# Position, Energy, and Transit Time Distributions in  a Hemispherical Deflector Analyzer with Position  Sensitive Detector

Authors: Omer Sise; Theo J. M. Zouros
Journal: Journal of Spectroscopy
 (2015)

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## Abstract

Practical analytic equations, for the ideal field, and numerical results from SIMION simulations, for the fringing field, are presented for the exit radiusrπ and transit time tπ of electrons in a hemispherical deflector analyzer (HDA) over a wide range of analyzer parameters. Results are presented for a typically dimensioned HDA with mean radius R-=101.6 mm and interradial separation ΔR=R2-R1=58.4 mm able to accommodate a 40 mm diameter position sensitive detector (PSD). Results for three different entry positions R0 are compared: R0=R- (the conventional central entry) and two displaced (paracentric) entries: R0=82.55 mm and R0=116 mm. Exit spreads Δrπ, Δtπ and base energy resolution ΔEB are computed for HDA pass energies E0=10, 100, 500, and 1000 eV, entry aperture sizes Δr0≤1.5 mm, entry angular spreads |αmax|≤5°, and an electron beam with relative energy spread δE/E0≤0.4%. Overall, under realistic conditions, both paracentric entries demonstrate near ideal field behavior and clear superiority over the conventional entry at R0=R-. The R0=82.55 mm entry has better absolute energy and time spread resolutions, while the R0=116 mm has better relative energy resolutions, both offering attractive alternatives for time-of-flight and coincidence applications where both energy and timing resolutions are important.


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## 1. Introduction

The hemispherical deflector analyzer (HDA) has become increasingly popular in electron spectroscopy (see [1–6] and references therein), due to several advantages including (a) superior energy resolution approaching the sub-meV level [6], (b) the use of flexible transfer lens systems [3, 6] that can be operated in different modes, optimizing selectively transmission, spatial resolution, or angular resolution, and (c) high efficiency by using a fast, high-resolution two-dimensional position sensitive detector (PSD). The simultaneous recording of hundreds of energy channels along the dispersive axis enables the acquisition of high-resolution electron spectra across a large (typically 10–15% of the tuning energy) kinetic energy window without the need to voltage sweep the analyzer, thus cutting down tremendously on overall data acquisition times (see [6, 7] and references therein). In addition, its high time-resolution capabilities enable measurements at the time-of-flight (TOF) spread limit of the HDA [8].For conventional electron spectroscopy, theflight time of electrons in an analyzer of this type is unimportant. However, if an HDA is used to record phenomena varying rapidly in time as in synchrotron and free-electron laser (FEL) investigations [8], or if it is coupled to another detector in an electron-electron coincidence setup [9, 10], the travel times of the electrons through the analyzer become an important point of consideration. The use of an HDA in this type of electron-electron coincidence experiments is possible under certain restrictions: Due to the time-of-flight spread in the HDA, a compromise between energy resolution and time resolution must be made, which renders the higher energy resolutions at low pass energies unusable. For coincidence experiments, pass energies of ~100 eV are typical, having energy resolutions of ~100 meV and time-of-flight spreads of ~4 ns [10].Recent interest in the use of the HDA has been generated primarily from fields such as angle-resolved photoemission spectroscopy (ARPES) and time-resolved-X-ray photoelectron spectroscopy (tr-XPS). In addition, technological advances in photon sources which can deliver bright, coherent, polarized, monochromatic, or ultrashort stroboscopic light including intense femtosecond lasers, 4th generation synchrotrons [11], and ultramodern FELs [12] such as FLASH [13], LCLS [14–16], and the soon coming European XFEL [17] have been particularly impressive over the last decade [13, 18]. Furthermore, the application of pump-probe techniques [19] in combination with these sources allows for novel and unique time-dependent investigations practically in the femto- [20–22] and more recently even attosecond [23] time domain. Time-domain X-ray spectroscopy techniques are drawing attention from a variety of scientific fields due to their potential to reveal details of fast dynamic processes in matter. The prospects to probe changes in local electronic structures and chemical transformations in real time are a major force in the development of laboratory-based [20–23] and accelerator-based [11, 12, 14–16] ultrafast X-ray techniques that complement both energy-domain X-ray spectroscopy efforts at synchrotrons as well as time-domain spectroscopy techniques using ultraviolet, visible, and infrared lasers.Motivation for the computation of flight times in HDAs has been provided by various publications as, for example, Imhof et al. [24], Caprari [25], Kugeler et al. [26], and Shavorskiy et al. [8]. In these studies the transit time is calculated using an approximate model, which ignores fringing fields at the boundaries. Suggestions to improve the time resolution in coincidence experiments have been made by Volkel and Sandner [10]. An extensive study of HDAs employing elliptical trajectories applicable also to paracentric HDAs including basic timing properties has been presented by Zouros and Benis [1].Biased paracentric HDAs represent a novel class of HDAs, which use the lensing action of the strong fringing fields at the HDA entry, to restore the 1st-order focus characteristics of the ideal HDAs in a controlled way. The use of this arrangement was first motivated by the work of [27, 28], who measured the energy distribution of electrons using an HDA with a mean radius R-=101.6mm, but with an entry displaced from the mean radius (i.e., paracentric) at R0=82.55mm. The advantage of using this type of unconventional paracentric configuration is that the effects of angular aberrations on the energy dispersion are minimized. This improvement in energy resolution (and transmission) is attained without the use ofany additional fringing field correction electrode schemes and therefore is of particular interest to analyzers using position sensitive detectors as most modern HDAs do nowadays. Measurements using such a paracentric HDA in the Auger electron spectroscopy of projectile ions in collisions with gas targets at the 7 MV tandem accelerator of Kansas State University (KSU) have been reported since 1998 [28]. Recently, the first differential cross sections measurements for electron scattering have also been performed using a biased paracentric system [29]. The biased paracentric HDA from KSU is presently in operation in the APAPES project [30, 31] for zero-degree Auger projectile electron spectroscopy at the new atomic physics experimental station located at the 5 MV tandem accelerator at the NCSR Demokritos in Athens.Methods of optimizing the intrinsic fringing fields to restore 1st-order focusing and improve energy resolution are given in a number of articles [32–35]. However, to the best of our knowledge, no systematic studies on the transit time spread of biased paracentric HDAs exist to date. In this work, we calculate the transit time and energy distributions of electrons within the biased paracentric HDAs for two paracentric entry positions R0=82.55mm and R0=116mm and compare them to those for entry R0=R-=101.6mm (the conventional HDA mean radius entry) using the computer-based finite-difference trajectory simulation software SIMION 8.1 [36]. The particular values of R0 in the two paracentric entry positions were discovered by previous systematic numerical search also using SIMION [34]. In addition, other such paracentric positions for a variety of HDAs with different mean radii R- and interradial separations ΔR were also presented in a previous systematic investigation [33].Here, we present simulations of spreads in both exit position and time as a function of entry aperture sizeΔr0 (typically controlled by the magnification of the injection lens), the HDA launching angle spread αmax, and the spread in energy δE at the fixed pass energy of E0=1 keV. The transit time and energy spreads are also reported for pass energies E0=10,100,500eV for particular practical initial conditions. In Section 2, the theoretical background is summarized based on the Kepler orbit equation of an ideal field HDA [1] and new results both in the time spreads and also in the radial spreads are presented in closed form analytic results (where possible). These results are then verified using SIMION simulations both for the ideal field and also for the fringing field HDA. In Section 3, the SIMION simulation procedure is described and in Section 4 our results are presented for both fringing and ideal field HDAs and discussed. Comparisons of both paracentric and conventional entry results are made and a figure of merit is obtained for the best combination of energy and transit time distributions.


## 2. Theoretical Background

The analytical solution of the equation of motion in theideal1/r field of an HDA is well documented in the extensive literature (see [1, 4, 5, 37–40] and references therein). In particular, [1] developed in detail the theoretical framework for analyzing the elliptical (Kepler) trajectories of charged particles in the ideal HDA and is extensively referenced. In this section, we only give a summary of the necessary theoretical results and present the final working equations for the energy resolution and the time-of-flight of electrons in anideal HDA. These equations are then used as a benchmark for the detailed intercomparison of the time-of-flight and energy resolution properties of the ideal HDA to those of the “real” (fringing field) HDA whose simulation in SIMION is presented in Sections 3 and 4.As shown in Figure1, the generic HDA consists of two concentric hemispherical electrodes of radii R1 and R2 (R1<R2). Typically, electrons enter the HDA with an energy E, an angular spread ±αmax (with respect to the optical axis), and a spatial spread Δr0 around the entry radius r0. After a deflection through 180° in the HDA field electrons exit at radius rπ after a transit time tπ and are recorded by a detector. The angular (±αmax) and spatial (Δr0) entry spreads and energy spreads δE (for a nonmonoenergetic source) contribute to the overall energy and transit time spreads ΔE and Δt, recorded at the HDA exit. In addition, most modern HDAs are typically equipped with a cylindrical electrostatic input lens whose symmetry axis is placed at R0 and a 1D or 2D position sensitive detector (PSD) at the exit. The lens system is used to collimate and focus the source emission onto the entry plane of the HDA and will not be considered in this investigation. The PSD offers multichannel detection capability decreasing substantially the data collection time since in this case the analyzer energy need not be scanned to record a spectrum. 2D capabilities may offer useful additional diagnostics or important angular information.Figure 1
HDA schematic consisting of two concentric hemispheres with radiiR1 and R2, an electrostatic lens at input, and a position sensitive detector at the exit. The HDA voltages are set to pass the reference trajectory (reddashed line) having nominal pass energy E0 and launching angle α=0, with specified entry radius R0 and exit radius Rπ. Two additional trajectories (red lines) of the same energy E0 are shown to originate from a point source, but with α=±αmax. Here, the entry is seen to be paracentric with R0 displaced from R-, the mean radius. The dash-dot line shows the conventional optical axis at R- for central entry R0=R-. A general trajectory is also shown (in blue) with pass energy E seen to start at entry radius r0 and launching angle α, exiting the HDA at radius rπ. The PSD records the exit positions of the trajectories.


