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Ion-induced tracks in Bi4Ge3O12 and Bi12GeO20 crystals
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 245 (2006) 243–245 www.elsevier.com/locate/nimb
Ion-induced tracks in Bi4Ge3O12 and Bi12GeO20 crystals G. Szenes a
a,*
´ . Pe´ter , D. Fink b, S. Klaumu¨nzer b, F. Pa´szti c, A
d
Department of General Physics, Eo¨tvo¨s University, P.O. Box 32, H-1518 Budapest, Hungary b Hahn-Meitner Institute, Berlin, Germany c Research Institute for Particle and Nuclear Physics, Budapest, Hungary d Research Institute for Solid State Physics and Optics, Budapest, Hungary Available online 4 January 2006
Abstract Single crystalline Bi4Ge3O12 and Bi12GeO20 samples were irradiated at room temperature with various fluences of O, Ne, Ar, Kr, Xe and Pb ions. The ion energies were 0.35, 1.0 and 1.7 MeV/u. The irradiated samples were investigated by Rutherford backscattering in channeling geometry. The track radii derived from the data depend on electronic stopping power and ion velocity. The average threshold values for track formation are (3.3 ± 0.5) and (2.6 ± 0.3) keV/nm for Bi4Ge3O12 and Bi12GeO20, respectively. The results are in good agreement with an analytical thermal spike model previously developed. 2005 Elsevier B.V. All rights reserved. PACS: 61.80.Az; 61.80.Jh; 61.82.Ms Keywords: Ion-irradiation; Tracks; Thermal spike; Velocity effect; Insulators
1. Introduction In spite of the fact that formation of ion tracks is long known [1] the relevant mechanisms are still under debate. The most prominent ones are the Coulomb explosion mechanism [1–4] and the thermal spike [5–8]. The Coulomb explosion mechanism is based on the idea that the ionized target atoms mutually repel each other in the wake of a fast ion, thus releasing a shock wave, which induces material changes. Temperature increases are thought to be unimportant. The alternative mechanism, the thermal spike, assumes that in the wake of the fast ion the atomic motion can be described as heat and material changes are due to heat-driven phase transformations. There exist essentially two thermal spike models for track formation. One version [6,7] is quite elaborated and describes the temperature evolution in the electronic and
*
Corresponding author. Tel.: +36 1 372 2821; fax: +36 1 372 2811. E-mail address: [email protected] (G. Szenes).
0168-583X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.11.140
the atomic system by classical heat conduction while the two systems are coupled by electron–phonon coupling. Using its strength as the only adjustable parameter this model requires considerable numerical computations and is in accord with numerous experiments although the existence of an atomic temperature on time-scales below 1 ps is highly questionable. This problem is bypassed by a much simpler analytical approach [8], which concentrates on the late phase of track formation where a Gaussian temperature distribution may be taken as a first approximation to the thermal spike. Besides its mathematical simplicity the appealing feature of this analytical model is an experimentally verified correlation between the thresholds for track formation Set, and the material parameters S et ¼ pqcT 0 a2 ð0Þ=g;
ð1Þ
where q denotes the mass density, c the specific heat calculated by the Dulong–Petit law, a(0) the width of the temperature distribution in the moment of the maximum peak temperature and T0 = Tm Tir (Tir and Tm are the irradiation and the melting temperatures, respectively). The efficiency parameter g characterizes that fraction gSe
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G. Szenes et al. / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 243–245
