New approach to energy loss measurements

Nuclear Instruments and Methods in Physics Research B 195 (2002) 147–165 www.elsevier.com/locate/nimb

New approach to energy loss measurements W.H. Trzaska a,b,*, V. Lyapin a,b, T. Alanko a, M. Mutterer c, J. R€ ais€ anen a, G. Tjurin a, M. Wojdyr a,d a

Department of Physics, University of Jyv€askyl€a, P.O. Box 35, FIN-40351, Jyv€askyl€a, Finland b Helsinki Institute of Physics, Finland c Technical University Darmstadt, Germany d Technical University Warsaw, Poland Received 16 November 2001; received in revised form 26 February 2002

Abstract A new approach to energy loss measurements is proposed. In the same experiment electronic stopping force (power) in gold, nickel, carbon, polycarbonate and Havar for 40 Ar, 28 Si, 16 O, 4 He and 1 H ions in the energy range 0.12–11 MeV/u has been measured. In this paper we give the full results for gold, nickel, and carbon and for 40 Ar, 16 O, 4 He and 1 H ions. Good agreement of the measured stopping force values for light ions with literature data is interpreted as the positive test of the experimental technique. The same technique used with heavy ions yields agreement with the published data only for energies above 1 MeV/u. At lower energies we observe progressively increasing discrepancy. This discrepancy is removed completely as soon as we neglect pulse height defect compensation. This observation makes us believe that the majority of the published results as well as semi-empirical calculations based on them (like the popular SRIM) may be in error at low ion energies. Procedures to evaluate foil quality and to determine pulse height defect for charged ions in semiconductor detectors are given. Our improved MCP-based time pick-off detector is described in detail.
2002 Elsevier Science B.V. All rights reserved. PACS: 34.50.Bw; 81.05.Bx Keywords: Stopping force; Stopping power; dE=dx; S; Pulse height defect; Time of flight (TOF); Microchannel plate (MCP); 40 Ar; 16 O; 4 He; 1 H; Heavy ions; Alpha-particles; Protons; Au; Ni; C

1. Introduction In our recent publication [1] we have proposed a new way to measure specific energy loss of charged particles in thin absorbers. The method combines * Corresponding author. Address: Department of Physics, University of Jyv€ askyl€ a, P.O. Box 35, FIN-40351, Jyv€askyl€a, Finland. Tel.: +358-14-260-2409; fax: +358-14-260-2351. E-mail address: [email protected].fi (W.H. Trzaska).

time-of-flight (TOF) detector with traditional, semiconductor-based set-up for transmission-type measurements. This combination should, as we have outlined in our previous paper [1], produce energy loss values for ions covering up to three decades of incident energy in a single measurement. In other words, one experiment should map the entire dE=dx peak including its maximum and both the low- and the high-energy tail. In this work we are demonstrating that our claim was

0168-583X/02/$ - see front matter
2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 2 ) 0 1 2 5 5 - 7

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fully justified. During just a few days of beam time we have measured electronic stopping force (power) values in gold, nickel, magnesium, carbon, polycarbonate, and Havar for 40 Ar ions in the energy range 0.12–11 MeV per nucleon and for 28 Si, 16 O, 4 He and 1 H ions in 0.13–6 MeV/u energy range. Such huge amount of data is impossible to present in one publication. Therefore we have decided to concentrate here only on the three absorbers that have been intensively studied in the past namely gold, nickel and carbon and on four ions: 40 Ar, 16 O, 4 He, 1 H. The rest of the results (polycarbonate and Havar) will be published together with theoretical treatment in a dedicated paper. Also the 28 Si data will be published separately as we would like to gain more statistics for the low energy part for that ion. The largest errors in transmission-type energy loss measurements arise from (i) problems with precise energy calibration and (ii) uncertainties in absorber thickness determination (and its uniformity). In this paper we are addressing both of these problems. For instance, we have described a procedure to determine pulse-height defect (PHD) corrections for energy detectors. This procedure is an inherent part of our measurement and does not require any additional equipment. To judge the uniformity of absorber foils we propose to compare line broadening to the straggling expected from a perfectly uniform foil. We have also made further improvements to our experimental set-up. In particular, we have developed a fully symmetric, twin-mirror time pick-off MCP detector for TOF measurements. The upgraded TOF system can now be used not only for tagging [1] but also to produce high quality spectrometric data. The new detector and its dedicated electronics are described here in detail.

2. Definitions Since the final results presented in this paper differ from many previous measurements, we felt obliged to state explicitly the basic terms used in this paper. We hope it will help the reader and assure him that the differences we report are real and not an artefact of analysis or nomenclature nuances.

It has been recently proposed to replace the old and well-established term stopping power by a more appropriate term stopping force [2]. We shall follow this recommendation but for the sake of clarity it is important to state that stopping force in the new notation is exactly the same as stopping power in the traditional notation. In case of doubt, publications by International Commission on Radiation Units and Measurements (ICRU) [http://www. icru.org/], like the latest REPORT 49 [3] are a valuable reference on units, exact definitions of commonly used terms and on data evaluations. The nominator in the stopping force expression stands for energy loss. In actual experiments there are at least three different values that can all be identified with energy loss: mean energy loss, median energy loss and the most probable energy loss. For Gaussian distribution all three values are identical. Generally, the more asymmetric the distribution, the more the three values diverge. In our data these differences are small (although measurable) as the energy slices resulting from TOF tagging [1] are nearly bell-shaped. Of the two physically meaningful parameters (mean and the most probable) we have chosen the mean energy loss as the measured quantity. The big advantage of using the mean (centroid) value as opposed to the most probable value is that it can be accurately determined even for peaks with very low number of counts. To eliminate the ambiguity in the integration limits and to allow for automatic analysis routine, we have chosen as the limits the points to the left and right of the peak where the number of counts falls down to about 10–15% of the maximum value. Therefore, throughout this paper by the energy loss we mean the difference between the centroids of the initial and the final peak calculated between such limits. In our case (almost fully symmetric peaks) the lowering of the limits (for instance down to 5% of the maximum) does not change the results for the points with good statistics. The only difference is for the low statistics points. There at 5% limit we observe sharp increase in random jitter as the centroid position becomes strongly distorted by the occasional background events. The energy loss in materials is caused by interaction with electrons and with nuclei, hence elec-