### 2.1. Kepler Orbits and General Radial Orbit Equation

For motion in the1/r potential of anideal HDA given by(1)Vr=-kr+c,thenonrelativistic trajectory of a particle with charge q (q=-e for electrons), kinetic energy E, and angular momentum L is given by (see equation (98) in [1])(2)rω=p1+ϵcos⁡ω-ωϵ.The particle enters at radius r0=rω=0, with launching angle α within the entry aperture of diameter Δr0 centered at R0. It then follows in general anelliptical trajectory r(ω) given as a function of the orbit angle ω and exits at rπ=r(ω=180°) after going through an orbit of π(Δω=180°). Thecentral trajectory (red dashed line in Figure 1) used asreference is defined as the trajectory of a particle which enters with energy E0 at r0=R0 with α=0° and exits at rπ=Rπ. At the exit either a slit or an aperture is typically placed with center at Rπ or alternatively a PSD is used. The conventional HDA arrangement typically utilizes R0=Rπ=R- in which case the central trajectory is acircle. By convention [1] the trajectory launched with α>0 is always anouter trajectory, while the one with α<0 is always aninner trajectory.The parametersp and ϵ in (2) are thelatus rectum andeccentricity of the resulting elliptical trajectory given by [1](3)p=L2mqk,ϵ=1+2EL2mq2k2with the kinetic energy E and (vector) angular momentum L inside the HDA given by(4)E=12mv2,(5)L=r×mv,where v is the velocity of the particle at radius r.The potential constantsk and c given in (1) are fixed by the voltages V1 and V2 supplied to the hemispherical electrodes and do not depend on the trajectory parameters of thearbitrary particle. These voltages are determined by thecentral trajectory (already defined above) and are discussed later in this section (see (10)). The trajectory described by (2) is in general part of an ellipse with one focus at the center of force r=0 with the constant ωϵ set by the initial conditions. The shape of the elliptical orbit is well defined by p and ϵ which are constants of the motion, since they are determined from the particle’s total energy Etot=E-qc (see equation (74) in [1]) and vector angular momentum L, both conserved quantities for motion in anypurely radial potential V(r)=V(r). We note that, in the fringing field HDA, even though the potential isnot radial [V(r)=V(r,θ); i.e., there is an additional dependence on the polar angle θ] and therefore the magnitude of the angular momentum L=|L| isnot conserved, the direction of the angular momentum L/Lis conserved and therefore motion remains planar.Assuming that the particle enters the HDA at timet=ω=0 with entry radius r0=r(ω=0) and launching angle α and then exits at orbit angle ω=π with exit radius rπ=r(ω=π) at t=tπ, thelatus rectum p and the eccentricity ϵ of the ellipse as a function of α and r0 are given by (see equation (40) of [1])(6a)p=aρ02-ρ0cos2⁡α,(6b)ϵ=sin2⁡α+1-ρ02cos2⁡αwith(6c)ρ0≡r0a,(6d)a≡aE=qk2qc-E,where a is the semimajor axis of the ellipse given in terms of constants q, k, and c and the particle kinetic energy E. Under the specified initial conditions, the elliptical trajectory can be further specified as(7)rω=pr0r0+p-r0cos⁡ω-psin⁡ωtan⁡αshowing it to have both odd and even (through p given in (6a), (6b), (6c), and (6d)) terms in the launching angle α.The values of the potential constantsk and c in (1) are specified so that the central trajectory used as reference enters the HDA at ω=0 with α=0 and r0=R0 and kinetic energy E0 and exits at ω=π with rπ=Rπ. Applying these conditions to (7) we obtain (see equations (86) and (87) of [1])(8a)qk=E0R0+RπγR0Rπ,(8b)qc=E01+γR0Rπ.It is seen from (6a), (6b), (6c), and (6d) and (8a)-(8b) that thereference trajectory has a semimajor axis a0 and eccentricity ϵ0 given by(8c)a0≡aE=E0=12Rπ+R0,(8d)ϵ0≡ϵα=0,r0=R0,E=E0=1-R0a0=Rπ-R0Rπ+R0which for the centric conventional HDA entry (Rπ=R0=R-, γ=1) is a circle with a-0≡R- and ϵ-0≡0 as expected.Using (8a)-(8b) and (1) we may next obtain the potential V(r) given by [1](9)qVr=E01-γR0RπR0+Rπr-1from which the HDA voltages can be immediately obtained:(10)qVi≡qVRi=E01-γR0RπR0+RπRi-1i=1,2.For the above voltages the general trajectory rω can then be readily shown to be given by(11)rω=2r0cos2⁡αγ1+ξ-r0/R0γ-ξτ-1cos⁡ω/γ1+ξ-r0/R0γ-ξτ-1+cos⁡2α+ω,where we have also introduced the short hand of the following [1]: ξ≡Rπ/R0 and τ≡E/E0. Also introduced in (8a), (8b), (8c), and (8d)–(11) is theentry bias parameter γ. Equation (11) is seen to satisfy the conditions of the central trajectory; that is, rω=π,α=0,r0=R0,τ=1=Rπindependent of the value of γ. However, γ does affect all other trajectories as seen from (11) and can therefore be used as an additional parameter to optimize the HDA performance. Setting r0=R0 and E=E0 in (11) we obtain the defining relation [1]: (12)γ≡1-qV0E0,where V0≡V(R0) is the value of the potential at the entry R0. We note that when γ=1, V0=0 and vice versa. Since the particle’s effective total energy Etotinside the HDA must always benegative for bound motion we must always have Etot=E-E01+γ/ξ<0 which leads to the restriction on γ: (13a)γ>ξτ-1which is also consistent with the requirement that the semimajor axis a>0 since using (8a)-(8b) a from (6d) can be expressed as(13b)a=aτ=R01+ξ2ξ1-τ+γ.In the past, we have reported on the properties of HDAs and their dependence on the entry parameters ξ and γ. Quite generally, we can identify two different entry classes as follows:(i)
the conventional entry HDA for whichR0=R-=Rπ, that is, ξ=1, and whose reference trajectory is acircle (ϵ=0);(ii)
the paracentric entry HDA for whichξ≠1, whose reference trajectory is anellipse (0<ϵ<1).For both types of entry, γ can be independently used to bias the entry. Thus, for conventional HDAs,zero bias is typically used having V0=0 (γ=1). For paracentric entries, however, bothpositive bias (V0>0 or γ>1) andnegative bias (V0<0 or γ<1) have been investigated [1, 7, 27, 29, 33, 34]. While the value of γ in anideal HDA does not affect its overall 1st-order focusing, in a realfringing field HDA, for which the conventional entry is well known to lose 1st-order focusing, specific combinations of optimal (ξ,γ) values have been found torestore 1st-order focusing [29, 33–35]. In Section 4, we will look at a concrete example of such an HDA having R1=72.4mm and R2=132.8mm (Rπ=R-=101.6mm) and investigate both the energy and the time resolution for paracentric entry R0=82.55mm with γ=1.4715 and R0=116mm with γ=0.5 in comparison to the conventional HDA with R0=R- and γ=1. These particular combinations of (ξ,γ) have been found [29, 33–35] to date, to restore 1st-order focusing. Such an HDA is being used in our laboratory for high-resolution Auger projectile electron spectroscopy [7, 27–29, 31, 35] with excellent energy resolution. In [33], we have also investigated numerically HDAs with different values of R1 and R2 and have also found corresponding optimal combinations of (ξ,γ) which restore 1st-order focusing.


