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How to specify the ion optical system of a time-of-flight mass spectrometer
International Journal 0168-l 176/94/$07.00
of Mass Spectrometry and Ion Processes 131 (1994) 21-41 0 1994 - Elsevier Science B.V. All rights reserved
How to specify the ion optical system of a time-of-flight mass spectrometer T. Bergmanna**,
T.P. Martinb
aBergmann Messgertite Entwicklung, Buchenweg 9a, 82441 Ohlstadt, Germany bMax-Planck-institut ftir FestkCrperforschung, Heisenbergstr. I, 70569 Stuttgart, Germany (Received 20 April 1993; accepted 11 August 1993) Abstract In order to make a qualified judgement of the ion optical properties of a time-of-flight system, it is necessary to have a method of presentation that does not require an extensive background in ion optical theory to understand it. The authors present such a method. This method gives a more clear understanding of state-of-the-art time-of-flight systems and gives a hint of what developments to expect in the future. Key words: Ion optical
system; Time of flight design
1. Introduction
At present, especially when it pertains to time-offlight mass-spectrometers with gridless reflectors, quality arguments and specifications are quite nebulous. Everybody has heard or read such statements before: “A mass resolution of 10000 is routine” or “The detection limit is 100 fmol”. For the resolution we usually find a plot of some mass peak, at which the quotient (total time)/(delta time) is demonstrated. For the detection limit we find a peak cropping out of the noise, perhaps with SNR = 10. Note that the above statements are performance examples and not specifications. Performance examples will always be of importance when demonstrating the benefits of some system. What we want to be able to do is judge the value of some performance example. * Corresponding
author.
SSDZ 0168-1176(93)03885-P
At present, the above statements are the only type that can be obtained from a time-of-flight manufacturer. These statements can mean one of the following. (i) They refer to a complete system. The user just puts some chemical substance into a sample holder, screws it into some port, and then obtains a mass spectrum at the terminal of his computer. Even though the above statements are usually given for some specific, usually favourable, substance, putting other substances into the sample holder will give comparable results. (ii) Somebody has purchased just the ion optics and the associated vacuum system to use it as a component in a larger system. This person may be seriously disappointed when trying to verify the above specifications. The first statement did not specify the volume of phase space used when demonstrating the resolution. The second statement may not tell enough about the mass resolution at low detection limits. (iii) Somebody wants to build his own optics. He
22
T. Bergmann and T.P. MartinlInt.
then will have to find the conditions under which the above statements should hold. In the second and third cases it is certainly desirable to have a method of presenting the ion optical properties of the system in such a way that the user can predict the behaviour of this system under the operating conditions he has chosen. This is not an article on how to design ion optics for a time-of-flight mass spectrometer, this article is about judging and evaluating different designs. It is the main intention of this article to introduce a method of presenting and specifying transport through the mass spectrometer. We hope that this method facilitates a clear comparison of different time-of-flight ion optics. 2. Splitting up the system First, it is mandatory to con~ptually split up the mass spectrometer system and separate out those parts that really have to do with ion optics. The main entities are: (i) Generation of the ions, i.e. placement of these ions into the phase space of the ion optical system. This means that we have to know the coordinate and velocity distribution which the ions have at the start time of the mass analysis. (ii) Movement through the ion optical system of the time-of-flight mass spectrometer. Specifying this movement and interpreting it in terms of sensitivity and mass resolution is the theme of this article. (iii) Everything that comes after the ions hit the surface that defines the end of the flight path. That surface can be e.g. a microchannel plate or the ion/ el~tron~onversion surface of a Daly converter. With the above separation it is possible to present the main properties of the ion optical system with a maximum of six plots, using very convenient units, on one piece of paper. This presentation will allow the prediction of the performance under most of the operating conditions encountered in practice. Above all, this presentation is so simple that performance of the complete mass spectrometer can be approxi-
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
mately judged by multiplying or dividing a few simple numbers. We will start by discussing the second entity of the system. The first and third entity are equally important; however, we will postpone their discussion until later because we want to discuss them with the units and definitions that we find convenient for the presentation of the second entity. 2.1. Moving through the ion opticaf system Just discussing the second conceptual entity of the mass spectrometer system, we only need to use one prerequisite and only need to answer one question. Before we look at the movement through the ion optical system we have to know from where and with what velocities the ions start. This is the prerequisite. Then we want to know, as a function of all starting positions and velocities, the final positions and velocities on the detector surface. This will be the answer to our question. The answer to that question gives us information on mass resolution and sensitivity. Firstly, we want to know from what initial coordinates and velocities we have paths ending on the detector surface. Secondly we want to know the associated time errors. The coordinate and velocity distributions are certainly different for many methods of starting ions on their flight path. For example, electron bombardment of neutral gas-phase particles gives initial velocity distributions in the energy range of 1 eV, while laser-ionizing molecules out of a supersonic expansion gives initial velocity distributions in the energy range of 1 meV. Some methods of starting ions on their flight path launch ions from very restricted coordinate regions in the ion source. Others produce ions in a very large region and thus it is very favourable if all ions from that large region can be transported to the detector.
2.1 .l. Coordinate systems The latter question has to be answered numeri-
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes I31 (1994) 21-41
tally. In order to do that we have to define coordinate systems within which to take measurements. We will define an unprimed coordinate system A somewhere in the middle of the region of the ion source where ions start on their flight path. From the center of this unprimed coordinate system with some median initial velocity (usually zero) we start a reference ion. We will orient the coordinate system A such that its z-axis points in the direction where the ion is accelerated along the symmetry axis of the ion source. We will put the center of a second, primed coordinate system R on the detector surface where the reference particle hits it. The z’-axis of this coordinate system should point in the direction where the ion came from. To calculate the time of flight for all other paths we take the time the reference particle needed on its path. The x- and x/-axis of the two-coordinate systems should be in the plane defined by the line connecting the ion source with the detector and the z- or z’-axis. 2.1.2. Just one set of f~~ctio~~ From classical mechanics we know the existence of the foIlowing functions: x; = x:(.$ Vi)
(1)
w; = Wi(Xj,Vi)
(2)
These functions give us the final coordinates and velocities as a function of initial coordinates and velocities. Of course, it is not possible to present the behaviour of any, perhaps complicated, six-dimensional function in just a few graphs. The following discussion will show, nevertheless, that it is possible to give sufficient information on the ion optical system of a time-of-flight mass spectrometer on just one sheet of paper. In fact, for most cases of interest, the function Z(Xj,vi) = x;(x~, Vi) has the necessary information. functions Plotting the six one-dimensional z/(x, O,O,O,O,O),z'(O, y,O,O,O,O), . . usually gives a sufficiently clear picture of the function z’(xj,‘uj).
23
These six plots are the ones given in the z’dispersion diagrams shown in this article. 2.1.3. The equivalent length s Before discussing an example, we need to find a convenient set of variables for our presentation. Thus, we do a “Gedankenexperiment”. Take two particles of mass ml ; we put one particle in the center of the ion source as a reference particle, and put the second at some other position in the ion source. Both particles are at rest. After the start time of the mass analysis we follow their paths through the mass spectrometer. After the reference particle hits the detector surface we stop the movement of the second and note its distance from the detector surface. (The second particle might hit the detector before the reference particle does: we just continue the movement of the second particle until the reference particle hits the surface.) We now start the second particle from all possible different locations in the ion source and, consequently, we get a function of final distances from the detector surface depending on the threedimensional variables of the initial position. (This is the left-hand side of the dispersion plots given later on.) Next, we repeat the same process that we just have done for particles of mass ml, for particles of mass m2. What do we get? Exactly the same function i.e. the same plots! The reason for this coincidence is that time scales with mass if ions move in time-independent electric fields. Knowing the distance of the second particle, we can divide the distance by its velocity and find a time error. Dividing the total time needed by the reference particle by the time error of the second particle we obtain information about mass resolution. We now assume that all other time errors in the mass spectrometer system, i.e. the time errors of the first and third entities, can be neglected. This would mean e.g. that the laser pulse for ionizing neutral particles is extremely short and that the detector and registration electronics is infinitely fast. We
24
T. Bergmann and T.P. MartinlInt.
then find that the quotient (time error)/(total time) is the same for all masses. Obviously, we must find some length by which we can divide the distances given in these plots. We will multiply the mass-dependent total time-offlight T(m) by the mass dependent velocity v,,(m) of the reference particle in the drift space of the mass spectrometer: S = w&n)T(m)
(3)
We will call S the total equivalent length of the flight path. This length is the same for all masses! We divide the length given as the result in the dispersion plots by this total equivalent length S and thus obtain info~ation on mass resolution for all masses that does not depend on mass. If we scale that information by the ratio of the velocity in drift space divided by the velocity of particles impinging on the detector surface (again a mass independent number), we have exactly the same values obtained when dividing (time error)/ (total time). In Eq. (3) we have defined the total equivalent length of the flight path. Likewise we can define a time variable: s = v&z)t
(4)
Of course, the above equation should read s(m)= ..* instead of s = . . . The fact that s is a coordinate-like variable allows us to completely neglect this mass dependence without losing any information: it just makes working with it easier. The equivalent length s has more convenient properties. Appendix A will give a mathematically-concise derivation of this normalization. Here we will list just a few of these properties. (i) Velocities: if we take & instead of dx’/dt, then the magnitude of this vector in drift space is unity for ions of all masses. In fact, we will define i& = &/ds and, as just stated, the equality l&l = 1 holds. (ii) v. ---t w,: by definition, vo(m) is mass dependent. By virtue of the definition of s, w. is not mass dependent any more.
