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Energy-loss and straggling of hydrogen and helium ions in selenium
Nuclear Instruments and Methods 205 (1983) 359-363 North-Holland Publishing Company
ENERGY-LOSS J. C O N R A D I E ,
AND STRAGGLING J. L O M B A A R D
359
OF HYDROGEN
AND HELIUM
IONS IN SELENIUM
a n d E. F R I E D L A N D
Department of Physics, University of Pretoria, South Africa Received 7 October 1981 and in revised form 23 April 1982 Using a transmission technique, the energy-loss and straggling of protons, deuterons and helium ions in Se were measured in the energy range between 0.3 and 2.5 MeV. The experimental stopping powers of the various ions are the same as the semiempirical values of Andersen and Ziegler, except in the vicinity of the stopping power maximum. The reduced straggling for the helium and hydrogen ions is energy independent above 0.25 MeV/amu with the helium values the same as predicted by Bohr but the hydrogen values are somewhat lower.
1. Introduction The shortage of reliable experimental results of the energy-loss and straggling of light ions in homogeneous solid targets, gave rise to a systematic study of the energy-loss and straggling of hydrogen and helium ions in different absorber foils [1-5]. In this paper we explore the energy-loss and straggling of these ions, with energies in the range from 300 keV to 2500 keV, in selenium. During the last decade the use of ion beams has proved itself a very powerful tool in the modification as well as in the analysis of the near surface region of materials [6]. In these analysis, it is important to know the exact behaviour of the ion in matter. When a beam of charged particles penetrates matter, the slowing-down is accompanied by a spreading of the beam energy due to statistical fluctuations in the number of collision processes and the energy transfer per collision. In many cases of practical interest, the distribution in energy loss is sufficiently close to a Gaussian that the spreading around the average value is completely characterized by the average square fluctuation in energy loss, also known as the energy straggling. Accurate information on both energy loss and energy straggling is important for example in the analysis of ion implantation profiles using backscattering or nuclear reactions [7]. In practice stopping values are usually obtained from the semiempirical values compiled by Andersen and Ziegler [8] for hydrogen and by Ziegler [9] for helium ions in all elements. In the case of Se, the semiempirical values are based on few experimental data points [10-14]. Some data on straggling are available [12]. Experimental straggling results at low energies on some solid targets [1-5] are higher than predicted by the 0167-5087/83/0000-0000/$03.00 © 1983 North-Holland
free electron gas theory of Lindhard and Scharff [15]. Besenbacher et al. [16] who analysed the straggling results for some gases at particle energies below 1 M e V / a m u , explained these deviations as due to additional straggling contributions from spatial atomic correlation effects and charge state fluctuations. They further stated that the correlation effects should be much smaller in a solid than in a gas and that the effect due to charge state fluctuations is negligible in a solid. Experimental straggling in a homogeneous solid target should therefore approximately be given by the LindhardScharff expression. However, for lithium ions with velocities around the Bohr velocity ( E = 175 keV), copper, selenium and silver seemed to yield straggling parameters in good agreement with gas data [17]. Preliminary measurements on selenium showed that these targets are easily to prepare without texture effects and foil inhomogeneities. Since selenium is a solid near the gas krypton in the periodic table, it would be of interest to compare the straggling of hydrogen and helium ions in selenium and in krypton [18].
2. Experimental The same experimental setup as described in ref. 1 was used. Since difficulties were experienced to produce self-supporting selenium foils, the absorber foils were prepared by vacuum deposition onto carbon backings (20 /Lg cm -2) mounted over the 6 mm opening of the absorber holder. With this method, inhomogeneities in the foils due to stresses produced in the foils during floatation processes, were eliminated. Different methods were used to measure the thickness of each absorber foil. First weighing of the foils was done by placing a thin microscope cover glass on a mask with a hole of
J. Conradie et al. / Energy loss and straggling
360
k n o w n dimensions next to the absorber holder during the evaporation process. The weight of the cover glass was determined with a microbalance before a n d after the evaporation. The thickness as well as the evaporation rate was controlled by using a quartz crystal monitor. F u r t h e r m o r e the thickness was independently calculated from the yield of 1 MeV alpha particles scattered through 150 ° using the well k n o w n Rutherford formula. The agreement between the different thickness determinations was within 6%. Foils of thicknesses of 139 /~g cm 2 and 210 /~g cm 2 were used. The final results were corrected for the c a r b o n backing by measuring the energy-loss and straggling through the backing independently.