### 2.2. HDA Exit Radiusrπ, Radial Spread Δrπ, and Base Energy Resolution ΔEB

Using (11), the exit radius rπ=r(ω=π) can be computed (see equation (99) and correction of equation (100) inErratum of [1]) and is given by (14)rπ≡rπr0,τ,α,γ=-r0+r01+cos2⁡αr0/R0+Rπ1+ξ/γ1-τ-1which may also be expanded in powers of α giving up to 5th order(15a)rπr0=-1+Δr01-α21+Δr0+α423-Δr053-Δr0+Oα6with(15b)Δ2≡DγR0+Rπγξτand where it is readily seen that(15c)rπr0,τ,α=0,γ=Δ-r0=R0+Rπ1+ξ/γ1-τ-r0.The energy dispersion Dγ is another useful analyzer parameter defined in general as (16)Dγ≡E∂rπ∂Eα=0=Dγτ=1+ξRπγ1+ξ/γ1-τ2τ.For slit HDAs, where τ=1 (i.e., E=E0), Dγ is a characteristic length of the HDA which determines the shift of the image position of a particle trajectory for an infinitesimal change in the particle energy E0 [5]. However, for HDAs with PSD its definition can be extended to any E as above. Equation (14) is also known as thebasic equation of the analyzer [1]. As can be verified in (14), the reference trajectory (r0=R0, τ=1, α=0)always exits at Rπ. We note that a similar equation can also be expressed in terms of α∗, the entry angle into the HDA (see equation (102) in [1] and Figure 1). The angles α and α∗ are related according to Snell’s law [1]. Here, our treatment is given in terms of the angle α inside the HDA, which is the launching angle typically used in simulations. A more complete treatment including the effect of the injection lens should be expressed in terms of the outside (incident) entry angle α∗ as determined by the lens and thus also include the effect of refraction due to the potential change at r0 in crossing the HDA entry plane. Because typically HDA entry apertures are small we in general have V(r0)≈V(R0) and therefore for conventional HDAs which use γ=1, Vr0≈0 and refraction can be neglected. This clearly is not the case for paracentric entry where optimum values of γ are typically different from 1 and therefore Vr0≠0.It is also obvious from (14) that only even powers of α are expected and therefore we should have(17)rπα=rπ-αindicating 1st-order focusing, a basic requirement for any energy analyzer. For monoenergetic beams of particles with energy E we may vary the input beam parameters Δr0 and α to obtain themaximum allowed exit radius variation Δrπ [41]. At the HDA exit plane this can be shown [7, 41] to be equivalent to the following calculation: (18)Δrπ=rπR0-Δr02,E,α=0-rπR0+Δr02,E,αmaxyielding(19)Δrπ=Δr0+R0+Rπ1+ξ/γ1-τ-1+Δr0/2R0R0+Rπ1+ξ-ξ-Δr0/2R0-1+Δr0/2R0ξ/γ1-τcos2⁡αwhich may be expanded in powers of Δr0 and αmax to give (20)Δrπ=∑i=0,j=0PijαmaxiΔr0j=P01Δr0+P10αmax+P20amax2+P21amax2Δr0+O4,where (21a)P01=M=1,(21b)P00=P10=P11=0,(21c)P20=Dγτ+γ-1τ,(21d)P21=-γ1+ξξτDγR0.Here, P10 and P20 are the well-known 1st- and 2nd-order α angular aberration coefficients, respectively, seen to depend on τ in the case of an HDA with PSD. For a slit HDA, however, we have τ=1 and these are characteristics of the analyzer. Finally, M is the analyzer magnification with M=-1 for an ideal HDA. For most analyzers P10=0, the well-known 1st-order focusing condition. For a conventional (ξ=γ=1, R0=R-) HDA we obtain the well-known [5] case P20=2R-.For most of electron spectroscopy and spectroscopic imaging applications, the relative energy resolution of an HDA,ΔE/E, is an important consideration; see [1–5, 42–45] and references therein. Theabsolute base energy resolution ΔEB, defined as “the range of energies over which amonoenergetic beam produces an output in an analyzer set at fixed deflection voltage,” can be related to the exit width Δrπ by [5](22)ΔEBE=δrπ+ΔrπDγ,where δrπ is the spatial resolution of the PSD (or the exit slit width) in the dispersion direction, while Δrπ is the maximal position spread at the exit for amonoenergetic beam of energy E. Using the results of (20) in (22) and keeping only up to 2nd-order terms, we finally obtain the well-known formula for the relative base energy resolution of anideal HDA:(23)ΔEBE=Δr0+δrπDγ+1+γ-1E/E0amax2.Equation (23) is particularly applicable to HDAs using a PSD for which in general E is different from the reference energy E0. For anyideal HDA with slits/apertures we need E=E0 and find P10=0, P20=γDγ, M=-1 with Dγ now given by (16) with τ=1 (E=E0):(24)Dγτ=1=R0+RπRπγR0SlitHDAfor which we then also have(25)ΔEBE0=γΔr0+δrπR0+RπR0Rπ+γαmax2SlitHDA.Finally, for the idealconventional HDA (R0=Rπ=R-, γ=1, E=E0) we obtain the well-known result:(26)ΔEBE0=Δr0+δrπ2R-+αmax2ConventionalSlitHDA.Inreal applications, ideal energy resolution formula equations (23), (25), and (26) are only used as a general guide, since it is well known that, without any additional fringing field correction, the real HDA has a substantialnonzero coefficient P10 and thus is no more 1st-order focusing [34]. Note, however, that the HDA electrode voltages are still typically specified by the ideal HDA voltages given by (10).Over the last 15 years, the authors and their collaborators have shown [27–30, 32–35] that the biased paracentric HDA does in fact overcome the fringing field problemwithout the need to introduceany additional electrodes [33, 34]. This is done by moving the entry point R0 to the so-called paracentric position and optimizing the entry bias γ so that the lensing action of the fringing fields restores 1st-order focusing conditions. For the present HDA these paracentric positions have been found through simulation to be at R0=82.55mm with γ=1.4715 and R0=116.0mm with γ=0.499 [34]. In general, while paracentric entries are thought to exist forany fringing field HDA (particularly for those using a large interradial distance in order to accommodate a PSD),no general scaling rules [34] have been discovered to date so their exact values have to be found by simulation in combination with the corresponding optimum value of γ.Very recently, the improvement of the energy resolution for the above paracentric entries found in simulation was shown to be valid also in experiment [29]. However, to date, there have been no investigations of the timing properties of paracentric entries. This is explored next in Section 2.3 for the ideal HDA and in Sections 3 and 4 for the fringing field via simulations using the charged particle optics code SIMION [36].


### 2.3. Time-of-Flighttπ and HDA Exit Time Spread Δtπ

As shown in Figure1, electrons entering the HDA with launching angles α follow elliptical trajectories in the radial electrostatic field. Due to the different initial conditions of their trajectories, a spread in their transit times accrues. Neglecting the fringing fields at the boundaries and other mechanical imperfections, the transit time, tπ, of a particle with mass m and charge q through an orbit of Δω=π in theideal HDA is computed following the treatment in [1]. Observing the rotational velocity to be, vω=rω˙=L/mr, where L is the conserved angular momentum, we then have(27)tπ=∫0πmr2ωLdω.Using (3)–(4) we can directly integrate (27) to obtain (using Wolfram Mathematica 9)(28)tπα,a,ρ0=T02ππ+2ArcTansin⁡α2ρ0-1-4ρ0/2-ρ0sin⁡αcos⁡2α-2+ρ0/2-ρ0,where T0 is the period of the full Kepler orbit given by(29a)T0=2πma3qk=πR01+ξγξγ+ξ1-τ3/2m2E0=πγ2Dγτγ+ξ1-τξm2E0which for the ideal centric conventional HDA entry (Rπ=R0=R-, γ=1) becomes(29b)T-0≡2πR-m2E0.It is practical to use m=5.68569eV-ns/mm for the electron mass so that when used with kinetic energy E0 in eV and distances in mm timing results are directly obtained in ns. We have done so in all our timing calculations in this paper. We note that expressions for tπ are also given in [25, 26], but (28) seems to be simpler and is found to be in agreement both with [25] and also withnonrelativistic SIMION for motion in the ideal field HDA.Equation (28) may also be expanded in powers of α as shown in [1]. However, the expression given there (equation (56) of [1]) has an error in the α5 term. Here we have performed again the evaluation (using Wolfram Mathematica 9) and obtained the new corrected result:(30)tπ=T02ππ+2ρ0-14α-283ρ0-1α3+33-208ρ0+288ρ02α530+Oα7.As can be seen clearly now from (30), the time-of-flight of a trajectory starting with negative launching angle α<0 will in general be smaller than that with α>0. This is primarily because the trajectory for α<0 will always be shorter. For α=0, (28) and (30) both give the correct half period T0/2. Furthermore, as known from [1] and also as discerned from Figure 2 we also have for the ideal field(31)tπα+tπ-α=T0=2tπ0which again is seen to be in agreement with the results of (28) and (30).Figure 2
Elliptical orbits with thesame kinetic energy E=E0 for α=±30° (a), α=±10° (b), and α=0° (c) with α>0 (red) and α<0 (blue). The field is central with center at O, one of the foci of the ellipse. The electron enters at the point P with radius r0 and angle α and exits at point M after going through a full 180°. All orbits with α>0 are always outer and therefore are always longer in both space and time with corresponding eccentricity vector ϵ lying in the opposite lower half plane. As α decreases the corresponding eccentricity vectors rapidly rotate towards the positive x-axis coalescing for α=0, in which case the orbits exactly overlap. The angular momentum L and eccentricity ϵ are even functions of α and are conserved in the ideal field. Therefore orbits with the same |α| have identical shape but a slightly different, but symmetric, tilt around the nodal line POM as shown. Due to this symmetry orbits with α>0 and α<0 are complementary giving rise to the condition of (31).
(a)
(b)
(c)Using (28) we investigate the spread in the transit time Δtπ for a monoenergetic beam of energy E for corresponding spreads in α and r0 as we did in the case of rπ. In this case the maximal transit variation can be seen to be given by(32)Δtπ≡tπR0+Δr02,E,αmax-tπR0-Δr02,E,-αmax,in accordance with the fact that tπ now depends on odd powers of α.Keeping only up to 3rd-order terms in the series expansion we obtain (using Wolfram Mathematica 9)(33)Δtπ=∑i=0,j=0Pij′αmaxiΔr0j=P10′αmax+P11′αmaxΔr0+P20′amax2+P30′amax3+O4with(34a)P00′=P01′=P11′=P20′,(34b)P10′=4γ+τ-1τDγm2E0=4γ+τ-1m2E0P20,(34c)P30′=-γ1+4ξ-3ξ1-τ6γ+ξ1-τP10′,and using (29a) and (29b) we can obtain an equation similar to that for Δrπ:(34d)ΔtπT0/πγ2ξγ+τ-1/γ+ξ1-τ=2αmax-13γ1+4ξ-3ξ1-τγ+ξ1-τamax3+O4.It is interesting to note that the lowest order dependence of Δtπ on α is 1st-order, compared to the 2nd-order α-dependence of Δrπ (equations (19)-(20)), thus possibly making it less susceptible to fringing field effects which seem to influence strongly the 1st-order dependence in α (destroy 1st-order focusing!) of Δrπ. Furthermore, the dependence of Δtπ on Δr0 is seen to be of higher order in Δtπ but correlated to the value of α, again showing Δtπ to be less sensitive to this parameter than Δrπ. This behavior is clearly demonstrated in the results of Figure 3.Figure 3
The time-of-flighttπ and the exit radius rπ are plotted as a function of launching angle α, for pass energy E0=1keV and for a point source (Γ=0), in both ideal and fringing field HDAs. In (d), the conventional (R0=R-=101.6mm) fringing field HDA (black squares) is seen to have a much larger and asymmetric spread in rπ as a function of α, far from 1st-order focusing conditions. The two paracentric entries (R0=82.55mm and R0=116 mm), however, have much narrower spreads in rπ demonstrating full 1st-order focusing and therefore much improved energy resolution. The analytical solutions valid for the ideal field HDA in (a) and (b) were obtained using (14) for rπ and (28) for tπ.
(a)
(b)
(c)
(d)Finally, a quantity related to the transit timetπ is the length of the trajectory sπ given by (35)sπ≡∫0πdsω=∫0πdxdω2+dydω2dω=∫0πrω2+drdω2dω.This can be simplified (using Wolfram Mathematica 9) to(36)sπα,a,ρ0=∫0πpaρ04aρ0p-aρ0cos⁡ω-1+2ppsec2⁡α-2aρ0sin⁡ωtan⁡α2aρ0+p-aρ0cos⁡ω-psin⁡ωtan⁡α2ⅆω.Εquation (36) is seen to include both odd (e.g., tan⁡α) and even (due to p from (3a)) powers of α and therefore a condition similar to that of (31) cannot be established also for sπ. Εquation (36) can only be evaluated numerically. However, we note that, for α=0, further simplifications ensue allowing for the following closed form result (using Wolfram Mathematica 9): (37)sπα=0=aρ0K41-ρ02-ρ02+2-ρ0E41-ρ02-ρ02given in terms of the complete elliptic integral of the first (K) and second (E) kind. The length of the trajectory sπ can be expected to be longer (shorter) for α>0 (α<0), with correspondingly longer (shorter) flight times tπ, respectively.In the presence of fringing fields, the exit radiusrπ, the flight time tπ, and the trajectory length sπ cannot be established in a simple analytical form. Thus, their calculations have to be carried out numerically. This is done here using the charge particle optics software SIMION [36]. Results are given in Sections 3 and 4, with comparison of ideal and fringing field results. To date, we have shown [33–35] that the fringing field biased paracentric HDA does in fact satisfy (17) to a very good approximation.


