Magnetic focusing of an intense slow positron beam for enhanced depth-resolved analysis of thin films and interfaces

Nuclear Instruments and Methods in Physics Research A 488 (2002) 478–492

Magnetic focusing of an intense slow positron beam for enhanced depth-resolved analysis of thin films and interfaces C.V. Falub*, S.W.H. Eijt, P.E. Mijnarends, H. Schut, A. van Veen Interfaculty Reactor Institute, Delft University of Technology, Mekelweg 15, NL-2629 JB Delft, The Netherlands Received 9 April 2001; accepted 12 February 2002

Abstract The intense reactor-based slow positron beam (POSH) at the Delft research reactor has been coupled to a TwoDimensional Angular Correlation of Annihilation Radiation (2D-ACAR) setup. The design is discussed with a new target chamber for the 2D-ACAR setup based on Monte Carlo simulations of the positron trajectories, beam energy distribution and beam transmission in an increasing magnetic field gradient. Numerical simulations and experiment show that when the slow positron beam with a FWHM of 11.6 mm travels in an increasing axial magnetic field created by a strong NdFeB permanent magnet, the intensity loss is negligible above B6 keV and a focusing factor of 5 in diameter is achieved. Monte Carlo simulations and Doppler broadening experiments in the target region show that in this configuration the 2D-ACAR setup can be used to perform depth sensitive studies of defects in thin films with a high resolution. The positron implantation energy can be varied from 0 to 25 keV before entering the non-uniform magnetic field. 2D-ACAR depth-profiling results in He-irradiated Si obtained with the new setup are presented. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.75.Fr; 78.70.Bj; 41.85.Lc; 68.35.Dv Keywords: Slow-positron beam; 2D-ACAR; Magnetic focusing; Doppler broadening

1. Introduction The study of defects is an important field of research in materials science. A better understanding of defects from atomic up to macroscopic scale is necessary for industry to improve the manufacture of materials. Owing to the high preference of positrons for open volume defects, positron annihilation spec*Corresponding author. Tel.: +31-15-278-9053; fax: +3115-278-6422. E-mail address: [email protected] (C.V. Falub).

troscopy (PAS) has been developed over the last decades into a successful non-destructive method for probing low atomic density regions (e.g. vacancies, clusters of vacancies, micro-cavities, open volumes) in materials in a wide range of depths, from the surface to depths of hundreds of nm [1]. Furthermore, PAS has been extensively applied to the study of thin films, layers of embedded nanoclusters and interfaces [2]. When injected into matter, the positron (eþ ) interacts with its surroundings prior to annihilating with an electron into gamma (g) rays. The characteristics of the emitted g rays are different

0168-9002/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 2 ) 0 0 5 6 6 - 1

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for positrons annihilating near a defect compared to positrons which annihilate in a defect-free material. By measuring the positron lifetime, the energy distribution and the angular correlation of the annihilation g0 s; one obtains information about the physical properties of the material under investigation. Positrons are produced either by nuclear reactions in radioactive isotopes or by pair production using high-energy electromagnetic radiation. Because of their broad energy spectrum, the fast positrons emitted by a radioactive bþ source (e.g. 22 Na) has long been used for bulk studies with the aid of conventional positron spectroscopies [3]. More recently, the development of radio-isotopebased slow mono-energetic positron beams has led to the study of a wealth of interesting surface and near surface phenomena [4]. By varying the initial energy of the implanted positrons one can study defects at different depths. The importance of depth-profiling studies becomes clear if one considers layered systems obtained by various deposition or implantation techniques. In these systems defects may appear at the interfaces of the layers because of low mobility of the deposited atoms, imperfect regrowth during crystallization, etc. However, in view of its low count rate, application of the high resolution Two Dimensional Angular Correlation of Annihilation Radiation (2DACAR) method to the study of surfaces and layered systems had to wait for the development of high-intensity slow positron beams. Nowadays, such beams are operational or under construction at a few laboratories in the world [5]. Depth profiling positron 2D-ACAR thin film studies have been performed for the first time at Brookhaven National Laboratory on the SiO2–Si system employing a strong 64Cu source [6]. Recently successful results have been obtained with the construction of a stable high intensity reactorbased slow positron beam, POSH (B108 e+/s), at the Delft 2 MW research reactor [7,8]. The positrons are obtained by pair-formation induced by g rays produced in the core of the reactor and extracted by means of electrostatic lenses. Transport of the positrons over a distance of B25 m is achieved by an axial magnetic field produced by a combination of solenoids and coils. Since March