3. Results and discussion
3.1. Energy-loss of hydrogen ions in Se Fig. 1 presents our results for the stopping power of protons and deuterons in Se. A very complete review on energy-loss of charged particles was given by Sigmund [19] in 1973. At low energies (v < Vo Z2/3, where v is the projectile velocity, v 0 the Bohr velocity and Z the atomic n u m b e r of the projectile) the electronic stopping power is found to be proportional to the projectile velocity. In
60--
t
I
the m e d i u m and low velocity region where the maxim u m energy loss occurs, no satisfactory theoretical model exists to describe the stopping of these ions in matter. The high energy behaviour of the stopping power is well described by the Bethe formula [20]. Andersen a n d Ziegler [8] used this Bethe stopping-power formula as the theoretical basis in the high-energy ( E / a m u > 600 keV) region of their semi-empirical stopping values. In the medium-energy region, from 600 keV a n d down, they mainly used the Varelas-Biersack [21] formula to fit the experimental data of 24 elements. To obtain stopping values for Se, a new fit which is considered to be accurate to a b o u t 5% at 500 keV, was made. At lower energies the accuracy deteriorates. Our experimental results for the stopping of hydrogen ions in Se are, within experimental error the same as the semiempirical values of A n d e r s e n and Ziegler [8] for projectile energies above 0.5 M e V / a m u . There is good agreement between our results and those of G r e e n et al. [10] and N a k a t a [11]. At the low energy side of our data near the stopping power maximum, our values are more than 14% higher than the semiempirical values, while results of Eckardt [12] are in agreement with these values. It must be pointed out that Eckhardt's results were used in obtaining the semiempirical values [8]. Eckardt's [12] experimental results of b o t h the energy-loss of hydrogen ions
I
I
A i
i= ,u
E
*÷
°C~l 0 0
2O
0,0--
I
I
I
0,5
1,0
1,5
I
2,0 1/~ 2,5 VELOCITY (MeV/Qmu)
Fig. 1. The experimental stopping power of hydrogen ions in Se; open squares: protons in 210 ~g cm -2 Se, full squares: protons in 139 ~g cm 2 Se, open circles: deuterons in 210 ,ug cm 2 Se, full circles: deuterons in 139 tLg cm 2 Se. The error bars are within the symbols if not indicated otherwise. The solid line represents the semiempirical values of Andersen and Ziegler [8]. The experimental results of Green et al. [10] (~7), Nakata [11] (A) and Eckardt [12] (v) are also shown.
361
J. Conradie et al. / Energy loss and straggling (fig. l) and the energy-straggling of hydrogen and helium ions (fig. 3) are lower than our results. The experimental results of the energy-loss of hydrogen ions in Se of Eckardt, also seem lower than those of Green et al. [10]. As stopping power depends only on the effective charge of the projectile, protons and deuterons should have the same stopping power at the same projectile velocity. The agreement between the proton and deuteron stopping values are clearly illustrated in fig. 1. 3.2. Energy-loss of helium ions in Se The stopping of helium ions is plotted in fig. 2. Within experimental error, our values are the same as the semiempirical values of Ziegler [9] except in the vicinity of the stopping power maximum where our results are about 8-10% higher. Experimental values of Nakata [13] are somewhat (4-7%) lower than the semi-empirical values and 8-13% lower than our experimental results. The experimental results of Lin et al. [14] are in good agreement with the semiempirical values. Ziegler [9] least-squared fit a curve through more than 10000 very much scattered data points to obtain a target independent ratio S H e / S n, of He to H stopping. This ratio can be used for scaling H to equivalent He stopping values and it is also defined as the ratio of the square of the effective charge of He to that of H in solids [9]. The best S H J S u values can be obtained by using ions of the same velocity on the same target with the same experimental setup. Table 1 gives the ratio of our experimental He to H stopping in Se at the same projectile velocity. For comparison the values of Ziegler
150
I
I
I
I
L
Table 1 Comparison of the experimental He to H stopping at the same velocity with the corresponding values of Ziegler [9]. Energy (MeV/amu)
SHe /SH experimental
SHe / S u Ziegler
0.13 0.18 0.25 0.32 0.50 0.55
2.6±0.2 3.2±0.1 3.4±0.1 3.6±0.2 3.9±0.3 4.0±0.2
2.8 3.1 3.4 3.6 3.8 3.9