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tronic and nuclear stopping force. The latter one becomes significant only at low ion energies. In practical terms the main difference between the two is that interaction with electrons, unlike with nuclei, does not make appreciable change to the ion trajectory. Since most authors make explicit reference to electronic values, we have chosen a relatively large absorber–detector distance to limit the maximum acceptable deflection angle and to assure in this way that only the electronic stopping force is measured. This choice effectively brings a low-energy limit for our data. By shifting the detector closer to the absorber, we would gain the number of events at low energies but at the same time we would lose the control of multiple scattering events and distinction between the electronic and nuclear energy loss. For shortness, we call the thin foils in which the energy loss is measured absorber foils as part of the initial energy of the ion is indeed absorbed in them. As there is no consensus in that respect, one may also talk of stopping or degrader foils meaning the same thing.

3. Experimental set-up Our latest measurement set-up is sketched in Fig. 1. The exact distances, diameters, etc. are listed in Table 1. The detection angle was 20 with respect to the beam direction. Throughout the

149

Table 1 The main parameters of the set-up Object

Distance from the beam spot (mm)

Diameter (mm) or size (mm
mm)

Thickness (material)

Beam spot Collimator MCP start MCP stop Absorber foil E detector

– 156 202 602 690 860

3 6 10 10 12 20
20

– 3 mm (Al) 30 lg/cm2 (C) 76 lg/cm2 (Ni) 0–2 mg/cm2 380 lm (Si)

measurement and for at least 48 h prior to it the vacuum in the chamber was maintained at or below 1
106 Torr range. The primary beam from the accelerator (in our case the K ¼ 130 cyclotron at the Department of Physics, University of Jyv€askyl€a) was partially scattered on a target before reaching the beam dump. We had a choice of one thin (0.5 mg/cm2 Au) and several thick scattering targets. In addition, the inclination angle gave the possibility to minimize or maximise the effective thickness of the scatterer as indicated in Fig. 1. Particles scattered on the thin target were nearly monoenergetic due to the well-defined geometry and minimal straggling in the thin, uniform target. We have used the thin scatterer to get a reference time peak for the TOF spectrometer and a calibration point for the energy detector as well as to acquire quickly high statistics at the most upper

Fig. 1. Experimental set-up. The exact sizes and distances are given in Table 1. The measuring position was 20 with respect to the beam direction. The position of the energy detector could be adjusted in a broad range.

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end of the energy range. This part of the measurement (with the thin scatterer) is nearly identical to the traditional transmission-type set-up. In this mode of operation the timing information is redundant. However, as soon as the thin target is replaced by a thick one, the energy spectrum of the scattered particles broadens. In the extreme case one gets all energies from zero up to the maximum value allowed by kinematics. Without TOF such a spectrum would be useless. With TOF, as illustrated in Fig. 2, the TOF vs. energy curves obtained with and without the absorber easily separate allowing for energy loss determination for all energies at once. Even at high energies, where the energy loss is the smallest, the separation is adequate (see inset in Fig. 2). In this way the two runs (one with and one without the absorber) are equivalent to hundreds of step-by-step measurements, evenly spaced between the minimum and the maximum energy.

Fig. 2. A typical TOF–energy plot from the measurement with 240 MeV 40 Ar ions and a thick scatterer. Of the total of 10 TOF-E plots obtained in each run only two are shown here: without the absorber and with a Havar absorber. The inset shows the top section enlarged in X and Y direction by a factor of 4.

4. MCP detector The main changes in our set-up compared to [1] include extension of the TOF base up to 40 cm and introduction of the new type of microchannel plate (MCP) based timing detectors as well as overall improvements in the electronics. These improvements allowed us to upgrade the function of the TOF from a simple tagging device [1] to the proper spectrometer giving the absolute TOF values (independent from the energy detector). In the simplest configuration the converter foil of the MCP detector is placed at 45 to the ion path. In this geometry there is no need for an electrostatic mirror to deflect secondary electrons towards MCP but the time signal is position dependent. This deficiency is removed by placing the foil perpendicular to the ion trajectory and using a 45 mirror. We have used such system in all of our earlier experiments [1]. Now we have added the second mirror to each of the detectors (Fig. 3). The fact that our MCP detectors are now fully symmetric (the same distance to the grounding grids on both sides of the converter) reduces electrostatic deflection of the foil allowing for TOF base accuracy of about 0.3 mm. The new detectors have also better overall performance (Table 2). Table 2 lists the efficiency values as obtained for 6 MeV/u particles using single and double mirror detectors at the same operating voltages. These

Fig. 3. Double-mirror MCP-based time pick-off detector.

W.H. Trzaska et al. / Nucl. Instr. and Meth. in Phys. Res. B 195 (2002) 147–165 Table 2 Efficiency of single- and double mirror MCP detectors as well as of the corresponding TOF spectrometers Efficiency (%) MCP double mirror MCP single mirror TOF double mirror TOF single mirror

Protons

4

12.5 10.8 1.6 1.2

43.0 40.6 18.5 16.5

He

16

O

88.2 66.5 77.9 44.2

40

Ar

98.0 53.6 96.1 28.7

The results were obtained with 6 MeV/u particles. Operating voltage was the same for each pair of measurements but different for different particles.

values should not be regarded as the upper limit but only as a comparison. For instance, we had no difficulties reaching 90% efficiency for 40 Ar also with a single mirror detector by simply increasing the operating voltage. This however is not always possible and it reduces the lifetime of the detector. In any case, as shown in Table 2, for any given operating voltage the twin-mirror detector has a higher efficiency and better time resolution than the single mirror detector. For instance 112 ps as opposed to 127 ps FWHM TOF for 7.7 MeV alpha line. The fact that in our twin mirror design the angle is not 45 but 39 (to allow for a smaller, less expensive MCP) adds (according to our estimates) about 10 ps to the time spread. Nevertheless, the overall gain in the time resolution is quite apparent. For in-beam measurements we routinely get 120 ps FWHM TOF for heavy ions and below 160 ps for protons. The MCP itself is a low-cost, engineeringgrade, matched chevron assembly, 36 mm diameter with 12.5 lm channel diameter, manufactured by Photonis (type G12-36ST/15/E/A). The fact that our fully symmetric, double-mirror MCP detectors work so well is in many ways counterintuitive. For instance, anticipating the forward–backward asymmetry in the electron emission from the converter foil, we have tried to vary the bias on the mirrors but the best results were with equal voltages. 5. MCP electronics MCP detectors are frequently used without amplifiers. Nevertheless, external amplification allows to use lower operating voltage or/and to match the