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
(iii) Units: the unit of s is the same as the unit used for coordinates in describing the geometry of the mass-spectrometer. (mm, cm, m, etc.) (iv) Estimating S: the total equivalent length from the ion source to the surface of the detector is approximately 20% more than the geometrical length of the ion path, or (usually) roughly twice the length of the vacuum housing. 2.1.4. Initial velocities The right-hand sides of the dispersion plots give the final distances of particles from the detector surface as a function of initial velocity. The initial velocity is given here in units of the drift velocity wo. These units are the same as the definition dZ/&, but to take it as a ratio to the drift velocity o. is even simpler. Usually, the parameter that is known about initial velocities is their energy. For example, one would expect energies of about 1 eV for ionization by electron bombardment, and thermal energies of about 1 meV (at 1OK) for laserionizing molecules in a supersonic expansion by a one-photon process, Taking the square root of the ratio (initial energy)/(energy in drift space) gives the magnitude of the initial velocities in units of no. Using this fo~ula for ions having an energy in drift space of 1 keV gives a magnitude for the initial velocities, in the case of electron bombardment, of M 0.03wo, and for an ionization without recoil in a supersonic expansion of m 0.001~~. Usually, some information about the form of the distribution of initial velocities is known, so together with the absolute values and the right-hand side of the dispersion plots, an easy judgement of mass resolution is possible. 2.1 S. Time-depen~nt potentials The examples in this article concern instruments with static electric fields. If the electric potential is static, the z’-dispersion plot holds for all masses. If the electric field is time dependent then there is one z’-dispersion plot for each mass. Usually, some typical behaviour is desired for all masses or at least for a large range of masses. One might give specifications for that situation by
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
showing one z’-dispersion plot for the middle of that mass range and two more plots for the lower and higher end of the mass range. 2.1.6. Numerical computation The intention of this article is to introduce a method of specifying time-of-flight mass-spectrometers, In order that everyone can produce specifications for his instrument, the method of calculating the specifications must be simple and straightforward. To calculate the dispersion plots shown in this article means calculating the electric potentials and ion paths for a number of different initial coordinates and velocities within these potentials. To underline the fact that this is the only thing done here each individual path calculated has one symbol entered into one of the dispersion plots. For clarity, symbols have been connected by cubic spline functions. The techniques to calculate electric potentials and paths are standard and will not be discussed here. An excellent review of all currently employed methods can be found in ref. 1. The method used for computing the paths of the dispersion plots shown in this article is given in ref. 2. Although this method is rather precise, such a method would not be necessary just for calculating the dispersion plots. We have used these programs simply because we had them already. The authors have not made any tests with the ion optical simulation program SIMION PC/PSZ [3], but probably this would also be sufficient. Most of the dispersion plots in this article have been given for optimized potentials and perhaps other optimized variables. Optimizing lenses of time-of-flight mass spectrometers is a rather involved procedure and will not be discussed in this article either. The interested reader may refer to some previous publications [2,4,5]. 2.2. EvaIuati~g performance Evaluating
perfo~an~e
examples means uni-
25
fying all three conceptual entities to get an answer to the question of perfo~ance of the instrument under specific circumstances. Assume now that we want an instrument with mass resolution above 10000. If the vacuum housing is 1 m long then the total equivalent path length will usually be around 2 m. This means that the range of path errors in the z’-dispersion plots should be less than 100 pm. This corresponds to a scattering of f5Opm around some median value. If we want a mass resolution of better than 10000, then we have to stay with path errors below these f50pm. To check if this is the case we have to do the following. (i) Obtain information on’initial coordinates and velocities at which ions are started in the ion source. (iii) To use that info~ation in the z’-dispersion plots we have to scale the initial velocities against the drift velocity vo_Section 2.1.4 shows how this is done. (iii) For judgement, use only those initial coordinates and velocities that have paths actually ending on the detector. (To improve resolution, some paths might be blocked by apertures.) (iv) Verify that the z’ value for all these paths stays within +50 pm. Now assume that the energy in the drift space is SOOeV and that the mass of the ion is 1000 u. The speed v. of that particle is roughly 10 000 m s-’ , so the time necessary for an equivalent path of 2 m is then roughly 200~s. This means that start time of mass analysis should be defined better than Ions. (Usually this time span is the temporal length of a pulse from an ionizing laser.) Likewise, the detector and registration electronics should also be faster than 10 ns. To be on the safe side, both times should be better than 5 ns. Knowing the variables for ions of mass 1000 u at an energy of 500eV it is easy to scale to different masses or energies. For example, ions of mass 40 u at 500eV need time definitions better than (lath) of the previously calculated values, which is 2 and 1 ns respectively.
26
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
2.2.1. Verifving specifications
Assume now that we have purchased the ion optical system of a time-of-flight mass spectrometer. First, we want to verify its specifications. To do that we have to measure the arrival times of some ion as a function of initial coordinates and velocities in the ion source. From the time errors we have to calculate the z/-errors by a trivial formula and then compare the results with the data given by the manufacturer. A first prerequisite for this measurement is that the laser pulse should be significantly shorter than the time differences that need to be determined. Likewise, the detector and registration electronics must be faster than the time differences to be measured. That means we need to pick an ion that is heavy enough or whose total time of flight is long enough so that small fractions of the total time of flight can be measured. Directly verifying all six z’-dispersion functions precisely and completely is usually not possible. Still, in many cases we can get a good idea. Assume we are creating the ions in the ion source by a short laser pulse. We can then translate the laser beam in the x-z-plane. What we are measuring then, will not be the function; 4x, 0, z, 0, 0, 0)
(5)
but the function
z*(x, z) =
JJJJ
z'(x,Y,
z, 21x7l-‘y, G-bd~,d~,d~z
(6) We can take care that the range of initial velocities stays much smaller than what is encountered during regular operation, e.g. by ionizing a gas out of thermal equilibrium or from a supersonic expansion. That simplifies our integral considerably: z*(x, z) =
J
z’(x, y, z, O,O,0)dy
(7)
We can measure z*(x,z) as in Eq. (7) for a small range of initial velocities by a soft ionization without any recoil. We can measure another function z*(x,z) as in Eq. (6) for a large range of initial
velocities just by ionizing molecules that fragment upon ionization. These two functions should allow enough conclusions about the behaviour of z/(x, y, z, w,, r+, wZ), including the answer to the question as to whether the instrument fulfills the manufacturers specifications or not. 2.2.2. Sensitivity and transmitted phase space One very important thing to remember is the Liouville theorem. This theorem states that the volume of phase space transmitted by any ion optical system stays constant on its path. It is also important to note that this statement holds, no matter how strongly the shape of the transported phase space is distorted on its path. This means that we can always distort the transported phase space volume such that it gives an optimal fit to the ions offered to the optical system. For that reason it is possible to equate the terms sensitivity and transmitted phase space. We assume that it is always possible to distort the transported phase space as needed. Under this prerequisite-which has very few exceptions - an instrument designer first has to find out where exactly in phase space the ions enter the instrument, and then has to design the ion optical system to fit these externally imposed conditions. He has to play with that six-dimensional product of velocities and coordinates, (keeping the value of the product constant), until he has found a configuration that transports the maximum number of particles. 3. Demonstration: standard instrument with twostage grid reflector
To substantiate the preceding discussion, we will now look at a well-known, time-of-flight design. Figure l(a) shows a standard gas-phase ion source and Fig. l(b) shows its axis potential with the lens operated in negative mode. The lens of the ion source can be adjusted to do one of the following. (i) The ion source can guide all ions that start on the axis at z = 0, but with different initial trans-
T. Bergmann and T.P. Martin/Int.
J. Mass Spectrom.
Ion Processes
21
131 (1994) 21-41
(bf
Fig. 1. (a) Standard gas phase ion source. The first three plates define the accelerating field, the thick fifth electrode is an Einzel lens. (b) The axis potential shows the Einzel lens operated in the negative mode.
versa1 velocities, to paths parallel to the axis. We will call this the velocity focussing mode. (ii) The ion source can guide all ions that start with zero initial velocity, but different locations around the axis at z = 0, to paths parallel to the axis. We will call this the coordinate focussing or telescope mode. The difference between these two adjustments is that the first has a more negative voltage on the lens electrode. Note that the phase space transmitted is the same for both modes. In the velocity focussing mode, paths with a larger range of initial velocities end on the detector surface, while in the coordinate focussing mode, paths with a larger range of initial coordinates end on the detector surface. All further
examples, and also the one in this section, operate the ion source in the velocity focussing mode. Section 2.2.2 discusses the possibilities of changing the shape of the transported phase space; this is what is done by changing the voltage of the lens electrode of the ion source. Figure 2 shows the z’-dispersion curves for a time-of-flight instrument with the above standard gas-phase ion source operated in the velocity focussing mode. The ion beam entering the reflector and the axis of the reflector subtend an angle of 4”. The reflector is of the ~amyrin type [6] having two regions of homogeneous electric field separated by grids. The detector is just a flat plate of 20mm diameter and can be rotated a few degrees around the y/-axis to give
28
0 =
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes I31 (1994) 21-41
’ z ’ (0.0, 2, 0, 0, 0)
-50
0 =
x = z (0.0. 2.0.0. -0 .OOl)
z ’ (0,0, 0, 0. 0, v J _rn
1 K”
Fig. 2. Dispersion curves of a standard ion source (from Fig. 1) and two-stage grid reflector. The graphs give the error in the final icoordinate as a function of all six initial phase space variables. Since a grid reflector is optically active only for velocity components parallel to its axis, it only affects the function ~'(0, 0,z, 0,0,O).The other five dispersion functions closely reflect the behaviour of the ion source in Fig. 1. The initial velocities are given in units of ~0, which is the velocity of the reference ion in the drift space of the mass spectrometer.
