Optical properties of the 127� cylindrical energy analyzerused in LEIS experiments

N. Bundaleski *, Z. Rako�ccevi�cc, I. Terzi�ccLaboratory of Atomic Physics, Vin�cca, Institute of Nuclear Sciences, P.O. Box 522, 11001 Belgrade, Yugoslavia

Received 15 May 2002; received in revised form 21 August 2002

Abstract

The optical properties of the 127� cylindrical energy analyzer used in low energy ion scattering (LEIS) experiments

are studied by means of SIMION 3D version 6.0 program. The dependence of the acceptance solid angle X on the target

plane coordinates ðx; yÞ and the relative particle energy e completely describes the optical properties of an analyzer. TheXðx; y; eÞ function is calculated from the computed trajectories of ions emitted from different points of the target plane.

The influence of spherical aberrations to the error in the energy measurement is determined experimentally. The ex-

perimental results agree very well with the results obtained using the numerical simulations as well as, with the results

obtained by means of the second order analytical approach. The optical properties are analyzed for different electrode

potential configurations i.e. for different deflection voltage modes defined according to the potentials of the inner and

the outer electrode. The applied deflection voltage mode does not change Xðx; y; eÞ significantly. However, there is animportant influence of the deflection voltage mode to the analyzer constant due to the acceleration of ions traversing

along the optical axis of the analyzer. The knowledge of Xðx; y; eÞ can be used to determine the dependence of the energyspectra on the optical properties of the analyzer as well as, on the primary beam profile. This is of particular interest in

the analysis of LEIS spectra, because deviation of spectra caused by the optics of the analyzer can be a source of

significant errors in quantitative surface composition analysis.

� 2002 Elsevier Science B.V. All rights reserved.

Keywords: Ion optics; Electrostatic energy analyzer; Low energy ion scattering

1. Introduction

Low energy ion scattering (LEIS) is a well-es-

tablished technique for analyzing the composition

and structure of a solid-state surface. When noble

gas ions are used as projectiles, information depth

is restricted practically to the first atomic mono-

layer. This makes LEIS a powerful tool for

studying the surface composition in different pro-

cesses such as heterogeneous catalysis, adhesive

and segregation processes [1]. The results of the

surface composition measurements using LEIS

cannot be directly compared with those obtained

by Auger electron spectrometry or X-ray photo-electron spectrometry due to different information

depths in these experiments. In many cases, surface

segregation can induce a difference between the

Nuclear Instruments and Methods in Physics Research B 198 (2002) 208–219

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*Corresponding author. Tel.: +38-11-455451; fax: +38-11-

3410100.

E-mail address: nenadbun@rt270.vin.bg.ac.yu (N. Bundal-

eski).

0168-583X/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0168 -583X(02)01470 -2

compositions of the outermost atomic monolayer

and the inner monolayers [2]. The surface com-

position analysis of the first monolayer can hardly

be performed without LEIS. Therefore, thequantitative composition analysis by this tech-

nique is of special interest in surface science.

The energy spectra of low energy noble gas ions

scattered from the solid-state surface consist of

peaks, which approximately correspond to the

single elastic-scattering event. Target atom masses

can be calculated from the peak positions, ac-

cording to the classical elastic binary collisionmodel [1]. Peak intensities give the information

about the concentration of the appropriate chem-

ical element in the first monolayer. However, a

standardless composition analysis using LEIS is

not possible mainly due to the problems concern-

ing the neutralization effects. The calibration can

be performed using pure elemental targets as

standards [3], bearing in mind that �matrix effects�are found in only a few cases [4,5]. In this manner,

relative sensitivity factors can be attributed to

different elements. Unfortunately, there is a strong

deviation in some cases between the relative sen-

sitivity factors obtained in different experimental

groups [6]. In order to understand the origin of

these deviations, a close insight into different as-

pects of LEIS experiments and especially into theenergy analysis of scattered ions should be per-

formed.