[9] are also given. Since the proton is fully stripped of its orbital electrons above 200 keV [9], the S H e / S . ratio directly gives the square of the effective charge of the He ion at a spesific energy, as it is moving through Se. If this effective charge is target independent, this ratio can be used for scaling H to He stopping or vice versa. At the energy of about 0.50 M e V / a m u the ratio of 4 indicates that the He ion is fully stripped of its orbital electrons. 3.3. Straggling of hydrogen and helium ions in Se The experimental results for the straggling of hydrogen- and helium ions in Se are plotted in fig. 3 as a function of projectile energy per nucleon. The results are normalized to the energy independent Bohr [22] value: $2~ = 4~Z2Z2e 4 N A R ,
I
o
o 100 o
t~
50
I
0,0
0,5
VELOCITY
1,0 1/2 (PleV/amu)
Fig. 2. The experimental stopping power of helium ions (circles) in a 139 p~g cm 2 Se-foil. The solid line represents the semiempirical values of Ziegler [9]. The experimental results of Nakata [13] (triangles) and Lin et al. [14] (squares) are also shown.
362
J. Conradie et al. / Energy loss and straggling
1,5
I
[
I
I
8 1,0
0,5 A,6~t~& ix
S I
0,0
0,5
I
1,0
I
1,5
I
2,0 ENERGY (PleV/clmu)
2,5
Fig. 3. Experimental normalized straggling of hydrogen and helium ions in Se; Open squares: protons in 210/~g cm 2 Se, full squares: protons in 139 /~g cm 2 Se, open circles: deuterons in 210 ~tg cm 2 Se, full circles: deuterons in 139/zg cm -2 Se, triangles: alpha particles in 139 /~g cm 2 Se. The prediction of the free electron gas model, as calculated by Chu (full line) is also shown. The following experimental results are also shown: ,x: Eckardt [12] (hydrogen ions in Se), A: Eckardt [12] (helium ions in Se), "~: Besenbacher et al. [18] (hydrogen ions in Kr), v; Besenbacher et al. [18] (helium ions in Kr).
Here N A R is the target thickness perpendicular to the ion beam in atoms per unit area. Lindhard and Scharff [15] extended Bohr's model and treated the atomic electron cloud as consisting of two independent free electron gases, viz. and outer part with Fermi velocity v v < v and an inner part with v v > v . This free electron gas theory has subsequently been further extended by Bonderup and Hvelplund [23] and by Chu [24]. As has already been pointed out by reference 3, this model is expected to lead to problems at low velocities, as the unrealistic assumption of independence of the different parts of the electronic cloud violates the Pauli principle. Above 0.25 M e V / a m u the normalized straggling for hydrogen and helium ions is energy independent, but have different values: at higher energies helium straggling is in agreement with the Bohr value while the hydrogen value is about 8% lower. Below 0.25 M e V / a m u a sharp drop in straggling is observed for all ions. The measured straggling in Se in the low energy region of our results is higher than predicted by the free electron gas theory. This is in general agreement with measurements on Ni [1,2], A1 [2], Au [2], Cu [3], Ge [4] and Ag [5] done by means of the same transmission technique. Besenbacher et al. [16], who analysed the straggling data for some gases at particle energies below 1 M e V / a m u , explained these deviations from the theoretical results by additional straggling contributions from spatial atomic correlation effects and charge state fluctuations. It is believed that spatial atomic correla-
tion effects and charge state fluctuations can be ignored [16] when dealing with solids. Andersen et al. [17] however, found that for lithium ions, the solid target Se and the gas target Kr behave very similar when straggling was considered. In fig. 3 our experimental straggling results of hydrogen and helium ions in Se, are within experimental error in agreement with the experimental results for Kr of Besenbacher et al. [18]. Factors that can have a significant effect on straggling measurements in solids are texture effects and foil inhomogeneity. Texture effects due to partially channelled ions can positively be excluded. Since the halfangle for channelling is proportional to ( Z I / M I v 2 ) 1 / 2 the influence of texture on straggling should be less for deuterons than for protons of the same velocity. This contribution due to texture effects should also be energy dependent. No such effects were observed in our experimental results. Foil inhomogeneity would result in a higher reduced straggling for helium than for hydrogen ions [16]. However, this effect is also dependent on the target thickness while our experimental results for hydrogen ions for two foils of different thicknesses agree. The most probable explanation for the discrepency between the normalized helium and hydrogen straggling which is also observed for some other heavy target materials [5] is that charge state fluctuations of the helium ions play a more important role in solids than is generally believed.