T. Bergtnann and T.P. Martin/h.
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
an optimum .z’-dispersion. The total equivalent length of the flight path is approximately 2m. The ionizing laser beam has a diameter of a 2.4mm. The length of the acceleration region is about 20 mm, and thus a 2.4mm width of the ionizing laser beam corresponds to an energy spread of z 316%. All free parameters are optimized using a procedure described in ref. 2, the resulting z’-dispersion being shown in Fig. 2. All six dispersion plots in Fig. 2 show the deviation parallel to the incoming path of the reference particle (along the $-axis). Since we have assumed 2 m for the total equivalent length of the flight path, a path error of AZ’ = 100 pm corresponds to a time error of 5 x 10e5 or an error in mass determination of 10-4. The plot in the lower left hand corner shows the dispersion as a function of the initial z-coordinate in the ion source. This dispersion corresponds to the standard solution of the Mamyrin equations f67. The circles show the dispersion for ions initially at rest and the crosses show the dispersion for ions with an initial velocity of -0.001~~. (Remember that v. is the velocity of the reference ion in the drift tube as defined in section 2.1.4.) The plot in the lower right-hand corner shows the z’-dispersion as a function of initial velocity v,. The slope of this function z’(O,O,O,O,0, v,) is inversly proportional to the accelerating field strength in the ion source, here we have w 0.5Us cm-‘. (U, is the voltage corresponding to the drift velocity 21s.) Note the scale: just one thousandth of the drift velocity is necessary to effect an error in path length of z’ = 4Opm! If we assume that the energy of a particle in the drift tube is 1 keV then this initial velocity corresponds to a recoil energy of only 1 meV. This sensitivity to recoil has been pointed out before [7,8]. The usual solution for this problem is that the initial velocity w, is restricted or reduced by some means. The upper left-hand plot shows the z’-dispersion as a function of the initial x-coordinate with the initial velocity w, as the parameter, while the upper right graph shows the z’-dispersion as a func-
29
tion of the initial velocity w, with the initial x-coordinate as the parameter. This is a coordinate or velocity deviation from the axis of the ion source in the plane defined by the spectrometer axis and the connecting line between the ion source and the detector. Both graphs very closely reflect the behaviour of the ion source in Fig. 1 itself. The two graphs in the middle show the z’-dispersion as a function of y or IJ,,.The behaviour shown by these graphs is very similar to the z’-dispersion as a function of x and v,. Because the spectrometer is assumed to be symmetric about the x-z-plane, only two instead of three parameters are plotted. The initial x- and y-coordinates are limited to 10.5mm by the magnifi~tion of the ion source optics and the diameter of the detector. The performance of such an instrument can be verified with the methods described in section 2.2.1. Figure 6 of ref. 2 shows the results of a measurement that can be used to determine z*(O,z)=
J
z’(O,y,z,O,O,O)dy
Scanning the laser in the x-z-plane given the function z*(x, z).
would have
4. Examples: ions from the gas phase This section addresses instruments that start the ions in the time-of-flight instrument out of the gas phase. Instruments of this category are used for all kinds of photoionization experiments. When electron impact is used, the interest is usually to have a mass-selective detector as in gas chromatographymass spectrometry. In secondary neutral mass spectrometry (SNMS) either a laser or electron pulse is used to ionize sputtered neutrals. This ionization event is then the start event for time-of-flight analysis. 4.1. State-of-the-art gridless rejectors The purpose of this section is to warn the reader about believing statements like “As it seems
30
T. Bergmann and T.P. MartinlInt.
logical, this reflector can do everything at the same time! It cannot only perform energy focussing but can also do space focussing. It can effect a significant enhancement of sensitivity at a superior mass resolution”. What is logical? The Liouville theorem always holds, even for gridless reflectors! State-of-the-art instruments with gridless reflectors presently sell for twice the price of regular instruments on the basis of the above arguments. Yet, their performance is much worse, as will be shown in this section. We will just pick out two examples, not calling them by name. The authors have looked at more gridless reflector designs to be found in the literature or in patents, all of which have a similar design and probably a comparable performance. Some newer designs have been announced, but no information about their detailed construction is given. So, if you think about buying an instrument with a gridless reflector, be sure to ask for the s~~ifi~tions! The reflector of patent A has four adjustable voltages, the ion paths entering the reflector subtend an angle of 4” with its axis. The total length of the drift path is again = 2 m. The ions are started in the standard ion source of Fig. 1. If we optimize the four adjustable voltages in a range of f2% of the values prescribed in the patent we get the dispersion curves shown in Fig. 3. Naturally, the function ~'(0,0,0,0,0,w,) shown in the lower right-hand corner is identical to the corresponding function in Fig. 2, since we use the same ion source with the same accelerating field strength. The z’-dispersion as a function of z is also comparable in quality. The z’-dispersions of initial variables perpendicular to the direction of acceleration are much worse for this gridless reflector than for a regular two-grid reflector as can be seen by comparing Fig. 2 and Fig. 3. We can set apertures behind the ion source such that paths with an error of z’ > 100 pm are blocked. Looking at the transmitted phase space in the x+,-plane we find that the two-grid reflector of Fig. 2 transmits a
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
trapezoid of area M 1.Omm x 0.06~ and the gridless reflector of Fig. 3 transmits a diamond of area c 0.5(0.5 mm x 0.06~~). This makes a factor of 4 in the x-v,-plane. The same situation is found in the y-+,-plane, such that, disregarding transmission losses by the meshes in a two-grid reflector, a gridless reflector after patent A transmits a phase space 16 times smaller than a two-grid reflector. If we take transmission losses of about 50% passing four times through a mesh into account, the transmission of this gridless reflector is still eight times smaller. Figure 4 shows the z’-dispersion curves of a reflector after patent A, if not the prescribed voltages are used, but an optimized, completely different set of values. The phase space transmission is close to a factor of 2 better than in Fig. 3, still it is a factor of 4 worse than a standard twogrid reflector. The reflector after patent B has a total of 17 independent voltage adjustments. The angle that ion paths entering the reflector subtend with the reflector axis is not given, so that it was freely varied for optimum performance. The total length of the drift path is not given so that it was used also as a free variable in the optimization. The detector and ion source are assumed to be 3cm off the spectrometer axis. The voltages are optimized in a f2% range around the prescribed values. One should think that this reflector gives a much better performance than the reflector after patent A. Figure 5 gives the resulting performance. It closely corresponds to Fig. 3 and Fig. 4. An instrument with this reflector does not have a higher phase space transmission than instruments using reflector A. 4.2. Can reflectors without grids have a superior transmission?
The authors do not want to convey the impression that gridless reflectors are worthless. The purpose of this section is to show that it is theoretically possible to construct gridless reflectors that can transmit a phase space volume a factor of 20
T. Bergmann and T.P. Martin/Int. J. Mass Spectrom. Ion Processes I31 (1994) 21-41
31
X x=z'(x,O,O,-0.02.0,0)
x = 2’ 60.2. 0. 0. v,, 0. 0)
o = 2’ (x. 0. 0, 0. 0, 01
o=2’l0.0,0,v,,0.01
cJ= 2’(x,0,0,0.02,0.0)
q
= 2’(0.2.0,0,v,.0.01
Z x-2'~0,0,2,0.0,-0.001) 0 = 2’
(0,0, 2,0,0.0)
Fig. 3. Dispersion curves of a standard ion source and gridless reflector after patent A. The reflector voltages are restricted to 12% of the values prescribed in the patent. The i-dispersion as a function of z and V, are comparable to the dispersions of a two-grid reflector shown in Fig. 2. The z’-dispersion as a function of x, y, V, and vYshow that the phase space transmission of this gridless reflector is a factor of 8 smaller than a two-grid reflector.
T. Bergmann and T.P. Martin/M
-oil
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
[mm1
0= XS
Fig. 4. Dispersion curves of a standard ion source and gridiess reflector after patent A. The reflector voltages have been optimized to give the lowest possible timing or z’-errors. The i-dispersion as a function of z is somewhat smoother than in Fig. 3. The z’-dispersions as a function of intial x and y are a bit wider. The transmitted phase space of this adjustment is close to a factor of 2 better than the adjustment of Fig. 3.
T. Bergmann and T.P. Martin/h.
33
J. Mass Spectrom. Ion Processes I31 (1994) 21-41
X x=
z'Ix.O,O,-0.02,0,0)
o= 2 ’ q
lx. 0. 0, 0. 0, 01
~=2’~-0.2~0,0.v~0,0)
-50
o=z’(o.o,o,v,o,ol
-50
= 2 * tx. 0, 0,o. 02. 0, 0)
_ _
(0,y. 0,o. 0,O)
05
2‘
X-
2 a (0. y. 0.0.0.02.0)
-50 t
0 = 2
’ (0,0.0. 0, v p 0)
-50
x=Z'(0.0.2,0,O,V~Of
Fig. 5. Dispersion curves of a standard ion source and gridless reflector after patent B. The reflector voltages are restricted to f2% of the values prescribed in the patent. The z’-dispersion functions show that this reflector gives a phase space transmission somewhere between Fig. 3 and Fig. 4
34
T. Bergmann and T.P. ~art~n~~nt. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
larger than a standard two-grid reflector. The following section will then show the performance of an instrument presently under construction that is expected to perform a factor of 7 better. This section shows one instrument which has a low accelerating field of M 0.5 U,, cm-’ in the ion source and another instrument with a high accelerating field of z 14 Uecm-’ in the ion source. Ue is the voltage corresponding to the drift velocity uo. Both of these examples will not be actual constructions with electrode shapes etc. but will just be defined through their axis potentials which are given by spline functions (see ref. 2). Whether an actual construction will achieve these limits depends very much on geometric limitations of electrode design. Fig. 6. shows the optimum performance of an instrument with a low accelerating field of x 0.5 U. cm-‘. By comparing Fig. 6 with Fig. 2 we see that this instrument transmits a phase space volume at least a factor of 20 larger than a standard instrument with two-grid reflector. (A factor of 10 for larger phase space and another factor of 2 because it uses no grids.) The z’-dispersion as a function of z is almost completely flat. That is not too surprising since the reflector potential is controlled by a multivariable spline function. The phase space volume in the X- and y-direction is so large that the question might arise as to whether it is possible at all to admit ions to fill this whole volume. The z’-dispersion as a function of w, is still as sensitive as in all the previous figures. For that reason one might try to change the shape of the transported phase space volume to allow for a larger variation of v,, perhaps reducing the phase space in x- and y-directions. This is what has been done in Fig. 7. The length of the transported z-coordinate is much shorter than in Fig. 6, but the accelerating field is also much higher: M 14 Uo cm-‘. Thus, the range of z is a factor of 5 less but the allowable range of w, is a factor of 28 higher. The transported phase in the X- and y- directions is also less, so that the total
transmitted phase space volume is again the same as in Fig. 6. Note that the allowable recoil upon ionization for this instrument is 0.032~~.If the energy of ions on the drift tube is 1 keV, this recoil corresponds to 0.9eV. This instrument should have a high mass resolution even for “hard” ionization. 4.3. SNMS-time-of-flight
instrument
Figure 8 shows the performance data of a completely designed SNMS time-of-flight instrument. All potentials, also that of the detector, are given by explicitly defined electrodes. The postacceleration in the detector is assumed to be 6Ko. (For ion energies of 1 keV this corresponds to 6 keV.) The accelerating field is = 10 U. cm-‘. The z’-dispersion plots in Fig. 8 have been scaled by the ratio of the velocity in drift space divided by the velocity of particles impinging on the detector surface, in this case l/m (see section 2.1.3). Thus, the data in Fig. 8 should be compared, as in all previous cases, to a total equivalent length ofS=2m. The origin of the unprimed coordinate system xi has been fixed to a repeller plate 0.5 mm behind the starting point of the reference path; the reference path is also assumed to start with a velocity of 0.05~~. This corresponds to what is expected for SNMS operation. Before the laser ionizes the sputtered neutrals they are desorbed from the analysed surface at z = 0 mm. When they pass through the volume around z = 0.5 mm, they have a velocity of z 0.05~~. The angle that the ion source electrodes above the sample may subtend is limited to 20” and thus it is not possible to freely model the field as in the theoretical calculations for Fig. 7. This partially explains why the total transmitted phase space volume, as shown in Fig. 8, is only a factor of 7 better than a standard two-grid reflector can handle and not a factor of 20 as for Fig. 7. That the instrument shown in Fig. 8 uses a detector while Fig. 7 just uses an imaginary