## 2.1. Kepler Orbits and General Radial Orbit Equation

For motion in the1/r potential of anideal HDA given by(1)Vr=-kr+c,thenonrelativistic trajectory of a particle with charge q (q=-e for electrons), kinetic energy E, and angular momentum L is given by (see equation (98) in [1])(2)rω=p1+ϵcos⁡ω-ωϵ.The particle enters at radius r0=rω=0, with launching angle α within the entry aperture of diameter Δr0 centered at R0. It then follows in general anelliptical trajectory r(ω) given as a function of the orbit angle ω and exits at rπ=r(ω=180°) after going through an orbit of π(Δω=180°). Thecentral trajectory (red dashed line in Figure 1) used asreference is defined as the trajectory of a particle which enters with energy E0 at r0=R0 with α=0° and exits at rπ=Rπ. At the exit either a slit or an aperture is typically placed with center at Rπ or alternatively a PSD is used. The conventional HDA arrangement typically utilizes R0=Rπ=R- in which case the central trajectory is acircle. By convention [1] the trajectory launched with α>0 is always anouter trajectory, while the one with α<0 is always aninner trajectory.The parametersp and ϵ in (2) are thelatus rectum andeccentricity of the resulting elliptical trajectory given by [1](3)p=L2mqk,ϵ=1+2EL2mq2k2with the kinetic energy E and (vector) angular momentum L inside the HDA given by(4)E=12mv2,(5)L=r×mv,where v is the velocity of the particle at radius r.The potential constantsk and c given in (1) are fixed by the voltages V1 and V2 supplied to the hemispherical electrodes and do not depend on the trajectory parameters of thearbitrary particle. These voltages are determined by thecentral trajectory (already defined above) and are discussed later in this section (see (10)). The trajectory described by (2) is in general part of an ellipse with one focus at the center of force r=0 with the constant ωϵ set by the initial conditions. The shape of the elliptical orbit is well defined by p and ϵ which are constants of the motion, since they are determined from the particle’s total energy Etot=E-qc (see equation (74) in [1]) and vector angular momentum L, both conserved quantities for motion in anypurely radial potential V(r)=V(r). We note that, in the fringing field HDA, even though the potential isnot radial [V(r)=V(r,θ); i.e., there is an additional dependence on the polar angle θ] and therefore the magnitude of the angular momentum L=|L| isnot conserved, the direction of the angular momentum L/Lis conserved and therefore motion remains planar.Assuming that the particle enters the HDA at timet=ω=0 with entry radius r0=r(ω=0) and launching angle α and then exits at orbit angle ω=π with exit radius rπ=r(ω=π) at t=tπ, thelatus rectum p and the eccentricity ϵ of the ellipse as a function of α and r0 are given by (see equation (40) of [1])(6a)p=aρ02-ρ0cos2⁡α,(6b)ϵ=sin2⁡α+1-ρ02cos2⁡αwith(6c)ρ0≡r0a,(6d)a≡aE=qk2qc-E,where a is the semimajor axis of the ellipse given in terms of constants q, k, and c and the particle kinetic energy E. Under the specified initial conditions, the elliptical trajectory can be further specified as(7)rω=pr0r0+p-r0cos⁡ω-psin⁡ωtan⁡αshowing it to have both odd and even (through p given in (6a), (6b), (6c), and (6d)) terms in the launching angle α.The values of the potential constantsk and c in (1) are specified so that the central trajectory used as reference enters the HDA at ω=0 with α=0 and r0=R0 and kinetic energy E0 and exits at ω=π with rπ=Rπ. Applying these conditions to (7) we obtain (see equations (86) and (87) of [1])(8a)qk=E0R0+RπγR0Rπ,(8b)qc=E01+γR0Rπ.It is seen from (6a), (6b), (6c), and (6d) and (8a)-(8b) that thereference trajectory has a semimajor axis a0 and eccentricity ϵ0 given by(8c)a0≡aE=E0=12Rπ+R0,(8d)ϵ0≡ϵα=0,r0=R0,E=E0=1-R0a0=Rπ-R0Rπ+R0which for the centric conventional HDA entry (Rπ=R0=R-, γ=1) is a circle with a-0≡R- and ϵ-0≡0 as expected.Using (8a)-(8b) and (1) we may next obtain the potential V(r) given by [1](9)qVr=E01-γR0RπR0+Rπr-1from which the HDA voltages can be immediately obtained:(10)qVi≡qVRi=E01-γR0RπR0+RπRi-1i=1,2.For the above voltages the general trajectory rω can then be readily shown to be given by(11)rω=2r0cos2⁡αγ1+ξ-r0/R0γ-ξτ-1cos⁡ω/γ1+ξ-r0/R0γ-ξτ-1+cos⁡2α+ω,where we have also introduced the short hand of the following [1]: ξ≡Rπ/R0 and τ≡E/E0. Also introduced in (8a), (8b), (8c), and (8d)–(11) is theentry bias parameter γ. Equation (11) is seen to satisfy the conditions of the central trajectory; that is, rω=π,α=0,r0=R0,τ=1=Rπindependent of the value of γ. However, γ does affect all other trajectories as seen from (11) and can therefore be used as an additional parameter to optimize the HDA performance. Setting r0=R0 and E=E0 in (11) we obtain the defining relation [1]: (12)γ≡1-qV0E0,where V0≡V(R0) is the value of the potential at the entry R0. We note that when γ=1, V0=0 and vice versa. Since the particle’s effective total energy Etotinside the HDA must always benegative for bound motion we must always have Etot=E-E01+γ/ξ<0 which leads to the restriction on γ: (13a)γ>ξτ-1which is also consistent with the requirement that the semimajor axis a>0 since using (8a)-(8b) a from (6d) can be expressed as(13b)a=aτ=R01+ξ2ξ1-τ+γ.In the past, we have reported on the properties of HDAs and their dependence on the entry parameters ξ and γ. Quite generally, we can identify two different entry classes as follows:(i)
the conventional entry HDA for whichR0=R-=Rπ, that is, ξ=1, and whose reference trajectory is acircle (ϵ=0);(ii)
the paracentric entry HDA for whichξ≠1, whose reference trajectory is anellipse (0<ϵ<1).For both types of entry, γ can be independently used to bias the entry. Thus, for conventional HDAs,zero bias is typically used having V0=0 (γ=1). For paracentric entries, however, bothpositive bias (V0>0 or γ>1) andnegative bias (V0<0 or γ<1) have been investigated [1, 7, 27, 29, 33, 34]. While the value of γ in anideal HDA does not affect its overall 1st-order focusing, in a realfringing field HDA, for which the conventional entry is well known to lose 1st-order focusing, specific combinations of optimal (ξ,γ) values have been found torestore 1st-order focusing [29, 33–35]. In Section 4, we will look at a concrete example of such an HDA having R1=72.4mm and R2=132.8mm (Rπ=R-=101.6mm) and investigate both the energy and the time resolution for paracentric entry R0=82.55mm with γ=1.4715 and R0=116mm with γ=0.5 in comparison to the conventional HDA with R0=R- and γ=1. These particular combinations of (ξ,γ) have been found [29, 33–35] to date, to restore 1st-order focusing. Such an HDA is being used in our laboratory for high-resolution Auger projectile electron spectroscopy [7, 27–29, 31, 35] with excellent energy resolution. In [33], we have also investigated numerically HDAs with different values of R1 and R2 and have also found corresponding optimal combinations of (ξ,γ) which restore 1st-order focusing.