J. Conradie et al. / Energy loss and straggling
4. S u m m a ~ The present measurements show that the stopping of hydrogen and helium ions is somewhat higher than the semiempirical values [8,9] in the vicinity of the stopping power maximum. In agreement with other results on straggling in solids [1-5] it can be concluded that the free electron gas model does not provide the correct reduced straggling in solids at low ion velocities. This discrepancy between theory and experiment cannot be explained by foil inhomogeneity or texture effects.
References [1] E. Friedland and J.M. Lombaard, Nucl. Instr. and Meth. 163 (1979) 523. [2] E. Friedland and J.M. Lombaard, Nucl. Instr. and Meth. 168 (1980) 25. [3] E. Friedland and C.P. Kotze, Nucl. Instr. and Meth. 191 (1981) 490. [4] J.B. Malherbe and H.W. Alberts, Nucl. Instr. and Meth. 192 (1982) 559. [5] J.M. Lombaard, J. Conradie and E. Friedland, to be published. [6] H.H. Andersen, J. Bottiger and H. Knudsen, eds., Ion beam analysis (North-Holland, Amsterdam, 1980).
363
[7] E. Friedland, Nucl. Instr. and Meth. 150 (1978) 301. [8] H.H. Andersen and J.F. Ziegler, Hydrogen stopping powers and ranges in all elements (Pergamon Press, New York, 1977). [9] J.F. Ziegler, Helium stopping powers and ranges in all elements (Pergamon Press, New York, 1977). [10] D.W. Green, J.N. Cooper and J.C. Harris, Phys. Rev. 98 (1955) 466. [11] H. Nakata, Phys. Rev. B3 (1971) 2847. [12] J.C. Eckardt, Phys. Rev. A18 (1978) 426. [13] H. Nakata, Can. J. Phys. 47 (1969) 2545. [14] W.K. Lin, H.G. Olson and D. Powers, Phys. Rev. B8 (1973) 1881. [15] J. Lindhard and M. Scharff, Dan. Vid. Selsk. Mat. Fys. Medd. 27 (1953) no. 15. [16] F. Besenbacher, J.U. Andersen and F. Bonderup, Nucl. Instr. and Meth. 168 (1980) 1. [17] H.H. Andersen, F. Besenbacher and P. Goddiksen, Nucl. Instr. and Meth. 168 (1980) 75. [18] F. Besenbacher, H.H. Andersen, P. Hvelplund and H. Knudsen, Dan. Vid. Selsk. Mat. Fys. Medd. 40 (1981) no. 9. [19] P. Sigmund, in Radiation damage processes in materials (Nordhoff, Leyden, 1975) p. 3. [20] U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. [21] C. Varelas and J.P. Biersack, Nucl. Instr. and Meth. 79 (1970) 213. [22] N. Bohr, Phil. Mag. 30 (1915) 581. [23] E. Bonderup and P. Hvelplund, Phys. Rev. A4 (1971) 562. [24] W.K. Chu, Phys. Rev. A13 (1976) 2057.
ENERGY-LOSS J. C O N R A D I E ,
AND STRAGGLING J. L O M B A A R D
359
OF HYDROGEN
AND HELIUM
IONS IN SELENIUM
a n d E. F R I E D L A N D
Department of Physics, University of Pretoria, South Africa Received 7 October 1981 and in revised form 23 April 1982 Using a transmission technique, the energy-loss and straggling of protons, deuterons and helium ions in Se were measured in the energy range between 0.3 and 2.5 MeV. The experimental stopping powers of the various ions are the same as the semiempirical values of Andersen and Ziegler, except in the vicinity of the stopping power maximum. The reduced straggling for the helium and hydrogen ions is energy independent above 0.25 MeV/amu with the helium values the same as predicted by Bohr but the hydrogen values are somewhat lower.