of the electronic stopping power Se, which appears as heat in the vicinity of the ion trajectory. Surprisingly, it has turned out that for many insulators a(0) = 4.5 nm and g = 0.4 holds for projectile energies E < 2.2 MeV/u [8– 10]. Furthermore, this model predicts that the effective track radii Re as a function of electronic stopping power Se start as R2e ¼ a2 ð0Þ lnðS e =S et Þ for Re 6 a(0) and R2e ¼ a2 ð0ÞS e =ð2:7S et Þ for Re > a(0). Apart from TeO2 [10,11] previous tests of Eq. (1) and the predicted track evolution Re(Se) comprised only insulators with relatively high melting temperatures [7,12,13]. It is the purpose of this work to examine the validity of the predictions of this analytical model for Bi4Ge3O12 and Bi12GeO20, two other oxides with low melting points. For irradiation with low velocity ions at room temperature Eq. (1) predicts Set = 2.8 keV/nm for Bi4Ge3O12 (q = 7120 kg/m3, c = 0.38 kJ/ kg K and Tm = 1323 K [14]) and Set = 2.4 keV/nm for Bi12GeO20 (q = 9220 kg/m3, c = 0.285 kJ/kg K and Tm = 1208 K [14]). 2. Experimental The Bi4Ge3O12 and Bi12GeO20 single crystals were irradiated by O, Ne, Ar, Kr and Xe beams at the ISL cyclotron of the Hahn-Meitner Institute (Berlin, Germany), with energies of about 1.7 MeV/u (initial energy) and 0.35 MeV/ u by the use of aluminum foils of appropriate thickness. Two samples, one without and one with an aluminum foil, were irradiated simultaneously with the same fluence. Some TeO2 and Bi12GeO20 samples were also irradiated in GANIL (Caen, France) by Pb ions of 4.5 MeV initial energy, which was reduced to 0.75 and 1.2 MeV/u by aluminum foils. The irradiations were performed at room temperature with various fluences Ut. The SRIM code was applied for the calculation of the stopping powers [15]. The flux was kept between 2 and 4 · 108 ions/cm2 s to avoid beam heating effects. The irradiated crystals were investigated by Rutherford backscattering in channeling geometry (c-RBS). The details of the measurements of the damaged fraction Fd were reported in [10,11]. Fd is linked to the damage cross section A ¼ pR2e by the Poisson law Fd = 1 exp{AUt}. The absolute experimental error of Re is about 10–15%. However, the relative accuracy is improved when samples are compared after simultaneous irradiation due to the elimination of the absolute errors in the fluence determination.
Fig. 1. Variation of R2e (Re – effective track radius) with the electronic stopping power Se in Bi4Ge3O12. The numbers in the legend show the specific ion energies in MeV/u.
Y3Fe5O12 a velocity dependence of Set is not expected in this range [16]. Although the fits provided Set = (2.8 ± 0.45) and (3.8 ± 0.55) keV/nm for E 0.35 and E 1.7 MeV/u, respectively, the Set value of 2.8 keV/nm calculated from Eq. (1) for Bi4Ge3O12 is still in reasonable agreement with the experiments. The data measured in Bi12GeO20 at low values of Se did not provide a reliable estimate for Set. However, it was shown in the analysis of tracks in TeO2 that Set can be estimated reliably from the slope of the linear R2e ðS e Þ curves at Re > 4.5 nm. This method provided good agreement with the estimate from the intersection with the horizontal axis [10,11]. The applied procedure is described in [10]. We made a linear fit for Bi12GeO20 including the two data obtained by Pb irradiation and obtained Set = (2.6 ± 0.3) keV/nm for E (0.35–1.7) MeV/u. The results nicely fit to previous data shown in Fig. 2 from [17] which are also mean values over a range of E.
3. Results and discussion The results of the c-RBS measurements on Bi4Ge3O12 versus the electronic stopping power Se are shown in Fig. 1. There is a separation between the track evolution curves measured at E 0.35 and E 1.7 MeV/u indicating a velocity effect. The solid curves are fits according to the analytical track model of Szenes [8]. The fitting parameters are the threshold electronic stopping powers for track formation Set. According to the analysis of experiments on
Fig. 2. Variation of the threshold electronic stopping power Set with the material parameters q, c and T0 denoting the density, specific heat and the difference between the melting point and the irradiation temperature, respectively. The error bars are within the symbol, when they are not shown. The solid line is given by Eq. (1) with a(0) = 4.5 nm and g = 0.4.
