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The use of the time–energy dispersion in an electron energy analyzer
Journal of Electron Spectroscopy and Related Phenomena 131–132 (2003) 105–116 www.elsevier.com / locate / elspec
The use of the time–energy dispersion in an electron energy analyzer a, b c c O.M. Artamonov *, S.N. Samarin , G. Paolicelli , G. Stefani a
St. Petersburg State University, Research Institute of Physics, Ulyanorskaya 1, Petrodvorez, 198504 St. Petersburg, Russia b University of Western Australia, Perth, Australia c Universita` di Roma Tre and INFM Unita` Roma 3, Rome, Italy Received 26 April 2002; received in revised form 10 April 2003; accepted 30 May 2003
Abstract The time–energy dispersion (TED) characteristics of a spherical electrostatic mirror are used to design an unusual electron energy analyzer. It is shown that TED of a retarding spherical field in an electron mirror configuration is positive and it increases with the electron kinetic energy. This is due to an increasing penetration of electrons with high kinetic energy in the retarding field and provides the basis for developing a new type of time-of-flight spectrometer (TOF). Two limiting cases for the geometry of the proposed spectrometer are considered and for both of them the energy resolution of the TOF device depends on the combined time resolution of electron detector and readout electronics. In the case of a 10-cm inner sphere radius (22 cm outer sphere radius) for the electrostatic mirror, the spectrometer is expected to have an energy resolution of about 0.5 eV/ ns and an acceptance solid angle of about 2.2 srad for electrons of 75 eV kinetic energy. A position sensitive detector allows retrieving the energy and the emission angle on the basis of the measured time-of-flight and detection point position. In the second case the inner sphere radius is 6 cm and the outer spherical segment has a radius of 120 cm. The energy resolution is expected to be about 160 meV/ ns at the electron kinetic energy of 1 keV and the acceptance solid angle is 0.1 srad. In both cases changing the retarding potential can easily change the energy range within which the TOF analysis is performed. 2003 Elsevier B.V. All rights reserved. Keywords: Electron spectroscopy; Time-of-flight; Coincidence spectroscopy
1. Introduction Over the past 20 years break-up processes studied by time correlated experiments have experienced a rapid expansion because of their ability to highlight fragmentation dynamics in atoms, molecules and *Corresponding author. Tel.: 17-812-356-1350; fax: 17-812428-7240. E-mail address: [email protected] (O.M. Artamonov).
solids. To be efficient, these experiments require low incident flux, large accepted solid angles, high energy resolution and high momentum resolution. In order to develop a single analyzer that satisfies all these conflicting requirements, several different attempts have been made ranging from various successful versions of the COLTRIMS spectrometer [1], to toroidal spectrometers [2] and to arrays of time of flight [3] or dispersing element [4] spectrometers. TOF spectrometers are very attractive because of
0368-2048 / 03 / $ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016 / S0368-2048(03)00129-4
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their mechanical simplicity but often find a limit to their application in the reduced accepted solid angle and in the moderate energy resolution achievable at intermediate and high energies. Furthermore, to avoid pile-up in the electron detector, often the incident flux must be kept to levels so low that the intrinsic high luminosity due to the multichannel capabilities of TOFs are fully exploited. These drawbacks, severe when TOFs are used in conjunction with third generation synchrotron radiation (SR) sources, might become unbearable in view of application with free electron laser (FEL) sources that are characterized by low repetition rate and will produce bursts of photoelectrons with very high peak flux. A recent paper suggested that combining a TOF with a spherical mirror electrostatic reflector will result in improved, rather than degraded, energy resolution at high energy and in unusually large accepted solid angle [5]. In the present paper we suggest that combining such a TOF reflector with a wide band pass energy selector (WBP) will result in a very flexible analyzer capable of operating either with large accepted angle and moderate energy resolution, or with moderate solid angle and high energy resolution. In either case, the WBP allows to detect only electrons that are relevant to the physical process under study by rejecting all the others before they reach the detector. For instance, detection and subsequent time of flight analysis will be performed only for Auger electrons within a pre-set window while secondary electrons and photoelectrons will be rejected. Most modern electron spectroscopies use the spatial dispersion of an electrostatic field (as a function of energy and direction) to select electrons of a certain kinetic energy and to focus electron trajectories. Usually, in a dispersive electrostatic analyzer high resolution is obtained at the expense of detection efficiency and vice versa. The energy dispersion provides a relative shift of electron trajectories of different kinetic energies, for example, along a slit of an aperture. The angular dispersion gives a convergent set of trajectories, as in the case of an axial electrostatic field. Furthermore, electrons with different kinetic energies will be detected at different times, i.e. will suffer time–energy dispersion (TED), that is the basis for time-of-flight (TOF) spectroscopy. The final aim of an electron spectrometer design is to get the desired energy res-
olution with as high as possible detection efficiency. TED offers great advantages in optimizing both properties of an electron energy analyzer. For example, the spatial dispersion of the electrostatic field provides the focusing properties, whereas the time– energy dispersion allows the electron energy selection. The time-reference signal needed to implement any TOF spectroscopy is usually provided by the time structure of a pulsed excitation source, be it electrons or photons. The quantities practically measured by the time-of-flight electron spectrometer are the electron flight time, from the specimen to the detector, and the electron impact position on the detector entrance plane (or impact position). It is important that there is only one electron trajectory, which satisfies the initial emission parameters (the electron energy and emission angle) and the measured parameters (the time-of-flight and detection position). In a potential field there exists an infinitely large number of electron trajectories connecting the emission and detection points and some of these trajectories have the same time-of-flight, so we have to choose the energy range and the emission angle range in such a way that starting and detection points can be connected just by only one trajectory. The existence of a unique trajectory permits to calculate the energy and the emission angle on the basis of the measured time-of-flight and detection point position [5]. Examples considered below satisfy that condition. In a conventional time-of-flight spectrometer one measures the flight time that an electron needs to travel from the target to the detector in a field-free space and the flight time is a function of the electron kinetic energy t(E). The absolute energy resolution DE depends on the time resolution Dt and on the time–energy dispersion of the analyzer dt / dE (TED), that in the case of a field-free space is dt / dE 5 2 CE 23 / 2 L, where C is a constant and L is the flight distance. The dispersion decreases when the electron energy increases, consequently. And so the energy resolution degrades with the increase of the energy, as the uncertainty of the energy determination DE 5 (dt / dE)21 Dt increases with the energy. For example, if L 5 10 cm and the flight time is measured with the precision of Dt(1 ns, then DE (10 eV)50.37 eV for the electron energy E 5 10 eV. Mirror electrostatic analyzers have been used to
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achieve high resolution spectroscopy [6] and to realize display type spectrometers that analyze kinetic energy as well as angular distribution of charged particles [7]. In this paper we propose to realize a TOF spectrometer using a spherical retarding electrostatic field with reasonable focusing properties instead of the more usual field-free space. In this case one can expect to get a better energy resolution because in a retarding field the penetration of electron trajectories into the field and, accordingly, the flight time, are roughly proportional to the electron kinetic energy. One considerable advantage of the hemispheric geometry over other electrostatic mirror geometry, is that it allows analytical solutions in calculating electron trajectories.
2. Theoretical considerations
2.1. Time–energy dispersion of a spherical field We consider here a spherical electrostatic field as a dispersive media for the TOF analyzer. A charged sphere with a radius R set at a potential 2U, establishes a central field in the space outside of the sphere denoted by the heavy circle in Fig. 1 while the inner region is a field-free one. The same central field is formed in a spherical condenser if the potential difference between spheres Ur satisfies the equation U 5 Ur /s1 2 R 1 /R 2d, where R 1 , R 2 are the spheres radii. The electron trajectory originating at point S enters the central field through the inner spherical surface at point 1 (see Fig. 1) and it is determined by the relative electron energy w 5 E /eU (where e is the electron charge and U is the effective potential) and by the angle b between the radius-vector and the electron velocity direction at point 1. Unless differently stated, henceforth length will be expressed in units of R, energy in units of w and time in units of t (the time needed by an electron with w51 to travel a unit distance). The electron trajectory is symmetric about the radius to the point r max , where r max is the maximum distance from the center (or the maximum penetration depth). The time-of-flight in the central field region (from point 1 to point 2) and the maximum penetration depth are described (in relative units) by the equations [5]:
Fig. 1. The figure shows a typical electron trajectory in the basis plane of the spherical field. The solid circle is the spherical grid electrode (see text for details). A spherically symmetric field is established in the space outside the sphere denoted by the heavy circle, while there is a field-free region inside the sphere. The solid line in the fourth quadrant is the detector location. The electron trajectory starts from point S, crosses the spherical electrode at point 1, then it is reflected by the spherical retarding field and crosses again the spherical electrode at point 2. The trajectory terminates at point 3 where it crosses the detector plane. The maximum penetration of the electron trajectory in the retarding field is r max . The whole set of equations describing the electron trajectory is reported in Ref. [5].