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2000 the POSH beam is coupled to a highresolution 2D-ACAR setup which employs two Anger cameras up to 23 m apart [9]. In this configuration it is the first 2D-ACAR setup using a continuous variable-energy slow positron beam of stable intensity. The first depth-profiling 2DACAR studies of nanocavities in He-irradiated Si and of nanoprecipitates in He and Li irradiated MgO with a high momentum resolution have been performed using this setup [10,11]. An important aspect of positron beam technology is the size of the beam. A small diameter positron beam improves the angular resolution of the 2D-ACAR setup. Beam size reduction can be achieved by remoderation of the beam, but only at the expense of a substantial (75%) loss of intensity. In this paper, we present a different approach of positron beam focusing using a high-gradient magnetic field. In order to improve the angular resolution of the 2D-ACAR apparatus, a focusing system has been designed that involves the POSH transport solenoids, a set of Helmholtz coils and a strong NdFeB permanent magnet. The resulting beam diameter is B3 mm FWHM. The paper is organized as follows. In Section 2 a brief discussion of the relevant theory is given. The energy distribution (important for the positron implantation profile), the beam profile and some aspects regarding the non-adiabatic effects of strong fields on the final beam parameters at the target are discussed. In Section 3 we present and discuss our results, while our conclusions are collected in Section 4.

2. Theory and calculation methods The kinetic energy of positrons guided by a magnetic field consists of a transversal and a longitudinal part, corresponding to the transversal and longitudinal components of the positron velocity, respectively. When one implants positrons in materials, the longitudinal energy determines the implantation. An ideal focusing system for a mono-energetic beam would reduce the beam diameter while conserving beam monochromaticity and offering the highest possible transmission.

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Fig. 1. The principle of positron beam focusing in a non-uniform magnetic field.

Therefore, one can ask: Is the quality of a monoenergetic beam influenced by a high-gradient magnetic field? The principle of positron beam focusing in a non-uniform magnetic field is presented in Fig. 1. A mono-energetic positron beam is guided from a region of uniform magnetic field into a convergent magnetic field of cylindrical symmetry. Those positrons that enter the uniform magnetic field region at a non-zero angle with respect to the direction of the field undergo helical motion around the lines of force. If the change in the magnetic field, experienced by a positron in the course of a revolution, can be neglected then both the flux through the particle orbit and the corresponding transversal momentum are adiabatic invariants [12]: Br2 ¼ const:

ð1Þ

p2T ¼ const: B

ð2Þ

where B is the magnetic field strength, r the radius of the positron orbit, and pT the transversal component of the positron momentum. These two equations can also be written as follows:
1=2 sin a2 B2 ¼ ð3Þ sin a1 B1


v2L Dv 1 

B2 2 sin a1 B1

1=2 ð4Þ

where B1 ; B2 ; a1 and a2 are the magnetic field strengths and the angles between the positron velocities and the magnetic field lines at the beginning of the non-uniform magnetic field region and at the target, respectively, and v2L is the longitudinal component of the positron velocity at the target (see Fig. 1). The small angle ~2 and ~ between B v 2L ; of the order of 11, has been neglected. From these equations one can easily deduce the following: (i) a positron spiraling along the lines of force in an orbit of decreasing radius converts more and more longitudinal energy into transversal energy; (ii) there is a limiting value of the magnetic field strength at which all longitudinal energy is transformed into transversal energy and the positrons begin to be reflected; (iii) while the positrons are being focused, their orbits are always bounded by the same field lines. Eqs. (1) and (2) are only valid in the adiabatic limit given by K ¼ ð2 p mv=eB2 Þd Bz =dz {1

ð5Þ

where K is the adiabaticity parameter1, v the positron velocity and Bz is the magnetic field 1 Other, more global, adiabaticity parameters are sometimes used (cf. [13]). The use of K permits the localization of regions contributing to the overall adiabaticity and therefore has our preference.