G. Szenes et al. / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 243–245
245
4. Conclusions
Fig. 3. Variation of R2e (Re – effective track radius) with the electronic stopping power Se in Bi4Ge3O12, Bi12GeO20 and TeO2. Se is scaled by qcT0 where q, c and T0 are the density, average specific heat and the difference between the melting point Tm and the irradiation temperature Tir. The numbers in the legend show the specific ion energies in MeV/u. In the case of Pb irradiations, the open and solid symbols refer to Bi12GeO20 and TeO2, respectively.
In Fig. 3 scaled plots are shown for Bi4Ge3O12 and Bi12GeO20. The data for TeO2 [10] are included for comparison. There is a separation of the track data for irradiations at E 0.35–1 MeV/u and at E 1.7 MeV/u. The solid line is calculated with a(0) = 4.5 nm and g = 0.4. It gives a good description for tracks induced by low velocity ions in Y3Fe5O12, LiNbO3 and SiO2, as well [10]. Although the experimental data do not follow a single curve in Fig. 3, the scaling feature is evident. Our opinion is that scaling is an inherent property of the mechanism of track formation. The material parameters of the actual solid appear only in the factor qcT0 in Eq. (1) and in the scaling plots. The physical meaning of this factor is the energy required to heat up the material in a unit volume from Tir to Tm. It is evident that material parameters other than q, c and Tm may affect considerably the track size only in those solids for which scaling does not hold. Both the scaling and the agreement of calculated and measured Set values can be considered as strong evidence for the thermal mechanism of track formation. Once more, the analytical thermal spike model turns out to provide useful expressions, which yield a reliable description of the track evolution as a function of electronic stopping power. We consider a real effect the small deviations between track evolutions at E = 0.35 and 1.7 MeV/u, because the relative error was reduced due to the simultaneous irradiation of samples at different ion energies with equal fluences. Similar effects were observed in three different materials and we ascribe this separation to the velocity effect [7].
The track sizes are systematically smaller for irradiations at E 1.7 MeV/u than at E 0.35 MeV/u in Bi4Ge3O12, Bi12GeO20 and TeO2. This finding is attributed to the velocity effect. The mean values of Set are (3.3 ± 0.5) keV/nm and (2.6 ± 0.3) keV/nm for Bi4Ge3O12 and Bi12GeO20, respectively, which are in good agreement with the predictions of the Szenes model. The scaled track evolution curves, R2e versus Se/qcT0 of Bi4Ge3O12 and Bi12GeO20 agree within experimental error with the theoretical prediction and with those of TeO2, Y3Fe5O12, LiNbO3 and SiO2. Those material parameters, which are not included into the scaling factor, can have only minor effect on the track size. The results support the validity of the thermal spike mechanism. Acknowledgments The authors are grateful to the staff of the ISL cyclotron (HMI, Berlin) and of the CIRIL (Caen), especially to Dr. E. Balanzat for their assistance in the experiments. The partial support of the National Scientific Research Fund (OTKA, Hungary) under Contracts Nos. T031756, T043247 and T046990 is acknowledged by G. Sz. and F.P. References [1] R.L. Fleischer, R.M. Walker, P.B. Price, Nuclear Tracks in Solids, University of California Press, 1975. [2] D. Lesueur, A. Dunlop, Radiat. Eff. Defects Solids 126 (1993) 163. [3] H. Dammak, A. Dunlop, D. Lesueur, A. Brunelle, S. Della-Negra