]]] 2 œ2w cos ( b ) t 12 5 ]]]] 1 ]]]] 12w (2(1 2 w))3 / 2
S
D
p 2w 2 1 3 ] 1 arcsin ]]]]]]] , ]]]]]]] 2 1 2 4w(1 2 w) sin 2 ( b )
œ
2w sin 2 ( b ) r max 5 ]]]]]]]] . ]]]]]]] 2 1 2œ1 2 4w(1 2 w) sin ( b ) Fig. 2a–c shows the flight time t 12 , the maximum penetration depth r max (from the center of the spherical field) and the time–energy dispersion TED 5 dt 12 / dE as a function of the relative kinetic energy w for the fixed angle b 5258. One can easily convert the relative time t to the absolute time T measured in seconds by the dimensional factor T 5 ] k t t, where k t 5 C1 R /ŒU (C1 (2.38 3 10 28 , if the
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electron in the spherical retarding field (expressed in units of the charged sphere radius) increases monotonically with the electron energy. As a consequence, the time-of-flight also increases monotonically with the energy (see Fig. 2a). Fig. 2c clearly shows the time–energy dispersion that, on the contrary to what is usually found for free space drift tubes, displays always a positive value. In other words, the flight time through the central field increases with increasing of the electron energy. In particular, the TED characteristic has a shallow minimum for relative energies between 0.05,w,0.2. At higher energies TED monotonically increases with the energy, i.e. the energy resolving power improves with energy. Within this shallow minimum, TED values are too low to be of practical use in building an analyzer. At lower relative energies (w,0.05), though TED sharply rises to sizeable values, still the reflector is not efficient for TOF analysis as t 12 is too short to be of practical use. It is only in the high energy regime (w.0.2) that both TED and t 12 exhibit values large enough to allow an efficient TOF energy discrimination. These observations suggest to use the geometry shown in Fig. 1 as the basis for realizing an efficient TOF spectrometer.
2.2. Focusing properties of the spherical field
Fig. 2. The main characteristics of electron motion in the spherical electrostatic field as a function of the relative electron kinetic energy w(r.u.) in relative units: (a) the flight time t 12 spent in the retarding field from point 1 to the point 2 (see Fig. 1), (b) the maximum penetration depth r max inside the retarding field, (c) the time energy dispersion TED 5 dt 12 / dw of the electron motion. The starting angle is b 5258 in all cases (see Fig. 1).
distance is expressed in cm, potential in V, energy in eV and time in ns). Fig. 2b shows that the penetration depth of an
The focusing properties of the spherical field are well known and they are described by the spatial dispersion of the field [8]. For the kinetic energy equal to one half of the charged sphere potential w50.5 the trajectories converge onto a conjugate point on the diameter line containing the point source. This is the case for any distance of the source point from the center up to the sphere radius R. The penetration depth for kinetic energy w50.5 depends on the radial component of the electron velocity and in all cases it is less than 2R. The focusing properties degrade if the electron energy does not satisfy the condition w50.5 and electron trajectories, in this case, form a finite size crossover area. This tendency increases as the point source moves along the radius from the center to the periphery. For any given source point, the focusing properties are best at the polar angles of emission close to u 5908 (see Fig. 1). The crossover area of the bundle of trajectories with average polar angle of emission different from 908 is
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shifted away from the diameter, towards positive Z (see Fig. 1) for u .908 and towards negative Z for u ,908.
2.3. Time–energy dispersion of the full analyzer As shown in Fig. 1, the electron trajectory from the sample to the detector includes two paths in the field-free space (i.e. from the sample to point 1 and from point 2 to the detector). Hence, to evaluate TOF performances of the type of analyzer sketched in Fig. 1, TED must be determined taking into account paths in the field-free region as well as the one within the reflecting retarding field. In the central field the trajectory parameter b (see Fig. 1) is defined by the source point position S on the X-axis and by the polar emission angle u. The detector is placed in the vicinity of the X-axis. The TED of the full device (from the sample to the detector) is the sum of TEDs of separate parts of the path. TED for the whole analyzer is negative at relatively low electron energy because the path in the spherical field is small relative to the path in the field-free space and the latter behaviour will dominate. TED changes the sign and increases very rapidly when the relative energy increases. We define the energy resolution F as the inverse TED F 5 dw / dt. Fig. 3 shows F (in relative
Fig. 3. The energy resolution of a spectrometer F (r.u.) as a function of the relative electron energy w. The function F is the inverse TED for the whole spectrometer (i.e. for the total electron path from the source point S to the detecting point 3 of Fig. 1). Y-scale is truncated at w50.1. The solid vertical line at about 0.24w defines the discontinuity of the function F at the energy where TED drops to zero. The function F tends to zero as w tends to 1, but the penetration depth r max increases infinitely at the same time. The curve reported in the figure has been computed for conditions of Fig. 1 and with angle b 5258.
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units) as a function of the electron energy (in relative units too), note that the Y-scale is truncated at w5 0.1. TED crosses zero at about w50.24 because at this point TED of the spherical field equals TED of the field-free path and, as a consequence, the F curve in Fig. 3 has a discontinuity at this energy. TED of the field-free space inside the sphere defines the resolution in the energy region w → 0, because the penetration depth becomes small. The resolution degrades as the energy increases in this region. Below this critical value of w, and for kinetic energies tending to zero, the analyzer behavior is largely dominated by paths in the field-free region. The penetration depth is always smaller than 1.3R, hence the resolution F degrades as the energy increases. Conversely, above the critical value w5 0.24 the energy resolution improves with increasing the relative energy. It is possible to estimate the energy resolution in the units eV/ s as a function of the energy in eV for the specific example. To do this one needs to define the effective potential of the single charged sphere, which depends on the ratio of the sphere radii and on the retarding potential difference, and then express the electron kinetic energy in eV. The Y-scale will be multiplied by the factor U /k t and the X-scale by the factor U. Let’s estimate, as an example, the resolution of a spectrometer similar to the one shown in Fig. 1 with R510 cm and U 5100 V (this is equivalent to R 2 /R 1 5 2 and the retarding potential difference Ur 5 50 V). We fix the absolute value of the resolution at the level of 0.05 and estimate its equivalent in eV/ s and the energy values at which this resolution is achieved. Fig. 3 shows that uF (r.u.)u 5 0.05 occurs at two relative energies of about w 1 5 0.53 and w 2 5 0.105. The value uF (r.u.)u 5 0.05 is equal to 0.21 eV/ ns and energies w 1 and w 2 are equivalent to 53 eV and 10.5 eV. Thus, in the considered example the energy resolution uFu 5 0.21 eV/ ns is realized at the energies E1 5 10.5 eV and E2 5 53 eV.
2.4. Some important limitations We have considered trajectories in the plane that contains the center of the spherical field and the point source of electrons. The line connecting the center of the spherical field and the source point is
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the axis of symmetry. One can then achieve full description of the bundle of trajectories originated from the point source S by full rotation of the trajectory plane around the symmetry axis. Anyhow, in practical use, it is convenient to limit the spherical field by a spherical segment with the same central point. So far, all calculations have been done in a perfect central field. In reality this ideal condition will be approached by terminating the spherical sector with a set of guard rings with proper voltages applied. Furthermore, the inner electrode will be a grid rather than a solid sphere and a micro-channel plate will be located at the detector place. By defining the basis plane that includes the symmetry axis and the vertex of the spherical segment one can define the electron trajectory by two angles—the polar angle u in the trajectory plane and the tilt angle c of the trajectory plane relative to the basis plane. The range of the u - and c -angles is limited by the size of the spherical segment as well as by the size of the detector. The detector position should be in the vicinity of the symmetry axis. One has to avoid the intersection of the symmetry axis with the detector plane because at the intersection point it is not possible to distinguish trajectories with different c -angle. This is considered in detail in Ref. [5], as well as the restrictions imposed on the energy and angle ranges by the necessity to have a unique solution of the problem of retrieving the initial energy and angle of emission from the measured flight time and detection position.
3.1. Spectrometer design with the radii ratio close to 2 Fig. 4a and b shows a set of electron trajectories in the spherical field established within two concentric hemispherical surfaces with the ratio of radii R 2 /R 1 5 2.2. Simulations performed at electron energies comprised within 610% about the mean electron energy w m 50.5, show that trajectories originating from a point source S converge at the conjugate point on the symmetry axis irrespective of the
3. Design of the TED spectrometer We shall present below two examples of how it is possible to make use of TED of a spherical electrostatic mirror for TOF analysis of electron energy within a given energy band. The first example is the scheme of a spectrometer with a moderate energy resolution and with a large acceptance solid angle. In this example we shall make use of the excellent focusing properties of the spherical field at w50.5. In the second example we consider a spectrometer with a higher energy resolution that is achieved at the expense of the acceptance solid angle.
Fig. 4. Electron trajectories in the basis plane: 458,u ,1408, (a) w 5 0.5, (b) w 5 0.55. The ratio of the radii of the spherical electrodes is R 2 /R 1 5 2.2 and R 1 5 1. The electron source and the detector are placed in the field-free region inside the inner sphere. The electron source S is placed at x5 20.5. The point of the crossover is exactly at the symmetrical point x50.5 in the case (a), and the spot of the crossover is close to the point x50.5 in the case (b).