## 2.2. HDA Exit Radiusrπ, Radial Spread Δrπ, and Base Energy Resolution ΔEB

Using (11), the exit radius rπ=r(ω=π) can be computed (see equation (99) and correction of equation (100) inErratum of [1]) and is given by (14)rπ≡rπr0,τ,α,γ=-r0+r01+cos2⁡αr0/R0+Rπ1+ξ/γ1-τ-1which may also be expanded in powers of α giving up to 5th order(15a)rπr0=-1+Δr01-α21+Δr0+α423-Δr053-Δr0+Oα6with(15b)Δ2≡DγR0+Rπγξτand where it is readily seen that(15c)rπr0,τ,α=0,γ=Δ-r0=R0+Rπ1+ξ/γ1-τ-r0.The energy dispersion Dγ is another useful analyzer parameter defined in general as (16)Dγ≡E∂rπ∂Eα=0=Dγτ=1+ξRπγ1+ξ/γ1-τ2τ.For slit HDAs, where τ=1 (i.e., E=E0), Dγ is a characteristic length of the HDA which determines the shift of the image position of a particle trajectory for an infinitesimal change in the particle energy E0 [5]. However, for HDAs with PSD its definition can be extended to any E as above. Equation (14) is also known as thebasic equation of the analyzer [1]. As can be verified in (14), the reference trajectory (r0=R0, τ=1, α=0)always exits at Rπ. We note that a similar equation can also be expressed in terms of α∗, the entry angle into the HDA (see equation (102) in [1] and Figure 1). The angles α and α∗ are related according to Snell’s law [1]. Here, our treatment is given in terms of the angle α inside the HDA, which is the launching angle typically used in simulations. A more complete treatment including the effect of the injection lens should be expressed in terms of the outside (incident) entry angle α∗ as determined by the lens and thus also include the effect of refraction due to the potential change at r0 in crossing the HDA entry plane. Because typically HDA entry apertures are small we in general have V(r0)≈V(R0) and therefore for conventional HDAs which use γ=1, Vr0≈0 and refraction can be neglected. This clearly is not the case for paracentric entry where optimum values of γ are typically different from 1 and therefore Vr0≠0.It is also obvious from (14) that only even powers of α are expected and therefore we should have(17)rπα=rπ-αindicating 1st-order focusing, a basic requirement for any energy analyzer. For monoenergetic beams of particles with energy E we may vary the input beam parameters Δr0 and α to obtain themaximum allowed exit radius variation Δrπ [41]. At the HDA exit plane this can be shown [7, 41] to be equivalent to the following calculation: (18)Δrπ=rπR0-Δr02,E,α=0-rπR0+Δr02,E,αmaxyielding(19)Δrπ=Δr0+R0+Rπ1+ξ/γ1-τ-1+Δr0/2R0R0+Rπ1+ξ-ξ-Δr0/2R0-1+Δr0/2R0ξ/γ1-τcos2⁡αwhich may be expanded in powers of Δr0 and αmax to give (20)Δrπ=∑i=0,j=0PijαmaxiΔr0j=P01Δr0+P10αmax+P20amax2+P21amax2Δr0+O4,where (21a)P01=M=1,(21b)P00=P10=P11=0,(21c)P20=Dγτ+γ-1τ,(21d)P21=-γ1+ξξτDγR0.Here, P10 and P20 are the well-known 1st- and 2nd-order α angular aberration coefficients, respectively, seen to depend on τ in the case of an HDA with PSD. For a slit HDA, however, we have τ=1 and these are characteristics of the analyzer. Finally, M is the analyzer magnification with M=-1 for an ideal HDA. For most analyzers P10=0, the well-known 1st-order focusing condition. For a conventional (ξ=γ=1, R0=R-) HDA we obtain the well-known [5] case P20=2R-.For most of electron spectroscopy and spectroscopic imaging applications, the relative energy resolution of an HDA,ΔE/E, is an important consideration; see [1–5, 42–45] and references therein. Theabsolute base energy resolution ΔEB, defined as “the range of energies over which amonoenergetic beam produces an output in an analyzer set at fixed deflection voltage,” can be related to the exit width Δrπ by [5](22)ΔEBE=δrπ+ΔrπDγ,where δrπ is the spatial resolution of the PSD (or the exit slit width) in the dispersion direction, while Δrπ is the maximal position spread at the exit for amonoenergetic beam of energy E. Using the results of (20) in (22) and keeping only up to 2nd-order terms, we finally obtain the well-known formula for the relative base energy resolution of anideal HDA:(23)ΔEBE=Δr0+δrπDγ+1+γ-1E/E0amax2.Equation (23) is particularly applicable to HDAs using a PSD for which in general E is different from the reference energy E0. For anyideal HDA with slits/apertures we need E=E0 and find P10=0, P20=γDγ, M=-1 with Dγ now given by (16) with τ=1 (E=E0):(24)Dγτ=1=R0+RπRπγR0SlitHDAfor which we then also have(25)ΔEBE0=γΔr0+δrπR0+RπR0Rπ+γαmax2SlitHDA.Finally, for the idealconventional HDA (R0=Rπ=R-, γ=1, E=E0) we obtain the well-known result:(26)ΔEBE0=Δr0+δrπ2R-+αmax2ConventionalSlitHDA.Inreal applications, ideal energy resolution formula equations (23), (25), and (26) are only used as a general guide, since it is well known that, without any additional fringing field correction, the real HDA has a substantialnonzero coefficient P10 and thus is no more 1st-order focusing [34]. Note, however, that the HDA electrode voltages are still typically specified by the ideal HDA voltages given by (10).Over the last 15 years, the authors and their collaborators have shown [27–30, 32–35] that the biased paracentric HDA does in fact overcome the fringing field problemwithout the need to introduceany additional electrodes [33, 34]. This is done by moving the entry point R0 to the so-called paracentric position and optimizing the entry bias γ so that the lensing action of the fringing fields restores 1st-order focusing conditions. For the present HDA these paracentric positions have been found through simulation to be at R0=82.55mm with γ=1.4715 and R0=116.0mm with γ=0.499 [34]. In general, while paracentric entries are thought to exist forany fringing field HDA (particularly for those using a large interradial distance in order to accommodate a PSD),no general scaling rules [34] have been discovered to date so their exact values have to be found by simulation in combination with the corresponding optimum value of γ.Very recently, the improvement of the energy resolution for the above paracentric entries found in simulation was shown to be valid also in experiment [29]. However, to date, there have been no investigations of the timing properties of paracentric entries. This is explored next in Section 2.3 for the ideal HDA and in Sections 3 and 4 for the fringing field via simulations using the charged particle optics code SIMION [36].