151

pulse shape to that required by the CFD. Further, at the given voltage the MCP þ amplifier combination gives higher efficiency. In our measurements we have used fast amplifiers and CFDs, designed and made by our group especially for this type of work. The amplifiers are similar to ORTEC 9306 and have the amplification factor of about 100. The CFDs are based on the dual, ultra-fast ECL comparator MAX 9687CSE. The minimum threshold is 10 mV and the typical operating threshold was 20 mV. The CFDs follow the conventional scheme (zero crossing of the difference between the attenuated and the delayed pulse) but use a very short delay – typically 0.6 ns that is about half of the pulse rise time. The attenuation coefficient is about 3. With this setting the CFDs work only on the first half of the pulse front extending the dynamic range toward higher amplitudes. In practice, this combination of amplifier and CFD works well for amplitudes at the input of the amplifier between approximately 0.2 and 100 mV even if the signals over 20 mV become subsequently saturated in the amplification stage. The start and stop timing pulses were processed in parallel by two high-quality time-to-amplitude converters (CANBERRA 2145 with 50 ns and ORTEC 566 with 200 ns full range). The output signals from both of them were then digitised in ORTEC AD413A CAMAC Quad 8k Analogueto-Digital Converter (ADC) and stored event by event. Also the output pulse of each MCP was digitised (LeCroy 4300B charge ADC) and stored with each event. This was done only for monitoring purposes, as no off-line corrections were needed due to the satisfactory walk characteristics of our CFDs as it was explained above.

6. Energy determination Our primary energy detector was a 20
20
0:380 mm3 PIN diode manufactured by Siemens. It was operated at full depletion voltage of þ120 V. The dead layer thickness of this detector (as checked with an alpha source) was 370 nm of Si equivalent. The energy resolution was 32 keV for the 7.7 MeV 226 Ra line and the intrinsic resolution

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was close to 22 keV. The output of the detector was connected to a dual fast/slow preamplifier CSTA. The timing output was further processed in TFA97 timing filter amplifier and connected to Phillips Scientific 715 CFD to give the trigger pulse for the data acquisition. Both CSTA and TFA97 were designed and build at Technical University Darmstadt and exhibit excellent energy and timing properties [4]. The energy pulse from the preamplifier was split into two and connected to two high-grade spectroscopy amplifiers (ORTEC 855) with different gains. Each output was then digitised by an 8k ADC (ORTEC AD413A). The stability of the energy channels was constantly monitored with the research pulser (ORTEC 448) operating at the rate of one pulse per second. To make energy calibration one needs particles of well-determined energy. In addition to radioactive sources one can also use magnetic devices and TOF. A simple calculation indicates that with 120 ps resolution and 40 cm distance the TOFbased energy determination is accurate to better than 2% for up to 6 MeV/u particles. The lower the energy the better the relative accuracy. In our measurements the largest observed time drift of the TOF value was below 60 ps. Also differential non-linearity did not exceed 60 ps over the entire dynamic range. Even if a hypothetical time shift (drift) would reach as much as 200 ps, the TOFbased energy determination would still remain below 2% up to 2 MeV/u, as illustrated in Fig. 4. Traditional, peak-based energy calibration has the opposite characteristic. Fig. 5 shows errors in the traditional peak-based energy calibration caused by small uncertainties in the intercept value. It is evident from that picture that below 1 MeV/u a reliable energy calibration based on the standard alpha sources or scattered particles is nearly impossible. For instance, a 10 keV/u energy shift equals to 40 keV for alpha particles ðA ¼ 4Þ. As seen from Table 3, this value is comparable to the typical energy loss of alphas inside a calibration source and to the typical FWHM resolution of a silicon detector. It is clear that even if the thickness of the source and of the dead layer of the detector have been carefully investigated, errors of the order of a few keV/u cannot be easily excluded from the conventional energy calibration.

Fig. 4. Influence of time-shift error on TOF-based energy calibration.

Fig. 5. Influence of intercept error on energy determination.

Table 3 Energy of

226

Ra alpha lines used for calibration

226

Ra literature value (keV)

Energy loss in the source (keV)

Actual energy (keV)

7686.9 6002.4 5489.5 4784.4

33.7 39.6 39.6 41.9

7653.2 5962.8 5449.9 4742.5

The conclusion from comparing Figs. 4 and 5 is quite obvious. At low energies (below about 1 MeV/u) TOF calibration should be used while at

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high energies calibration from well-determined peaks gives a better accuracy. Thus the only way to obtain a reasonable accuracy of energy determination (below 2%) valid over a wide range is to use both types of calibration and to check them against each other. This is exactly what we have done. The final argument against extrapolating of energy calibration to low energies and to other ions is the pulse height defect, as it will be discussed later. 6.1. TOF-based energy calibration One way to extract the absolute TOF value from the relative time difference between the start and the stop detector is to make a step-by-step calibration for all the studied particles and for all the energies. The advantage of this method is that it does not have to assume anything about the energy dependence of the time response of the TOF detectors and electronics. The major disadvantage of this approach is the need to have the set of monoenergetic particles with very well defined energy. As the lowest energy produced by our cyclotron is 2 MeV/u, we could not make such a calibration in the most important energy range, which is below that value. A particle flying through the TOF system (like the one on Fig. 1) crosses the start detector at the time t1 and the stop detector at t2 . The response of the detectors is not instantaneous. It takes some time s before the electrons from the converter foil reach the surface of the MCP, produce a pulse that is in turn processed by the electronics. In principle this additional time s may be different for the start and stop detectors and may depend on the energy, mass and charge of the particle. Therefore, ignoring some constant delays, one can write tstart ¼ t1 þ s1 and tstop ¼ t2 þ s2 : Since, by definition, tTOF ¼ t2  t1 ; therefore,