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outgoing polar angles u. Fig. 4a shows 30 electron trajectories at mean energy w m 50.5 in the polar angle range from u 5458 up to u 51408. We assume the c -angle to vary in approximately the same range, 2508, c ,508 (as it is in the case of a conical spherical segment). This corresponds to an acceptance solid angle of about 2.2 srad. The point source is placed at 20.5R 1 and the crossover point is at 0.5R 1 for this energy. It is possible to increase the polar angle range, but one needs for this to increase the detector size and / or to move the detector position closer to the X-axis. Fig. 4b shows the electron trajectories for the same angular range but with the electron energy equal to the upper limit of the chosen electron energy range, i.e. w50.55. Electrons penetrate more deeply in the spherical field up to the outer sphere radius. The focusing is still fairly good, but the crossover area has now a finite size and it is shifted down below the X-axis. The trajectory with the lower and higher polar angles (u 5458 and u 51408) crosses the X-axis at the points 0.54R 1 , 0.46R 1 , respectively. The crossover area for trajectories with w50.45 is shifted above the X-axis and the trajectory with the lower and higher polar angles (u 5458 and u 51408) crosses the X-axis at the points 0.46R 1 , 0.54R 1 , respectively. This effect is a manifestation of the spatial energy dispersion of this device. In particular, the crossover position will move with the energy along the X-axis. As a consequence, by placing a slit along the X-axis close to 0.5R 1 the intrinsic bandpass filter can be properly exploited. Whenever the physics of the problem under investigation requires to analyze only a given interval of the electron kinetic energy, the combined band-pass and TED characteristics will allow to perform the experiment with higher incoming flux than by a TOF spectrometer alone and avoid overloading of the detector by unwanted events. Trajectories are not dependent on the angle c, hence all the conclusions drawn from simulations done in the plane c 50 are valid for any plane with c ± 0. Nevertheless, care should be taken in determining the intercept of a trajectory and the detector plane, as it changes with the changing of c [5]. We estimate now the main parameters of the
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spectrometer. The energy resolution for the chosen ratio of radii depends on the mean energy, or (which is the same) on the effective potential, and on the absolute size of the sphere. We chose, as an example, the inner sphere radius R 1 5 10 cm. Fig. 5 shows the energy resolution F in units of eV/ ns as a function of the electron energy E in eV for three energy intervals Em 610% (in all cases Em 5 0.5eU ), calculated for conditions roughly corresponding to the design shown in Fig. 4. Three curves show the absolute energy resolution for three different values of mean energy: Em 5 50 eV, Em 5 75 eV and Em 5 100 eV and a chosen interval of 610%. To increase the mean energy for a fixed value of the radii ratio, the potential U must be also increased and consequently the resolution becomes worse with increasing of the mean electron energy as the penetration depth is fixed by the condition Em 5 0.5eU. Inside each individual interval of the energy about the chosen mean value, the energy resolution behaves according to Fig. 3, i.e. it improves with increasing energy. The estimation of the energy resolution reported in Fig. 4 was performed for an individual trajectory and for a specified u angle. A realistic evaluation must take into account that the electrons transmitted through the analyzer may be emitted at various angles u inside an interval corresponding to the full
Fig. 5. The energy resolution F (eV/ ns) of the spectrometer (designed as shown in Fig. 4 with R 1 5 10 cm) in absolute units as a function of the electron kinetic energy E (eV) for three different mean energies Em . The values of the mean energy are shown close to the curves. The energy interval of the analyzed electrons is Em 610% and the penetration depth is limited by the condition eU 5 2Em .
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acceptance solid angle. This spread in accepted polar angles results in a time-of-flight spread but it does not limit the energy resolution provided the impact position is measured by a suitable position-sensitive detector (i.e. if the analyzer is operated in the socalled differential mode). It has been shown already [5] that by knowing the time of flight and the impact point on the detector, the initial kinetic energy and the polar angle can be unequivocally retrieved. Should the detector be insensitive to the impact position (so-called integral mode) the aforementioned time spread limits the energy resolution. The mean flight time for E545 eV calculated over 30 trajectories is 173 ns and the first deviation is about 3.3 ns, that is less than 2% of the mean value. This means, in the integral mode the absolute energy resolution DE is limited by the flight time spread to about DE51.2 eV. In the differential mode it is possible to calculate analytically the electron trajectory, hence to introduce corrections for different flight times of electrons with the same kinetic energy. In this latter case the absolute energy resolution will be limited by the time resolution of the detector and readout electronics, which can be estimated by a realistic value of about 0.1 ns.
3.2. Design of a spectrometer characterized by a large radii ratio A spectrometer with a large radii ratio provides a good energy resolution and a reasonable, though reduced with respect to the previous case, acceptance solid angle. Fig. 6a and b shows a set of electron trajectories in the spherical field with the ratio of radii R 2 /R 1 5 20. The energy range chosen for the simulation is 60.5% about the mean electron energy that in this case is chosen equal to w50.943. Fig. 6a shows 10 electron trajectories with energy w50.938 and Fig. 6b shows trajectories of electrons with energy w50.948. The polar angle ranges from 1108 to 1208. We assume that c is in the range of 658, the acceptance solid angle will be about 0.1 srad. The point source is always located at 2 0.5R 1 . Fig. 6c shows a blow-up of the trajectory plane in the vicinity of the inner sphere. The crossover of trajectories takes place below the x-axis, but it is still narrow due to the relatively small solid angle accepted.
Fig. 6. Electron trajectories in the basis plane, 1108,u ,1208: (a) w ¯ 0.938, (b) w ¯ 0.948, (c) a part of (b). The ratio of the radii of the spherical electrodes is R 2 /R 1 5 20 and R 1 5 1 (r.u.). The spherical retarding field exists between the spherical electrodes. The electron source and the detector are placed in the field-free region. The electron source S is placed at x5 20.5.
In accordance with Fig. 2c, TED increases rapidly in the relative energy region above about w50.8 due to the deep penetration of electron trajectories in the spherical field. Choosing R 1 5 6 cm and R 2 /R 1 5 20 the resolution is about 160 meV/ ns at an electron energy of 1 keV. One can improve resolution at the same energy or can retain this resolution and extend the mean energy to the higher energy region by increasing the radii ratio. The mean flight time for this set of trajectories is about 510 ns and the first deviation of this value is about 90 ps. The deviation of the flight time is less than the expected time resolution of the detector and readout electronics. This means there is no need to measure the detection position and the micro-channel plate detector can be used in the integral mode without loss of resolution.
3.3. The influence of the Earth’ s magnetic field The Earth’s magnetic field can disturb the electron trajectory thus affecting TOF and detection position. The magnetic field effect is maximum when the magnetic field vector is normal to the trajectory plane. In this case, the radius of the circular trajectory of the 1 eV energy electron in the uniform magnetic field of the Earth (the magnetic field is about 0.4 Gauss) is about R58.44 cm. Hence
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screening of any magnetic field is mandatory for the analyser to work properly. We estimate the influence of a uniform magnetic field by numerical simulation. The electric and magnetic fields were calculated by ‘Simion 7.0’ in the frame of a regular grid with step size of 0.25 mm. We consider a TED spectrometer geometry similar to that shown in Fig. 4a with a uniform magnetic field applied normal to the trajectory plane. Calculations were performed for the grid electrode radius of 100 mm, an outer electrode radius of 220 mm, UR 5 290 V, Ekin 575 eV and a point source located at Xs 5 250 mm. The obtained values of TOF and X (detection position) for seven magnetic field strength values are presented in Table 1 together with their first derivations. The numerical simulation of the electric field of the TED spectrometer with spherical electrodes leads to a finite uncertainty in the TOF and in the detection position relative to the analytical calculations. To estimate this uncertainty we use the helpful property of the TED to have an ideal focusing point on the symmetry axis for the electron of a specified kinetic energy (see Fig. 4a). The spread in detection position observed in the absence of a magnetic field, is ascribed to numerical uncertainties of the simulation. During the numerical calculations the TOF was estimated with an error of about 60.1 ns (except the case of B5 20.4 Gauss). One can assume that this error defines the simulation accuracy. The situation
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with the accuracy of the detection position simulation is similar to that in the TOF case. The uniform magnetic field influences the electron trajectory mostly in the region near the reversal point r max in Fig. 1 where the velocity of the electron is minimum. The magnetic field is parallel or antiparallel to the normal to the trajectory plane and the magnetic field shifts the reversal point clockwise or anti-clockwise. The trajectory in the magnetic field becomes non-symmetrical relative to the line from the center to the r max . As a result, the focus point shifts up or down relative to the X-axis. We calculate the TOF and the X-position at the moment when the trajectory crosses the X-axis. We chose a set of trajectories which start from the point source in the angular range 808,u ,1008 (see Fig. 1) with the kinetic energy 75 eV. Table 1 shows the mean value and the deviation of the TOF and the X-position for this set of trajectories calculated for various magnetic fields. The columns named DTOF / TOF(0) and DX /X(0) give the relative change of the TOF and the X-position relative to that in the first line, that is, relative to the TOF and the X-position in the absence of the magnetic field. The uniform magnetic field of B 5 60.4 Gauss changes the mean TOF by about 8–10% and it changes the X-position by about 1– 11% relative to the values in the absence of the magnetic field. These relative changes due to the magnetic field influence exceed the relative error of the simulation in the absence of the magnetic field
Table 1 The magnetic field influence on the TED spectrometer resolution a Magnetic field (Gauss)
TOF (ns)
DTOF / TOF(0) (%)
Detection position X (mm)
DX /X(0) (%)
B50 B 5 2 0.4 B 5 0.4 B 5 2 0.04 B 5 0.04 B 5 2 0.004 B 5 0.004
110.7660.10 120.1260.25 99.1760.10 111.8460.10 109.6660.10 110.8760.10 110.6560.10
–
50.260.5 55.762.9 50.961.0 50.360.7 50.160.4 50.260.6 50.260.5
– 11.1 1.4 0.3 20.2 0.04 20.02
a
8.4 210.5 1.0 21.0 0.1 20.1
The spectrometer design is as in Fig. 4a, the grid electrode radius is 100 mm, the outer electrode radius is 220 mm, UR 5 290 V, Ekin 575 eV and the point source position is Xs 5 250 mm. The simulation has been made by ‘Simion 7.0’, electric and magnetic fields were calculated in the points separated by the step of 0.25 mm. The set of trajectories were selected in the polar angle range of 808,u ,1008 (see Fig. 1). The uniform magnetic field is directed normally to the trajectory plane. In the columns DTOF / TOF(0) and DX /X(0) the deviations (in %) of the TOF (or X) calculated for the corresponding B from the TOF at B 5 0 are given.