Electrostatic energy analyzers (ESA) are irre-

placeable as high-resolution monochromators and

energy spectrometers of ions and electrons in

various atomic collision experiments, as well as in

almost every experimental set-up for surface

characterization [7]. The schematic of a cylindricalESA analyzer is given in Fig. 1. For a defined

deflection voltage, a charged particle traversing

along the optical axis and passing through the

centers of the entrance and exit slits of an analyzer

possesses the tuning energy Ea. This trajectory is

usually termed main path [8]. The main path co-

incides with the optical axis in the frame of the

analytical approach. The tuning energy is directlyproportional to the deflection voltage US defined

as the voltage applied between the electrodes:

Ea ¼ kUS, where k is the analyzer constant. As a

first approximation, it is assumed that any particle

detected for a specific deflection voltage has the

same energy, equal to corresponding tuning energy

Ea. However, this is not the case. For the fixed

tuning energy Ea, particles having energies in a fi-nite energy range will get to the detector. In LEIS

experiments, ions entering the analyzer are coming

from finite surface (Fig 1). For the fixed point on

the target plane ðx; yÞ from which the ion is scat-

tered, the ion energy E, and the tuning energy Ea,

the acceptance solid angle Xðx; y;EÞ can be definedas a solid angle in which a particle should enter in

order to pass through the analyzer and get to thedetector placed behind the exit slit. It is usually

assumed that there is an area in the target plane

with a maximum and approximately constant

magnitude of the acceptance solid angle. This area

is referred to as the acceptance region [9]. The

emitting surface represents an area from which

ions are scattered. It is defined by the primary ion

beam spot on the target plane. The maximum peakintensity will be obtained if the emitting surface

lies inside the acceptance region. A systematic er-

ror in quantitative LEIS analysis will generally

arise if the position of the emitting surface relative

to the acceptance region is not the same in every

measurement. This effect is identified as a major

problem in quantitative surface analysis using

LEIS [6].

Fig. 1. The definition of the acceptance solid angle Xðx; y;EÞ;U1 and U2 are potentials of the inner and the outer electrode,

respectively; US is the deflection voltage. Dashed line represents

the optical axis of the system.

N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219 209

Besides the peak intensity, the primary beam

profile also influences the peak position and width

due to the oblique incidence of ions into the ana-

lyzer. This deteriorates the properties of LEIS set-up for the qualitative composition analysis. All

these problems could be lowered if the primary ion

beam profile is controlled and if the acceptance

region is defined. In order to estimate the trans-

parency and the position of the acceptance region,

it is necessary to compute the acceptance solid

angle Xðx; y;EÞ. However, the function Xðx; y; eÞ,where e ¼ E=Ea, is of particular interest. It will beshown in this article that this function does not

only represent the transparency of an analyzer; it

also includes all of its optical properties. The

knowledge of Xðx; y; eÞ gives us an opportunity to

compute the influence of the analyzer and the

primary beam profile to the energy spectra and to

obtain more realistic energy distribution from the

experimental results. However, to the best of ourknowledge, determining of Xðx; y;EÞ or Xðx; y; eÞ israre in experimental practice (a similar kind of

calculation is performed in [11]). In most of the

cases, only the magnitude of the acceptance solid

angle in the acceptance region is given.

One of the most popular ESA analyzers is the

127� cylindrical energy analyzer. Although it has

lower transparency than, for instance, 180�spherical ESA analyzer with the same resolution, it

is still widely used due to its simplicity and sig-

nificantly lower price. The optical properties of

cylindrical analyzers were calculated in detail by

means of the second order analytical approach

[10–14]. Discrepancies between the experimental

results and the theoretical predictions were ac-

counted mainly to errors in manufacturing ormounting the system. However, the first computer

calculations showed that particle trajectories in the

fringing field region are not calculated properly

using analytical approach [15]. Besides the greater

precision of the numerical simulations as com-

pared to the analytical methods, the former ap-

proach has two more advantages. Analytical

methods are developed for specific analyzer ge-ometries and it is supposed that the optical axis is

on the earth potential – the potentials of the inner

electrode U1 and the outer electrode U2 are �US=2and US=2, respectively (see Fig. 1). This kind of the

electrode potential configuration is usually re-

ferred to as the antisymmetrical deflection voltage

mode. On the other hand, numerical simulations

are not limited concerning the analyzer geometryand the electrode potential configuration. This

allows studying the optical properties of analyzers

with non-standard geometries as well as, of those

that work in deflection voltage modes that are

more convenient in some cases.