153

tstop  tstart ¼ tTOF þ ðs2  s1 Þ: In our case the start and the stop detectors (and their electronics) were identical so one should not expect any major differences between s2 and s1 due to the geometry and electronics. The linearity of the TAC and ADC was checked and found to be sufficiently good. Also the hypothetical energy dependence of s should not be big since the energy of the particle impinging into the converter foil of the start detector is nearly identical to the energy of that particle when it reaches the stop detector. This difference, even for the lowest energy of 40 Ar ions (0.15 MeV/u) is below 10%. In other words, one can assume that in our case (large TOF base and very thin converter foils in TOF detectors), s2  s1  tTOF ; and consequently, tstop  tstart tTOF : With these assumptions the extraction of the absolute time of flight value from the relative time difference between the start and the stop detector becomes straightforward. To get the TOF from the channel number of the time spectrum one needs just two parameters: the slope and the intercept. The most accurate way to extract the slope of the time calibration is by using a high precision time calibrator. We have used ORTEC 462 that gives 10 ns spaced pulses with 10 ps accuracy. For the offset (intercept) value we have compared the calculated TOF value with the time peak position produced by the scattered 6 MeV/u particles. To use the TOF values (measured between the converter foils of the start and stop detector) to calibrate the silicon detector (E) we had to account for the energy loss in the converter foil of the stop detector. To do that with sufficient precision, we had placed a similar foil on the rotating wheel among the other absorbers. In this way, by having measured the energy loss in the single (no absorber foil just the stop detector) and in the double foil (the foil in the stop detector and the nearly identical one on the absorber wheel), we have extracted a reliable correction for the energy loss in the stop detector. The final thickness ratio between the foils was determined with 226 Ra source and yielded

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0:76  0:02. Since the nominal thickness of the thicker Ni foil was 0.100 mg/cm2 , the thickness of the foil in the stop MCP detector was around 76 lg/cm2 . With this final, small correction we had a direct correspondence between the pulse height from the E detector and the TOF value for the particle that has produced this pulse. This correspondence was used to make the TOF-based energy calibration of the silicon detector. 6.2. Peak-based energy calibration The standard way of calibrating energy detectors is by the use of monoenergetic particles from radioactive sources or accelerators. We call this peak-based calibration as each monoenergetic particle produces a distinct peak in the spectrum. For alpha particles, in addition to the 226 Ra source with well-known effective thickness (see Table 3), we have used the scattered 4 He beam of 24 MeV. Thus we could make a reliable energy calibration for alpha particles in the energy range of 4.7–24 MeV. For the other particles (40 Ar, 28 Si, 16 O and 1 H) we could only use the scattered peak at 6 MeV/u. This gave us the highest reference point close to 240 MeV (Ar) and the lowest at about 6 MeV (H). These additional points allowed us to extend (and verify) the linearity of the pulse height output up to 240 MeV. The use of different particles for this purpose was justified in this case because, at 6 MeV/u the pulse height defect is already small, even for 40 Ar ions, as we shall show below. The beam energy of the cyclotron was known to about 0.5% accuracy.

strongest manifestation of PHD is for heavy ions (HI) at low energies while for alpha particles and for light ions as well as at high energies it becomes negligible. Having a wide-range peak-based energy calibration, we were in a good position to compare it with the TOF-based energy calibration. We made this comparison independently for alphas, protons, 16 O and 40 Ar ions. The results of the comparison are shown in Fig. 6. The ratio of the TOF-based energy value to the peak-based energy value (using the reference calibration) is plotted as a function of the pulse height output of the PIN diode. For convenience, the energy scale is given in MeV/u (using again the reference calibration). To interpret Fig. 6 one should consider each particle separately. For alpha particles there is very little difference between the TOF- and the peak-based calibration. Throughout the entire energy range (0.2–6 MeV/u) the ETOF =EPIN ratio remains equal to 1 within the 2% limit consistent with the overall accuracy of our energy calibration. The possibility of reaching this agreement accidentally can be excluded as extremely unlikely. Therefore we can conclude that both the peakbased and the TOF-based energy calibration for alphas are good to better than 2%. For the peakbased calibration this was expected as we had several alpha peaks between 1 and 6 MeV/u. The fact that also the TOF-based energy calibration is

7. Pulse height defect It is a well-known fact that not all of the kinetic energy of an ion is transferred into the pair creation in a semiconductor detector. This phenomenon is known as pulse height defect (PHD) [5–7]. Several semi-empirical formulas have been proposed to account for PHD but to get a reliable correction and not just an estimate, one needs to determine PHD for each individual detector. The

Fig. 6. Relative pulse height defect determined from the ratio of TOF-based energy to the energy obtained from the traditional alpha calibration.

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good gives assurance of the accuracy of our TOF spectrometer and confirms the correctness of the assumptions made in Section 6.1. For all the other particles the peak-based energy calibration was fixed (confirmed) by just one high-energy point – the scattered peak at 6 MeV/u. Not surprisingly, the energy of the scattered peak is in a good agreement with the alpha calibration extrapolated to this energy. At 6 MeV/u the ion penetration depth into the detector is already sufficiently large (even for Ar) to minimise various non-linear effects occurring mostly close to the detector surface. (For the detail study of pulse shape dependence on energy, charge and mass of the ion see [4] and the references therein.) Indeed, at 6 MeV/u the ETOF =EPIN ratio equals 1 for all the studied particles (within the 2% accuracy discussed above). At lower energies, especially below 2 MeV/ u, the lines start to diverge noticeably, which we interpret as the onset of PHD. To be precise, we detect only a relative PHD because we have used alpha particles as the reference. In principle, it would be possible (albeit difficult) to obtain the absolute energy calibration for instance with highresolution calorimetric low-temperature detectors [8]. However, such an approach would have been far beyond the scope of this paper. The main conclusion from our PHD study summarised in Fig. 6 is that at higher energies (above 2 MeV/u) the linearity of pulse height dependence on particle energy is very good. In that region PHD is negligible (below 2%) also for 40 Ar so the use of the peak-based calibration described above is fully justified. Below 1–2 MeV/u the use of such calibration may lead to the progressively increasing errors. Therefore, at low energies we have used only the TOF-based energy calibration. We are confident that with this approach we know the absolute value of energy at each energy point with the accuracy better than 2%.