## 2.3. Time-of-Flighttπ and HDA Exit Time Spread Δtπ

As shown in Figure1, electrons entering the HDA with launching angles α follow elliptical trajectories in the radial electrostatic field. Due to the different initial conditions of their trajectories, a spread in their transit times accrues. Neglecting the fringing fields at the boundaries and other mechanical imperfections, the transit time, tπ, of a particle with mass m and charge q through an orbit of Δω=π in theideal HDA is computed following the treatment in [1]. Observing the rotational velocity to be, vω=rω˙=L/mr, where L is the conserved angular momentum, we then have(27)tπ=∫0πmr2ωLdω.Using (3)–(4) we can directly integrate (27) to obtain (using Wolfram Mathematica 9)(28)tπα,a,ρ0=T02ππ+2ArcTansin⁡α2ρ0-1-4ρ0/2-ρ0sin⁡αcos⁡2α-2+ρ0/2-ρ0,where T0 is the period of the full Kepler orbit given by(29a)T0=2πma3qk=πR01+ξγξγ+ξ1-τ3/2m2E0=πγ2Dγτγ+ξ1-τξm2E0which for the ideal centric conventional HDA entry (Rπ=R0=R-, γ=1) becomes(29b)T-0≡2πR-m2E0.It is practical to use m=5.68569eV-ns/mm for the electron mass so that when used with kinetic energy E0 in eV and distances in mm timing results are directly obtained in ns. We have done so in all our timing calculations in this paper. We note that expressions for tπ are also given in [25, 26], but (28) seems to be simpler and is found to be in agreement both with [25] and also withnonrelativistic SIMION for motion in the ideal field HDA.Equation (28) may also be expanded in powers of α as shown in [1]. However, the expression given there (equation (56) of [1]) has an error in the α5 term. Here we have performed again the evaluation (using Wolfram Mathematica 9) and obtained the new corrected result:(30)tπ=T02ππ+2ρ0-14α-283ρ0-1α3+33-208ρ0+288ρ02α530+Oα7.As can be seen clearly now from (30), the time-of-flight of a trajectory starting with negative launching angle α<0 will in general be smaller than that with α>0. This is primarily because the trajectory for α<0 will always be shorter. For α=0, (28) and (30) both give the correct half period T0/2. Furthermore, as known from [1] and also as discerned from Figure 2 we also have for the ideal field(31)tπα+tπ-α=T0=2tπ0which again is seen to be in agreement with the results of (28) and (30).Figure 2
Elliptical orbits with thesame kinetic energy E=E0 for α=±30° (a), α=±10° (b), and α=0° (c) with α>0 (red) and α<0 (blue). The field is central with center at O, one of the foci of the ellipse. The electron enters at the point P with radius r0 and angle α and exits at point M after going through a full 180°. All orbits with α>0 are always outer and therefore are always longer in both space and time with corresponding eccentricity vector ϵ lying in the opposite lower half plane. As α decreases the corresponding eccentricity vectors rapidly rotate towards the positive x-axis coalescing for α=0, in which case the orbits exactly overlap. The angular momentum L and eccentricity ϵ are even functions of α and are conserved in the ideal field. Therefore orbits with the same |α| have identical shape but a slightly different, but symmetric, tilt around the nodal line POM as shown. Due to this symmetry orbits with α>0 and α<0 are complementary giving rise to the condition of (31).
(a)
(b)
(c)Using (28) we investigate the spread in the transit time Δtπ for a monoenergetic beam of energy E for corresponding spreads in α and r0 as we did in the case of rπ. In this case the maximal transit variation can be seen to be given by(32)Δtπ≡tπR0+Δr02,E,αmax-tπR0-Δr02,E,-αmax,in accordance with the fact that tπ now depends on odd powers of α.Keeping only up to 3rd-order terms in the series expansion we obtain (using Wolfram Mathematica 9)(33)Δtπ=∑i=0,j=0Pij′αmaxiΔr0j=P10′αmax+P11′αmaxΔr0+P20′amax2+P30′amax3+O4with(34a)P00′=P01′=P11′=P20′,(34b)P10′=4γ+τ-1τDγm2E0=4γ+τ-1m2E0P20,(34c)P30′=-γ1+4ξ-3ξ1-τ6γ+ξ1-τP10′,and using (29a) and (29b) we can obtain an equation similar to that for Δrπ:(34d)ΔtπT0/πγ2ξγ+τ-1/γ+ξ1-τ=2αmax-13γ1+4ξ-3ξ1-τγ+ξ1-τamax3+O4.It is interesting to note that the lowest order dependence of Δtπ on α is 1st-order, compared to the 2nd-order α-dependence of Δrπ (equations (19)-(20)), thus possibly making it less susceptible to fringing field effects which seem to influence strongly the 1st-order dependence in α (destroy 1st-order focusing!) of Δrπ. Furthermore, the dependence of Δtπ on Δr0 is seen to be of higher order in Δtπ but correlated to the value of α, again showing Δtπ to be less sensitive to this parameter than Δrπ. This behavior is clearly demonstrated in the results of Figure 3.Figure 3
The time-of-flighttπ and the exit radius rπ are plotted as a function of launching angle α, for pass energy E0=1keV and for a point source (Γ=0), in both ideal and fringing field HDAs. In (d), the conventional (R0=R-=101.6mm) fringing field HDA (black squares) is seen to have a much larger and asymmetric spread in rπ as a function of α, far from 1st-order focusing conditions. The two paracentric entries (R0=82.55mm and R0=116 mm), however, have much narrower spreads in rπ demonstrating full 1st-order focusing and therefore much improved energy resolution. The analytical solutions valid for the ideal field HDA in (a) and (b) were obtained using (14) for rπ and (28) for tπ.
(a)
(b)
(c)
(d)Finally, a quantity related to the transit timetπ is the length of the trajectory sπ given by (35)sπ≡∫0πdsω=∫0πdxdω2+dydω2dω=∫0πrω2+drdω2dω.This can be simplified (using Wolfram Mathematica 9) to(36)sπα,a,ρ0=∫0πpaρ04aρ0p-aρ0cos⁡ω-1+2ppsec2⁡α-2aρ0sin⁡ωtan⁡α2aρ0+p-aρ0cos⁡ω-psin⁡ωtan⁡α2ⅆω.Εquation (36) is seen to include both odd (e.g., tan⁡α) and even (due to p from (3a)) powers of α and therefore a condition similar to that of (31) cannot be established also for sπ. Εquation (36) can only be evaluated numerically. However, we note that, for α=0, further simplifications ensue allowing for the following closed form result (using Wolfram Mathematica 9): (37)sπα=0=aρ0K41-ρ02-ρ02+2-ρ0E41-ρ02-ρ02given in terms of the complete elliptic integral of the first (K) and second (E) kind. The length of the trajectory sπ can be expected to be longer (shorter) for α>0 (α<0), with correspondingly longer (shorter) flight times tπ, respectively.In the presence of fringing fields, the exit radiusrπ, the flight time tπ, and the trajectory length sπ cannot be established in a simple analytical form. Thus, their calculations have to be carried out numerically. This is done here using the charge particle optics software SIMION [36]. Results are given in Sections 3 and 4, with comparison of ideal and fringing field results. To date, we have shown [33–35] that the fringing field biased paracentric HDA does in fact satisfy (17) to a very good approximation.