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Fig. 2. The POSH-ACAR electromagnetic configuration near the 2D-ACAR setup simulated using the CERN-POISSON program package. The component elements are: (1) positron beam; (2) solenoid; (3) inner guiding tube; (4) accelerator; (5) set of Helmholtz coils; (6) sample; (7) NdFeB magnet. The contours are lines of constant vector (black) and electrostatic (grey) potential, respectively. The dimensions are given in cm.

strength along the z-axis [13]. If relation (3) is not valid because of the particle velocity being too high or the gradient of the magnetic field being too large, the final spot on the target after focusing will be affected by non-adiabatic aberrations [13]. The magnetic focusing system close to the target of the 2D-ACAR setup, presented in Fig. 2, has been calculated using the POISCR program package, a set of computer programs developed at CERN which solves the Poisson or Laplace equation in two-dimensional regions with linear or non-linear properties [14]. The trajectories of the positrons, their longitudinal energy and the beam size at the target have been determined using the Monte Carlo simulation program POEM (polarized beam simulator in electric and magnetic fields), developed by Kumita [15]. This program, based on GEANT (software developed at CERN simulating the passage of elementary particles through matter), calculates the positron trajectory and its spin motion in electric and magnetic fields by solving the relativistic equation of motion and spin precession. Basically, we first calculate the electric and magnetic fields in our system by means of program POISCR and then input these results into POEM.

3. Results and discussion 3.1. Electric and magnetic field configuration In this section we present the calculated magnetic and electric field configuration of the focusing system near the target of the 2D-ACAR setup. The positrons, created at 1.5 keV and guided by a uniform magnetic field of 100 G, are accelerated to the desired energy in a simple accelerator consisting of 11 equidistant electrostatic lenses. In order to maintain a constant magnetic field, the accelerator is surrounded by a system of Helmholtz coils. A strong NdFeB permanent magnet of 1.37 T produces a convergent high-gradient magnetic field necessary for the final focusing of the positron beam. The fields produced by different parts of the system, as determined from the POISCR simulations, are characterized in Fig. 2 by their equipotential lines (V ðrÞ ¼ const; AðrÞ ¼ const:). In order to avoid the electric field penetration at the end of the inner guide tube near the sample, the magnet, the last lens of the accelerator, the inner guide tube and the sample are at the same potential. In Fig. 3 we show the magnetic field and the adiabaticity parameter (K) for different values of the positron

C.V. Falub et al. / Nuclear Instruments and Methods in Physics Research A 488 (2002) 478–492 0.4 Bz vs. z K ; E = 1.5 keV K ; E = 3 keV) K ; E = 10 keV) K ; E = 20 keV) Bz x 20 vs. z

2.0

0.3 Magnetic field imperfections

Bz x 20

1.5

0.2

K=1

1.0

POSH

Magnet

K - adiabaticity parameter (dimensionless)

2.5

accel.

Bz (Tesla)

482

1690 Gauss

0.1

0.5

100 Gauss

0.0 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

z - axis (m)

0.7

0.8

0.9

Target

Fig. 3. The adiabaticity parameter for a few values of the positron energy after the acceleration stage. The magnetic field along the section is also presented.

energy. One observes that the magnetic field along the beam axis increases from a constant value of 100 G along the POSH tube and the accelerator to 1690 G at the sample. A 20 times magnification shows that at the end of the POSH tube and in the region of the Helmholtz coils the magnetic field exhibits some imperfections. It is clear that for positron energies higher than 1.5 keV, deviations from the adiabatic limit take place in front of the target. Owing to the small fluctuations of the magnetic field, small deviations from the adiabatic limit also occur at the end of the POSH tube and in the region of the Helmholtz coils. These deviations may influence the beam profile at the target and are responsible for non-adiabatic aberrations [13]. 3.2. Positron trajectory calculations Using the simulation program POEM, we have calculated the trajectories of the positrons, their longitudinal energy and the beam size at the target for the system presented in the previous subsection. At the entrance to the magnetic field system we consider a Gaussian-shaped positron beam with FWHM=11.6 mm, which constitutes a good description of the POSH beam profile. In order to study the effect of the transversal energy on the transport and focusing properties of the beam, we