Deflection ESA analyzers used in LEIS experi-

ments generally work in the antisymmetrical de-

flection voltage mode. In practice, it is simpler tocontrol the potential of only one of the electrodes,

while the other is on the earth potential. There are

two possibilities: U1 ¼ 0 V and U2 ¼ US or

U1 ¼ �US and U2 ¼ 0 V. These operating regimes

are termed the positive and the negative deflection

voltage modes, respectively. If antisymmetrical

deflection voltage mode is not applied, the poten-

tial of the optical axis will not be 0 V, i.e. theparticle traversing along the main path is going to

be accelerated and decelerated inside the analyzer.

The most important consequence of this effect is

that the analyzer constant will be different as

compared to the case of the antisymmetrical de-

flection voltage mode. Thus, we are introducing

here three different analyzer constants corre-

sponding to the antisymmetrical, the negative andthe positive deflection voltage modes, respectively:

k0, k� and kþ. Other optical properties of the an-alyzer should also be influenced by the type of the

applied operating regime. It is important to stress

that these effects can be efficiently analyzed only by

the use of numerical computations.

In the present article, we are studying in detail

the optical properties of the real 127� cylindricalenergy analyzer. This energy analyzer is used in

LEIS and direct recoil spectrometry (DRS) ex-

periments [16]. Our analysis has been performed

by means of numerical simulations. These simu-

lations are realized using SIMION 3D program,

version 6.0 [17]. From the numerically obtained

trajectories, acceptance solid angle Xðx; y; eÞ is

computed. Xðx; y; eÞ is determined for differenttypes of the deflection voltage mode. The oblique

incidence into the analyzer influences the analyzer

constant and consequently contributes to an error

in the energy measurement. This error is

210 N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219

experimentally determined. Experimental results

are compared to those obtained using the numer-

ical simulations, and the second order analytical

approach [8]. The influence of the applied deflec-tion voltage mode on the analyzer constant is de-

termined and discussed. The analyzer in our

experimental set-up works in the negative deflec-

tion voltage mode. Thus, the difference between

the optical properties of the analyzer working in

the antisymmetrical and negative deflection volt-

age modes is particularly analyzed. Finally, the

obtained results and corresponding consequenceson the shape, the intensity and the position of the

peaks in LEIS spectra are discussed.

2. Numerical simulations

The 127� ESA analyzer used in LEIS and DRS

experiments has the following geometric parame-ters (Fig. 2). The radii of the inside and the outside

electrodes are r1 ¼ 112:5 mm and r2 ¼ 127:5 mm,

respectively. The electrodes are 75 mm high. The

real sector angle of the analyzer is 127�. The dis-tances between the apertures and the electrodes are

both, d ¼ 4 mm. The widths of the entrance andthe exit apertures are 2.8 and 4.3 mm, respectively.

The positions of the slits define the main path,

which coincides with the equipotential line UðrrÞ inthe framework of the analytical approach. Radius

rr ¼ 120 mm, is slightly greater than the radius of

the optical axis inside the analyzer �r0 ¼ 119:76mm [8]. Thus, UðrrÞ is not exactly equal to the

earth potential in case of the antisymmetrical de-flection voltage mode. The distance between the

slits and the electrodes is a1 ¼ 6:5 mm (entrance

slit), i.e. a2 ¼ 8:6 mm (exit slit). The widths of the

entrance and the exit slits are 0.7 and 1 mm, re-

spectively; the height of the slits is h ¼ 10 mm.

In order to compute the 3D trajectories of ions

traversing the analyzer, SIMION 6.0 program was

used for modeling the real 127� analyzer [17]. 2Dfield distribution was computed, because the ratio

between the heights of the slits and the analyzer

Fig. 2. The schematic cross section of the non-ideal 127� cylindrical energy analyzer: (1) inner electrode; (2) outer electrode; (3) en-

trance aperture; (4) exit aperture; (5) entrance slit; and (6) exit slit. The radii of the inner and the outer electrode are denoted by r1 andr2, respectively; d is the distance between the electrodes and the entrance or the exit aperture, a1 and a2 are distances between the

electrodes and the entrance and the exit slit, respectively; the distance between the target and the entrance into the analyzer is denoted

by l. The main path and the trajectory of the particle entering the analyzer with the incident angle ae are also presented.