1. Introduction The shortage of reliable experimental results of the energy-loss and straggling of light ions in homogeneous solid targets, gave rise to a systematic study of the energy-loss and straggling of hydrogen and helium ions in different absorber foils [1-5]. In this paper we explore the energy-loss and straggling of these ions, with energies in the range from 300 keV to 2500 keV, in selenium. During the last decade the use of ion beams has proved itself a very powerful tool in the modification as well as in the analysis of the near surface region of materials [6]. In these analysis, it is important to know the exact behaviour of the ion in matter. When a beam of charged particles penetrates matter, the slowing-down is accompanied by a spreading of the beam energy due to statistical fluctuations in the number of collision processes and the energy transfer per collision. In many cases of practical interest, the distribution in energy loss is sufficiently close to a Gaussian that the spreading around the average value is completely characterized by the average square fluctuation in energy loss, also known as the energy straggling. Accurate information on both energy loss and energy straggling is important for example in the analysis of ion implantation profiles using backscattering or nuclear reactions [7]. In practice stopping values are usually obtained from the semiempirical values compiled by Andersen and Ziegler [8] for hydrogen and by Ziegler [9] for helium ions in all elements. In the case of Se, the semiempirical values are based on few experimental data points [10-14]. Some data on straggling are available [12]. Experimental straggling results at low energies on some solid targets [1-5] are higher than predicted by the 0167-5087/83/0000-0000/$03.00 © 1983 North-Holland
free electron gas theory of Lindhard and Scharff [15]. Besenbacher et al. [16] who analysed the straggling results for some gases at particle energies below 1 M e V / a m u , explained these deviations as due to additional straggling contributions from spatial atomic correlation effects and charge state fluctuations. They further stated that the correlation effects should be much smaller in a solid than in a gas and that the effect due to charge state fluctuations is negligible in a solid. Experimental straggling in a homogeneous solid target should therefore approximately be given by the LindhardScharff expression. However, for lithium ions with velocities around the Bohr velocity ( E = 175 keV), copper, selenium and silver seemed to yield straggling parameters in good agreement with gas data [17]. Preliminary measurements on selenium showed that these targets are easily to prepare without texture effects and foil inhomogeneities. Since selenium is a solid near the gas krypton in the periodic table, it would be of interest to compare the straggling of hydrogen and helium ions in selenium and in krypton [18].
2. Experimental The same experimental setup as described in ref. 1 was used. Since difficulties were experienced to produce self-supporting selenium foils, the absorber foils were prepared by vacuum deposition onto carbon backings (20 /Lg cm -2) mounted over the 6 mm opening of the absorber holder. With this method, inhomogeneities in the foils due to stresses produced in the foils during floatation processes, were eliminated. Different methods were used to measure the thickness of each absorber foil. First weighing of the foils was done by placing a thin microscope cover glass on a mask with a hole of
J. Conradie et al. / Energy loss and straggling
360
k n o w n dimensions next to the absorber holder during the evaporation process. The weight of the cover glass was determined with a microbalance before a n d after the evaporation. The thickness as well as the evaporation rate was controlled by using a quartz crystal monitor. F u r t h e r m o r e the thickness was independently calculated from the yield of 1 MeV alpha particles scattered through 150 ° using the well k n o w n Rutherford formula. The agreement between the different thickness determinations was within 6%. Foils of thicknesses of 139 /~g cm 2 and 210 /~g cm 2 were used. The final results were corrected for the c a r b o n backing by measuring the energy-loss and straggling through the backing independently.