T. Bergmann and T.P. Martin/Int.
J. Mass Spectrom.
Ion Processes
131 (1994) 21-41
35
X
0 = 2’
lx. 0. 0, 0. 0.01
0 = z’ lx, 0, 0. 0.04, 0. 0)
50
_ ’ 0 = z (0. y, 0, 0. 0, 0) x=z’(0,y.0,0.0.03,0)
-50
_
E
1.0 [mm1
J o= X=
2’ 50
[pl i
[pm1
Fig. 6. Dispersion curves showing the optimum performance of a time-of-flight instrument with a low accelerating field of = 0.5 lJO cm-‘. The transmitted phase space volume for this instrument is at least 20 times as large as for an instrument with a twogrid reflector shown in Fig. 2. Note that the behaviour in the x-z-plane for this instrument is different than in the y-z-plane.
36
T. Bergmann and T.P. Martin/h.
X
-0.04
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
-0.02
0.02
0 - z ’ (0.0,0. vX.0.0) q=z’(o.4,o,o,v,.o.o)
0.04
vx
Iv01
~=z’~-0.4,0.0,v,.0,0)
-50
Fig. 7. Dispersion curves showing the optimum performance of a time-of-flight instrument with a high accelerating field of = 14 U, cm-‘. Because the accelerating field is higher by a factor of 28, the range of transported w, is also higher by a factor of 28. The ranges of the remaining five phase space variables multiply to about l/30, so that the total transmitted phase space volume for this instrument is the same as in Fig. 6.
T. Bergmann and T.P. Martin/Int. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
37
X x= z* (x.0,0.5. -0.02.0.0.05) *= 2’~x.0.0.5,0,0,0.05~
~=z’~0.2,0.0.5,v,.0.0.05~ +O
o= z’ (x, 0.0.5,0.02,0.0.05)
~=z’10,0,0.5,v,0.0.05~
-50
x=z’k0.2.0.0.5,v,O.0.05~
7’
Fig. 8. Dispersion curves showing the performance of a fully designed SNMS-time-of-flight mass-spectrometer. The origin of the coordinate system xi has been fixed to a repeller plate 0.5mm behind the starting point of the reference path; the reference path is also assumed to start with a velocity of 0.05~s. This instrument has a high accelerating field of w 10 U,,cm-’ to achieve a high mass resolution in spite of the expected recoil upon ionization. The total transmitted phase space volume is higher by a factor of 7 compared to the standard instrument with two-grid retkctor of Fig. 2.
38
T. Be~gmann and T.P. ~urtin~Int. J. Mass Spectrom. Ian Processes 131 (1994) 21-41
plate might account for another part of the difference. This instrument is presently under construction. 5. Examples: ions from surfaces This section addresses those instruments that create ions at surfaces from where they are drawn into the time-of-flight instrument. Typical examples of this type are secondary ion (SIMS) and laser desorption (LD) mass spectrometers. In SIMS instruments a short pulse of primary ions sputters off material from the analysed surface. The ionized part of this material is drawn into the time-of-flight mass spectrometer. In LD instruments the ions are desorbed from a surface by a short and intense laser pulse. In the case of matrix assisted laser desorption the molecules of interest are imbedded in a matrix of light-absorbing material which is heated by the laser pulse and which then evaporates and carries the analyte along into the time-of-flight massspectrometer. In both cases the z-coordinate of the initial phase space is fixed to the analysed surface (z = 0). Since it is desired to obtain information from different locations of the surface of the sample, the x- and y-coordinates may vary. Likewise, the ions have a finite spread of initial velocities in all three directions. Thus we have to consider five and
not six phase space variables in the following discussion. Figure 9 shows the design of a typical SIMS ion source as can be seen for example in Fig. 1 of ref. 9. The ions start at z = 0 and around x, y = 0 and are accelerated in the positive z-direction. The thick ring at z = 5.8 cm is the active electrode of a focussing lens. The ion source taken from ref. 9 is just meant as an example and the measures in Fig. 9 are only approximate!, simifar designs can be found in any publication on SIMS-time-of-flight massspectrometers. Figure 10 gives the performance of a system comprising the standard SIMS ion source of Fig. 9, a single-grid-refl~tor starting 7Ocm above the sample surface, and a standard detector design (not shown). The distance of the extraction lens to the sample surface has been adjusted, such that the accelerating field is M 10 U,, cm-‘. This has been done to provide a better comparison with Fig. 11. Figure 11 gives the performance data of the instrument shown in Fig. 8 operated in SIMS mode. Because the accelerating field above the sample surface is also x 10 Us cm-’ the d-dispersion as a function of v, is roughly the same as in Fig. 10. It is important to note that the z’dispersions in the plane perpendicular to the zaxis are signi~cantly better than for the standard instrument of Fig. 10. This is due to the gridless reflector.
X 2.0
km1 1.0
0.0
Fig. 9. Standard SIMS ion source [9]. The ions start at z = 0 and around x,y = 0 and are accelerated in the positive z-direction. The thick ring at z c 5.81x1 is the active electrode of a focussing Iens.
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
39
2’
2’
V
-0.2 x = 2’
-0.1
(x. 0, 0, -0.02.0, 0.051
o=r’~X.O.0.0,0.0.05l n = 2’ (x, 0,o. 0.02,0,0.05)
hml -50
n
0.2
0.1
-0.03
~=~‘~-o.l,o~o‘v~,o, o~z’~o.o,o.v,.o,
~=z’~o.l~o,o,v~‘o~
-0.01 (0.05_~~“~ (o.05-vyy50
(O.O~-v~~‘~
2’
Z’ Z’
Fig. 10. Dispersion curves showing the performance of a standard SIMS inst~ment with a single-grid reflector. This inst~ent ion source of Fig. 9, a single-grid reflector and a standard detector design.
References 1 P.W. Hawkes and E. Kasper, Principles of Electron Optics, Academic Press, London, 1989. 2 T. Bergmann, T.P. Martin and H. Schaber, Rev. Sci. Instrum., 61(10), (1990) 2592.
3
uses the
D.A. Dahl and J.E. Delmore, SIMION PC/PSZ, Version 4.0, EGG-C%7233 Rev. 2, April, 1988. 4 M. Szilagyi, Proc. IEEE, 73(3) (1985) 412. 5 J.P. Adriaanse, H.W.G. van der Steen and J.E. Barth, J. Vat Sci. Technol. B, 7(4) (1989) 651. 6 B.A. Mamyrin, V.I. Karataev, D.V. Shmikk and V.A. Zag&in, Sov. Phys. JETP, 37 (1973) 45.
T. Bergmann and T.P. MartinlInt.
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
2’
2’
X x = 2’
x= 2’ Ix. 0.0. -0.02,0.0.05) 0=2’(x.0.0.0,0.0.051
-50
q-2’(x,0.0.0.02.0.0.05)
0.05 “x
-0.05
(-0.15, 0, 0. v “, 0. (0.05-vg “?
o-2’(0,0.0,v,,0.
q=z’(o.15,o,o,v,,o,
(0.05-v$“?
-w
(0.05-v~“q
2’
2’
“Y
Y 0=2’~0,y.0.0.0.0.05)
-50
x = z’ (0. y. o,o, 0.02. 0.05)
0 = 2’ x =z
(0,0, 0, 0. v “’ IO.05-q
I”, _5()
* (0,O.15. 0. 0, v “, (0.05-v$
“‘,
Fig. 11. Dispersion curves showing the performance of the SNMS instrument of Fig. 8 operated in SIMS mode. The exceptional i-dispersion as a function ofx, y, V, and vYis the result of the combined ion optical properties of the ion source and the gridless reflector. U. Boesl, J. Grotemeyer, K. Walter and E.W. Schlag, Anal. Instrum., 16(l) (1987) 151. U. Boesl, R. Weinkauf and E.W. Schlag, Int. J. Mass Spectrom. Ion Processes, 112 (1992) 121. E. Niehuis. T. Heller, H. Feld and A. Benninghoven, J. Vat. Sci. Tech&l. A, 5(4) (1987) 1243.