have performed the calculations for initial angles of the positron velocity from 01 up to 201 with respect to the beam axis. The focusing factor is determined by defining two characteristic spot dimensions at the target: the first one is a circle of radius R1 containing 50% of the positrons and the second one a circle of radius R2 containing 90% of the positrons. This is necessary because of the nonadiabatic aberrations which create halos and ringshaped profiles strongly deviating from Gaussians. The distribution of longitudinal energies at the target has been determined by summing the contributions for a range of equidistant (11) angles. Due to the fact that all simulations were performed with the same initial total number of positrons one has to weigh every angular contribution in accordance with the POSH angular distribution. 3.3. Experimental determination of the angular distribution of the POSH beam The retarding field experiments for determining the angular distribution were performed as following. We installed in the POSH beamline a system consisting of three grids, the first and the last one being grounded while a variable bias is applied to the second one (see the inset of Fig. 4). Thus, positrons produced at about 1.1 keV are stopped

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Fig. 4. POSH beam retarding electric field measurements: (a) the count rate measured by a Ge detector as a function of the grid bias U. The inset shows a schematic diagram of the experimental setup; (b) the angular distribution of the POSH beam derived from these measurements.

in a variable electrostatic field2. A Ge detector is used to monitor the g flux obtained when the positrons are stopped in a plate located behind the last grid. Due to the non-zero angle of the positron velocity vector with respect to the beam axis, the longitudinal energy is smaller than the energy at which the positron is produced. Hence, by determining the voltage at which the positrons are stopped (cf. Fig. 4(a)) we can obtain the angle between the positron velocity ffi and the beam axis pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from a ¼ arccosð eU=Etot Þ; where U is the voltage at which the positron is stopped and Etot the total positron energy. The smoothed curve in Fig. 4(a) represents the fit of the left-hand side of the experimental data with a combination of three Gaussians centered at 1.1 keV with FWHM of 63, 276 and 1535 V, respectively. The extracted angular distribution of the POSH beam is shown in Fig. 4(b). Thus, we determined that 50% of the positrons in the POSH beam have angles up to 7.71 while 90% of the positrons have an angle o301. 3.4. The beam quality at the 2D-ACAR target We have performed our simulations for two implantation energies, 3 and 10 keV, respectively. In Table 1 the results of the POEM simulations of the positron beam spot sizes at the target for different initial angles and positron energies are 2

In this measurement only one section of the POSH source has been used. When all the sections are used the positrons are produced at 1.5 keV.

Table 1 The target spot size for implantation energies of 3 and 10 keV and different values of the initial angle. R1 and R2 are the radii of the spots containing 50% and 90% of the positrons, respectively Initial

Target 3 keV

R1 (mm) R2 (mm)

5.8 8.9

10 keV

01

51

81

181

01

51

81

181

1.2 2.0

1.2 2.0

1.6 2.6

1.8 3.1

1.2 1.7

1.1 1.8

1.0 2.0

2.2 3.7

presented. One observes that, up to angles of at least 81, a target spot is obtained with an approximately five times smaller diameter. For larger initial angles positrons may spiral far away from the symmetry axis of the system. In doing so they enter regions where the magnetic field imperfections are more pronounced or the electric field responsible for acceleration is not parallel anymore to the magnetic field. This, together with the non-adiabatic aberrations, creates halos and ring-shaped beam profiles (see Fig. 5). As expected, the influence of the non-adiabatic aberrations is more evident for 10 keV than for 3 keV (see Table 1). Taking into account all the angles, our estimate of the beam size at the target is 3.2 mm FWHM. This estimate is in good agreement with the experimental value of 3.3 mm FWHM which was obtained by intercepting the 3.5 keV positron beam at the target position with a moveable thin wire. The wire was used to determine the positron

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Fig. 5. POEM simulations of the positron beam profile: (a) The generated initial beam profile is a two-dimensional Gaussian with FWHM=11.6 mm; (b) and (c) the beam profile at the target for 3 keV positrons and 01 and 81 initial angles, respectively; (d) the beam profile at the target for 10 keV positrons and an initial angle of 161.

beam profile in two ways: (i) by measuring the small current (a few pA) which appears when the wire is hit by the positrons; (ii) by measuring the intensity of the g rays coming from positrons annihilating in the wire with a Ge detector (a narrow lead slit was used in order to observe only annihilations coming from the wire).