3. Results and discussion
3.1. Energy-loss of hydrogen ions in Se Fig. 1 presents our results for the stopping power of protons and deuterons in Se. A very complete review on energy-loss of charged particles was given by Sigmund [19] in 1973. At low energies (v < Vo Z2/3, where v is the projectile velocity, v 0 the Bohr velocity and Z the atomic n u m b e r of the projectile) the electronic stopping power is found to be proportional to the projectile velocity. In
60--
t
I
the m e d i u m and low velocity region where the maxim u m energy loss occurs, no satisfactory theoretical model exists to describe the stopping of these ions in matter. The high energy behaviour of the stopping power is well described by the Bethe formula [20]. Andersen a n d Ziegler [8] used this Bethe stopping-power formula as the theoretical basis in the high-energy ( E / a m u > 600 keV) region of their semi-empirical stopping values. In the medium-energy region, from 600 keV a n d down, they mainly used the Varelas-Biersack [21] formula to fit the experimental data of 24 elements. To obtain stopping values for Se, a new fit which is considered to be accurate to a b o u t 5% at 500 keV, was made. At lower energies the accuracy deteriorates. Our experimental results for the stopping of hydrogen ions in Se are, within experimental error the same as the semiempirical values of A n d e r s e n and Ziegler [8] for projectile energies above 0.5 M e V / a m u . There is good agreement between our results and those of G r e e n et al. [10] and N a k a t a [11]. At the low energy side of our data near the stopping power maximum, our values are more than 14% higher than the semiempirical values, while results of Eckardt [12] are in agreement with these values. It must be pointed out that Eckhardt's results were used in obtaining the semiempirical values [8]. Eckardt's [12] experimental results of b o t h the energy-loss of hydrogen ions
I
I
A i
i= ,u
E
*÷
°C~l 0 0
2O
0,0--
I
I
I
0,5
1,0
1,5
I
2,0 1/~ 2,5 VELOCITY (MeV/Qmu)
Fig. 1. The experimental stopping power of hydrogen ions in Se; open squares: protons in 210 ~g cm -2 Se, full squares: protons in 139 ~g cm 2 Se, open circles: deuterons in 210 ,ug cm 2 Se, full circles: deuterons in 139 tLg cm 2 Se. The error bars are within the symbols if not indicated otherwise. The solid line represents the semiempirical values of Andersen and Ziegler [8]. The experimental results of Green et al. [10] (~7), Nakata [11] (A) and Eckardt [12] (v) are also shown.
361
J. Conradie et al. / Energy loss and straggling (fig. l) and the energy-straggling of hydrogen and helium ions (fig. 3) are lower than our results. The experimental results of the energy-loss of hydrogen ions in Se of Eckardt, also seem lower than those of Green et al. [10]. As stopping power depends only on the effective charge of the projectile, protons and deuterons should have the same stopping power at the same projectile velocity. The agreement between the proton and deuteron stopping values are clearly illustrated in fig. 1. 3.2. Energy-loss of helium ions in Se The stopping of helium ions is plotted in fig. 2. Within experimental error, our values are the same as the semiempirical values of Ziegler [9] except in the vicinity of the stopping power maximum where our results are about 8-10% higher. Experimental values of Nakata [13] are somewhat (4-7%) lower than the semi-empirical values and 8-13% lower than our experimental results. The experimental results of Lin et al. [14] are in good agreement with the semiempirical values. Ziegler [9] least-squared fit a curve through more than 10000 very much scattered data points to obtain a target independent ratio S H e / S n, of He to H stopping. This ratio can be used for scaling H to equivalent He stopping values and it is also defined as the ratio of the square of the effective charge of He to that of H in solids [9]. The best S H J S u values can be obtained by using ions of the same velocity on the same target with the same experimental setup. Table 1 gives the ratio of our experimental He to H stopping in Se at the same projectile velocity. For comparison the values of Ziegler
150
I
I
I
I
L
Table 1 Comparison of the experimental He to H stopping at the same velocity with the corresponding values of Ziegler [9]. Energy (MeV/amu)
SHe /SH experimental
SHe / S u Ziegler
0.13 0.18 0.25 0.32 0.50 0.55
2.6±0.2 3.2±0.1 3.4±0.1 3.6±0.2 3.9±0.3 4.0±0.2
2.8 3.1 3.4 3.6 3.8 3.9
[9] are also given. Since the proton is fully stripped of its orbital electrons above 200 keV [9], the S H e / S . ratio directly gives the square of the effective charge of the He ion at a spesific energy, as it is moving through Se. If this effective charge is target independent, this ratio can be used for scaling H to He stopping or vice versa. At the energy of about 0.50 M e V / a m u the ratio of 4 indicates that the He ion is fully stripped of its orbital electrons. 3.3. Straggling of hydrogen and helium ions in Se The experimental results for the straggling of hydrogen- and helium ions in Se are plotted in fig. 3 as a function of projectile energy per nucleon. The results are normalized to the energy independent Bohr [22] value: $2~ = 4~Z2Z2e 4 N A R ,
I
o
o 100 o
t~
50
I
0,0
0,5
VELOCITY
1,0 1/2 (PleV/amu)
Fig. 2. The experimental stopping power of helium ions (circles) in a 139 p~g cm 2 Se-foil. The solid line represents the semiempirical values of Ziegler [9]. The experimental results of Nakata [13] (triangles) and Lin et al. [14] (squares) are also shown.