N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219 211

electrodes is low. The deflection voltage was kept

constant. It was assumed that the particles are

emitted from the surface positioned in front of the

entrance slit of the analyzer. The distance betweenthe entrance slit and the target plane (cf. Fig. 2) is

l ¼ 75 mm in our LEIS experiment [16]. The tra-

jectories were calculated for different ion energies

E and target plane coordinates ðx; yÞ, as parame-ters. An incident angle in the deflection plane aecan be attributed to the x coordinate. It is definedas the angle between the particle trajectory in the

deflection plane, and the main path on the en-trance into the analyzer: ae ¼ arctgðx=lÞ (Fig. 2).Detailed calculations were performed for both, the

antisymmetrical and the negative deflection volt-

age modes. Some computations were done for the

positive deflection voltage mode, too. The grid

density for computing the electrical field distribu-

tion was 0.2 mm. In order to estimate the error of

the computations, some simulations were per-formed with resolution of 0.1 mm. Relative devi-

ation between the corresponding results obtained

using different resolution is less than 0.15%.

3. Experimental

The experimental set-up for measuring the in-

fluence of the oblique incidence to the analyzer

constant is presented in Fig. 3. Nþ ions were ac-

celerated, mass analyzed, and focused to the en-

trance slit of the 127� cylindrical analyzer. Theangular width of the ion beam is about �0.3�.The width of the ion beam profile is 3 mm [16].

The analyzer can be rotated round the axis that is

perpendicular to the drawing plane. Different parts

of the ion beam pass through the entrance slit, by

rotating the analyzer. The ion energy distributions

were determined for different incident angles, in

the range ae 2 ½�3�; 3�. It was assumed that theion energy distribution does not change across the

ion beam profile. Thus, the discrepancies between

the ion energies for the different incident angles

can only be attributed to the energy measurement

error due to the oblique incidence of ions into the

energy analyzer. The experiments were performed

with four different ion energies: 1000, 1500, 2000

and 2500 eV, in the negative deflection voltagemode – U1 ¼ �US, U2 ¼ 0 V (Fig. 2).

4. Results

The influence of the x coordinate and the par-

ticle energy E on the acceptance solid angle

Xðx; y;EÞ was calculated from the computed tra-jectories. The tuning energy Ea was fixed. Accep-

tance solid angle was calculated for the xcoordinate in the [)5, 5] mm range. The compu-

tations were performed for the antisymmetrical

and the negative deflection voltage modes. For a

fixed point ðx; yÞ, transfer function can be defined

as the acceptance solid angle versus particle en-

ergy. Typical transfer function of this analyzer,shown in Fig. 4, has a triangular shape. The

maximum of this function Emðx; yÞ as well as its

full width half-maximum DEðx; yÞ can be calcu-

lated. For the point ð0; 0Þ, Em Ea was obtained.

From this result, the analyzer constant for differ-

ent operating modes was calculated. The results

are given in Table 1, together with appropriate

experimental result for negative deflection voltagemode and analytical result according to the second

Fig. 3. The experimental set-up for determining the influence of

oblique incidence to the error of the energy measurement.

212 N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219

order approximation for the antisymmetrical

voltage mode.