Appendix A: Notation
To simplify the notation we redefine the variable for energy, potential, time and velocity. If K is the total energy of an ion and & is the average energy
41
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes 131 (1994) 2X-41
of all ions, then let e be defined by K= (1+ E)KQ
(Al)
If U is the electric potential, then u is given by U=u&
(A2)
a very close relationship to the length dimensions of the problem. The velocities are always given as the s derivative of coordinates. Newton’s equations now become d2
-&fXiCs)
The location of ionization for ions with t = 0 and with w, = 0 will be at z = 0. The average velocity of ions at ground potential U = 0 is given by v. = ,,/m. Let s be defined by s = vat
(A3)
We give s the name “equivalent length”. In solving Newton’s equations we use s instead of t as the independent variable. As can easily be seen, s has
=
-~&“(xj(s))
i,j=
1,2,3
I
(A4)
Defining dxi/ds = wi, it follows that (WZ+ w; + WI) = 1 f 6
(A5)
In the dispersion curves we always define & and v. as the energy and velocity of a particle on the reference path at ground potential. In the examples discussing time-of-flight instruments, ground potential is defined as the potential on the drift paths.
of Mass Spectrometry and Ion Processes 131 (1994) 21-41 0 1994 - Elsevier Science B.V. All rights reserved
How to specify the ion optical system of a time-of-flight mass spectrometer T. Bergmanna**,
T.P. Martinb
aBergmann Messgertite Entwicklung, Buchenweg 9a, 82441 Ohlstadt, Germany bMax-Planck-institut ftir FestkCrperforschung, Heisenbergstr. I, 70569 Stuttgart, Germany (Received 20 April 1993; accepted 11 August 1993) Abstract In order to make a qualified judgement of the ion optical properties of a time-of-flight system, it is necessary to have a method of presentation that does not require an extensive background in ion optical theory to understand it. The authors present such a method. This method gives a more clear understanding of state-of-the-art time-of-flight systems and gives a hint of what developments to expect in the future. Key words: Ion optical
system; Time of flight design
1. Introduction
At present, especially when it pertains to time-offlight mass-spectrometers with gridless reflectors, quality arguments and specifications are quite nebulous. Everybody has heard or read such statements before: “A mass resolution of 10000 is routine” or “The detection limit is 100 fmol”. For the resolution we usually find a plot of some mass peak, at which the quotient (total time)/(delta time) is demonstrated. For the detection limit we find a peak cropping out of the noise, perhaps with SNR = 10. Note that the above statements are performance examples and not specifications. Performance examples will always be of importance when demonstrating the benefits of some system. What we want to be able to do is judge the value of some performance example. * Corresponding
author.
SSDZ 0168-1176(93)03885-P
At present, the above statements are the only type that can be obtained from a time-of-flight manufacturer. These statements can mean one of the following. (i) They refer to a complete system. The user just puts some chemical substance into a sample holder, screws it into some port, and then obtains a mass spectrum at the terminal of his computer. Even though the above statements are usually given for some specific, usually favourable, substance, putting other substances into the sample holder will give comparable results. (ii) Somebody has purchased just the ion optics and the associated vacuum system to use it as a component in a larger system. This person may be seriously disappointed when trying to verify the above specifications. The first statement did not specify the volume of phase space used when demonstrating the resolution. The second statement may not tell enough about the mass resolution at low detection limits. (iii) Somebody wants to build his own optics. He
22
T. Bergmann and T.P. MartinlInt.
then will have to find the conditions under which the above statements should hold. In the second and third cases it is certainly desirable to have a method of presenting the ion optical properties of the system in such a way that the user can predict the behaviour of this system under the operating conditions he has chosen. This is not an article on how to design ion optics for a time-of-flight mass spectrometer, this article is about judging and evaluating different designs. It is the main intention of this article to introduce a method of presenting and specifying transport through the mass spectrometer. We hope that this method facilitates a clear comparison of different time-of-flight ion optics. 2. Splitting up the system First, it is mandatory to con~ptually split up the mass spectrometer system and separate out those parts that really have to do with ion optics. The main entities are: (i) Generation of the ions, i.e. placement of these ions into the phase space of the ion optical system. This means that we have to know the coordinate and velocity distribution which the ions have at the start time of the mass analysis. (ii) Movement through the ion optical system of the time-of-flight mass spectrometer. Specifying this movement and interpreting it in terms of sensitivity and mass resolution is the theme of this article. (iii) Everything that comes after the ions hit the surface that defines the end of the flight path. That surface can be e.g. a microchannel plate or the ion/ el~tron~onversion surface of a Daly converter. With the above separation it is possible to present the main properties of the ion optical system with a maximum of six plots, using very convenient units, on one piece of paper. This presentation will allow the prediction of the performance under most of the operating conditions encountered in practice. Above all, this presentation is so simple that performance of the complete mass spectrometer can be approxi-
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
mately judged by multiplying or dividing a few simple numbers. We will start by discussing the second entity of the system. The first and third entity are equally important; however, we will postpone their discussion until later because we want to discuss them with the units and definitions that we find convenient for the presentation of the second entity. 2.1. Moving through the ion opticaf system Just discussing the second conceptual entity of the mass spectrometer system, we only need to use one prerequisite and only need to answer one question. Before we look at the movement through the ion optical system we have to know from where and with what velocities the ions start. This is the prerequisite. Then we want to know, as a function of all starting positions and velocities, the final positions and velocities on the detector surface. This will be the answer to our question. The answer to that question gives us information on mass resolution and sensitivity. Firstly, we want to know from what initial coordinates and velocities we have paths ending on the detector surface. Secondly we want to know the associated time errors. The coordinate and velocity distributions are certainly different for many methods of starting ions on their flight path. For example, electron bombardment of neutral gas-phase particles gives initial velocity distributions in the energy range of 1 eV, while laser-ionizing molecules out of a supersonic expansion gives initial velocity distributions in the energy range of 1 meV. Some methods of starting ions on their flight path launch ions from very restricted coordinate regions in the ion source. Others produce ions in a very large region and thus it is very favourable if all ions from that large region can be transported to the detector.
2.1 .l. Coordinate systems The latter question has to be answered numeri-
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes I31 (1994) 21-41
tally. In order to do that we have to define coordinate systems within which to take measurements. We will define an unprimed coordinate system A somewhere in the middle of the region of the ion source where ions start on their flight path. From the center of this unprimed coordinate system with some median initial velocity (usually zero) we start a reference ion. We will orient the coordinate system A such that its z-axis points in the direction where the ion is accelerated along the symmetry axis of the ion source. We will put the center of a second, primed coordinate system R on the detector surface where the reference particle hits it. The z’-axis of this coordinate system should point in the direction where the ion came from. To calculate the time of flight for all other paths we take the time the reference particle needed on its path. The x- and x/-axis of the two-coordinate systems should be in the plane defined by the line connecting the ion source with the detector and the z- or z’-axis. 2.1.2. Just one set of f~~ctio~~ From classical mechanics we know the existence of the foIlowing functions: x; = x:(.$ Vi)
(1)
w; = Wi(Xj,Vi)
(2)
These functions give us the final coordinates and velocities as a function of initial coordinates and velocities. Of course, it is not possible to present the behaviour of any, perhaps complicated, six-dimensional function in just a few graphs. The following discussion will show, nevertheless, that it is possible to give sufficient information on the ion optical system of a time-of-flight mass spectrometer on just one sheet of paper. In fact, for most cases of interest, the function Z(Xj,vi) = x;(x~, Vi) has the necessary information. functions Plotting the six one-dimensional z/(x, O,O,O,O,O),z'(O, y,O,O,O,O), . . usually gives a sufficiently clear picture of the function z’(xj,‘uj).
23
These six plots are the ones given in the z’dispersion diagrams shown in this article. 2.1.3. The equivalent length s Before discussing an example, we need to find a convenient set of variables for our presentation. Thus, we do a “Gedankenexperiment”. Take two particles of mass ml ; we put one particle in the center of the ion source as a reference particle, and put the second at some other position in the ion source. Both particles are at rest. After the start time of the mass analysis we follow their paths through the mass spectrometer. After the reference particle hits the detector surface we stop the movement of the second and note its distance from the detector surface. (The second particle might hit the detector before the reference particle does: we just continue the movement of the second particle until the reference particle hits the surface.) We now start the second particle from all possible different locations in the ion source and, consequently, we get a function of final distances from the detector surface depending on the threedimensional variables of the initial position. (This is the left-hand side of the dispersion plots given later on.) Next, we repeat the same process that we just have done for particles of mass ml, for particles of mass m2. What do we get? Exactly the same function i.e. the same plots! The reason for this coincidence is that time scales with mass if ions move in time-independent electric fields. Knowing the distance of the second particle, we can divide the distance by its velocity and find a time error. Dividing the total time needed by the reference particle by the time error of the second particle we obtain information about mass resolution. We now assume that all other time errors in the mass spectrometer system, i.e. the time errors of the first and third entities, can be neglected. This would mean e.g. that the laser pulse for ionizing neutral particles is extremely short and that the detector and registration electronics is infinitely fast. We
24
T. Bergmann and T.P. MartinlInt.
then find that the quotient (time error)/(total time) is the same for all masses. Obviously, we must find some length by which we can divide the distances given in these plots. We will multiply the mass-dependent total time-offlight T(m) by the mass dependent velocity v,,(m) of the reference particle in the drift space of the mass spectrometer: S = w&n)T(m)
(3)
We will call S the total equivalent length of the flight path. This length is the same for all masses! We divide the length given as the result in the dispersion plots by this total equivalent length S and thus obtain info~ation on mass resolution for all masses that does not depend on mass. If we scale that information by the ratio of the velocity in drift space divided by the velocity of particles impinging on the detector surface (again a mass independent number), we have exactly the same values obtained when dividing (time error)/ (total time). In Eq. (3) we have defined the total equivalent length of the flight path. Likewise we can define a time variable: s = v&z)t
(4)
Of course, the above equation should read s(m)= ..* instead of s = . . . The fact that s is a coordinate-like variable allows us to completely neglect this mass dependence without losing any information: it just makes working with it easier. The equivalent length s has more convenient properties. Appendix A will give a mathematically-concise derivation of this normalization. Here we will list just a few of these properties. (i) Velocities: if we take & instead of dx’/dt, then the magnitude of this vector in drift space is unity for ions of all masses. In fact, we will define i& = &/ds and, as just stated, the equality l&l = 1 holds. (ii) v. ---t w,: by definition, vo(m) is mass dependent. By virtue of the definition of s, w. is not mass dependent any more.