Concerning the energy spread of the beam we observed that the maximum of the longitudinal positron energy distribution at the target moves towards lower energies if the initial angle of the positron velocity is increased, consistent with Eq. (4). Moreover, the width of the energy distribution increases for large values of the initial

C.V. Falub et al. / Nuclear Instruments and Methods in Physics Research A 488 (2002) 478–492

angle, which is clearly shown in Figs. 6(a) and (b) for initial angles of 01 and 81, respectively. In these figures, the simulated transversal positions of the positrons in the beam at the entrance and the target are represented versus the longitudinal energies of the positrons at the target. The histograms of the longitudinal energy distributions are also shown. For an initial angle of 01, the forward energy distribution at the target follows a parabolic law with respect to the initial and final

transversal positions of the positron in the beam (see Fig. 6(a)). This behavior can be explained in the adiabatic limit as being due to the imperfections in the magnetic field at the end of POSH and in the region of the Helmholtz coils. Since in these regions the velocity vector and the magnetic field lines are not parallel, the positron velocity vector gains a small angle which is proportional to the transversal position of the positron in the beam, as shown by the simulations. This angle becomes less

180

f ri vs. Elong

(a)

16

f rf vs. Elong

14

Histogram of the energy distribution Smoothed energy distribution Model (adiabatic approx.)

160 140

12

120

10

100

8

80

6

60

4

40

2

20

Intensity

r (mm)

18

485

0

0

2.84

2.88

2.92

2.96

3.00

f Elong (keV)

180

18

(b)

16

f rf vs. Elong

14

Histogram of the energy distribution Smoothed energy distribution Model (adiabatic approximation)

160 140

12

120

10

100

8

80

6

60

4

40

2

20

0 1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

Intensity

r (mm)

f ri vs. Elong

0 2.8

f Elong (keV)

Fig. 6. Simulated longitudinal energy distribution at the target for different angles of the positron velocity: (a) 01; (b) 81. The positron energy at the entrance is 3 keV. The total number of positrons simulated is 1000.

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in the acceleration section but is then amplified in the region of non-uniform magnetic field. For nonzero initial angles this supplementary angle in the region of the magnetic field imperfections may be subtracted or added to the initial angle. Since the positron velocity and the additional velocity gained due to the magnetic field imperfections are combined vectorially, the distributions of the initial and final transversal positions of the positrons in the beam versus the longitudinal energy are of the type presented in Fig. 6(b). One observes that all the points of these distributions are distributed between two extremes due to the vectorial summation of the velocities, which we could describe (see the dashed lines in Fig. 6(b)) in the adiabatic formalism presented in Section 2. The small angle gained (or lost) by the positron velocity at the end of the POSH tube is da= arctan (Br =Bz ), where Br and Bz are the radial and longitudinal components of the magnetic field. We found that this angle is linearly dependent on the transversal position of the positron in the beam. In the acceleration stage da is reduced depending on how much the positrons are accelerated. Finally, the positrons enter the non-uniform magnetic field where they can gain or loose a small angle due to

the convergence of the field. By variation of the initial angle we determined that up to 121 the POEM simulations are well described by the adiabatic formalism. For larger angles we could not describe the two extremes of the simulated distributions. This is due to the positrons moving far away from the beam axis and entering regions where the magnetic and the electric field are not parallel. Also, we could not describe the distributions for high values of the positron energy when the adiabatic formalism is no longer valid. For a better understanding of the influence of the positron velocity angle on the beam focusing we present in Fig. 7 the simulated final positron positions at the target versus the initial positions in the beam for different initial angles. We observe that for 01 the (ri ; rf ) points are nearly distributed on the line rf ¼ ri (Bi =Bf Þ1=2 ; as expected from the adiabatic formalism (see Section 2), where Bi and Bf are the magnetic field strengths at the entrance (B100 G) and the target (B1700 G), respectively. The very small deviations from the central line are due to the small imperfections of the magnetic field at the end of the POSH tube and the region of the accelerator. One observes that for angles different from zero the (ri ; rf ) points are distributed around