It can be seen from Fig. 4 that Em 6¼ Ea. This is a

consequence of the spherical aberrations. The in-

fluence of the oblique incidence to the error of the

energy measurement computed using the numericalsimulations is expressed as ðEmðaeÞ � Emð0�ÞÞ=Emð0�Þ ¼ ðkðaeÞ � kð0�ÞÞ=kð0�Þ (the deflection

voltage US was constant in these calculations) and

presented in Fig. 5. These results were obtained for

the antisymmetrical deflection voltage mode as well

as, for the negative deflection voltage mode. The

deviation among the results is found to be negli-

gible. The experimental results concerning the in-fluence of the oblique incidence to the error of the

energy measurement is also given in Table 2 as well

as, in Fig. 5. The energy of the primary ion beam Ewas constant. However, the magnitude of US pro-

viding maximum signal, depends on the entrance

angle ae. Thus, in this case, Dk�ðaeÞ=k�ð0�Þ ¼

�DUSðaeÞ=USð0�Þ. The obtained data are also

compared to the results of the second order ana-

lytical approach, for the case of the antisymmetri-

cal deflection voltage mode [8]. Appropriate results

are also presented in Fig. 5.As it was expected, there is no change of com-

puted trajectories if the particle energy and the

deflection voltage are multiplied by the same

constant. This is the well-known characteristic of

charged particles deflected in the electrostatic field.

Two consequences of this fact are the linear de-

pendences of the tuning energy Ea and the width of

the transfer function DE on the deflection voltage.It would be useful in practice to describe optical

properties of an analyzer using a function that

does not depend on the deflection voltage i.e. on

the tuning energy. This can be done by introducing

the relative particle energy e: e ¼ E=Ea. Accep-

tance solid angle defined as a function of the target

Fig. 5. The relative discrepancy of the analyzer constant versus

entrance angle ae. Open marks correspond to the experimental

results (the negative deflection voltage mode); solid squares are

numerical simulation results obtained for the negative deflec-

tion voltage mode; the result obtained by the second order

analytical approach (the antisymmetrical deflection voltage

mode) is presented with the solid line.

Fig. 4. The instrument function of the energy analyzer for

ae ¼ �0:74� in the case of the negative deflection voltage;

Ea ¼ 352:20 eV; Em ¼ 351:77. The magnitude of entrance angle

corresponds to the point (x ¼ �1 mm, y ¼ 0 mm) in the target

plane.

Table 1

The analyzer constant determined by the numerical simulations, the second order analytical approach and the experiment

Type of calculation Numerical simulation Analytical approach Experiment

Deflection voltage Negative Antisymmetrical Positive Antisymmetrical Negative

k 3.522 4.022 4.521 4.011 3.409

The numerical simulation results are obtained for different types of the deflection voltage mode.

N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219 213

plane coordinates and the relative particle energyXðx; y; eÞ, entirely determines optical properties of

an analyzer and does not depend on the tuning

energy. The dependence of the acceptance solid

angle on x and e for y ¼ 0 mm in the case of the

negative deflection voltage mode is given in Fig. 6.

It has a shape of a ridge, which is folded due to the

spherical aberrations. Positions of the local max-

ima of the acceptance solid angle in the x–e planecorrespond to the dependence of the relative error

in the energy measurement on the entrance angle,

given in Fig. 5. The shape of Xðx; eÞy¼0 does not

depend qualitatively on the deflection voltage

mode. It is generally wider with respect to the ecoordinate, if the negative deflection voltage mode

is applied. The average relative difference between

the energy widths is 5.7%. The maximum of

Xðx; eÞy¼0 is about 5% greater in the case of the

negative deflection voltage mode. Thus, applying

the negative deflection voltage mode lowers theenergy resolution and increases the transparency

as compared to case of the antisymmetrical de-

flection voltage mode. The acceptance solid angle

in case of for the negative deflection voltage mode

versus the target plane coordinates for three

magnitudes of e as a parameter is shown in Fig. 7.

5. Discussion

The equipotential surfaces inside the analyzer,

far from the ends of the electrodes, are cylinders.

Particle trajectories obtained using the numerical

simulations are close to the main path. This is

valid even if the antisymmetrical deflection voltage

mode is not applied. However, UðrrÞ is changedwith the change of the deflection voltage mode.

This means that the particle energy is changed

along the trajectory, if the deflection voltage mode

is not antisymmetrical. It is clear that this effect

has strong influence to the analyzer constant.

Three different analyzer constants that are most

interesting in practice are already introduced in

Section 1:

• Antisymmetrical deflection voltage mode U1 ¼�US=2, U2 ¼ US=2, k0 ¼ E0=US; U 0ðr0Þ � 0 V.

