion time-of-flight and then calculating flight times as a function of the remaining variable. The results can be used to produce plots such as those shown in Figure 1. With these, one can examine the flight time spread expected as a result of the remaining variable. This spread is then used as a figure of merit for the optimization of instrumental parameters, such as, but not restricted to, the delay time t. Optimization may then be accomplished by simple parameter variation or by any of a variety of established numerical optimization routines (i. e. Simplex). Since the two distributions are related, both spatial and velocity focusing can be accomplished simultaneously.

SVCF has a number of advantages over the time-lag focusing algorithm presented in 1955. In particular, it involves none of the assumptions discussed above and can be used to optimize any or all of the instrumental parameters. These are the most important factors differentiating it from time-lag focusing. SVCF does, however, require other assumptions. Foremost among these is the presupposition that the spatial and velocity distributions at the time of extraction are the primary factors limiting resolution. Other possible contributions such as metastable decay have not yet been included in the model. One must also select a range of initial velocities (or positions) over which focusing is to be calculated. Although this is preferred over picking a single velocity, as in time-lag focusing, the proper choice of values is still important to the results obtained. Finally, the approach requires that a relationship between the initial spatial and velocity distributions can be found that is valid for the relevant range of conditions. Typically, a simple analytical relation such as X₀ = v₀t has been used to represent this dependence. It requires that all ions are formed in the same location, do not fragment, and do not undergo collisions that change their velocity. These assumptions, made in the dynamic environment of the desorption, are more tenuous than those made when v₀t is used to represent the distance an ion moves in the gas phase as is done in time-lag focusing [11]. Fortunately, if the distributions can be measured, they can be incorporated in the SVCF approach either numerically or as an analytical expression. A full description of the implementation of SVCF is presented in Reference 6.

Conclusion

We have examined two methods for calculating optimum conditions in delayed extraction TOFMS. The assumptions required in each are summarized in Table 1. It is clear that the application of Wiley and McLaren's time-lag focusing algorithum to samples desorbed from a surface must be made with caution. Many of the assumptions made in this approach are questionable when the sample does not originate in the gas phase. In addition, it is clear that it is not sufficient to optimize only the extraction delay. Other instrumental parameters are critical to the mass resolution attained. While the time-lag focusing algorithm has been used for many years, these factors prevented the full benefit of delayed extraction of desorbed samples from being realized until 1994 [1-3]. In contrast, the SVCF approach requires fewer assumptions and may be applied to a larger variety of experimental conditions. SVCF is, however, more complicated than time-lag focusing. It also involves a variety of iterative calculations that are best performed on a computer. We suggest that one carefully consider the limitations of both of these methods when determining optimal conditions for delayed extraction TOFMS.

References

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