Y. LeBeyec, Phys. Rev. Lett. 74 (1995) 1135. [4] H. Dammak, A. Dunlop, D. Lesueur, Nucl. Instr. and Meth. B 107 (1996) 204. [5] F. Seitz, J.S. Koehler, Solid State Phys. 2 (1956) 305. [6] M. Toulemonde, E. Paumier, C. Dufour, Radiat. Eff. Defects Solids 126 (1993) 201. [7] A. Meftah, F. Brisard, J.M. Costantini, E. Dooryhee, M. Hage-Ali, M. Hervieu, J.P. Stoquert, F. Studer, M. Toulemonde, Phys. Rev. B 49 (1994) 12457. [8] G. Szenes, Phys. Rev. B 51 (1995) 8026. [9] G. Szenes, Phys. Rev. B 60 (1999) 3140. ´ . Pe´ter, A.I. Popov, Nucl. Instr. and Meth. B [10] G. Szenes, F. Pa´szti, A 166–167 (2000) 949. ´ . Pe´ter, D. Fink, Nucl. Instr. and Meth. B 191 [11] G. Szenes, F. Pa´szti, A (2002) 186. [12] A. Meftah, J.M. Costantini, M. Djebara, N. Khalfaoui, J.P. Stoquert, F. Studer, M. Toulemonde, Nucl. Inst. and Meth. B 122 (1997) 470. [13] B. Canut, S.M.M. Ramos, R. Brenier, P. Thevenard, J.L. Louvet, M. Toulemonde, Nucl. Instr. and Meth. B 107 (1996) 194. [14] K. Shigematsu, Y. Anzai, K. Omote, S. Kimura, J. Cryst. Growth 137 (1994) 509. [15] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon Press, New York, 1985. [16] G. Szenes, Nucl. Instr. and Meth. B 146 (1998) 420. [17] G. Szenes, J. Nucl. Mater. 336 (2005) 81.
Nuclear Instruments and Methods in Physics Research B 245 (2006) 243–245 www.elsevier.com/locate/nimb
Ion-induced tracks in Bi4Ge3O12 and Bi12GeO20 crystals G. Szenes a
a,*
´ . Pe´ter , D. Fink b, S. Klaumu¨nzer b, F. Pa´szti c, A
d
Department of General Physics, Eo¨tvo¨s University, P.O. Box 32, H-1518 Budapest, Hungary b Hahn-Meitner Institute, Berlin, Germany c Research Institute for Particle and Nuclear Physics, Budapest, Hungary d Research Institute for Solid State Physics and Optics, Budapest, Hungary Available online 4 January 2006
Abstract Single crystalline Bi4Ge3O12 and Bi12GeO20 samples were irradiated at room temperature with various fluences of O, Ne, Ar, Kr, Xe and Pb ions. The ion energies were 0.35, 1.0 and 1.7 MeV/u. The irradiated samples were investigated by Rutherford backscattering in channeling geometry. The track radii derived from the data depend on electronic stopping power and ion velocity. The average threshold values for track formation are (3.3 ± 0.5) and (2.6 ± 0.3) keV/nm for Bi4Ge3O12 and Bi12GeO20, respectively. The results are in good agreement with an analytical thermal spike model previously developed. 2005 Elsevier B.V. All rights reserved. PACS: 61.80.Az; 61.80.Jh; 61.82.Ms Keywords: Ion-irradiation; Tracks; Thermal spike; Velocity effect; Insulators
1. Introduction In spite of the fact that formation of ion tracks is long known [1] the relevant mechanisms are still under debate. The most prominent ones are the Coulomb explosion mechanism [1–4] and the thermal spike [5–8]. The Coulomb explosion mechanism is based on the idea that the ionized target atoms mutually repel each other in the wake of a fast ion, thus releasing a shock wave, which induces material changes. Temperature increases are thought to be unimportant. The alternative mechanism, the thermal spike, assumes that in the wake of the fast ion the atomic motion can be described as heat and material changes are due to heat-driven phase transformations. There exist essentially two thermal spike models for track formation. One version [6,7] is quite elaborated and describes the temperature evolution in the electronic and
*
Corresponding author. Tel.: +36 1 372 2821; fax: +36 1 372 2811. E-mail address: [email protected] (G. Szenes).