• Positive deflection voltage mode U1 ¼ 0 V,

U2 ¼ US, kþ ¼ Eþ=US; Uþðr0Þ � UþS =2.

• Negative deflection voltage mode U1 ¼ �US,

U2 ¼ 0 V, k� ¼ E�=US; U�ðr0Þ � �U�S =2.

In case of the positive (negative) deflection

voltage mode, particles are decelerated (acceler-

ated) at the entrance into the analyzer. Their en-

ergy inside the analyzer is approximately E � US=2ðE þ US=2Þ. At the exit from the analyzer, particles

are going to be accelerated (decelerated), and their

final energy will be equal to the starting energy E.If the deflection voltage is equal for any of theapplied deflection voltage modes (deflecting field is

the same), the energy of the particle traversing

along the main path will approximately be

the same: E0 � Eþ � US=2 � E� þ US=2. After

Table 2

Experimentally determined dependence of the analyzer constant

k� on the incident angle ae

Dk�ðaeÞ=k�ð0�Þ (%)ae (�) E0 ¼

1000 eV

E0 ¼1500 eV

E0 ¼2000 eV

E0 ¼2500 eV

)2.67 )0.2 )0.09 )0.47 )0.48)1.83 )0.27 )0.22 )0.43 )0.51)1.0 )0.03 )0.09 )0.20 )0.270.0 0 0 0 0

0.83 0.2 0.22 0.23 0.13

1.67 0.47 0.54 0.44 0.41

2.5 0.74 0.85 0.64 0.68

Fig. 6. The acceptance solid angle as a function of the target

plane coordinate x and the relative particle energy e, in case of

the negative deflecting voltage mode; y ¼ 0 mm.

214 N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219

dividing this expression by US the following rela-tion is obtained: k0 � kþ � 0:5 � k� þ 0:5. This

rough estimation is remarkably good according to

the results of the numerical simulation (cf. Table

1). The difference between the k� magnitudes ob-

tained experimentally and by use of the computer

simulations are most probably due to the errors in

manufacturing or mounting the system.

In some cases, the optical properties of an an-alyzer can contribute to serious errors during the

measurement of the energy spectra. The deviations

of spectra can generally be manifested as errors in

peak position, peak intensity, and peak shape. The

errors are directly connected to the shape of the

emitting surface i.e. to the primary beam profile on

the target plane. Let us firstly discuss two extreme

cases in order to clarify possible deviations of en-

ergy spectra of scattered ions due to different pri-mary beam profiles.

(a) The case of the narrow primary ion beam pro-

file; if the maximum of the primary ion beam

profile does not coincide with the center of

the target (point ð0; 0Þ), the majority of scat-

tered ions will enter the analyzer with

ae 6¼ 0�. The oblique incidence into the ana-lyzer will contribute to the systematic error in

the energy measurement due to the spherical

aberrations.

(b) The case of the wide primary ion beam profile;

if the emitting surface is wider than the accep-

tance region, only a part of the primary ion

current will contribute to the intensity of LEIS

peak obtained in the spectrum. Thus, LEIS

Fig. 7. The acceptance solid angle as a function of the target plane coordinates Xðx; yÞ in case of the negative deflection voltage mode;relative particle energy e is a parameter: (a) e ¼ 0:9920; (b) e ¼ 0:9993 and (c) e ¼ 1:0106.

N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219 215

peak intensity may not be directly propor-

tional to the total primary ion current, if the

primary ion beam profile is not constant [6].Wide primary beam profile i.e. large emitting

area can also contribute to the significant peak

broadening.

A very good agreement between the experi-

mental and the numerical results concerning the

influence of the oblique incidence to the error in

energy measurement (Fig. 5), indicates that per-formed numerical simulation approximates the

real analyzer very well. It can also be seen that the

results of the second order analytical approach

give the same results as the numerical simulation.