8. Absorber foils Before and after each run we have inspected all the absorber foils with alpha particles using 226 Ra source. The source was placed and removed from the measuring position without breaking of the

Fig. 7. Energy loss in each foil as measured with

155

226

Ra source.

vacuum. An example of such data is shown in Figs. 7 and 8. Fig. 7 shows the energy loss measured for each foil on the rotating wheel. For every foil there are four points corresponding to the four major alpha lines of 226 Ra. Such measurements give an independent check of the foil thickness and of the thickness ratio (if several absorbers of the same material were used). For each peak we have also registered the full width at half maximum (FWHM) and plotted it as a function of the energy loss (Fig. 8). For an ideally uniform absorber the broadening would be the smallest. The larger the non-uniformity of the absorber, the larger the deviation from the expected FWHM. The FWHM expected from the uniform absorber was calculated as the square root of the sum of the squares of the detector resolution (taken from the measurement without the absorber) and straggling (typically about 7–8% of the energy loss value). As can be seen in Fig. 8, only one of our absorbers (Havar) yields FWHM expected from the uniform foil. Most foils yield a linear dependence on the energy loss (thickness) and two are clearly inferior from the rest. Indeed the two foils observed under the microscope showed visible roughness of the

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9. Measurements

Fig. 8. Measured FWHM for each alpha line from 226 Ra plotted as a function of energy loss in the absorber. The dashed line was calculated assuming 8% straggling and 32 keV detector resolution. It can be clearly seen that one of our absorbers (four square symbols corresponding to four 226 Ra peaks) was excellent in uniformity and two (four circles and four rhombus) were bad. The others follow a linear trend as shown by the solid line.

surface. Needless to say, they were not used for the final analysis. The same results emerged (even more clearly) from the irradiation of the foils with 240 MeV 40 Ar ions (Fig. 9).

Fig. 9. Same as in Fig. 8 but using scattered beam of 240 MeV 40 Ar ions instead of alpha particles from the source. The thin dashed line was calculated assuming 7% straggling.

As it was explained in [1] a single measurement is in principle sufficient to map about three decades of energy of the dE=dx curve. However, as nobody has made such measurements before and as our results clearly deviate from the majority of the experimental data for low energies, we have made three independent runs during the fourmonth period. The first run used 11 MeV/u 40 Ar, in the second we have used a 6 MeV/u cocktail consisting of 40 Ar, 28 Si and 16 O. In the third run we had again a 6 MeV/u cocktail but this time it consisted of 40 Ar, 16 O, 4 He and 1 H. The last run, in addition to delivering energy loss of light ions in the same absorbers, was also aimed at increasing the number of events at low energies for heavy ions. During that run a special care was given to the long-term stability of all the energy and timing channels. Fig. 10 shows an example of the raw data without any corrections or manipulations other than taking the difference between the TOF-E curves collected with and without the absorber, as it was illustrated in Fig. 2 and explained in detail in [1]. The three sets of data correspond to 2.0, 0.5

Fig. 10. Energy loss of 40 Ar ions in gold foils of three different thickness values: 2.0, 0.5 and 0.2 mg/cm2 . By making the curves overlap one can very precisely get the relative thickness ratio. This method is more accurate than using plots from Fig. 7.

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and 0.2 mg/cm2 Au foils bombarded with 40 Ar ions up to 6 MeV/u. All three curves have identical statistics. The wheel with the absorbers (Fig. 1) was turned automatically after each 2000 events and made at least 100 full turns during each subrun. Nevertheless, the curve obtained with the thickest foil is much smoother than the other curves. The explanation of this feature is quite simple. The energy loss in the thin foil is small. To determine it reliably one needs good statistics. Therefore, if several foils of the same material are used, one should preferably use the thick foil data where the energy loss is the largest [1]. The upper acceptable limit for the maximum energy loss is roughly Ef > 0:6Ei which is equivalent to dE < 0:5E. If the energy loss exceeds 40% of the initial energy or (50% of the mean energy) the linear approximation that assigns dE ¼ Ei  Ef to the energy point E ¼ ðEi þ Ef Þ=2 is no longer valid. To use such data one has to apply a correction. Nevertheless, for the final results presented in this paper no correction was needed as the limit of dE=E < 50% was not even approached and the largest used dE=E ratio was only 23% (0.12 MeV/u 40 Ar in gold). Our approach of using several absorber foils of the same material with different thickness allows both to verify the maximum acceptable energy loss and to measure the needed correction if higher energy losses must be used. It can be done directly from the Fig. 10. The data on Fig. 10 is plotted in the logarithmic scale to demonstrate that by simply shifting the curves (that is by multiplying each point by a constant value) one can get them to overlap very well. Apart from the larger statistical jitter of the lower curve and a slight dE > 50% effect in the lowest energy part of the upper curve, the three curves are identical. In this way (shifting the spectra in Fig. 10 till they overlap) we have also extracted foil thickness ratio to better than 1% and, as the result, determined the areal density of the thinner foil. Foil thickness information collected in Table 4 has been obtained in this way. Using broad-energy spectra (Fig. 10) gives better accuracy and reliability than the data from the source (Fig. 7). Thin foils (like the 0.5 and 0.2 mg/ cm2 ) could not be directly weighted with the accuracy of a few percent.