362
J. Conradie et al. / Energy loss and straggling
1,5
I
[
I
I
8 1,0
0,5 A,6~t~& ix
S I
0,0
0,5
I
1,0
I
1,5
I
2,0 ENERGY (PleV/clmu)
2,5
Fig. 3. Experimental normalized straggling of hydrogen and helium ions in Se; Open squares: protons in 210/~g cm 2 Se, full squares: protons in 139 /~g cm 2 Se, open circles: deuterons in 210 ~tg cm 2 Se, full circles: deuterons in 139/zg cm -2 Se, triangles: alpha particles in 139 /~g cm 2 Se. The prediction of the free electron gas model, as calculated by Chu (full line) is also shown. The following experimental results are also shown: ,x: Eckardt [12] (hydrogen ions in Se), A: Eckardt [12] (helium ions in Se), "~: Besenbacher et al. [18] (hydrogen ions in Kr), v; Besenbacher et al. [18] (helium ions in Kr).
Here N A R is the target thickness perpendicular to the ion beam in atoms per unit area. Lindhard and Scharff [15] extended Bohr's model and treated the atomic electron cloud as consisting of two independent free electron gases, viz. and outer part with Fermi velocity v v < v and an inner part with v v > v . This free electron gas theory has subsequently been further extended by Bonderup and Hvelplund [23] and by Chu [24]. As has already been pointed out by reference 3, this model is expected to lead to problems at low velocities, as the unrealistic assumption of independence of the different parts of the electronic cloud violates the Pauli principle. Above 0.25 M e V / a m u the normalized straggling for hydrogen and helium ions is energy independent, but have different values: at higher energies helium straggling is in agreement with the Bohr value while the hydrogen value is about 8% lower. Below 0.25 M e V / a m u a sharp drop in straggling is observed for all ions. The measured straggling in Se in the low energy region of our results is higher than predicted by the free electron gas theory. This is in general agreement with measurements on Ni [1,2], A1 [2], Au [2], Cu [3], Ge [4] and Ag [5] done by means of the same transmission technique. Besenbacher et al. [16], who analysed the straggling data for some gases at particle energies below 1 M e V / a m u , explained these deviations from the theoretical results by additional straggling contributions from spatial atomic correlation effects and charge state fluctuations. It is believed that spatial atomic correla-
tion effects and charge state fluctuations can be ignored [16] when dealing with solids. Andersen et al. [17] however, found that for lithium ions, the solid target Se and the gas target Kr behave very similar when straggling was considered. In fig. 3 our experimental straggling results of hydrogen and helium ions in Se, are within experimental error in agreement with the experimental results for Kr of Besenbacher et al. [18]. Factors that can have a significant effect on straggling measurements in solids are texture effects and foil inhomogeneity. Texture effects due to partially channelled ions can positively be excluded. Since the halfangle for channelling is proportional to ( Z I / M I v 2 ) 1 / 2 the influence of texture on straggling should be less for deuterons than for protons of the same velocity. This contribution due to texture effects should also be energy dependent. No such effects were observed in our experimental results. Foil inhomogeneity would result in a higher reduced straggling for helium than for hydrogen ions [16]. However, this effect is also dependent on the target thickness while our experimental results for hydrogen ions for two foils of different thicknesses agree. The most probable explanation for the discrepency between the normalized helium and hydrogen straggling which is also observed for some other heavy target materials [5] is that charge state fluctuations of the helium ions play a more important role in solids than is generally believed.
J. Conradie et al. / Energy loss and straggling
4. S u m m a ~ The present measurements show that the stopping of hydrogen and helium ions is somewhat higher than the semiempirical values [8,9] in the vicinity of the stopping power maximum. In agreement with other results on straggling in solids [1-5] it can be concluded that the free electron gas model does not provide the correct reduced straggling in solids at low ion velocities. This discrepancy between theory and experiment cannot be explained by foil inhomogeneity or texture effects.
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