A peak shift due to very fine effects, such as in-

elastic energy losses in LEIS experiments, can be

less than 1% in some cases [18]. The relative error

of the energy measurement due to the oblique in-cidence of scattered ions can easily be of the same

order (cf. Table 2) and even greater if the distance lis small. The ideal sector angle equals about 127.6�that is very close to the optimal magnitude needed

for the first order focusing )127.3� [8]. Unfortu-nately, the relative error is increased to some ex-

tent because the positions of the entrance and the

exit slits do not coincide with the ideal fieldboundaries.

The dependence of the acceptance solid angle

on the x coordinate and the relative particle energye (Fig. 6) is qualitatively similar to the equivalent

results given in [11] for the same type of the ana-

lyzer. The difference between the Xðx; eÞy¼0 for

different deflection voltage modes is a consequence

of the particle acceleration on the entrance into theanalyzer in the case of the negative deflection

voltage mode – the circular component of the

particle velocity is increased, which contributes to

folding the trajectory and decreasing the absolute

magnitude of the angle between the trajectory and

the optical axis. The particle acceleration increases

the maximum of Xðx; eÞy¼0 function due to the

trajectory folding in the non-dispersive plane,while the broadening of Xðx; eÞy¼0 along the energyaxis is not significant. The trajectory folding in the

dispersive plane contributes to the broadening of

Xðx; eÞy¼0 along the energy axis. The maximum of

Xðx; eÞy¼0 cannot be further increased by the tra-

jectory folding in the dispersive plane – it is limited

by the entrance slit width.

The dependence of the acceptance solid angle

on the target plane coordinates and the relativeparticle energy Xðx; y; eÞ (Fig. 7) is the one that hasmajor practical significance. As it was mentioned

already, this function includes all the optical

properties of the analyzer used for measuring the

energy spectra of charged particles emitted from

the target plane. There is a strong dependence of

the acceptance solid angle on the relative particle

energy. This complicates determining the accep-tance region i.e. estimating the influence of the

primary ion beam profile on the peak intensity.

However, the knowledge of Xðx; y; eÞ allows us tocompute the influence of the beam profile on the

position, the intensity and the shape of the peaks

obtained in the LEIS spectra. A possibility to

precisely calculate the energy distribution of scat-

tered ions from the experimentally obtained energyspectra, if primary ion beam profile is measured,

should also be considered.

All these effects are important in every single

experiment in which an ESA analyzer is used.

However, additional effects are present in LEIS

experiments due to the dependence of the scatter-

ing angle on the target plane coordinates

�h ¼ hðx; yÞ. As the energy of scattered ions inLEIS experiment directly depends on the scatter-

ing angle h according to the elastical binary colli-

sion model [1], the deviation of h will contribute tothe deviation of the energy of scattered ions i.e. to

the LEIS peak position and/or broadening. Ac-

cording to the mentioned model, a relative devia-

tion of the energy of scattered ions DE=E due to

the deviation of the scattering angle Dh is definedby the expression

DEE

� �¼ 2 sin hDhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2 � sin2 hp ; ð1Þ

where A represents target mass to projectile mass

ratio. The dependence of DE=E on h for several

magnitudes of A as a parameter is given in Fig. 8.

As in case of the oblique incidence into the ana-

lyzer, this effect can contribute to the peak positionerror and/or to the peak broadening. We find that

the magnitude of Dh ¼ 1:5� is realistic for the

216 N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219

typical LEIS systems, since it also includes the

angular spread of the primary ion beam. It can be

seen that in the case of small target mass to pro-jectile mass ratio this effect contributes to greater

deviations of the energy spectra than an analyzer

with reasonable optical properties.

Generally speaking, systematic errors concern-

ing the peak positions and peak intensities rapidly

increase with the decrease of the distance between

the target and the entrance slit of the analyzer, l(cf. Fig. 2). The decrease of l contributes to theincrease of the acceptance solid angle. Neverthe-

less, the area seen by the detector is decreased, the

spherical aberrations are more pronounced, and

the shift of the beam spot from the center of the

target contributes to greater error in the energy

measurement. The distance l is greater in our case

than in the standard devices of similar type. This

lowers the importance of the mentioned effects inour measurements as compared to other devices.