0168-583X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.11.140
the atomic system by classical heat conduction while the two systems are coupled by electron–phonon coupling. Using its strength as the only adjustable parameter this model requires considerable numerical computations and is in accord with numerous experiments although the existence of an atomic temperature on time-scales below 1 ps is highly questionable. This problem is bypassed by a much simpler analytical approach [8], which concentrates on the late phase of track formation where a Gaussian temperature distribution may be taken as a first approximation to the thermal spike. Besides its mathematical simplicity the appealing feature of this analytical model is an experimentally verified correlation between the thresholds for track formation Set, and the material parameters S et ¼ pqcT 0 a2 ð0Þ=g;
ð1Þ
where q denotes the mass density, c the specific heat calculated by the Dulong–Petit law, a(0) the width of the temperature distribution in the moment of the maximum peak temperature and T0 = Tm Tir (Tir and Tm are the irradiation and the melting temperatures, respectively). The efficiency parameter g characterizes that fraction gSe
244
G. Szenes et al. / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 243–245
of the electronic stopping power Se, which appears as heat in the vicinity of the ion trajectory. Surprisingly, it has turned out that for many insulators a(0) = 4.5 nm and g = 0.4 holds for projectile energies E < 2.2 MeV/u [8– 10]. Furthermore, this model predicts that the effective track radii Re as a function of electronic stopping power Se start as R2e ¼ a2 ð0Þ lnðS e =S et Þ for Re 6 a(0) and R2e ¼ a2 ð0ÞS e =ð2:7S et Þ for Re > a(0). Apart from TeO2 [10,11] previous tests of Eq. (1) and the predicted track evolution Re(Se) comprised only insulators with relatively high melting temperatures [7,12,13]. It is the purpose of this work to examine the validity of the predictions of this analytical model for Bi4Ge3O12 and Bi12GeO20, two other oxides with low melting points. For irradiation with low velocity ions at room temperature Eq. (1) predicts Set = 2.8 keV/nm for Bi4Ge3O12 (q = 7120 kg/m3, c = 0.38 kJ/ kg K and Tm = 1323 K [14]) and Set = 2.4 keV/nm for Bi12GeO20 (q = 9220 kg/m3, c = 0.285 kJ/kg K and Tm = 1208 K [14]). 2. Experimental The Bi4Ge3O12 and Bi12GeO20 single crystals were irradiated by O, Ne, Ar, Kr and Xe beams at the ISL cyclotron of the Hahn-Meitner Institute (Berlin, Germany), with energies of about 1.7 MeV/u (initial energy) and 0.35 MeV/ u by the use of aluminum foils of appropriate thickness. Two samples, one without and one with an aluminum foil, were irradiated simultaneously with the same fluence. Some TeO2 and Bi12GeO20 samples were also irradiated in GANIL (Caen, France) by Pb ions of 4.5 MeV initial energy, which was reduced to 0.75 and 1.2 MeV/u by aluminum foils. The irradiations were performed at room temperature with various fluences Ut. The SRIM code was applied for the calculation of the stopping powers [15]. The flux was kept between 2 and 4 · 108 ions/cm2 s to avoid beam heating effects. The irradiated crystals were investigated by Rutherford backscattering in channeling geometry (c-RBS). The details of the measurements of the damaged fraction Fd were reported in [10,11]. Fd is linked to the damage cross section A ¼ pR2e by the Poisson law Fd = 1 exp{AUt}. The absolute experimental error of Re is about 10–15%. However, the relative accuracy is improved when samples are compared after simultaneous irradiation due to the elimination of the absolute errors in the fluence determination.
Fig. 1. Variation of R2e (Re – effective track radius) with the electronic stopping power Se in Bi4Ge3O12. The numbers in the legend show the specific ion energies in MeV/u.
Y3Fe5O12 a velocity dependence of Set is not expected in this range [16]. Although the fits provided Set = (2.8 ± 0.45) and (3.8 ± 0.55) keV/nm for E 0.35 and E 1.7 MeV/u, respectively, the Set value of 2.8 keV/nm calculated from Eq. (1) for Bi4Ge3O12 is still in reasonable agreement with the experiments. The data measured in Bi12GeO20 at low values of Se did not provide a reliable estimate for Set. However, it was shown in the analysis of tracks in TeO2 that Set can be estimated reliably from the slope of the linear R2e ðS e Þ curves at Re > 4.5 nm. This method provided good agreement with the estimate from the intersection with the horizontal axis [10,11]. The applied procedure is described in [10]. We made a linear fit for Bi12GeO20 including the two data obtained by Pb irradiation and obtained Set = (2.6 ± 0.3) keV/nm for E (0.35–1.7) MeV/u. The results nicely fit to previous data shown in Fig. 2 from [17] which are also mean values over a range of E.