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and they reflect the magnetic field influence. The energy resolution is defined by the TOF accuracy measurements and for the B 5 u0.4u Gauss its average value is about 0.17 ns according to Table 1. The relative energy resolution for the kinetic energy of 75 eV is about 0.5 eV/ ns (see Fig. 5). Thus the absolute energy resolution in this case is about 85 meV. Screening or compensating of the magnetic field decreases the deviation of the TOF or the detection position. The trajectory curvature radius in the magnetic field for the constant velocity is inversely proportional to the magnetic field strength. Thus changing the TOF and the detection position should be roughly inversely proportional to the magnetic field, too. As a consequence of this, the relative variation of the TOF and the X-position should be proportional to the relative magnetic field variation. The result of the simulation in Table 1 confirms this conclusion in the main. The relative variations of the TOF and the detection position as it is illustrated by DTOF / TOF(0) and DX /X(0) columns decrease with the uniform magnetic field decreasing. The magnetic field of B 5 u0.04u Gauss produces the relative TOF variation of about 1% or about 1.1 ns in the absolute scale. The magnetic field of B 5 u0.004u Gauss produces the relative TOF variation of about 0.1% or about of 0.11 ns. This is roughly equal to the intrinsic accuracy of the simulation, so the real influence of the magnetic field should be less. The variation of the TOF in the last case is close to the experimental limit of the time measurements. The above simulation allows to make a conclusion that the screening (or compensating) of the magnetic field of the Earth by 50–100 times up to 0.004 Gauss decreases the influence of the magnetic field on the energy resolution of the TED spectrometer in the considered geometry to the appropriate value. The energy resolution depends mainly on the accuracy of the TOF measurement. The variation of the detection position with the magnetic field affects the angular resolution.
4. Discussion The TED spectrometer can analyze the electron energy in some energy range. The spectrometer geometry and the potential setting define the energy
range and the mean energy of electrons. For the fixed geometry the retarding potential setting between spherical electrodes changes the mean analyzed energy Em 5 meUr /s1 2 R 1 /R 2d, where m is a coefficient. One can scan step by step the whole spectrum of photoelectrons, for example, or select a desired part of the spectrum by varying or selecting the retarding potential. In the first example considered above, we chose m 5 0.5 to satisfy the best focusing conditions and to achieve large acceptance solid angle. In the second example we chose m 5 0.943 to optimize the energy resolution. The focusing conditions degrade in the latter case and the spectrometer acceptance angle gets reduced. In both considered cases, the spectrometer can work either in the integral mode or in the positionsensitive mode. The mode depends on the type of the electron detector used, that is, it depends on whether the detector allows only to measure the total electron current or it possesses position sensitivity. In the latter case (the position sensitive mode) the spectrometer produces the highest possible energy resolution. The TOF and the detection position are measured for each electron separately. The electron energy and the emission angle are retrieved from the measured data by solving the inverse problem [5] on the basis of the analytical expression of the electron trajectory. This mode allows us to measure with the high energy resolution and in the large acceptance solid angle. In the integral mode one measures the average TOF of all electrons with the same energy in the total acceptance solid angle and the energy resolution depends on the acceptance solid angle. In this mode one does not need to retrieve each electron trajectory. The relation between the measured TOF and the electron energy is based either on the analytical expression or on the calculated table, which has been obtained by the integration of all electron trajectories inside the specified solid angle. In the second case considered above, the electron spectrometer of high relative energy resolution and with relatively low acceptance angle can work successfully in the integral mode. The two limiting cases discussed in this paper give us an idea of how to optimize an electron spectrometer to different needs based on the spherical retarding field. Further development of this idea might consider retarding fields with geometries different from
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the spherical one. The elliptical retarding field, for example, seems to be very attractive because it should give more freedom in the relative position of source and detector and allows to reduce the linear size of the spectrometer. All above estimations were made in the ideal case of a point source and negligible fringing field effects on the trajectories at the grid limiting the retarding field region. Therefore, one can consider the results as some approximation of the reality. On the other hand, the finite size of the source should produce a symmetric increase of the crossover area due to the central symmetry of all trajectories. And also, the minimum size of the inner sphere cannot be less than 5–6 cm in order to accommodate a sample holder and a detector. Thus, the point source approximation seems to be realistic up to a source size of about 1 mm. The micro-channel plate detector is suitable for the TED spectrometer, because it allows to measure time signal with the precision of about 100 ps. The use of a position sensitive anode is mandatory in the case of a high acceptance angle spectrometer in order to get both angular resolution (initial polar and azimuth angle of emission) and to unambiguously determine the initial kinetic energy of the detected electron. In the case of the low acceptance angle spectrometer and when the angular parameters of emission are fixed the spatial resolution in the detector plane is not necessary because of the small spread in flight time generated by the variation of the flight angle within the limits of the acceptance solid angle. TED and TOF spectrometers alike, need a time reference to define the flight time. In some cases a time-structured incident beam provides such a reference signal. For example, in a photoemission experiment with synchrotron radiation, operating in the single bunch mode, a pulsed photon beam with the pulse width of about 100 ps excites the target. In photoelectron spectroscopy with a pulsed laser excitation the pulse width can be made even shorter. In electron spectroscopy with electron excitation the incident beam can be easily pulsed with the pulse width below 1 ns. In the case of coincidence spectroscopy, if only the difference of the flight time is measured, the signal from a narrow band pass spectrometer can be used as a reference. The great advantage of the described spectrometer
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is the possibility to measure simultaneously energy and angular distributions of emitted electrons (‘in parallel’). It means that whatever is the energy of a detected electron and whatever is its emission angle (both are, of course, within certain limits), the energy and the emission angle can be traced back using the electron arrival time and the impact position. It seems to us that the TED spectrometer could be useful in a wide range of applications. For example, in spin-polarized electron spectroscopy, where the detection efficiency is very important (because of the low intensity of spin-polarized incident current and low efficiency of spin detectors) the TED spectrometer with large acceptance angle could offer a very high detection efficiency. Since the TED spectrometer can operate with a very low electron current it could be useful for studying dielectrics where charging up of the sample is one of the most severe limitations to application of electron spectroscopy.
5. Conclusion We have shown in this paper that TED of the spherical electrostatic field can be used to design a new class of electron spectrometers. The exploitation of the time–energy dispersion together with the spatial dispersion can give advantages and additional freedom in the construction of electron spectrometers with desirable properties. In contrast to the conventional time-of-flight spectrometer (with a field-free flight tube), where the energy resolution decreases for higher electron energies, the TED spectrometer has higher energy resolution for higher electron energy. Indeed, electrons with relatively high energy penetrate deeper in the spherical field, the spherical field, in turn, becomes weaker and TED increases. We considered briefly two limiting cases of the use of a retarding spherical field in the electron spectrometer design. The first one is a spectrometer with a medium energy resolution and large acceptance angle. In the case of a 10-cm inner sphere radius (22 cm outer sphere radius) the spectrometer is expected to have an energy resolution of about 0.5 eV/ ns and an acceptance solid angle of about 2.2 srad for an electron kinetic energy of 75 eV. We believe that modern electronic devices can provide a time resolution of about 100 ps and this sets a limit
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for the absolute energy resolution to about DE550 meV in this energy range. The second example is a spectrometer with a relatively small acceptance angle and high energy resolution. The inner sphere radius is 6 cm and the outer spherical segment has a radius of 120 cm. Theenergy resolution is expected to be about 160 meV/ ns for electron kinetic energy of 1 keV and the acceptance solid angle is 0.1 srad. In both cases the energy range can be easily selected by tuning the retarding potential. We have considered the properties of the TED of the retarding spherical field using an analytical approach and propose the conceptual design of TED analyzers. Construction of the prototypes is in progress. We hope that this paper will stimulate the appearance of new TED spectrometers.
Acknowledgements One of the authors (O.M.A.) thanks the Universita` di Roma Tre and INFM Unita` Roma 3 for their hospitality.