5

rf (mm)

4

3

2

o

18 8o o 5 o 0 1/ 2 rf = ri (Bi /Bf )

1

0 0

2

4

6

8

10

12

14

16

18

ri (mm) Fig. 7. Influence of the initial angle on the positron distribution at the target. Bi and Bf represent the magnetic field strengths at the entrance (B100 G) and the target (B1700 G), respectively, while ri and rf denote the radii of the positron trajectories at their positions.

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the central line rf ¼ ri ðBi =Bf Þ1=2 ; but that they are limited by two extremes that are nearly parallel to the central line (the dashed lines in Fig. 7). Clearly, the larger the initial angle, the larger the spread of the (ri ; rf ) points around the central line. It is interesting to note that, for example, in the case of an initial angle of 81 positrons entering the system on the axis arrive at the target at 0.8 mm off axis. Similarly, positrons entering the system at a distance of B3 mm from the beam axis arrive at the target in the interval of 0–1.6 mm. For larger angles (e.g. 181), the (ri ; rf ) points are distributed far away from the central line and the limits of the distribution cannot be described anymore by the adiabatic formalism.

acceleration stage the quality of the beam is affected during the transport in the non-uniform magnetic field. However, when positrons are implanted deeper in materials the drawback of a wider energy distribution of the beam is compensated by the broad implantation profile. The broadening of the implantation profile is due to the elastic and inelastic collisions of the positrons with electrons, nuclei and phonons, resulting in a distribution of the annihilating slow positrons over a range of depths. In conclusion, the simulations show that the system presented has a reasonably sharp final longitudinal energy distribution, allowing depth-sensitive 2D-ACAR experiments as we shall show later in this section.

3.5. Energy distribution at the target

3.6. Results obtained with the focused variable energy positron beam

The simulated longitudinal energy distributions at the target for 3 and 10 keV positrons are presented in Fig. 8 and given in Table 2. Thus, we determined that for 3 keV, 50% of the positrons at the target have energies between 2.8 and 3 keV and 90% of them have energies between 1.8 and 3 keV. For 10 keV positrons these energy intervals are 8.9–10 and 5.7–10 keV, respectively. Thus, by increasing the positron energy in the

Based on these simulations, we have built a new vacuum chamber for the 2D-ACAR setup shown in Fig. 9 and have carried out our first depthresolved 2D-ACAR measurements on annealed helium-irradiated silicon [10]. Cross-section transmission electron microscopy images of the subsurface region on similar annealed helium-irradiated silicon samples show the presence of a well defined

500 3 keV 10 keV

Intensity

400

300

200

100

0 0

1

2

3

4

5

6

7

8

9

10

11

Positron energy (keV) Fig. 8. Simulated POSH energy distributions for 3 and 10 keV (positrons with an energy of 1.5 keV accelerated by 1.5 and 8.5 keV, respectively; the area under the curves represents 1000 positrons).

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Table 2 The widths of the energy distribution for implantation energies of 3 and 10 keV, respectively. DE1 and DE2 are the widths of the energy distribution for 50% and 90% of the positrons, respectively

DE1 (keV) DE2 (keV)

3 keV

10 keV

0.20 1.18

1.10 4.25

0.64 VEP experiment POSH experiment

0.62

S parameter

488

0.60

0.58

0.56

0.54

2D-ACAR

2D-ACAR

0.52 0

5

10

15

20

25

Positron energy (keV)

Fig. 10. Doppler broadening depth-profiling experiments for He implanted Si using the high-resolution 22Na-based positron beam VEP and the reactor-based POSH beam. The values of the positron energy at which the 2D-ACAR experiments were performed are indicated.