## 3. Calculation Method

Simulation of the electron motion in electrostatic fields is a subject of growing interest and stimulating research as can be ascertained by the numerous mention of charged particle optics (CPO) software usage at poster exhibitions in conferences. An interesting historical perspective on the evolution of SIMION itself is provided by David Dahl, “father” of SIMION [46], written in 2000. Since 2000, PCs have undergone tremendous improvements in memory, disk space, and speed making CPO programs increasingly more accurate and capable of more sophisticated calculations (see, e.g., [47–49]). Following the theoretical presentation in Section 2, the numerical simulations described in the present work have been realized using the computer program SIMION 8.1, which is a ray-tracing code for three-dimensional simulations. SIMION solves the Laplace equation in the HDA for a given geometry utilizing the finite-difference method. This software has been chosen because of its ease in controlling quickly several parameters such as the pass energy, the initial beam parameters (position, pencil angle, etc.), and the voltages of the electrodes. The accuracy of SIMION 7 has been discussed in the past [50] (SIMION 8.1 is much improved) and also been compared to the more accurate BEM approach [33].In particular, the aim of the transit time calculations is to investigate the suitability of the biased paracentric HDA forsimultaneous high energy resolution spectroscopy and time-of-flight measurements within the same HDA. We have chosen the HDA parameters provided by [34]. The three entry positions, R0=82.55mm, 101.6 mm, and 116 mm, already discussed, were studied. The nominal pass energy was set at E0=1000eV for most of the investigations; however at the end of Section 4 we also present results for E0=10,100,500eV. For each of the three values of R0, the HDA voltages depended on both R0 and γ according to (10). The optimal values of γ that achieve the best focusing effect were determined in a previous publication on the specific paracentric HDA: γ≈1.5 for R0=82.55mm and γ=0.5 for R0=116mm [35]. In Table 1, we list the values of the most important parameters used in our simulations.Table 1
HDA parameters with their symbols and values. VoltagesVi have been computed from (10).
Symbol
Value
Parameter
R
1 (mm)
72.4
Inner HDA radius
R
2 (mm)
130.8
Outer HDA radius
R
- (mm)
101.6
Mean radius
R
π (mm)
101.6
Exit reference radius
R
0 (mm)
82.55
101.6
116.0
Paracentric entry radius
γ
1.4715
1.0
0.499
Entry bias
ξ
1.2308
1.0
0.8759
Paracentricity
V
1 (V)
845.4089
806.6298
142.5965
Inner hemisphere voltage
V
2 (V)
−512.3477
−446.4832
−621.9260
Outer hemisphere voltage
V
0 (V)
471.5
0.000
−501.0
Nominal entry voltageHere, the time resolution is determined by the transit time spread of trajectories passing through the analyzer without the input lens. As in the case of energy resolution, the relative time resolution of the HDA (Δtπ/tπ) depends on three main factors: (a) the dimensions such as the diameter of the entry aperture Δr0, the mean radius R-, and the entry position R0, (b) the angular spread of the electron velocity vectors at entry (±αmax), and (c) the input electron beam energy spread δE around the pass energy. We assume that the input lens produces an image spot with a Gaussian distribution at the HDA entry plane; the entry beam extent is therefore generally specified in terms of the half width, Γ, of this distribution. Again we model the energy of the electrons having a Gaussian shape centered around 1 keV and its FWHM is defined by δE. Our computations cover the ranges αmax≤5°, Γ≤1.5mm, and δE/E0≤0.4%, so that the exit position of the electrons lies within the values defined by the detector area. The time spectrum was obtained by ray-tracing calculations where 50,000 electrons were used in each run. Each curve has been normalized to its maximum value.The lens system at the entry of the HDA (injection lens) can also influence significantly the temporal resolution of the analyzer by modifying the angular spread with which the electrons enter the HDA [8]. According to Liouville’s theorem, the volume occupied in phase space by the beam is constant for conservative systems. Thus, the size of the beam can be reduced by the lens but only at the expense of angular divergence [51]. There is also a strong dependence of the electron time spread on the biasing parameter γ. However, in the present work, this effect was not investigated since γ was kept constant at the corresponding values for each of the three entry positions R0 shown in Table 1.Space charge effects have also been neglected in this work. Other simulation software programs such as CPO, LORENTZ, OPERA, and IBSIMU do have capabilities to handle space charge effects to some extent. Even the newest version of SIMION 8.1 has a Poisson solver and can now also handle some types of space charge problems. For HDAs, space charge effects can be particularly important in cases where the HDA is used as a monochromator [52, 53] (typically to limit the energy width of e-guns, not addressed in this paper) rather than as an analyzer/spectrometer. However, in most cases where the HDA is used as a spectrometer (as in our present treatment) analyzed electron currents are rather low (≲pA) as count rates are typically less than 100 kHz. In particular, in coincidence experiments, where the transit times are of particular interest, count rates are even smaller in an effort to maintain an acceptable true/random coincidence ratio. Therefore, in both these applications space charge effects can be expected to be negligible and for this reason we have not considered them in our presentation.Finally, we note some points relating to the accuracy of the simulations. As the HDA system has curved electrodes that do not perfectly align to square grid points, we have used the “surface enhancement” feature of SIMION 8.1 to improve the accuracy of the surface regions modeled [36]. The improvement in the field accuracy is expected to be at least an order of magnitude over that reported in [50]. In addition, since relativistic effects are not considered in the classical theory of motion presented, we have usednonrelativistic trajectories (an option provided by SIMION in its otherwise fully relativistic treatment of the motion) in the ideal field calculations shown in the contour plots of Figures 4–7 and the results extracted from Figure 7 and shown in Table 4.Figure 4
Calculated spot size on the detector plane (xy-plane) and corresponding time-energy distributions at the exit of the HDA for three different values of the HDA entry radius R0: (a) 82.55 mm, (b) 101.6 mm, and (c) 116 mm, and for a monoenergetic parallel beam: δE=0eV, αmax=0, and E0=1keV. Simulations were performed for source extent: Γ=0.5, 1.0, and 1.5 mm. Normalized line profiles resulting from vertical and horizontal projections of the electron distributions are also shown. The detector plane is assumed to coincide with the HDA exit plane.
(a)
R
0
=
82.55 mm
(b)
R
0
=
101.6 mm
(c)
R
0
=
116 mmFigure 5
This is the same as Figure4 but for electrons having uniform and random launching angles α≤|αmax| with αmax=1°, 3°, and 5° and δE=0eV, Γ=0.5mm, and E0=1keV.
(a)
R
0
=
82.55 mm
(b)
R
0
=
101.6 mm
(c)
R
0
=
116 mmFigure 6
This is the same as Figure4 but for nonmonoenergetic beams with given source energy spread (FWHM) δE=2, 3, and 4 eV for source extent Γ=0.5mm, αmax=0, and E0=1keV.
(a)
R
0
=
82.55 mm
(b)
R
0
=
101.6 mm
(c)
R
0
=
116 mmFigure 7
This is the same as Figure6 but with finite extent: Γ=0.5mm, δE=2eV, αmax=3°, and E0=1keV. In addition, the ideal field calculation is also shown in (a) for R0=101.6mm. The two paracentric entries for the fringing field HDA are seen in (c) and (d) to have near ideal field HDA distributions.
(a)
Ideal HDA
(b)
R
0
=
101.6mm
(c)
R
0
=
82.55 mm
(d)
R
0
=
116 mm