References [1] R. Dorner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, H. Schmidt-Bocking, Phys. Rep. 330 (2000) 95. [2] J.P. Wightman, S. Cvejanovic, T.J. Reddish, J. Phys. B 31 (1998) 1753. [3] R. Wehlitz, J. Viefhaus, K. Wieliczek, B. Langer, S.B. Whitfield, U. Becker, Nucl. Instrum. Methods Phys. Res. Sect. B99 (1995) 257. [4] R. Gotter, A. Ruocco, A. Morgante, D. Cvetko, L. Floreano, F. Tommasini, G. Stefani, Nucl. Instrum. Methods Phys. Res. Sect. A467–468 (2001) 1468; R.R. Blyth, R. Delaunay, M. Zitnik, J. Krempasky, R. Krempaska, J. Slezak, K.C. Prince, R. Richter, M. Vondracek, R. Camilloni, L. Avaldi, M. Coreno, G. Stefani, C. Furlani, M. de Simone, S. Stranges, M.-Y. Adam, J. Electron Spectrosc. Relat. Phenom. 101–103 (1999) 959. [5] O.M. Artamonov, S.N. Samarin, A.O. Smirnov, J. Electron Spectrosc. Relat. Phenom. 120 (2001) 11. [6] R. Prange, Rev. Sci. Instrum. 46 (1975) 278. [7] H. Daimon, Rev. Sci. Instrum. 59 (1988) 545. [8] V.V. Zashkvara, L.S. Yurchak, A.F. Bylinkin, Sov. Phys.Tech. Phys. (USA) 33 (1988) 1218.
The use of the time–energy dispersion in an electron energy analyzer a, b c c O.M. Artamonov *, S.N. Samarin , G. Paolicelli , G. Stefani a
St. Petersburg State University, Research Institute of Physics, Ulyanorskaya 1, Petrodvorez, 198504 St. Petersburg, Russia b University of Western Australia, Perth, Australia c Universita` di Roma Tre and INFM Unita` Roma 3, Rome, Italy Received 26 April 2002; received in revised form 10 April 2003; accepted 30 May 2003
Abstract The time–energy dispersion (TED) characteristics of a spherical electrostatic mirror are used to design an unusual electron energy analyzer. It is shown that TED of a retarding spherical field in an electron mirror configuration is positive and it increases with the electron kinetic energy. This is due to an increasing penetration of electrons with high kinetic energy in the retarding field and provides the basis for developing a new type of time-of-flight spectrometer (TOF). Two limiting cases for the geometry of the proposed spectrometer are considered and for both of them the energy resolution of the TOF device depends on the combined time resolution of electron detector and readout electronics. In the case of a 10-cm inner sphere radius (22 cm outer sphere radius) for the electrostatic mirror, the spectrometer is expected to have an energy resolution of about 0.5 eV/ ns and an acceptance solid angle of about 2.2 srad for electrons of 75 eV kinetic energy. A position sensitive detector allows retrieving the energy and the emission angle on the basis of the measured time-of-flight and detection point position. In the second case the inner sphere radius is 6 cm and the outer spherical segment has a radius of 120 cm. The energy resolution is expected to be about 160 meV/ ns at the electron kinetic energy of 1 keV and the acceptance solid angle is 0.1 srad. In both cases changing the retarding potential can easily change the energy range within which the TOF analysis is performed. 2003 Elsevier B.V. All rights reserved. Keywords: Electron spectroscopy; Time-of-flight; Coincidence spectroscopy
1. Introduction Over the past 20 years break-up processes studied by time correlated experiments have experienced a rapid expansion because of their ability to highlight fragmentation dynamics in atoms, molecules and *Corresponding author. Tel.: 17-812-356-1350; fax: 17-812428-7240. E-mail address: [email protected] (O.M. Artamonov).
solids. To be efficient, these experiments require low incident flux, large accepted solid angles, high energy resolution and high momentum resolution. In order to develop a single analyzer that satisfies all these conflicting requirements, several different attempts have been made ranging from various successful versions of the COLTRIMS spectrometer [1], to toroidal spectrometers [2] and to arrays of time of flight [3] or dispersing element [4] spectrometers. TOF spectrometers are very attractive because of
0368-2048 / 03 / $ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016 / S0368-2048(03)00129-4
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their mechanical simplicity but often find a limit to their application in the reduced accepted solid angle and in the moderate energy resolution achievable at intermediate and high energies. Furthermore, to avoid pile-up in the electron detector, often the incident flux must be kept to levels so low that the intrinsic high luminosity due to the multichannel capabilities of TOFs are fully exploited. These drawbacks, severe when TOFs are used in conjunction with third generation synchrotron radiation (SR) sources, might become unbearable in view of application with free electron laser (FEL) sources that are characterized by low repetition rate and will produce bursts of photoelectrons with very high peak flux. A recent paper suggested that combining a TOF with a spherical mirror electrostatic reflector will result in improved, rather than degraded, energy resolution at high energy and in unusually large accepted solid angle [5]. In the present paper we suggest that combining such a TOF reflector with a wide band pass energy selector (WBP) will result in a very flexible analyzer capable of operating either with large accepted angle and moderate energy resolution, or with moderate solid angle and high energy resolution. In either case, the WBP allows to detect only electrons that are relevant to the physical process under study by rejecting all the others before they reach the detector. For instance, detection and subsequent time of flight analysis will be performed only for Auger electrons within a pre-set window while secondary electrons and photoelectrons will be rejected. Most modern electron spectroscopies use the spatial dispersion of an electrostatic field (as a function of energy and direction) to select electrons of a certain kinetic energy and to focus electron trajectories. Usually, in a dispersive electrostatic analyzer high resolution is obtained at the expense of detection efficiency and vice versa. The energy dispersion provides a relative shift of electron trajectories of different kinetic energies, for example, along a slit of an aperture. The angular dispersion gives a convergent set of trajectories, as in the case of an axial electrostatic field. Furthermore, electrons with different kinetic energies will be detected at different times, i.e. will suffer time–energy dispersion (TED), that is the basis for time-of-flight (TOF) spectroscopy. The final aim of an electron spectrometer design is to get the desired energy res-
olution with as high as possible detection efficiency. TED offers great advantages in optimizing both properties of an electron energy analyzer. For example, the spatial dispersion of the electrostatic field provides the focusing properties, whereas the time– energy dispersion allows the electron energy selection. The time-reference signal needed to implement any TOF spectroscopy is usually provided by the time structure of a pulsed excitation source, be it electrons or photons. The quantities practically measured by the time-of-flight electron spectrometer are the electron flight time, from the specimen to the detector, and the electron impact position on the detector entrance plane (or impact position). It is important that there is only one electron trajectory, which satisfies the initial emission parameters (the electron energy and emission angle) and the measured parameters (the time-of-flight and detection position). In a potential field there exists an infinitely large number of electron trajectories connecting the emission and detection points and some of these trajectories have the same time-of-flight, so we have to choose the energy range and the emission angle range in such a way that starting and detection points can be connected just by only one trajectory. The existence of a unique trajectory permits to calculate the energy and the emission angle on the basis of the measured time-of-flight and detection point position [5]. Examples considered below satisfy that condition. In a conventional time-of-flight spectrometer one measures the flight time that an electron needs to travel from the target to the detector in a field-free space and the flight time is a function of the electron kinetic energy t(E). The absolute energy resolution DE depends on the time resolution Dt and on the time–energy dispersion of the analyzer dt / dE (TED), that in the case of a field-free space is dt / dE 5 2 CE 23 / 2 L, where C is a constant and L is the flight distance. The dispersion decreases when the electron energy increases, consequently. And so the energy resolution degrades with the increase of the energy, as the uncertainty of the energy determination DE 5 (dt / dE)21 Dt increases with the energy. For example, if L 5 10 cm and the flight time is measured with the precision of Dt(1 ns, then DE (10 eV)50.37 eV for the electron energy E 5 10 eV. Mirror electrostatic analyzers have been used to
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achieve high resolution spectroscopy [6] and to realize display type spectrometers that analyze kinetic energy as well as angular distribution of charged particles [7]. In this paper we propose to realize a TOF spectrometer using a spherical retarding electrostatic field with reasonable focusing properties instead of the more usual field-free space. In this case one can expect to get a better energy resolution because in a retarding field the penetration of electron trajectories into the field and, accordingly, the flight time, are roughly proportional to the electron kinetic energy. One considerable advantage of the hemispheric geometry over other electrostatic mirror geometry, is that it allows analytical solutions in calculating electron trajectories.
2. Theoretical considerations
2.1. Time–energy dispersion of a spherical field We consider here a spherical electrostatic field as a dispersive media for the TOF analyzer. A charged sphere with a radius R set at a potential 2U, establishes a central field in the space outside of the sphere denoted by the heavy circle in Fig. 1 while the inner region is a field-free one. The same central field is formed in a spherical condenser if the potential difference between spheres Ur satisfies the equation U 5 Ur /s1 2 R 1 /R 2d, where R 1 , R 2 are the spheres radii. The electron trajectory originating at point S enters the central field through the inner spherical surface at point 1 (see Fig. 1) and it is determined by the relative electron energy w 5 E /eU (where e is the electron charge and U is the effective potential) and by the angle b between the radius-vector and the electron velocity direction at point 1. Unless differently stated, henceforth length will be expressed in units of R, energy in units of w and time in units of t (the time needed by an electron with w51 to travel a unit distance). The electron trajectory is symmetric about the radius to the point r max , where r max is the maximum distance from the center (or the maximum penetration depth). The time-of-flight in the central field region (from point 1 to point 2) and the maximum penetration depth are described (in relative units) by the equations [5]:
Fig. 1. The figure shows a typical electron trajectory in the basis plane of the spherical field. The solid circle is the spherical grid electrode (see text for details). A spherically symmetric field is established in the space outside the sphere denoted by the heavy circle, while there is a field-free region inside the sphere. The solid line in the fourth quadrant is the detector location. The electron trajectory starts from point S, crosses the spherical electrode at point 1, then it is reflected by the spherical retarding field and crosses again the spherical electrode at point 2. The trajectory terminates at point 3 where it crosses the detector plane. The maximum penetration of the electron trajectory in the retarding field is r max . The whole set of equations describing the electron trajectory is reported in Ref. [5].