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Table 4 Areal density of absorber foils used in the measurements Material

Direct measurement (mg/cm2 )

Measured with alphas (mg/cm2 )

Thickness ratio

Final value (mg/ cm2 )

C

–

0.1

1

0.1

Ni

– –

0.393

1 0.24

0.393 0.0943

Au

2:09  0:02 – –

1 0.251 0.1056

2.09 0.525 0.221

10. Results and discussion Although the main goal of this work was the energy loss of heavy ðA > 16Þ ions, we have paid considerable attention to the lighter ions (protons and alphas) as well. The reason is threefold. For heavy ions, especially 40 Ar, experimental stopping force data are scarce and precise energy determination of such ions causes problems, mostly due to PHD. We wanted to be able to compare our results with a set of reliable experimental points extending over a wide energy range. For HI such comparison was impossible precisely because of the lack of good reference data [9] and because of problems with PHD. The second reason for measuring also lighter ions is that in many ways stopping force for alpha particles is the yardstick by which the stopping force for all other ions is measured. Parameterisation used by Paul [9] is a good example of this approach. It represents the stopping force as the ratio of dE=dx value for the given ion to the dE=dx value in the same material for alphas. There is a linear dependence on the foil thickness (and therefore also on the thickness error) both in the nominator and in the denominator. Since we provide both dE=dx curves (HI and alpha), measured on the very same foil, the error connected with the absolute foil thickness cancels out. Finally, nearly everybody uses energy loss of alphas (typically from a radioactive source) as the means to determine the foil thickness. By providing this value for all the conceivable alpha energies that could be produced by a source or by a small

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accelerator, we give the reference value to be compared with any other measurement, past or present, which uses alphas for absorber thickness determination. 10.1. Light ions Figs. 11 and 12 show the stopping force in C, Ni, and Au for protons and alphas respectively. There are no surprises in the data. The experimental points, especially for protons, follow very well SRIM 2000 [10] prediction and agree well with the literature data [9,11]. Unfortunately, our short proton measurement did not allow us to reach the energies below the dE=dx peak where discrepancies would be the most likely to occur. Also for 4 He (Fig. 12) the agreement of the high-energy slope with SRIM 2000 is perfect. Please note the highest energy point for each absorber. These points are very reliable as they are

Fig. 12. Stopping force for alphas in carbon, nickel and gold. The solid points were measured in this work. The open circles mark the literature data. The solid line represents SRIM 2000 prediction.

Fig. 11. Stopping force for protons in carbon, nickel and gold. The solid points were measured in this work. The open circles mark the literature data. The solid line represents SRIM 2000 prediction.

based on high statistics from the run with the thin scatterer. At the lowest energy data, close to the maximum of the dE=dx peak we observe a small deviation from SRIM 2000. In our data the values are consistently higher. Since otherwise our data extend in perfect agreement with SRIM up to 24 MeV (6 MeV/u), there is no way to attribute this small deviation (about 4–5%) to the thickness error of the absorber. Neither a small calibration error could explain such effect. Unfortunately, the spread of the literature data [9,12–18], especially for carbon, is quite large and our data does not extend below the maximum of the dE=dx peak. It would therefore be unwise to speculate about this small deviation before we can produce better data. It is clear that it would be worthwhile to address the energy loss of light ions in a dedicated highprecision experiment. Our present measurement was not intended to challenge or replace the wellestablished data for light ions. The goal was to

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verify that our technique works for particles with energies between 0.1 and 10 MeV/u. This check could not have been done with HI data alone since there the literature data are scarce and/or highly inconsistent [9]. Figs. 11 and 12 indicate that we have achieved the intended goal. 10.2. Heavy ions Figs. 13–15 show stopping force for 16 O ions and Figs. 16–18 for 40 Ar. In each figure we show SRIM 2000 prediction, previous measurements [9] and the results of two of our experimental series taken with 6 MeV/u beams. The data from the first run is displayed without PHD correction. In other words, for the data from the first run full linearity between the pulse height output of the detector and the kinetic energy of the ion was assumed. It was further assumed that the same energy gives the same pulse height independent on mass and atomic number of the ion. Such assumptions, as it was discussed in the chapter on PHD, are unsubstantiated so the data from the first run, displayed without PHD correction, serves as an illustration of a simplified and certainly incorrect approach. The data from the second run is shown already corrected for PHD. It is important to stress that there is full consistency between all of our measurements. By switching the PHD correction off

Fig. 14. Stopping force for 16 O in nickel. Solid line represents SRIM prediction. Data from previous measurements were taken from Paul [9].

Fig. 15. Stopping force for 16 O in gold. Solid line represents SRIM prediction. Data from previous measurements were taken from Paul [9].

Fig. 13. Stopping force for 16 O in carbon. Solid line represents SRIM prediction. Data from previous measurements were taken from Paul [9].

(that is by using alpha source and scattered peaks calibration only) the data of all of our runs overlap perfectly for each ion/absorber combination as demonstrated in Fig. 19. It could be compared with Fig. 16 (dE=dx of 40 Ar in C). Above 30 MeV for 40 Ar (Figs. 16–18) and above 10 MeV for 16 O (Figs. 13–15) there is full agreement between the first run (no PHD correction) and the second run (PHD corrected). This is consistent with PHD properties illustrated in Fig. 6. In that energy range there is also a good agreement

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Fig. 16. Stopping force for 40 Ar in carbon. Solid line represents SRIM prediction. Data from previous measurements were taken from Paul [9].

Fig. 18. Stopping force for 40 Ar in gold. Solid line represents SRIM prediction. Data from previous measurements were taken from Paul [9].

Fig. 17. Stopping force for 40 Ar in nickel. Solid line represents SRIM prediction. Data from previous measurements were taken from Paul [9].

Fig. 19. Stopping force for 40 Ar in carbon. The same data as in Fig. 16 but here the PHD correction was not applied. Both of our measurements are in perfect agreement with each other.

with the previous measurements. The only major exception is for argon on nickel where our data are consistently above SRIM around the dE=dx peak. As soon as the peak is cleared the high-energy slope predicted by SRIM is very close to our data. This discrepancy is not surprising since there were no previous measurements [9] between 1 and 100 MeV for argon ions in nickel. Similar deficiency of SRIM parameterisation, although much smaller, is also present for argon ions in carbon.