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
(iii) Units: the unit of s is the same as the unit used for coordinates in describing the geometry of the mass-spectrometer. (mm, cm, m, etc.) (iv) Estimating S: the total equivalent length from the ion source to the surface of the detector is approximately 20% more than the geometrical length of the ion path, or (usually) roughly twice the length of the vacuum housing. 2.1.4. Initial velocities The right-hand sides of the dispersion plots give the final distances of particles from the detector surface as a function of initial velocity. The initial velocity is given here in units of the drift velocity wo. These units are the same as the definition dZ/&, but to take it as a ratio to the drift velocity o. is even simpler. Usually, the parameter that is known about initial velocities is their energy. For example, one would expect energies of about 1 eV for ionization by electron bombardment, and thermal energies of about 1 meV (at 1OK) for laserionizing molecules in a supersonic expansion by a one-photon process, Taking the square root of the ratio (initial energy)/(energy in drift space) gives the magnitude of the initial velocities in units of no. Using this fo~ula for ions having an energy in drift space of 1 keV gives a magnitude for the initial velocities, in the case of electron bombardment, of M 0.03wo, and for an ionization without recoil in a supersonic expansion of m 0.001~~. Usually, some information about the form of the distribution of initial velocities is known, so together with the absolute values and the right-hand side of the dispersion plots, an easy judgement of mass resolution is possible. 2.1 S. Time-depen~nt potentials The examples in this article concern instruments with static electric fields. If the electric potential is static, the z’-dispersion plot holds for all masses. If the electric field is time dependent then there is one z’-dispersion plot for each mass. Usually, some typical behaviour is desired for all masses or at least for a large range of masses. One might give specifications for that situation by
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
showing one z’-dispersion plot for the middle of that mass range and two more plots for the lower and higher end of the mass range. 2.1.6. Numerical computation The intention of this article is to introduce a method of specifying time-of-flight mass-spectrometers, In order that everyone can produce specifications for his instrument, the method of calculating the specifications must be simple and straightforward. To calculate the dispersion plots shown in this article means calculating the electric potentials and ion paths for a number of different initial coordinates and velocities within these potentials. To underline the fact that this is the only thing done here each individual path calculated has one symbol entered into one of the dispersion plots. For clarity, symbols have been connected by cubic spline functions. The techniques to calculate electric potentials and paths are standard and will not be discussed here. An excellent review of all currently employed methods can be found in ref. 1. The method used for computing the paths of the dispersion plots shown in this article is given in ref. 2. Although this method is rather precise, such a method would not be necessary just for calculating the dispersion plots. We have used these programs simply because we had them already. The authors have not made any tests with the ion optical simulation program SIMION PC/PSZ [3], but probably this would also be sufficient. Most of the dispersion plots in this article have been given for optimized potentials and perhaps other optimized variables. Optimizing lenses of time-of-flight mass spectrometers is a rather involved procedure and will not be discussed in this article either. The interested reader may refer to some previous publications [2,4,5]. 2.2. EvaIuati~g performance Evaluating
perfo~an~e
examples means uni-
25
fying all three conceptual entities to get an answer to the question of perfo~ance of the instrument under specific circumstances. Assume now that we want an instrument with mass resolution above 10000. If the vacuum housing is 1 m long then the total equivalent path length will usually be around 2 m. This means that the range of path errors in the z’-dispersion plots should be less than 100 pm. This corresponds to a scattering of f5Opm around some median value. If we want a mass resolution of better than 10000, then we have to stay with path errors below these f50pm. To check if this is the case we have to do the following. (i) Obtain information on’initial coordinates and velocities at which ions are started in the ion source. (iii) To use that info~ation in the z’-dispersion plots we have to scale the initial velocities against the drift velocity vo_Section 2.1.4 shows how this is done. (iii) For judgement, use only those initial coordinates and velocities that have paths actually ending on the detector. (To improve resolution, some paths might be blocked by apertures.) (iv) Verify that the z’ value for all these paths stays within +50 pm. Now assume that the energy in the drift space is SOOeV and that the mass of the ion is 1000 u. The speed v. of that particle is roughly 10 000 m s-’ , so the time necessary for an equivalent path of 2 m is then roughly 200~s. This means that start time of mass analysis should be defined better than Ions. (Usually this time span is the temporal length of a pulse from an ionizing laser.) Likewise, the detector and registration electronics should also be faster than 10 ns. To be on the safe side, both times should be better than 5 ns. Knowing the variables for ions of mass 1000 u at an energy of 500eV it is easy to scale to different masses or energies. For example, ions of mass 40 u at 500eV need time definitions better than (lath) of the previously calculated values, which is 2 and 1 ns respectively.
26
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
2.2.1. Verifving specifications
Assume now that we have purchased the ion optical system of a time-of-flight mass spectrometer. First, we want to verify its specifications. To do that we have to measure the arrival times of some ion as a function of initial coordinates and velocities in the ion source. From the time errors we have to calculate the z/-errors by a trivial formula and then compare the results with the data given by the manufacturer. A first prerequisite for this measurement is that the laser pulse should be significantly shorter than the time differences that need to be determined. Likewise, the detector and registration electronics must be faster than the time differences to be measured. That means we need to pick an ion that is heavy enough or whose total time of flight is long enough so that small fractions of the total time of flight can be measured. Directly verifying all six z’-dispersion functions precisely and completely is usually not possible. Still, in many cases we can get a good idea. Assume we are creating the ions in the ion source by a short laser pulse. We can then translate the laser beam in the x-z-plane. What we are measuring then, will not be the function; 4x, 0, z, 0, 0, 0)
(5)
but the function
z*(x, z) =
JJJJ
z'(x,Y,
z, 21x7l-‘y, G-bd~,d~,d~z
(6) We can take care that the range of initial velocities stays much smaller than what is encountered during regular operation, e.g. by ionizing a gas out of thermal equilibrium or from a supersonic expansion. That simplifies our integral considerably: z*(x, z) =
J
z’(x, y, z, O,O,0)dy
(7)
We can measure z*(x,z) as in Eq. (7) for a small range of initial velocities by a soft ionization without any recoil. We can measure another function z*(x,z) as in Eq. (6) for a large range of initial
velocities just by ionizing molecules that fragment upon ionization. These two functions should allow enough conclusions about the behaviour of z/(x, y, z, w,, r+, wZ), including the answer to the question as to whether the instrument fulfills the manufacturers specifications or not. 2.2.2. Sensitivity and transmitted phase space One very important thing to remember is the Liouville theorem. This theorem states that the volume of phase space transmitted by any ion optical system stays constant on its path. It is also important to note that this statement holds, no matter how strongly the shape of the transported phase space is distorted on its path. This means that we can always distort the transported phase space volume such that it gives an optimal fit to the ions offered to the optical system. For that reason it is possible to equate the terms sensitivity and transmitted phase space. We assume that it is always possible to distort the transported phase space as needed. Under this prerequisite-which has very few exceptions - an instrument designer first has to find out where exactly in phase space the ions enter the instrument, and then has to design the ion optical system to fit these externally imposed conditions. He has to play with that six-dimensional product of velocities and coordinates, (keeping the value of the product constant), until he has found a configuration that transports the maximum number of particles. 3. Demonstration: standard instrument with twostage grid reflector
To substantiate the preceding discussion, we will now look at a well-known, time-of-flight design. Figure l(a) shows a standard gas-phase ion source and Fig. l(b) shows its axis potential with the lens operated in negative mode. The lens of the ion source can be adjusted to do one of the following. (i) The ion source can guide all ions that start on the axis at z = 0, but with different initial trans-
T. Bergmann and T.P. Martin/Int.
J. Mass Spectrom.
Ion Processes
21
131 (1994) 21-41
(bf
Fig. 1. (a) Standard gas phase ion source. The first three plates define the accelerating field, the thick fifth electrode is an Einzel lens. (b) The axis potential shows the Einzel lens operated in the negative mode.
versa1 velocities, to paths parallel to the axis. We will call this the velocity focussing mode. (ii) The ion source can guide all ions that start with zero initial velocity, but different locations around the axis at z = 0, to paths parallel to the axis. We will call this the coordinate focussing or telescope mode. The difference between these two adjustments is that the first has a more negative voltage on the lens electrode. Note that the phase space transmitted is the same for both modes. In the velocity focussing mode, paths with a larger range of initial velocities end on the detector surface, while in the coordinate focussing mode, paths with a larger range of initial coordinates end on the detector surface. All further
examples, and also the one in this section, operate the ion source in the velocity focussing mode. Section 2.2.2 discusses the possibilities of changing the shape of the transported phase space; this is what is done by changing the voltage of the lens electrode of the ion source. Figure 2 shows the z’-dispersion curves for a time-of-flight instrument with the above standard gas-phase ion source operated in the velocity focussing mode. The ion beam entering the reflector and the axis of the reflector subtend an angle of 4”. The reflector is of the ~amyrin type [6] having two regions of homogeneous electric field separated by grids. The detector is just a flat plate of 20mm diameter and can be rotated a few degrees around the y/-axis to give
28
0 =
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes I31 (1994) 21-41
’ z ’ (0.0, 2, 0, 0, 0)
-50
0 =
x = z (0.0. 2.0.0. -0 .OOl)
z ’ (0,0, 0, 0. 0, v J _rn
1 K”
Fig. 2. Dispersion curves of a standard ion source (from Fig. 1) and two-stage grid reflector. The graphs give the error in the final icoordinate as a function of all six initial phase space variables. Since a grid reflector is optically active only for velocity components parallel to its axis, it only affects the function ~'(0, 0,z, 0,0,O).The other five dispersion functions closely reflect the behaviour of the ion source in Fig. 1. The initial velocities are given in units of ~0, which is the velocity of the reference ion in the drift space of the mass spectrometer.