]]] 2 œ2w cos ( b ) t 12 5 ]]]] 1 ]]]] 12w (2(1 2 w))3 / 2
S
D
p 2w 2 1 3 ] 1 arcsin ]]]]]]] , ]]]]]]] 2 1 2 4w(1 2 w) sin 2 ( b )
œ
2w sin 2 ( b ) r max 5 ]]]]]]]] . ]]]]]]] 2 1 2œ1 2 4w(1 2 w) sin ( b ) Fig. 2a–c shows the flight time t 12 , the maximum penetration depth r max (from the center of the spherical field) and the time–energy dispersion TED 5 dt 12 / dE as a function of the relative kinetic energy w for the fixed angle b 5258. One can easily convert the relative time t to the absolute time T measured in seconds by the dimensional factor T 5 ] k t t, where k t 5 C1 R /ŒU (C1 (2.38 3 10 28 , if the
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electron in the spherical retarding field (expressed in units of the charged sphere radius) increases monotonically with the electron energy. As a consequence, the time-of-flight also increases monotonically with the energy (see Fig. 2a). Fig. 2c clearly shows the time–energy dispersion that, on the contrary to what is usually found for free space drift tubes, displays always a positive value. In other words, the flight time through the central field increases with increasing of the electron energy. In particular, the TED characteristic has a shallow minimum for relative energies between 0.05,w,0.2. At higher energies TED monotonically increases with the energy, i.e. the energy resolving power improves with energy. Within this shallow minimum, TED values are too low to be of practical use in building an analyzer. At lower relative energies (w,0.05), though TED sharply rises to sizeable values, still the reflector is not efficient for TOF analysis as t 12 is too short to be of practical use. It is only in the high energy regime (w.0.2) that both TED and t 12 exhibit values large enough to allow an efficient TOF energy discrimination. These observations suggest to use the geometry shown in Fig. 1 as the basis for realizing an efficient TOF spectrometer.
2.2. Focusing properties of the spherical field
Fig. 2. The main characteristics of electron motion in the spherical electrostatic field as a function of the relative electron kinetic energy w(r.u.) in relative units: (a) the flight time t 12 spent in the retarding field from point 1 to the point 2 (see Fig. 1), (b) the maximum penetration depth r max inside the retarding field, (c) the time energy dispersion TED 5 dt 12 / dw of the electron motion. The starting angle is b 5258 in all cases (see Fig. 1).
distance is expressed in cm, potential in V, energy in eV and time in ns). Fig. 2b shows that the penetration depth of an
The focusing properties of the spherical field are well known and they are described by the spatial dispersion of the field [8]. For the kinetic energy equal to one half of the charged sphere potential w50.5 the trajectories converge onto a conjugate point on the diameter line containing the point source. This is the case for any distance of the source point from the center up to the sphere radius R. The penetration depth for kinetic energy w50.5 depends on the radial component of the electron velocity and in all cases it is less than 2R. The focusing properties degrade if the electron energy does not satisfy the condition w50.5 and electron trajectories, in this case, form a finite size crossover area. This tendency increases as the point source moves along the radius from the center to the periphery. For any given source point, the focusing properties are best at the polar angles of emission close to u 5908 (see Fig. 1). The crossover area of the bundle of trajectories with average polar angle of emission different from 908 is
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shifted away from the diameter, towards positive Z (see Fig. 1) for u .908 and towards negative Z for u ,908.
2.3. Time–energy dispersion of the full analyzer As shown in Fig. 1, the electron trajectory from the sample to the detector includes two paths in the field-free space (i.e. from the sample to point 1 and from point 2 to the detector). Hence, to evaluate TOF performances of the type of analyzer sketched in Fig. 1, TED must be determined taking into account paths in the field-free region as well as the one within the reflecting retarding field. In the central field the trajectory parameter b (see Fig. 1) is defined by the source point position S on the X-axis and by the polar emission angle u. The detector is placed in the vicinity of the X-axis. The TED of the full device (from the sample to the detector) is the sum of TEDs of separate parts of the path. TED for the whole analyzer is negative at relatively low electron energy because the path in the spherical field is small relative to the path in the field-free space and the latter behaviour will dominate. TED changes the sign and increases very rapidly when the relative energy increases. We define the energy resolution F as the inverse TED F 5 dw / dt. Fig. 3 shows F (in relative
Fig. 3. The energy resolution of a spectrometer F (r.u.) as a function of the relative electron energy w. The function F is the inverse TED for the whole spectrometer (i.e. for the total electron path from the source point S to the detecting point 3 of Fig. 1). Y-scale is truncated at w50.1. The solid vertical line at about 0.24w defines the discontinuity of the function F at the energy where TED drops to zero. The function F tends to zero as w tends to 1, but the penetration depth r max increases infinitely at the same time. The curve reported in the figure has been computed for conditions of Fig. 1 and with angle b 5258.
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units) as a function of the electron energy (in relative units too), note that the Y-scale is truncated at w5 0.1. TED crosses zero at about w50.24 because at this point TED of the spherical field equals TED of the field-free path and, as a consequence, the F curve in Fig. 3 has a discontinuity at this energy. TED of the field-free space inside the sphere defines the resolution in the energy region w → 0, because the penetration depth becomes small. The resolution degrades as the energy increases in this region. Below this critical value of w, and for kinetic energies tending to zero, the analyzer behavior is largely dominated by paths in the field-free region. The penetration depth is always smaller than 1.3R, hence the resolution F degrades as the energy increases. Conversely, above the critical value w5 0.24 the energy resolution improves with increasing the relative energy. It is possible to estimate the energy resolution in the units eV/ s as a function of the energy in eV for the specific example. To do this one needs to define the effective potential of the single charged sphere, which depends on the ratio of the sphere radii and on the retarding potential difference, and then express the electron kinetic energy in eV. The Y-scale will be multiplied by the factor U /k t and the X-scale by the factor U. Let’s estimate, as an example, the resolution of a spectrometer similar to the one shown in Fig. 1 with R510 cm and U 5100 V (this is equivalent to R 2 /R 1 5 2 and the retarding potential difference Ur 5 50 V). We fix the absolute value of the resolution at the level of 0.05 and estimate its equivalent in eV/ s and the energy values at which this resolution is achieved. Fig. 3 shows that uF (r.u.)u 5 0.05 occurs at two relative energies of about w 1 5 0.53 and w 2 5 0.105. The value uF (r.u.)u 5 0.05 is equal to 0.21 eV/ ns and energies w 1 and w 2 are equivalent to 53 eV and 10.5 eV. Thus, in the considered example the energy resolution uFu 5 0.21 eV/ ns is realized at the energies E1 5 10.5 eV and E2 5 53 eV.
2.4. Some important limitations We have considered trajectories in the plane that contains the center of the spherical field and the point source of electrons. The line connecting the center of the spherical field and the source point is
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the axis of symmetry. One can then achieve full description of the bundle of trajectories originated from the point source S by full rotation of the trajectory plane around the symmetry axis. Anyhow, in practical use, it is convenient to limit the spherical field by a spherical segment with the same central point. So far, all calculations have been done in a perfect central field. In reality this ideal condition will be approached by terminating the spherical sector with a set of guard rings with proper voltages applied. Furthermore, the inner electrode will be a grid rather than a solid sphere and a micro-channel plate will be located at the detector place. By defining the basis plane that includes the symmetry axis and the vertex of the spherical segment one can define the electron trajectory by two angles—the polar angle u in the trajectory plane and the tilt angle c of the trajectory plane relative to the basis plane. The range of the u - and c -angles is limited by the size of the spherical segment as well as by the size of the detector. The detector position should be in the vicinity of the symmetry axis. One has to avoid the intersection of the symmetry axis with the detector plane because at the intersection point it is not possible to distinguish trajectories with different c -angle. This is considered in detail in Ref. [5], as well as the restrictions imposed on the energy and angle ranges by the necessity to have a unique solution of the problem of retrieving the initial energy and angle of emission from the measured flight time and detection position.
3.1. Spectrometer design with the radii ratio close to 2 Fig. 4a and b shows a set of electron trajectories in the spherical field established within two concentric hemispherical surfaces with the ratio of radii R 2 /R 1 5 2.2. Simulations performed at electron energies comprised within 610% about the mean electron energy w m 50.5, show that trajectories originating from a point source S converge at the conjugate point on the symmetry axis irrespective of the
3. Design of the TED spectrometer We shall present below two examples of how it is possible to make use of TED of a spherical electrostatic mirror for TOF analysis of electron energy within a given energy band. The first example is the scheme of a spectrometer with a moderate energy resolution and with a large acceptance solid angle. In this example we shall make use of the excellent focusing properties of the spherical field at w50.5. In the second example we consider a spectrometer with a higher energy resolution that is achieved at the expense of the acceptance solid angle.
Fig. 4. Electron trajectories in the basis plane: 458,u ,1408, (a) w 5 0.5, (b) w 5 0.55. The ratio of the radii of the spherical electrodes is R 2 /R 1 5 2.2 and R 1 5 1. The electron source and the detector are placed in the field-free region inside the inner sphere. The electron source S is placed at x5 20.5. The point of the crossover is exactly at the symmetrical point x50.5 in the case (a), and the spot of the crossover is close to the point x50.5 in the case (b).