At low energies PHD correction introduces a dramatic effect. For instance, Fig. 16 (argon in carbon) indicates that only one previous experimental series [19] agrees with our data. It is a relatively old work (1978) but, to our knowledge, the only one that mentions explicitly making PHD corrections. By turning PHD off we get very good agreement with all the other measurements (Fig. 19). The final results presented in Figs. 20 and 21 supersede our preliminary results quoted in our

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Fig. 20. Our final stopping force data for 16 O in carbon, nickel and gold.

first instrumental paper [1] where no PHD correction was applied. Otherwise (with PHD correction turned off), there is a good agreement between the two sets of data. For convenience, the data from Figs. 20 and 21 is also represented by the set of reference points in Tables 5 and 6. Before the controversy with PHD is finally settled we recommend treating the low energy part of our data only as the lower limit [21]. At the same time we recommend to treat the published values as the upper limit of the actual dE=dx curve. The reasons are discussed below. 10.3. Errors Never before one short experimental session has yielded stopping force values simultaneously for several absorber/ion combinations all covering two decades of ion energy. Can such results be taken seriously? Are they more reliable than painstaking,

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Fig. 21. Our final stopping force data for 40 Ar in carbon, nickel and gold.

step-by-step approach? In our opinion the clear answer is yes. Measuring all the data points at once protects against small drifts in amplification, steadily progressing foil deterioration, etc. Any such effect would influence all the TOF-E curves simultaneously and in a similar way. Such smearing effect, if reasonably small, would only reduce the ability to resolve small energy losses obtained with low statistics but otherwise it would not change appreciably the absolute value of the extracted energy loss. This is true as long as the change was accounted for by the energy calibration. In our case, registering TOF gave the possibility to verify energy calibration nearly event-by-event. The fact that all of our stopping curves reach up to high energies solves automatically the first major problem of dE=dx determination – uncertainty in absorber thickness. Only at high energies thick absorbers can be used. The areal density of such

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Table 5 Stopping force of

16

O ions in gold, nickel and carbon

Energy (MeV/u)

dE=dx in Au (MeV/(mg/cm2 ))

Error (%)

Energy (MeV/u)

dE=dx in Ni (MeV/(mg/cm2 ))

Error (%)

Energy (MeV/u)

dE=dx in C (MeV/ (mg/cm2 ))

Error (%)

0.145 0.183 0.229 0.278 0.351 0.446 0.619 0.800 1.043 1.382 1.736 2.448 2.985 3.760 4.410 5.001 5.776

1.25 1.55 1.82 2.07 2.30 2.51 2.59 2.58 2.49 2.37 2.28 2.14 2.04 1.89 1.77 1.67 1.56

12.9 8.0 5.8 3.7 2.8 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

0.157 0.205 0.252 0.316 0.436 0.624 0.880 1.277 1.804 2.843 3.889 4.536 5.030 5.846

2.86 3.47 3.97 4.43 4.89 5.11 5.09 4.79 4.33 3.65 3.17 2.93 2.77 2.51

9.2 6.5 4.1 2.6 2.1 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

0.135 0.161 0.184 0.217 0.276 0.367 0.510 0.717 1.001 1.449 2.283 3.389 4.614 5.865

5.98 7.43 8.16 9.05 10.08 10.59 10.37 9.62 8.82 7.89 6.77 5.58 4.71 3.89

18.5 8.8 6.9 4.3 2.3 2.0 2.1 2.1 2.1 2.0 2.0 2.0 2.0 2.0

The error was calculated assuming 2% uncertainty in the absolute foil thickness and 2% uncertainty in the energy determination. The given errors represent thus the maximum errors consistent with our measurement.

Table 6 Stopping force of Energy (MeV/u)

0.129 0.145 0.215 0.355 0.584 0.818 1.040 1.351 1.734 2.119 2.507 2.889 3.528 4.380 4.895 5.570 11.044

40

Ar ions in gold, nickel and carbon

dE=dx in Au (MeV/(mg/cm2 ))

1.63 2.08 3.25 4.76 6.11 6.81 7.13 7.31 7.32 7.23 7.13 7.00 6.73 6.39 6.16 5.95 4.70

Error (%)

33.8 16.4 6.9 3.2 2.2 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

Energy (MeV/u)

0.159 0.178 0.211 0.253 0.334 0.445 0.632 1.010 1.390 1.655 2.111 2.595 3.300 4.175 4.888 5.819 11.131

dE=dx in Ni (MeV/(mg/cm2 ))

5.07 5.74 6.54 7.77 9.39 11.09 12.93 14.64 15.21 15.06 14.43 13.69 12.67 11.67 11.00 10.00 7.23

Error (%)

Energy (MeV/u)

dE=dx in C (MeV/(mg/cm2 ))

Error (%)

14.5 8.7 9.3 5.5 3.8 2.7 2.1 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

0.129 0.152 0.201 0.238 0.252 0.279 0.321 0.374 0.459 0.624 0.864 1.160 1.583 2.412 3.353 4.224 4.907 5.847 11.154

8.35 10.55 14.65 16.84 17.88 19.49 20.93 22.72 24.36 25.74 26.09 25.98 25.36 23.54 21.28 19.35 18.06 16.32 9.91

23.2 15.8 8.4 9.0 6.9 4.1 3.8 2.6 2.1 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

The error was calculated assuming 2% uncertainty in the absolute foil thickness and 2% uncertainty in the energy determination. The given errors represent thus the maximum errors consistent with our measurement.

absorbers is easily and accurately measured. For thinner foils the thickness ratio will be obtained automatically (Fig. 10). Therefore, one can say

that our measurements do not suffer from foil thickness error because, even if we did not have a thick foil with well determined areal density

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(C and Ni), it is easy to shift the entire curve up or down to normalise it to a well determined point somewhere at high energies. This claim is further strengthened by the fact that for all the measured absorbers we give dE=dx values for 4 He as well. Anybody using an alpha source can re-normalise our results to his without introducing additional errors. The strongest argument indicating that we have used correct values for areal density of the absorbers is the consistency of our results. The highenergy part of our stopping force curves (where no PHD corrections are needed) agree very well with previous measurements. If there was a mistake in the areal density for one of the absorbers, all four curves (1 H, 4 He, 16 O and 40 Ar) should be either above or below the expected values. This is not the case. Overall, our data have good statistics. For each ion the largest statistical jitter is for carbon and the smallest for gold (Figs. 20 and 21). This is the effect of foil thickness. As seen in Table 4 our carbon foil was 0.1 mg/cm2 while the thickest gold foil was 2 mg/cm2 . The second serious source of error is naturally the energy determination. This is probably the main error that influences the shape of dE=dx curves in our measurements. In this respect we are as vulnerable as all the other measurements. A precise energy calibration can be reached with the use of electrostatic accelerators and magnetic separators but maintaining of the same energy calibration over a long period of time is very difficult if not impossible. Measurements that take a long time have to cope with serious stability problems. As our measurements are short, the likelihood of instabilities is proportionally reduced. Therefore we feel justified to claim that energy determination in our approach is not worse than that used by any other method. As a confirmation of that claim we can point to the very good agreement between peak-based and TOF-based calibration for 4 He (Fig. 6). As we have already indicated, the absolute accuracy of our energy determination (accounting for PHD) is 2%. This value is roughly the same for all ions and for the entire energy range. The 2% uncertainty in the energy calibration translates to