Fig. 9. Accelerator section and target chamber of the POSHACAR system.

layer of nanocavities with an average diameter ( for annealing temperatures above larger than 85 A 7001C [16]. The positron implantation energies for the angular correlation experiments were selected by performing Doppler-broadening depth profiling experiments at the 2D-ACAR target chamber employing a Ge detector with a resolution of 1.2 keV. In Fig. 10 we present the results of these Doppler broadening measurements on heliumirradiated silicon performed with the VEP beam

line (a 22Na-based slow positron beam) [17] and with the POSH beam [10]. Basically, an increase of the S parameter means that positrons annihilate in defects via formation of a positronium atom, the bound state between an electron and a positron [18,19]. The results show that the S parameter exhibits a maximum attributed to the layer of nanocavities obtained by implanting He in Si. The different maximum values for the S parameter are due to different Ge detectors used in these two experiments. One observes that, although the trend of the curves is the same in both experiments, the maxima of S are located at different energies, 6.5 keV for VEP and B7 keV for POSH, respectively. This difference is due to the conversion of the longitudinal energy into transversal energy for positrons entering the strong magnetic field of the POSH-ACAR system. Thus, the positrons need about 0.5 keV higher energy to be optimally implanted in the layer of cavities. A similar figure is also obtained by interpolation between 3 and 10 keV in the first line of Table 2. In conclusion, although the VEP has a better defined energy distribution (with a width of a few eV), the quality of the POSH beam after focusing is good enough to determine small variations of the S parameter with implantation depth. Figs. 11(a) and (b) show the 2D-ACAR spectra for positron

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Fig. 11. 2D-ACAR spectra of helium-irradiated Si(1 0 0) for (a) 7.5 keV and (b) 16.5 keV slow positrons. The spectrum in (c), obtained with fast positrons generated by a 22Na source, is representative of bulk annihilations in Si(1 0 0).

beam energies of 7.5 and 16.5 keV, while the bulk reference 2D-ACAR spectrum is presented in Fig. 11(c). The spectra were normalized to the total number of counts. One observes that the 7.5 keV spectrum exhibits an enhanced peak reflecting the contribution of the layer containing the nanocavities, while at 16.5 keV the spectrum resembles that of the bulk. These results show the capability of the POSH beam for 2D-ACAR depth profiling studies. 3.7. Intensity losses The transmission of the POSH-ACAR system was determined by measuring the coincidence count rate as a function of the implantation energy (see Fig. 12). One observes that for small values of the implantation energy the coincidence count rate is low, which means that many positrons are

reflected. These are positrons that have a large angle between the velocity vector and the beam axis, and on entering the convergent magnetic field can be reflected. For implantation energies higher than 6 keV, the coincidence count rate reaches the saturation level (B45 counts/s) corresponding to a transmission factor of the positron beam of 100%. Using the experimentally determined POSH angular distribution and the POEM simulations we estimated the number of positrons that are reflected for 3 and 10 keV at 20% and 0.5%, respectively. The difference of 10% between the simulated and experimental fraction of the reflected positrons at 3 keV could be explained if one takes into account that the angular distribution of POSH has been determined about 12 m before the POSH-ACAR target. Due to the small imperfections in the axial magnetic field at the positions where the long solenoids of the POSH beam are

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490

100 40 80 30 60 20

40 exp eri ment al

10

POEM simu lati on s

0

Transmission (%)

Coin ci dence count rate (cou nts)

50

20 0

0

3

6

9 E (keV)

12

15

18

Fig. 12. The transmission of the positron beam POSH in the POSH-ACAR system determined by performing coincidence 2D-ACAR measurements. The simulated values of the transmission for 3 and 10 keV are indicated.

interrupted,3 the angular distribution of the positron beam becomes wider. Therefore one expects to find in reality a higher fraction of reflected positrons than predicted by the simulations. 3.8. Alternative approach for focusing At an early stage of our positron beam research we have tried a different magnetic focusing configuration (see Fig. 13). Basically, in this system the positrons accelerated at 30 keV are decelerated to the desired energy in front of the target. In order to have a uniform magnetic field in the acceleration region we considered a special yoke consisting of two iron pieces, one of them surrounding the system and the other one, behind the sample, having a concave shape. Varying the dimensions of the yoke we could simulate a uniform magnetic field of 1640 G in a region of B6 cm length and B12 cm in diameter in front of the sample. However, after testing the configuration by means of the POEM software, this system proved to have a low transmission (about 25%) and a poor longitudinal energy resolution, 3 The POSH beam line consists of a few long pipes connected by vacuum chambers where the ion getter pumps are placed. In order to avoid strong fluctuations in the magnetic field at the position of the vacuum chambers Helmholtz coils are used.