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outgoing polar angles u. Fig. 4a shows 30 electron trajectories at mean energy w m 50.5 in the polar angle range from u 5458 up to u 51408. We assume the c -angle to vary in approximately the same range, 2508, c ,508 (as it is in the case of a conical spherical segment). This corresponds to an acceptance solid angle of about 2.2 srad. The point source is placed at 20.5R 1 and the crossover point is at 0.5R 1 for this energy. It is possible to increase the polar angle range, but one needs for this to increase the detector size and / or to move the detector position closer to the X-axis. Fig. 4b shows the electron trajectories for the same angular range but with the electron energy equal to the upper limit of the chosen electron energy range, i.e. w50.55. Electrons penetrate more deeply in the spherical field up to the outer sphere radius. The focusing is still fairly good, but the crossover area has now a finite size and it is shifted down below the X-axis. The trajectory with the lower and higher polar angles (u 5458 and u 51408) crosses the X-axis at the points 0.54R 1 , 0.46R 1 , respectively. The crossover area for trajectories with w50.45 is shifted above the X-axis and the trajectory with the lower and higher polar angles (u 5458 and u 51408) crosses the X-axis at the points 0.46R 1 , 0.54R 1 , respectively. This effect is a manifestation of the spatial energy dispersion of this device. In particular, the crossover position will move with the energy along the X-axis. As a consequence, by placing a slit along the X-axis close to 0.5R 1 the intrinsic bandpass filter can be properly exploited. Whenever the physics of the problem under investigation requires to analyze only a given interval of the electron kinetic energy, the combined band-pass and TED characteristics will allow to perform the experiment with higher incoming flux than by a TOF spectrometer alone and avoid overloading of the detector by unwanted events. Trajectories are not dependent on the angle c, hence all the conclusions drawn from simulations done in the plane c 50 are valid for any plane with c ± 0. Nevertheless, care should be taken in determining the intercept of a trajectory and the detector plane, as it changes with the changing of c [5]. We estimate now the main parameters of the
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spectrometer. The energy resolution for the chosen ratio of radii depends on the mean energy, or (which is the same) on the effective potential, and on the absolute size of the sphere. We chose, as an example, the inner sphere radius R 1 5 10 cm. Fig. 5 shows the energy resolution F in units of eV/ ns as a function of the electron energy E in eV for three energy intervals Em 610% (in all cases Em 5 0.5eU ), calculated for conditions roughly corresponding to the design shown in Fig. 4. Three curves show the absolute energy resolution for three different values of mean energy: Em 5 50 eV, Em 5 75 eV and Em 5 100 eV and a chosen interval of 610%. To increase the mean energy for a fixed value of the radii ratio, the potential U must be also increased and consequently the resolution becomes worse with increasing of the mean electron energy as the penetration depth is fixed by the condition Em 5 0.5eU. Inside each individual interval of the energy about the chosen mean value, the energy resolution behaves according to Fig. 3, i.e. it improves with increasing energy. The estimation of the energy resolution reported in Fig. 4 was performed for an individual trajectory and for a specified u angle. A realistic evaluation must take into account that the electrons transmitted through the analyzer may be emitted at various angles u inside an interval corresponding to the full
Fig. 5. The energy resolution F (eV/ ns) of the spectrometer (designed as shown in Fig. 4 with R 1 5 10 cm) in absolute units as a function of the electron kinetic energy E (eV) for three different mean energies Em . The values of the mean energy are shown close to the curves. The energy interval of the analyzed electrons is Em 610% and the penetration depth is limited by the condition eU 5 2Em .
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acceptance solid angle. This spread in accepted polar angles results in a time-of-flight spread but it does not limit the energy resolution provided the impact position is measured by a suitable position-sensitive detector (i.e. if the analyzer is operated in the socalled differential mode). It has been shown already [5] that by knowing the time of flight and the impact point on the detector, the initial kinetic energy and the polar angle can be unequivocally retrieved. Should the detector be insensitive to the impact position (so-called integral mode) the aforementioned time spread limits the energy resolution. The mean flight time for E545 eV calculated over 30 trajectories is 173 ns and the first deviation is about 3.3 ns, that is less than 2% of the mean value. This means, in the integral mode the absolute energy resolution DE is limited by the flight time spread to about DE51.2 eV. In the differential mode it is possible to calculate analytically the electron trajectory, hence to introduce corrections for different flight times of electrons with the same kinetic energy. In this latter case the absolute energy resolution will be limited by the time resolution of the detector and readout electronics, which can be estimated by a realistic value of about 0.1 ns.
3.2. Design of a spectrometer characterized by a large radii ratio A spectrometer with a large radii ratio provides a good energy resolution and a reasonable, though reduced with respect to the previous case, acceptance solid angle. Fig. 6a and b shows a set of electron trajectories in the spherical field with the ratio of radii R 2 /R 1 5 20. The energy range chosen for the simulation is 60.5% about the mean electron energy that in this case is chosen equal to w50.943. Fig. 6a shows 10 electron trajectories with energy w50.938 and Fig. 6b shows trajectories of electrons with energy w50.948. The polar angle ranges from 1108 to 1208. We assume that c is in the range of 658, the acceptance solid angle will be about 0.1 srad. The point source is always located at 2 0.5R 1 . Fig. 6c shows a blow-up of the trajectory plane in the vicinity of the inner sphere. The crossover of trajectories takes place below the x-axis, but it is still narrow due to the relatively small solid angle accepted.
Fig. 6. Electron trajectories in the basis plane, 1108,u ,1208: (a) w ¯ 0.938, (b) w ¯ 0.948, (c) a part of (b). The ratio of the radii of the spherical electrodes is R 2 /R 1 5 20 and R 1 5 1 (r.u.). The spherical retarding field exists between the spherical electrodes. The electron source and the detector are placed in the field-free region. The electron source S is placed at x5 20.5.
In accordance with Fig. 2c, TED increases rapidly in the relative energy region above about w50.8 due to the deep penetration of electron trajectories in the spherical field. Choosing R 1 5 6 cm and R 2 /R 1 5 20 the resolution is about 160 meV/ ns at an electron energy of 1 keV. One can improve resolution at the same energy or can retain this resolution and extend the mean energy to the higher energy region by increasing the radii ratio. The mean flight time for this set of trajectories is about 510 ns and the first deviation of this value is about 90 ps. The deviation of the flight time is less than the expected time resolution of the detector and readout electronics. This means there is no need to measure the detection position and the micro-channel plate detector can be used in the integral mode without loss of resolution.
3.3. The influence of the Earth’ s magnetic field The Earth’s magnetic field can disturb the electron trajectory thus affecting TOF and detection position. The magnetic field effect is maximum when the magnetic field vector is normal to the trajectory plane. In this case, the radius of the circular trajectory of the 1 eV energy electron in the uniform magnetic field of the Earth (the magnetic field is about 0.4 Gauss) is about R58.44 cm. Hence
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screening of any magnetic field is mandatory for the analyser to work properly. We estimate the influence of a uniform magnetic field by numerical simulation. The electric and magnetic fields were calculated by ‘Simion 7.0’ in the frame of a regular grid with step size of 0.25 mm. We consider a TED spectrometer geometry similar to that shown in Fig. 4a with a uniform magnetic field applied normal to the trajectory plane. Calculations were performed for the grid electrode radius of 100 mm, an outer electrode radius of 220 mm, UR 5 290 V, Ekin 575 eV and a point source located at Xs 5 250 mm. The obtained values of TOF and X (detection position) for seven magnetic field strength values are presented in Table 1 together with their first derivations. The numerical simulation of the electric field of the TED spectrometer with spherical electrodes leads to a finite uncertainty in the TOF and in the detection position relative to the analytical calculations. To estimate this uncertainty we use the helpful property of the TED to have an ideal focusing point on the symmetry axis for the electron of a specified kinetic energy (see Fig. 4a). The spread in detection position observed in the absence of a magnetic field, is ascribed to numerical uncertainties of the simulation. During the numerical calculations the TOF was estimated with an error of about 60.1 ns (except the case of B5 20.4 Gauss). One can assume that this error defines the simulation accuracy. The situation
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with the accuracy of the detection position simulation is similar to that in the TOF case. The uniform magnetic field influences the electron trajectory mostly in the region near the reversal point r max in Fig. 1 where the velocity of the electron is minimum. The magnetic field is parallel or antiparallel to the normal to the trajectory plane and the magnetic field shifts the reversal point clockwise or anti-clockwise. The trajectory in the magnetic field becomes non-symmetrical relative to the line from the center to the r max . As a result, the focus point shifts up or down relative to the X-axis. We calculate the TOF and the X-position at the moment when the trajectory crosses the X-axis. We chose a set of trajectories which start from the point source in the angular range 808,u ,1008 (see Fig. 1) with the kinetic energy 75 eV. Table 1 shows the mean value and the deviation of the TOF and the X-position for this set of trajectories calculated for various magnetic fields. The columns named DTOF / TOF(0) and DX /X(0) give the relative change of the TOF and the X-position relative to that in the first line, that is, relative to the TOF and the X-position in the absence of the magnetic field. The uniform magnetic field of B 5 60.4 Gauss changes the mean TOF by about 8–10% and it changes the X-position by about 1– 11% relative to the values in the absence of the magnetic field. These relative changes due to the magnetic field influence exceed the relative error of the simulation in the absence of the magnetic field
Table 1 The magnetic field influence on the TED spectrometer resolution a Magnetic field (Gauss)
TOF (ns)
DTOF / TOF(0) (%)
Detection position X (mm)
DX /X(0) (%)
B50 B 5 2 0.4 B 5 0.4 B 5 2 0.04 B 5 0.04 B 5 2 0.004 B 5 0.004
110.7660.10 120.1260.25 99.1760.10 111.8460.10 109.6660.10 110.8760.10 110.6560.10
–
50.260.5 55.762.9 50.961.0 50.360.7 50.160.4 50.260.6 50.260.5
– 11.1 1.4 0.3 20.2 0.04 20.02
a
8.4 210.5 1.0 21.0 0.1 20.1
The spectrometer design is as in Fig. 4a, the grid electrode radius is 100 mm, the outer electrode radius is 220 mm, UR 5 290 V, Ekin 575 eV and the point source position is Xs 5 250 mm. The simulation has been made by ‘Simion 7.0’, electric and magnetic fields were calculated in the points separated by the step of 0.25 mm. The set of trajectories were selected in the polar angle range of 808,u ,1008 (see Fig. 1). The uniform magnetic field is directed normally to the trajectory plane. In the columns DTOF / TOF(0) and DX /X(0) the deviations (in %) of the TOF (or X) calculated for the corresponding B from the TOF at B 5 0 are given.