T. Bergtnann and T.P. Martin/h.
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
an optimum .z’-dispersion. The total equivalent length of the flight path is approximately 2m. The ionizing laser beam has a diameter of a 2.4mm. The length of the acceleration region is about 20 mm, and thus a 2.4mm width of the ionizing laser beam corresponds to an energy spread of z 316%. All free parameters are optimized using a procedure described in ref. 2, the resulting z’-dispersion being shown in Fig. 2. All six dispersion plots in Fig. 2 show the deviation parallel to the incoming path of the reference particle (along the $-axis). Since we have assumed 2 m for the total equivalent length of the flight path, a path error of AZ’ = 100 pm corresponds to a time error of 5 x 10e5 or an error in mass determination of 10-4. The plot in the lower left hand corner shows the dispersion as a function of the initial z-coordinate in the ion source. This dispersion corresponds to the standard solution of the Mamyrin equations f67. The circles show the dispersion for ions initially at rest and the crosses show the dispersion for ions with an initial velocity of -0.001~~. (Remember that v. is the velocity of the reference ion in the drift tube as defined in section 2.1.4.) The plot in the lower right-hand corner shows the z’-dispersion as a function of initial velocity v,. The slope of this function z’(O,O,O,O,0, v,) is inversly proportional to the accelerating field strength in the ion source, here we have w 0.5Us cm-‘. (U, is the voltage corresponding to the drift velocity 21s.) Note the scale: just one thousandth of the drift velocity is necessary to effect an error in path length of z’ = 4Opm! If we assume that the energy of a particle in the drift tube is 1 keV then this initial velocity corresponds to a recoil energy of only 1 meV. This sensitivity to recoil has been pointed out before [7,8]. The usual solution for this problem is that the initial velocity w, is restricted or reduced by some means. The upper left-hand plot shows the z’-dispersion as a function of the initial x-coordinate with the initial velocity w, as the parameter, while the upper right graph shows the z’-dispersion as a func-
29
tion of the initial velocity w, with the initial x-coordinate as the parameter. This is a coordinate or velocity deviation from the axis of the ion source in the plane defined by the spectrometer axis and the connecting line between the ion source and the detector. Both graphs very closely reflect the behaviour of the ion source in Fig. 1 itself. The two graphs in the middle show the z’-dispersion as a function of y or IJ,,.The behaviour shown by these graphs is very similar to the z’-dispersion as a function of x and v,. Because the spectrometer is assumed to be symmetric about the x-z-plane, only two instead of three parameters are plotted. The initial x- and y-coordinates are limited to 10.5mm by the magnifi~tion of the ion source optics and the diameter of the detector. The performance of such an instrument can be verified with the methods described in section 2.2.1. Figure 6 of ref. 2 shows the results of a measurement that can be used to determine z*(O,z)=
J
z’(O,y,z,O,O,O)dy
Scanning the laser in the x-z-plane given the function z*(x, z).
would have
4. Examples: ions from the gas phase This section addresses instruments that start the ions in the time-of-flight instrument out of the gas phase. Instruments of this category are used for all kinds of photoionization experiments. When electron impact is used, the interest is usually to have a mass-selective detector as in gas chromatographymass spectrometry. In secondary neutral mass spectrometry (SNMS) either a laser or electron pulse is used to ionize sputtered neutrals. This ionization event is then the start event for time-of-flight analysis. 4.1. State-of-the-art gridless rejectors The purpose of this section is to warn the reader about believing statements like “As it seems
30
T. Bergmann and T.P. MartinlInt.
logical, this reflector can do everything at the same time! It cannot only perform energy focussing but can also do space focussing. It can effect a significant enhancement of sensitivity at a superior mass resolution”. What is logical? The Liouville theorem always holds, even for gridless reflectors! State-of-the-art instruments with gridless reflectors presently sell for twice the price of regular instruments on the basis of the above arguments. Yet, their performance is much worse, as will be shown in this section. We will just pick out two examples, not calling them by name. The authors have looked at more gridless reflector designs to be found in the literature or in patents, all of which have a similar design and probably a comparable performance. Some newer designs have been announced, but no information about their detailed construction is given. So, if you think about buying an instrument with a gridless reflector, be sure to ask for the s~~ifi~tions! The reflector of patent A has four adjustable voltages, the ion paths entering the reflector subtend an angle of 4” with its axis. The total length of the drift path is again = 2 m. The ions are started in the standard ion source of Fig. 1. If we optimize the four adjustable voltages in a range of f2% of the values prescribed in the patent we get the dispersion curves shown in Fig. 3. Naturally, the function ~'(0,0,0,0,0,w,) shown in the lower right-hand corner is identical to the corresponding function in Fig. 2, since we use the same ion source with the same accelerating field strength. The z’-dispersion as a function of z is also comparable in quality. The z’-dispersions of initial variables perpendicular to the direction of acceleration are much worse for this gridless reflector than for a regular two-grid reflector as can be seen by comparing Fig. 2 and Fig. 3. We can set apertures behind the ion source such that paths with an error of z’ > 100 pm are blocked. Looking at the transmitted phase space in the x+,-plane we find that the two-grid reflector of Fig. 2 transmits a
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
trapezoid of area M 1.Omm x 0.06~ and the gridless reflector of Fig. 3 transmits a diamond of area c 0.5(0.5 mm x 0.06~~). This makes a factor of 4 in the x-v,-plane. The same situation is found in the y-+,-plane, such that, disregarding transmission losses by the meshes in a two-grid reflector, a gridless reflector after patent A transmits a phase space 16 times smaller than a two-grid reflector. If we take transmission losses of about 50% passing four times through a mesh into account, the transmission of this gridless reflector is still eight times smaller. Figure 4 shows the z’-dispersion curves of a reflector after patent A, if not the prescribed voltages are used, but an optimized, completely different set of values. The phase space transmission is close to a factor of 2 better than in Fig. 3, still it is a factor of 4 worse than a standard twogrid reflector. The reflector after patent B has a total of 17 independent voltage adjustments. The angle that ion paths entering the reflector subtend with the reflector axis is not given, so that it was freely varied for optimum performance. The total length of the drift path is not given so that it was used also as a free variable in the optimization. The detector and ion source are assumed to be 3cm off the spectrometer axis. The voltages are optimized in a f2% range around the prescribed values. One should think that this reflector gives a much better performance than the reflector after patent A. Figure 5 gives the resulting performance. It closely corresponds to Fig. 3 and Fig. 4. An instrument with this reflector does not have a higher phase space transmission than instruments using reflector A. 4.2. Can reflectors without grids have a superior transmission?
The authors do not want to convey the impression that gridless reflectors are worthless. The purpose of this section is to show that it is theoretically possible to construct gridless reflectors that can transmit a phase space volume a factor of 20
T. Bergmann and T.P. Martin/Int. J. Mass Spectrom. Ion Processes I31 (1994) 21-41
31
X x=z'(x,O,O,-0.02.0,0)
x = 2’ 60.2. 0. 0. v,, 0. 0)
o = 2’ (x. 0. 0, 0. 0, 01
o=2’l0.0,0,v,,0.01
cJ= 2’(x,0,0,0.02,0.0)
q
= 2’(0.2.0,0,v,.0.01
Z x-2'~0,0,2,0.0,-0.001) 0 = 2’
(0,0, 2,0,0.0)
Fig. 3. Dispersion curves of a standard ion source and gridless reflector after patent A. The reflector voltages are restricted to 12% of the values prescribed in the patent. The i-dispersion as a function of z and V, are comparable to the dispersions of a two-grid reflector shown in Fig. 2. The z’-dispersion as a function of x, y, V, and vYshow that the phase space transmission of this gridless reflector is a factor of 8 smaller than a two-grid reflector.
T. Bergmann and T.P. Martin/M
-oil
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
[mm1
0= XS
Fig. 4. Dispersion curves of a standard ion source and gridiess reflector after patent A. The reflector voltages have been optimized to give the lowest possible timing or z’-errors. The i-dispersion as a function of z is somewhat smoother than in Fig. 3. The z’-dispersions as a function of intial x and y are a bit wider. The transmitted phase space of this adjustment is close to a factor of 2 better than the adjustment of Fig. 3.
T. Bergmann and T.P. Martin/h.
33
J. Mass Spectrom. Ion Processes I31 (1994) 21-41
X x=
z'Ix.O,O,-0.02,0,0)
o= 2 ’ q
lx. 0. 0, 0. 0, 01
~=2’~-0.2~0,0.v~0,0)
-50
o=z’(o.o,o,v,o,ol
-50
= 2 * tx. 0, 0,o. 02. 0, 0)
_ _
(0,y. 0,o. 0,O)
05
2‘
X-
2 a (0. y. 0.0.0.02.0)
-50 t
0 = 2
’ (0,0.0. 0, v p 0)
-50
x=Z'(0.0.2,0,O,V~Of
Fig. 5. Dispersion curves of a standard ion source and gridless reflector after patent B. The reflector voltages are restricted to f2% of the values prescribed in the patent. The z’-dispersion functions show that this reflector gives a phase space transmission somewhere between Fig. 3 and Fig. 4
34
T. Bergmann and T.P. ~art~n~~nt. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
larger than a standard two-grid reflector. The following section will then show the performance of an instrument presently under construction that is expected to perform a factor of 7 better. This section shows one instrument which has a low accelerating field of M 0.5 U,, cm-’ in the ion source and another instrument with a high accelerating field of z 14 Uecm-’ in the ion source. Ue is the voltage corresponding to the drift velocity uo. Both of these examples will not be actual constructions with electrode shapes etc. but will just be defined through their axis potentials which are given by spline functions (see ref. 2). Whether an actual construction will achieve these limits depends very much on geometric limitations of electrode design. Fig. 6. shows the optimum performance of an instrument with a low accelerating field of x 0.5 U. cm-‘. By comparing Fig. 6 with Fig. 2 we see that this instrument transmits a phase space volume at least a factor of 20 larger than a standard instrument with two-grid reflector. (A factor of 10 for larger phase space and another factor of 2 because it uses no grids.) The z’-dispersion as a function of z is almost completely flat. That is not too surprising since the reflector potential is controlled by a multivariable spline function. The phase space volume in the X- and y-direction is so large that the question might arise as to whether it is possible at all to admit ions to fill this whole volume. The z’-dispersion as a function of w, is still as sensitive as in all the previous figures. For that reason one might try to change the shape of the transported phase space volume to allow for a larger variation of v,, perhaps reducing the phase space in x- and y-directions. This is what has been done in Fig. 7. The length of the transported z-coordinate is much shorter than in Fig. 6, but the accelerating field is also much higher: M 14 Uo cm-‘. Thus, the range of z is a factor of 5 less but the allowable range of w, is a factor of 28 higher. The transported phase in the X- and y- directions is also less, so that the total
transmitted phase space volume is again the same as in Fig. 6. Note that the allowable recoil upon ionization for this instrument is 0.032~~.If the energy of ions on the drift tube is 1 keV, this recoil corresponds to 0.9eV. This instrument should have a high mass resolution even for “hard” ionization. 4.3. SNMS-time-of-flight
instrument
Figure 8 shows the performance data of a completely designed SNMS time-of-flight instrument. All potentials, also that of the detector, are given by explicitly defined electrodes. The postacceleration in the detector is assumed to be 6Ko. (For ion energies of 1 keV this corresponds to 6 keV.) The accelerating field is = 10 U. cm-‘. The z’-dispersion plots in Fig. 8 have been scaled by the ratio of the velocity in drift space divided by the velocity of particles impinging on the detector surface, in this case l/m (see section 2.1.3). Thus, the data in Fig. 8 should be compared, as in all previous cases, to a total equivalent length ofS=2m. The origin of the unprimed coordinate system xi has been fixed to a repeller plate 0.5 mm behind the starting point of the reference path; the reference path is also assumed to start with a velocity of 0.05~~. This corresponds to what is expected for SNMS operation. Before the laser ionizes the sputtered neutrals they are desorbed from the analysed surface at z = 0 mm. When they pass through the volume around z = 0.5 mm, they have a velocity of z 0.05~~. The angle that the ion source electrodes above the sample may subtend is limited to 20” and thus it is not possible to freely model the field as in the theoretical calculations for Fig. 7. This partially explains why the total transmitted phase space volume, as shown in Fig. 8, is only a factor of 7 better than a standard two-grid reflector can handle and not a factor of 20 as for Fig. 7. That the instrument shown in Fig. 8 uses a detector while Fig. 7 just uses an imaginary