Another general advantage of our system is that a

small shift of the beam spot in the non-dispersive

plane (along the y-axis) will not contribute to the

significant error concerning the transparency and

the energy measurement, because the magnitude of

the acceptance solid angle is almost constant in a

wide range of y (cf. Fig. 7). This is in contrast to

the analyzers having focusing action in both

planes. Thus, in spite of using the analyzer withmodest optical properties, our system is quite

convenient for performing high quality experi-

ments.

6. Conclusion

In this work, we present a numerical and ex-perimental study of the 127� energy analyzer used

in the LEIS experiments. The main results are the

following:

(a) The dependence of the analyzer acceptance

solid angle on the target plane coordinates and the

relative particle energy X ¼ Xðx; y; eÞ is identifiedas a function that completely describes the realistic

optical characteristics of an energy analyzer. Thisfunction is numerically computed in detail for the

127� cylindrical energy analyzer used in LEIS ex-

periments by means of SIMION program.

(b) The oblique incidence of ions into the en-

ergy analyzer contributes to the error of the energy

Fig. 8. The relative deviation of the energy of scattered ions DE=E versus the scattering angle h. The error of the energy measurement isdue to the deviation of the scattering angle Dh ¼ 1:5�; target mass to projectile mass ratio A is a parameter.

N. Bundaleski et al. / Nucl. Instr. and Meth. in Phys. Res. B 198 (2002) 208–219 217

measurement. The dependence of this error on the

incident angle is experimentally measured. The

same error is calculated from the numerically de-

termined function Xðx; y; eÞ, as well as by means ofthe second order analytical approach. All these

results agree very well.

(c) The influence of the applied deflection volt-

age mode on the optical properties of the analyzer

is computed, too. The major influence is on the

analyzer constant, which can be explained very

well by considering the particle acceleration along

the optical axis. A simple relation between theanalyzer constants for different deflection voltage

modes is introduced: k0 ¼ k� þ 0:5 ¼ kþ � 0:5.This relation is valid for the magnitudes of nu-

merically obtained analyzer constants. Other char-

acteristics of the analyzer are not significantly

changed using different deflection voltage modes.

The energy resolution is decreased and the trans-

parency increased when the negative deflectionvoltage mode is applied instead of the antisym-

metrical deflection voltage mode.

(d) An additional peak broadening and even-

tual peak position shift is present in LEIS experi-

ments due to the scattering angle dependence on

the target plane coordinates. In case of low target

mass to projectile mass ratio, this effect can have

greater influence on the energy spectra than theabove-mentioned effects. This phenomenon is

specific for LEIS as well as, for other scattering

experiments.

The primary beam profile and the relative po-

sition of the target and the analyzer are very im-

portant in surface characterization. This problem

is especially present in the LEIS technique: beam

control and focusing is more difficult in the case oflow energy ion beams as compared to electron

beams and medium or high energy ion beams. A

systematic error in the energy measurement will be

made if the primary beam is not focused on the

center of the target and/or if the optical axis of the

analyzer does not intersect the center of the target.

Thus, in case of fine measurements, such as de-

termination of inelastic energy losses in LEIS ex-periments, extremely good determination of the

experimental set-up geometry and of the primary

beam profile is obligatory. The problem is also

present in the case of the quantitative analysis:

primary ion beam profile will not contribute to the

systematic errors as long as the beam spot is

smaller than the acceptance region [6,9]. However,

it is not simple to define this area – it can stronglydepend on the relative particle energy (cf. Fig. 7).

The most reliable way to quantitatively determine

the influence of the analyzer on the energy spectra

is to compute the spectra obtained by the analyzer

for the defined primary beam profile and the en-

ergy distribution of particles emitted from the

surface, using the knowledge of Xðx; y; eÞ. Unfor-tunately, modeling the energy distribution ofemitted particles is generally a problem. The

problem is increased in the case of LEIS, because

distribution (i.e. scattering angle) depends on the

target plane coordinates. Nevertheless, this type of

computations can provide important information

and further improve the analysis of the experi-

mental results. The investigation of these problems

is in progress.

Acknowledgements

This work has been supported by the Project

2018 from the Ministry of Development, Science

and Technology, Republic of Serbia.

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