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and they reflect the magnetic field influence. The energy resolution is defined by the TOF accuracy measurements and for the B 5 u0.4u Gauss its average value is about 0.17 ns according to Table 1. The relative energy resolution for the kinetic energy of 75 eV is about 0.5 eV/ ns (see Fig. 5). Thus the absolute energy resolution in this case is about 85 meV. Screening or compensating of the magnetic field decreases the deviation of the TOF or the detection position. The trajectory curvature radius in the magnetic field for the constant velocity is inversely proportional to the magnetic field strength. Thus changing the TOF and the detection position should be roughly inversely proportional to the magnetic field, too. As a consequence of this, the relative variation of the TOF and the X-position should be proportional to the relative magnetic field variation. The result of the simulation in Table 1 confirms this conclusion in the main. The relative variations of the TOF and the detection position as it is illustrated by DTOF / TOF(0) and DX /X(0) columns decrease with the uniform magnetic field decreasing. The magnetic field of B 5 u0.04u Gauss produces the relative TOF variation of about 1% or about 1.1 ns in the absolute scale. The magnetic field of B 5 u0.004u Gauss produces the relative TOF variation of about 0.1% or about of 0.11 ns. This is roughly equal to the intrinsic accuracy of the simulation, so the real influence of the magnetic field should be less. The variation of the TOF in the last case is close to the experimental limit of the time measurements. The above simulation allows to make a conclusion that the screening (or compensating) of the magnetic field of the Earth by 50–100 times up to 0.004 Gauss decreases the influence of the magnetic field on the energy resolution of the TED spectrometer in the considered geometry to the appropriate value. The energy resolution depends mainly on the accuracy of the TOF measurement. The variation of the detection position with the magnetic field affects the angular resolution.
4. Discussion The TED spectrometer can analyze the electron energy in some energy range. The spectrometer geometry and the potential setting define the energy
range and the mean energy of electrons. For the fixed geometry the retarding potential setting between spherical electrodes changes the mean analyzed energy Em 5 meUr /s1 2 R 1 /R 2d, where m is a coefficient. One can scan step by step the whole spectrum of photoelectrons, for example, or select a desired part of the spectrum by varying or selecting the retarding potential. In the first example considered above, we chose m 5 0.5 to satisfy the best focusing conditions and to achieve large acceptance solid angle. In the second example we chose m 5 0.943 to optimize the energy resolution. The focusing conditions degrade in the latter case and the spectrometer acceptance angle gets reduced. In both considered cases, the spectrometer can work either in the integral mode or in the positionsensitive mode. The mode depends on the type of the electron detector used, that is, it depends on whether the detector allows only to measure the total electron current or it possesses position sensitivity. In the latter case (the position sensitive mode) the spectrometer produces the highest possible energy resolution. The TOF and the detection position are measured for each electron separately. The electron energy and the emission angle are retrieved from the measured data by solving the inverse problem [5] on the basis of the analytical expression of the electron trajectory. This mode allows us to measure with the high energy resolution and in the large acceptance solid angle. In the integral mode one measures the average TOF of all electrons with the same energy in the total acceptance solid angle and the energy resolution depends on the acceptance solid angle. In this mode one does not need to retrieve each electron trajectory. The relation between the measured TOF and the electron energy is based either on the analytical expression or on the calculated table, which has been obtained by the integration of all electron trajectories inside the specified solid angle. In the second case considered above, the electron spectrometer of high relative energy resolution and with relatively low acceptance angle can work successfully in the integral mode. The two limiting cases discussed in this paper give us an idea of how to optimize an electron spectrometer to different needs based on the spherical retarding field. Further development of this idea might consider retarding fields with geometries different from
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the spherical one. The elliptical retarding field, for example, seems to be very attractive because it should give more freedom in the relative position of source and detector and allows to reduce the linear size of the spectrometer. All above estimations were made in the ideal case of a point source and negligible fringing field effects on the trajectories at the grid limiting the retarding field region. Therefore, one can consider the results as some approximation of the reality. On the other hand, the finite size of the source should produce a symmetric increase of the crossover area due to the central symmetry of all trajectories. And also, the minimum size of the inner sphere cannot be less than 5–6 cm in order to accommodate a sample holder and a detector. Thus, the point source approximation seems to be realistic up to a source size of about 1 mm. The micro-channel plate detector is suitable for the TED spectrometer, because it allows to measure time signal with the precision of about 100 ps. The use of a position sensitive anode is mandatory in the case of a high acceptance angle spectrometer in order to get both angular resolution (initial polar and azimuth angle of emission) and to unambiguously determine the initial kinetic energy of the detected electron. In the case of the low acceptance angle spectrometer and when the angular parameters of emission are fixed the spatial resolution in the detector plane is not necessary because of the small spread in flight time generated by the variation of the flight angle within the limits of the acceptance solid angle. TED and TOF spectrometers alike, need a time reference to define the flight time. In some cases a time-structured incident beam provides such a reference signal. For example, in a photoemission experiment with synchrotron radiation, operating in the single bunch mode, a pulsed photon beam with the pulse width of about 100 ps excites the target. In photoelectron spectroscopy with a pulsed laser excitation the pulse width can be made even shorter. In electron spectroscopy with electron excitation the incident beam can be easily pulsed with the pulse width below 1 ns. In the case of coincidence spectroscopy, if only the difference of the flight time is measured, the signal from a narrow band pass spectrometer can be used as a reference. The great advantage of the described spectrometer
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is the possibility to measure simultaneously energy and angular distributions of emitted electrons (‘in parallel’). It means that whatever is the energy of a detected electron and whatever is its emission angle (both are, of course, within certain limits), the energy and the emission angle can be traced back using the electron arrival time and the impact position. It seems to us that the TED spectrometer could be useful in a wide range of applications. For example, in spin-polarized electron spectroscopy, where the detection efficiency is very important (because of the low intensity of spin-polarized incident current and low efficiency of spin detectors) the TED spectrometer with large acceptance angle could offer a very high detection efficiency. Since the TED spectrometer can operate with a very low electron current it could be useful for studying dielectrics where charging up of the sample is one of the most severe limitations to application of electron spectroscopy.
5. Conclusion We have shown in this paper that TED of the spherical electrostatic field can be used to design a new class of electron spectrometers. The exploitation of the time–energy dispersion together with the spatial dispersion can give advantages and additional freedom in the construction of electron spectrometers with desirable properties. In contrast to the conventional time-of-flight spectrometer (with a field-free flight tube), where the energy resolution decreases for higher electron energies, the TED spectrometer has higher energy resolution for higher electron energy. Indeed, electrons with relatively high energy penetrate deeper in the spherical field, the spherical field, in turn, becomes weaker and TED increases. We considered briefly two limiting cases of the use of a retarding spherical field in the electron spectrometer design. The first one is a spectrometer with a medium energy resolution and large acceptance angle. In the case of a 10-cm inner sphere radius (22 cm outer sphere radius) the spectrometer is expected to have an energy resolution of about 0.5 eV/ ns and an acceptance solid angle of about 2.2 srad for an electron kinetic energy of 75 eV. We believe that modern electronic devices can provide a time resolution of about 100 ps and this sets a limit
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for the absolute energy resolution to about DE550 meV in this energy range. The second example is a spectrometer with a relatively small acceptance angle and high energy resolution. The inner sphere radius is 6 cm and the outer spherical segment has a radius of 120 cm. Theenergy resolution is expected to be about 160 meV/ ns for electron kinetic energy of 1 keV and the acceptance solid angle is 0.1 srad. In both cases the energy range can be easily selected by tuning the retarding potential. We have considered the properties of the TED of the retarding spherical field using an analytical approach and propose the conceptual design of TED analyzers. Construction of the prototypes is in progress. We hope that this paper will stimulate the appearance of new TED spectrometers.
Acknowledgements One of the authors (O.M.A.) thanks the Universita` di Roma Tre and INFM Unita` Roma 3 for their hospitality.
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