although the focusing factor was B5 in diameter. Moreover, this system would have been more difficult and costly to build. Its poor performance may be due to the fact that positrons begin to feel the deceleration potential before entering the uniform magnetic field. In the same region the gradient of the magnetic field is very high which is expected to generate additional problems due to the non-adiabatic aberrations. 3.9. Comparison with a superconductor magnetic focusing system In order to obtain microprobes from positron beams, the use of ultra-high magnetic fields generated by superconducting magnets has been suggested [20]. Basically, in Ref. [20] a B1 cm diameter positron beam in a 50 G guide field is focused to B267 mm in a 7 T field generated by the superconducting magnet. However, there are a few important aspects that were not considered. We mentioned in Section 2 that in a converging magnetic field the positrons can be reflected due to the conversion of longitudinal energy into transversal energy. For the system presented in Ref. [20], in the adiabatic limit the maximum angle for the positron velocity with respect to the symmetry axis of the magnetic field (above which all positrons are reflected) is 1.51. Positrons entering the system off axis loose their longitudinal energy more rapidly than those traveling along the axis (see Fig. 6). Hence, these positrons can be reflected even at zero angle, and all positrons traveling at a distance larger than 9 mm from the axis are indeed reflected. If one assumes that the incident positron beam is POSH one can estimate that the transmission of this system will be very low, B8%, while a focusing factor of B40 is obtained. Therefore, the system based on a superconducting magnet would have a low transmission and would not be efficient for 2D-ACAR. However, this transmission factor could be very much improved if one uses a very low-divergence positron beam, which could be obtained by remoderation of the POSH beam. However, as pointed out earlier the resulting intensity at the target will be much lower. Another important aspect is that for such a system the longitudinal

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Fig. 13. Sketch of the alternative magnetic focusing design. Positrons with energy of 30 keV, entering the system on the z-axis at 0.05 m, are decelerated in front of the target. A special yoke is used to create a uniform magnetic field in the deceleration region. The electric and magnetic field are characterized by equipotential lines.

energy distribution of the beam at the target is much broader, which leads to a large uncertainty in the depth at which the positrons annihilate. Owing to the radial component of the magnetic field, the longitudinal energy of the positrons which spiral around field lines parallel to, but at some distance from, the axis, is smaller than the longitudinal energy of positrons that perform helices of a very small radius around the axis. For positrons moving far from the symmetry axis, a strongly convergent magnetic field will lead to smaller forward energies at the target and there-

fore to a broad longitudinal energy distribution. In conclusion, the system proposed in Ref. [20] could be appropriate for focusing moderated particle beams but is not suitable for depth-profiling applications using the 2D-ACAR technique which requires high intensity.

4. Conclusions A new positron beam focusing system has been designed for use in 2D-ACAR experiments. It

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involves the reactor-based positron beam POSH, a set of Helmholtz coils and a strong (1.37 T) permanent magnet. It is shown that with the aid of this magnet a linear focusing factor of about 5 can be achieved, which is independent of the initial positron angle up to 81. The dependence of the longitudinal energy distribution at the target (which determines the penetration depth into the sample) on the positron angle shows a substantial broadening for angles larger than about 51. Because of the angular distribution of POSH, which is of the order of 7.71 FWHM, and the moderate gradient of the magnetic field, the nonadiabatic aberrations, which affect the beam profile at the target, can be neglected for our system. Therefore, a well-focused target spot is achieved which to a large extent conserves the intensity and monochromaticity of the beam. This system allows the study of small samples with good statistics. In general, however, the intensity and the monochromaticity of a beam are reduced because of the conversion of longitudinal energy into energy of rotation when a mono-energetic beam is focused in a high gradient magnetic field. Therefore, a positron beam focusing system, in general, requires finding a compromise between the focusing factor, the transmission and the energy distribution of the beam.

Acknowledgements We are greatly indebted to Dr. Kumita for making available the POEM software and assisting us in its use.

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