T. Bergmann and T.P. Martin/Int.
J. Mass Spectrom.
Ion Processes
131 (1994) 21-41
35
X
0 = 2’
lx. 0. 0, 0. 0.01
0 = z’ lx, 0, 0. 0.04, 0. 0)
50
_ ’ 0 = z (0. y, 0, 0. 0, 0) x=z’(0,y.0,0.0.03,0)
-50
_
E
1.0 [mm1
J o= X=
2’ 50
[pl i
[pm1
Fig. 6. Dispersion curves showing the optimum performance of a time-of-flight instrument with a low accelerating field of = 0.5 lJO cm-‘. The transmitted phase space volume for this instrument is at least 20 times as large as for an instrument with a twogrid reflector shown in Fig. 2. Note that the behaviour in the x-z-plane for this instrument is different than in the y-z-plane.
36
T. Bergmann and T.P. Martin/h.
X
-0.04
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
-0.02
0.02
0 - z ’ (0.0,0. vX.0.0) q=z’(o.4,o,o,v,.o.o)
0.04
vx
Iv01
~=z’~-0.4,0.0,v,.0,0)
-50
Fig. 7. Dispersion curves showing the optimum performance of a time-of-flight instrument with a high accelerating field of = 14 U, cm-‘. Because the accelerating field is higher by a factor of 28, the range of transported w, is also higher by a factor of 28. The ranges of the remaining five phase space variables multiply to about l/30, so that the total transmitted phase space volume for this instrument is the same as in Fig. 6.
T. Bergmann and T.P. Martin/Int. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
37
X x= z* (x.0,0.5. -0.02.0.0.05) *= 2’~x.0.0.5,0,0,0.05~
~=z’~0.2,0.0.5,v,.0.0.05~ +O
o= z’ (x, 0.0.5,0.02,0.0.05)
~=z’10,0,0.5,v,0.0.05~
-50
x=z’k0.2.0.0.5,v,O.0.05~
7’
Fig. 8. Dispersion curves showing the performance of a fully designed SNMS-time-of-flight mass-spectrometer. The origin of the coordinate system xi has been fixed to a repeller plate 0.5mm behind the starting point of the reference path; the reference path is also assumed to start with a velocity of 0.05~s. This instrument has a high accelerating field of w 10 U,,cm-’ to achieve a high mass resolution in spite of the expected recoil upon ionization. The total transmitted phase space volume is higher by a factor of 7 compared to the standard instrument with two-grid retkctor of Fig. 2.
38
T. Be~gmann and T.P. ~urtin~Int. J. Mass Spectrom. Ian Processes 131 (1994) 21-41
plate might account for another part of the difference. This instrument is presently under construction. 5. Examples: ions from surfaces This section addresses those instruments that create ions at surfaces from where they are drawn into the time-of-flight instrument. Typical examples of this type are secondary ion (SIMS) and laser desorption (LD) mass spectrometers. In SIMS instruments a short pulse of primary ions sputters off material from the analysed surface. The ionized part of this material is drawn into the time-of-flight mass spectrometer. In LD instruments the ions are desorbed from a surface by a short and intense laser pulse. In the case of matrix assisted laser desorption the molecules of interest are imbedded in a matrix of light-absorbing material which is heated by the laser pulse and which then evaporates and carries the analyte along into the time-of-flight massspectrometer. In both cases the z-coordinate of the initial phase space is fixed to the analysed surface (z = 0). Since it is desired to obtain information from different locations of the surface of the sample, the x- and y-coordinates may vary. Likewise, the ions have a finite spread of initial velocities in all three directions. Thus we have to consider five and
not six phase space variables in the following discussion. Figure 9 shows the design of a typical SIMS ion source as can be seen for example in Fig. 1 of ref. 9. The ions start at z = 0 and around x, y = 0 and are accelerated in the positive z-direction. The thick ring at z = 5.8 cm is the active electrode of a focussing lens. The ion source taken from ref. 9 is just meant as an example and the measures in Fig. 9 are only approximate!, simifar designs can be found in any publication on SIMS-time-of-flight massspectrometers. Figure 10 gives the performance of a system comprising the standard SIMS ion source of Fig. 9, a single-grid-refl~tor starting 7Ocm above the sample surface, and a standard detector design (not shown). The distance of the extraction lens to the sample surface has been adjusted, such that the accelerating field is M 10 U,, cm-‘. This has been done to provide a better comparison with Fig. 11. Figure 11 gives the performance data of the instrument shown in Fig. 8 operated in SIMS mode. Because the accelerating field above the sample surface is also x 10 Us cm-’ the d-dispersion as a function of v, is roughly the same as in Fig. 10. It is important to note that the z’dispersions in the plane perpendicular to the zaxis are signi~cantly better than for the standard instrument of Fig. 10. This is due to the gridless reflector.
X 2.0
km1 1.0
0.0
Fig. 9. Standard SIMS ion source [9]. The ions start at z = 0 and around x,y = 0 and are accelerated in the positive z-direction. The thick ring at z c 5.81x1 is the active electrode of a focussing Iens.
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes 131 (1994) 21-41
39
2’
2’
V
-0.2 x = 2’
-0.1
(x. 0, 0, -0.02.0, 0.051
o=r’~X.O.0.0,0.0.05l n = 2’ (x, 0,o. 0.02,0,0.05)
hml -50
n
0.2
0.1
-0.03
~=~‘~-o.l,o~o‘v~,o, o~z’~o.o,o.v,.o,
~=z’~o.l~o,o,v~‘o~
-0.01 (0.05_~~“~ (o.05-vyy50
(O.O~-v~~‘~
2’
Z’ Z’
Fig. 10. Dispersion curves showing the performance of a standard SIMS inst~ment with a single-grid reflector. This inst~ent ion source of Fig. 9, a single-grid reflector and a standard detector design.
References 1 P.W. Hawkes and E. Kasper, Principles of Electron Optics, Academic Press, London, 1989. 2 T. Bergmann, T.P. Martin and H. Schaber, Rev. Sci. Instrum., 61(10), (1990) 2592.
3
uses the
D.A. Dahl and J.E. Delmore, SIMION PC/PSZ, Version 4.0, EGG-C%7233 Rev. 2, April, 1988. 4 M. Szilagyi, Proc. IEEE, 73(3) (1985) 412. 5 J.P. Adriaanse, H.W.G. van der Steen and J.E. Barth, J. Vat Sci. Technol. B, 7(4) (1989) 651. 6 B.A. Mamyrin, V.I. Karataev, D.V. Shmikk and V.A. Zag&in, Sov. Phys. JETP, 37 (1973) 45.
T. Bergmann and T.P. MartinlInt.
J. Mass Spectrom. Ion Processes 131 (1994) 21-41
2’
2’
X x = 2’
x= 2’ Ix. 0.0. -0.02,0.0.05) 0=2’(x.0.0.0,0.0.051
-50
q-2’(x,0.0.0.02.0.0.05)
0.05 “x
-0.05
(-0.15, 0, 0. v “, 0. (0.05-vg “?
o-2’(0,0.0,v,,0.
q=z’(o.15,o,o,v,,o,
(0.05-v$“?
-w
(0.05-v~“q
2’
2’
“Y
Y 0=2’~0,y.0.0.0.0.05)
-50
x = z’ (0. y. o,o, 0.02. 0.05)
0 = 2’ x =z
(0,0, 0, 0. v “’ IO.05-q
I”, _5()
* (0,O.15. 0. 0, v “, (0.05-v$
“‘,
Fig. 11. Dispersion curves showing the performance of the SNMS instrument of Fig. 8 operated in SIMS mode. The exceptional i-dispersion as a function ofx, y, V, and vYis the result of the combined ion optical properties of the ion source and the gridless reflector. U. Boesl, J. Grotemeyer, K. Walter and E.W. Schlag, Anal. Instrum., 16(l) (1987) 151. U. Boesl, R. Weinkauf and E.W. Schlag, Int. J. Mass Spectrom. Ion Processes, 112 (1992) 121. E. Niehuis. T. Heller, H. Feld and A. Benninghoven, J. Vat. Sci. Tech&l. A, 5(4) (1987) 1243.
Appendix A: Notation
To simplify the notation we redefine the variable for energy, potential, time and velocity. If K is the total energy of an ion and & is the average energy
41
T. Bergmann and T.P. MartinlInt. J. Mass Spectrom. Ion Processes 131 (1994) 2X-41
of all ions, then let e be defined by K= (1+ E)KQ
(Al)
If U is the electric potential, then u is given by U=u&
(A2)
a very close relationship to the length dimensions of the problem. The velocities are always given as the s derivative of coordinates. Newton’s equations now become d2
-&fXiCs)
The location of ionization for ions with t = 0 and with w, = 0 will be at z = 0. The average velocity of ions at ground potential U = 0 is given by v. = ,,/m. Let s be defined by s = vat
(A3)
We give s the name “equivalent length”. In solving Newton’s equations we use s instead of t as the independent variable. As can easily be seen, s has
=
-~&“(xj(s))
i,j=
1,2,3
I
(A4)
Defining dxi/ds = wi, it follows that (WZ+ w; + WI) = 1 f 6
(A5)
In the dispersion curves we always define & and v. as the energy and velocity of a particle on the reference path at ground potential. In the examples discussing time-of-flight instruments, ground potential is defined as the potential on the drift paths.