## 4. Results and Discussion

The comparison of the three entry positions,R0=82.55 mm, 101.6 mm, and 116 mm, involved calculations of electron trajectories having otherwise identical analyzer parameters. From these simulations, a detailed comparison of the electrons’ spatial (x and y) distributions and kinetic energy variations with time was made. In Table 2 we present a summary of the electron optical properties. As can be seen from Table 2, in the fringing field HDAs the 1st-order focusing is achieved (aberration coefficient P10≈0) in both paracentric cases. Note that 2nd-order focusing is not possible in any type of HDA (i.e., P20≠0) but only in the full spherical electrostatic capacitor for complete orbits [54].Table 2
Comparison of the electron optical properties for fringing and ideal field cases for each of the three entries computed by SIMION. For all cases,Rπ=R-=101.6 mm and E0=1 keV. Also given in parentheses are the results obtained using the closed form equations (see text). All results are nonrelativistic.
Positive biasParacentric HDA
Conventional HDA
Negative biasParacentric HDA
R
0 (mm)
82.55
101.6
116.0
γ
1.4715
1.00
0.499
V
0 (V)
471.5
0.000
−501.0
Ideal
Fringing
Ideal
Fringing
Ideal
Fringing
r
π
M
−1.000(−1.000)
−0.882
−1.000(−1.000)
−1.018
−1.000(−1.000)
−1.500
D
γ (mm)
154.026(154.024)
159.559
203.205(203.200)
206.686
381.957(381.939)
320.596
P
10 (mm/rad)
0.011(0.000)
0.030
0.009(0.000)
106.07
0.007(0.000)
0.033
P
20 (mm/rad2)
225.96(226.65)
216.71
202.68(203.20)
−5.4729
190.14(190.59)
130.01
t
π
P
10
′ (ns/rad)
39.848(39.848)
38.355
43.338(43.337)
43.927
57.542(57.541)
45.546
P
30
′ (ns/rad2)
−39.4739(−39.34)
−30.260
−36.0783(−36.11)
−38.169
−42.8798(−43.19)
−38.532Figure3 shows the time-of-flight tπ and the exit radius rπ as a function of the maximum pencil angle for the three different entry positions. As also shown in Table 3, the time-of-flight values for the central trajectory (α=0) are 14.300 ns (R0=82.55mm), 19.892 ns (R0=101.6mm), and 21.880 ns (R0=116mm), respectively, at a pass energy of E0=1keV. The time-of-flight values along the central (α=0) trajectories given by (28) in the ideal HDA are shorter than that in the fringing field HDA. For R0=116mm with the entry bias γ=0.499, the time peaks shift to longer times. In this case, the time-of-flight increases, since the electrons are slowed down in the analyzer for γ<1.0 and because their paths sπ are longer than for the other entry positions. For R0=82.55mm with γ≈1.5 the electron velocity increases since they are now accelerated by the entry bias, shortening the electron drift time across the region with electrostatic field.Table 3
This is as in Table2 but for the time-of-flight tπ and the length of the flown trajectory sπ of electrons starting with different launching angles α.
Positive biasParacentric HDA
Conventional HDA
Negative biasParacentric HDA
R
0 (mm)
82.55
101.6
116.0
γ
1.4715
1.00
0.499
V
0 (V)
471.5
0.000
−501.0
Ideal
Fringing
Ideal
Fringing
Ideal
Fringing
t
π  (ns)
α
=
-
5
°
12.379(12.379)
12.651
15.139(15.139)
17.945
21.648(21.648)
19.867
α
=
0
°
14.105(14.105)
14.300
17.018(17.018)
19.892
24.145(24.145)
21.880
α
=
+
5
°
15.831(15.831)
15.978
18.898(18.898)
21.753
26.641(26.641)
23.816
s
π  (mm)
α
=
-
5
°
270.270(270.270)
272.259
300.915(300.914)
327.439
323.121(323.120)
314.594
α
=
0
°
288.487(288.487)
289.688
319.186(319.186)
344.409
341.432(341.431)
330.043
α
=
+
5
°
305.611(305.611)
306.1560
336.244(336.244)
359.511
358.446(358.445)
343.745Table 4
Calculated values of the time spreadΔtπ (in ns), radial width variation Δrπ (in mm), and energy spread ΔE (in eV) at 1% and at 50% of the peak height for the three different entry values R0, for nonmonoenergetic input source beams of finite extent with Γ=0.5 mm, δE/E0=0.2%, and αmax⁡=3°. The absolute and relative product of FWHM resolutions are computed in the figures of merit g≡ΔE(50%)·Δtπ(50%) and G≡g/(E0tπ). The energy of the electrons E is equal to the tuning energy; that is, E=E0=1000 eV.
E
0
Δ
t
π(50%) (ns)
Δ
t
π(1%) (ns)
Δ
r
π(50%) (mm)
Δ
r
π(1%) (mm)
Δ
E(50%) (eV)
Δ
E(1%) (eV)
t
π (ns)
g (ns-eV)
G (×10−8)
(
Δ
t
π
(
50
%
)
/
t
π
) (%)
(
Δ
E
(
50
%
)
/
E
0
) (%)
R
0
=
101.6 mm (ideal HDA)
10
19.50
23.50
0.71
1.86
0.03
0.09
170.18
0.68
3.97
11.46%
0.35%
100
6.10
7.43
0.72
1.87
0.36
0.92
53.82
2.19
4.07
11.33%
0.36%
500
2.79
3.32
0.73
1.88
1.83
4.51
24.07
5.12
4.25
11.60%
0.37%
1000
1.96
2.35
0.72
1.87
3.59
9.25
17.02
7.05
4.14
11.5%
0.36%
R
0
=
101.6 mm (fringing HDA)
10
19.91
23.96
4.22
6.49
0.20
0.32
198.91
4.06
20.42
10.01%
2.04%
100
6.18
7.70
4.27
6.50
2.01
3.16
62.90
12.43
19.76
9.82%
2.01%
500
2.76
3.46
4.24
6.49
10.30
15.91
28.13
28.39
20.18
9.80%
2.06%
1000
1.98
2.39
4.14
6.46
20.30
31.29
19.89
40.19
20.21
9.95%
2.03%
R
0
=
82
.
55 mm (fringing HDA)
10
17.10
22.48
0.65
1.75
0.04
0.11
142.99
0.69
4.85
11.96%
0.41%
100
5.37
7.21
0.66
1.71
0.42
1.07
45.21
2.26
5.00
11.88%
0.42%
500
2.38
3.21
0.65
1.71
2.08
5.34
20.21
4.95
4.90
11.79%
0.42%
1000
1.70
2.26
0.65
1.71
4.07
10.74
14.30
6.92
4.84
11.89%
0.41%
R
0
=
116 mm (fringing HDA)
10
19.87
28.10
1.03
2.64
0.03
0.08
218.80
0.64
2.92
9.08%
0.32%
100
6.32
8.99
1.01
2.64
0.33
0.85
69.19
2.06
2.98
9.14%
0.33%
500
2.81
3.96
0.99
2.61
1.56
4.17
30.94
4.38
2.83
9.09%
0.31%
1000
1.98
2.84
0.98
2.61
3.17
8.35
21.88
6.28
2.87
9.05%
0.32%The exit position for entryR0=101.6mm in Figure 3(d) is seen to be asymmetric with respect to the central trajectory (α=0), greatly increasing its spread in position across the detector plane. For the paracentric cases (with the optimized values of R0 and γ) the electron trajectories show near ideal performance and the exit positions are seen to be symmetric in the exit plane. In this case, the exit positions are nearly ideal compared with the same traces in a truly ideal field in Figure 3(b).In Figures4–7 we show our HDA results in the form of contour plots for each of the three entry radii R0. The calculated exit spot sizes are shown on the detector plane (xy-plane) as time and energy distributions. The changes in the time-of-flight depend on the entry beam size Γ, the source energy spread δE, and the limiting launching angle αmax. The calculated time-of-flight line shapes for a parallel monoenergetic input beam (δE=0 and αmax=0) are shown in Figure 4 for source extent Γ=0.5, 1.0, and 1.5 mm. The time spread of the electrons was found to increase linearly as Γ increases. For R0=101.6mm, the time-of-flight deviations due to the finite source extent are seen not to contribute significantly to the time spread, because the differences in electron optical paths are negligible. The narrow base width in the tπ distribution for R0 = 101.6 mm can be understood by inspection of the nominal entry bias potential V(R0), which is zero, and therefore electrons travel along or very near the zero-equipotential line. For the two paracentric entries, however, for which VR0≠0, the effect of Γ on tπ cannot be ignored.We have also investigated the contributions of the launching angle to the time spread, calculating the flight time through the HDA for an electron of energyE0 with varying launching angle between -αmax and +αmax. The time profile of electrons for HDAs for the three different values of R0 and a range of pencil angles are shown in Figure 5. Clearly, the width of the time-of-flight distribution varies significantly with the entrance angle, even in the case of a monoenergetic electron beam. With the angles varying from 1° to 5°, the aberrations degrade the time resolution of the analyzer in all three cases. The simulations show that even though the focusing becomes poorer for R0=101.6mm, the time spread is not much affected by the fringing fields. In addition, one can see from Figure 5 that the time spread increases rapidly with increasing pencil angle, but the full width at half maximum (Δtπ (50%)) is not increased as much as the base width in either case, which can lead to significant misconceptions about the true time spread for the HDAs.Another limitation of the temporal resolution of the HDA is given by the energy spread of the source. In Figure6, the same calculations have been repeated as those shown in Figures 4 and 5 where the electrons were monoenergetic. However, now the energy of the electrons has been smeared out around the pass energy by δE=2 eV, 3 eV, and 4 eV but assuming no angular deviations. The electrons with a given initial energy spread δE (FWHM) were generated using a Gaussian distribution. Higher energy electrons penetrate more deeply into the field of the HDA, leading to a longer path length and different arrival times at the exit of the analyzer. It is clear that (i) the time spread increases strongly with δE and (ii) the biased paracentric HDA with R0=116mm has larger time spreads compared to the other two entries due to its higher energy dispersion Dγ [33]. Overall, a small deviation in the energy spread is seen to cause but a slight increase in the time spread for all cases.In Figure7, an electron source with both energy and angular distribution is considered, and four data sets are taken with finite source extent: Γ=0.5mm, δE=2eV, and αmax=3°. The ideal field calculation for R0=101.6mm is given in Figure 7(a) for reference. The simulation parameters and calculated resolutions are also given in Table 4. Since the line shapes in Figure 7 are mostly Gaussian, base widths are not considered, but the widths at 1% (Δtπ (1%)) and at 50% (Δtπ (50%)) of the peak height are presented. This comparison shows that an improvement in time and energy spreads is obtained for the biased paracentric case with R0=82.55mm compared to the conventional centric HDA. The time and energy FWHM for the HDA with R0=82.55mm are Δtπ (50%) = 1.70 ns and ΔE (50%) = 4.07 eV, respectively, while those for R0=116mm are 1.98 ns and 3.17 eV and for R0=101.6mm they are 1.98 ns and 20.3 eV, respectively.In Table4, the variation of the time and energy spreads is tabulated for different pass energies E0, assuming that both the initial angular and the spatial distributions are kept constant. The time spread is seen to be inversely proportional to the square root of the pass energy, that is, Δt∝E0-1/2. This means that the reduction of the time-of-flight spread can be achieved by increasing the pass energy. However, an increase of the pass energy leads to a decrease in the energy resolution. Therefore, a suitable compromise between good energy resolution (favoring smaller E0) and good time resolution (favoring larger E0) must be found. With this in mind, we have introduced two figures of merits, g≡ΔE50%·Δtπ50% and G≡g/E0tπ, for the detailed intercomparison of the joint time and energy resolution properties of the HDAs. In Figure 8, both g and G are plotted for four values of E0. For each R0, we find an almost constant G because the relative resolutions are constant. In the case of fringing field HDAs, the two paracentric entries give a smaller value of g and G than the conventional centric entry.Figure 8
Variation of the figures of meritg≡ΔE50%·Δtπ50% and G≡g/E0tπ with pass energy E0. In addition, the ideal field results are also shown for R0=101.6mm. The two fringing field paracentric entries are seen to have near ideal field behavior with the entry at R0=116mm showing a slight superiority in both figures of merit.
(a)
(b)


## 5. Summary and Conclusions

The biased paracentric HDA differs from the conventional HDA in two important ways: (a) the entry distanceR0 is paracentric, that is, either larger or smaller than the mean HDA radius used in a conventional HDA, and (b) the two hemispherical electrode voltages are set so that the HDA entry bias potential V(R0) is nonzero (biased), compared to conventional HDA usage in which VR0=0. For very particular value combinations of R0 and VR0, empirically found through simulations, we have shown in the past that substantial improvements in energy resolution can be attained, with 1st-order focusing conditions practically restored without the use of any additional fringing field correction electrodes, as typically done for conventional HDAs.Here, we have explored for the first time the time-of-flight properties of the biased paracentric HDA for the two special entry positions already investigated in the past as to their energy resolution and compared them to those of the conventional HDA. The trajectory related time spreads for electrons were calculated numerically since their flight times are not predictable analytically in the fringing field HDA. Our analysis quantified the effect of varying the controllable input variables of entry position, source extent, launching angle, and energy spread on selected output variables of time-of-flight and time spread.When one considers a realistic distribution with energy spread, angular spread, and finite source extent, both biased paracentric entries were found to have near ideal field behavior with theR0=82.55 mm entry showing betterabsolute energy and time distributions and the R0=116 mm entry, showing betterrelative energy and time distributions and their products, the figures of merit g and G (see Table 4). Thus, overall, both biased paracentric entries should prove themselves to be superior to the conventional HDA entry both in timing and in energy resolution applications offering an attractive alternative for combined timing and energy resolution applications such as coincidence experiments, where electrons detected in the HDA with high resolution must also be related to events in other detectors for the same event. It is our hope that these results will stimulate further interest in the construction and testing of such HDAs.



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*Source: 153513-2015-06-01.xml*