163

the stopping force error in the following way. Assuming that a point with energy E is on the increasing slope of the dE=dx curve, the largest dE=dx value consistent with our 2% energy calibration error is dE=dxðEÞmax ¼ dE=dxð1:02EÞ; that is dE=dx taken at the point 2% higher in energy. Analogically, dE=dxðEÞmin ¼ dE=dxð0:98EÞ: For the decreasing slope the formulas for Emax and Emin are naturally reversed. The last major source of error is the material composition of the foil. While we could reliably check the uniformity of our foils (Figs. 8 and 9) we did not check their chemical composition. With C, Ni and Au this should not be a major problem. Nevertheless, it should be stated that we did not verify it in a dedicated measurement. Nuclear methods give several possibilities to check the composition of the material. At this stage however we do not see an urgent need to do so. A slight oxidisation or some other small impurity in the absorber foil cannot account for a serious disagreement with nearly all of the previous lowenergy HI data while maintaining a very good agreement at high energies and for lighter ions throughout the measured energy range. The errors quoted in Tables 5 and 6 have been calculated assuming 2% uncertainty in the energy determination and 2% uncertainty in the absolute foil thickness determination. As the set of the representative points is based on a large statistical sample shown in Figs. 20 and 21, the statistical error was considered negligible. Finally, one should consider whether it is possible that despite the good results for light ions over the full velocity range (Figs. 11 and 12) our method gives a systematic error that is amplified only for slow HI. For example, such an error would appear if the assumption from chapter 6.1 ðtstop  tstart tTOF Þ is not fulfilled for such ions. We did not have monoenergetic low-energy HI to verify such possibility directly but we shall address this problem in a dedicated measurement in the future [21].

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11. Conclusions and outlook Without any doubt, the major benefit of this work is pointing out to the fact that the present knowledge on energy losses needs to be considerably improved and that our method is able to do this task properly and more effectively than the previous methods. For all light ions and for swift HI our new approach can be considered already fully tested and producing data of excellent quality and consistency. The main finding of this work is that the new data progressively deviates from the published data at energies below about 0.5 MeV/u for 16 O and below about 1 MeV/u for 40 Ar. In this ‘‘controversial region’’, we could show that we are able to reproduce the old data well by neglecting the pulse high defect. As there is no doubt about the presence of PHD in silicon detectors, the majority of the published results as well as semi-empirical calculations based on them (like the popular SRIM) must be in error at low energies for HI. At this stage of the investigation there is still a possibility that, for the slowest HI, we may be slightly overcompensating for PHD as we were not able to make a completely independent check of the linearity of the TOF spectrometer. Considerable inconsistencies in the previous data (Figs. 15 and 16) or the lack of it (Fig. 17) do not help either. Therefore, as a precaution, one should treat the new data in the controversial low-energy region as the realistic lower limit [21]. At the same time we have proven that the previous data should now be considered only as the upper limit. The main advantages of our new approach to energy loss measurements are: • very wide energy range • possibility to cover the entire dE=dx peak • insensitivity to the problem of accurate determination of the absorber thickness • possibility to make direct comparisons with any other experiment that has used alphas to determine absorber thickness • quick uniformity check of each sample • high overall precision and consistency of the data • short collection times

• simultaneous data for several ion/absorber combinations. The main preconditions to apply our method are: • a well-tuned and equipped experimental set-up • good software tools • a cyclotron with the provision to deliver beam cocktails • good set of high quality absorber foils. As the main tasks for the future we consider: • to determine the response of the TOF spectrometer to low energy HI • to measure dE=dx curves for low energy HI • to make a very precise run for light ions that might solve the 2r discrepancy in the Risø/ Aarhus vs. Kyoto data [11] • to extend the measurements to other heavy ions and absorber materials • to measure energy loss for fission fragments. We welcome further suggestions and invite collaborators who could contribute know-how, ideas, equipment and materials (like the highquality absorber foils) to this challenging project.

Note added in proof Our latest measurements [21] indicate that the possible distortions of the measured TOF caused by some unaccounted – for, non-linear effects in our TOF spectrometer are small, of the order of 60 ps. This value holds even for ions as diverse in mass and atomic charge as alpha particles and fission fragments. At 400 mm TOF base a 60 ps uncertainty in time is practically negligible as it gives less than 0.5% energy error even at 6 MeV/u (see Fig. 4). To make a significant (2% or more) error in TOF-based energy at and below 0.5 MeV/ u, one would need at least 400 ps deviation in TOF. This is certainly not the case. Therefore, we are now confident that there was no overcompensation for the pulse height defect. As a result, the new energy loss data presented here can be

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taken as the exact value and not just as a safe lower limit. Acknowledgements We are thankful to the participants of the STOP01 conference [20] for stimulating discussions and for pointing out many weak points in our initial work. This feedback has helped us to make considerable improvements in our experimental approach. We are also grateful to our colleagues for helping us to collect the necessary absorber foils and to the staff of the Jyv€ askyl€ a cyclotron for providing excellent beam cocktails. This work was supported by the Access to Large Scale Facility program under the Training and Mobility of Researchers program of the European Union, the Access to Large Scale Facility program under the TMR program of the EU, and by the Academy of Finland under the Finnish Centre of Excellence Programme 2000–2005 (project no. 44875, Nuclear and Condensed Matter Physics Programme at JYFL). References [1] W.H. Trzaska et al., Nucl. Instr. and Meth. B 183 (2001) 203.

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