3. Results and discussion The results of the c-RBS measurements on Bi4Ge3O12 versus the electronic stopping power Se are shown in Fig. 1. There is a separation between the track evolution curves measured at E 0.35 and E 1.7 MeV/u indicating a velocity effect. The solid curves are fits according to the analytical track model of Szenes [8]. The fitting parameters are the threshold electronic stopping powers for track formation Set. According to the analysis of experiments on
Fig. 2. Variation of the threshold electronic stopping power Set with the material parameters q, c and T0 denoting the density, specific heat and the difference between the melting point and the irradiation temperature, respectively. The error bars are within the symbol, when they are not shown. The solid line is given by Eq. (1) with a(0) = 4.5 nm and g = 0.4.
G. Szenes et al. / Nucl. Instr. and Meth. in Phys. Res. B 245 (2006) 243–245
245
4. Conclusions
Fig. 3. Variation of R2e (Re – effective track radius) with the electronic stopping power Se in Bi4Ge3O12, Bi12GeO20 and TeO2. Se is scaled by qcT0 where q, c and T0 are the density, average specific heat and the difference between the melting point Tm and the irradiation temperature Tir. The numbers in the legend show the specific ion energies in MeV/u. In the case of Pb irradiations, the open and solid symbols refer to Bi12GeO20 and TeO2, respectively.
In Fig. 3 scaled plots are shown for Bi4Ge3O12 and Bi12GeO20. The data for TeO2 [10] are included for comparison. There is a separation of the track data for irradiations at E 0.35–1 MeV/u and at E 1.7 MeV/u. The solid line is calculated with a(0) = 4.5 nm and g = 0.4. It gives a good description for tracks induced by low velocity ions in Y3Fe5O12, LiNbO3 and SiO2, as well [10]. Although the experimental data do not follow a single curve in Fig. 3, the scaling feature is evident. Our opinion is that scaling is an inherent property of the mechanism of track formation. The material parameters of the actual solid appear only in the factor qcT0 in Eq. (1) and in the scaling plots. The physical meaning of this factor is the energy required to heat up the material in a unit volume from Tir to Tm. It is evident that material parameters other than q, c and Tm may affect considerably the track size only in those solids for which scaling does not hold. Both the scaling and the agreement of calculated and measured Set values can be considered as strong evidence for the thermal mechanism of track formation. Once more, the analytical thermal spike model turns out to provide useful expressions, which yield a reliable description of the track evolution as a function of electronic stopping power. We consider a real effect the small deviations between track evolutions at E = 0.35 and 1.7 MeV/u, because the relative error was reduced due to the simultaneous irradiation of samples at different ion energies with equal fluences. Similar effects were observed in three different materials and we ascribe this separation to the velocity effect [7].
The track sizes are systematically smaller for irradiations at E 1.7 MeV/u than at E 0.35 MeV/u in Bi4Ge3O12, Bi12GeO20 and TeO2. This finding is attributed to the velocity effect. The mean values of Set are (3.3 ± 0.5) keV/nm and (2.6 ± 0.3) keV/nm for Bi4Ge3O12 and Bi12GeO20, respectively, which are in good agreement with the predictions of the Szenes model. The scaled track evolution curves, R2e versus Se/qcT0 of Bi4Ge3O12 and Bi12GeO20 agree within experimental error with the theoretical prediction and with those of TeO2, Y3Fe5O12, LiNbO3 and SiO2. Those material parameters, which are not included into the scaling factor, can have only minor effect on the track size. The results support the validity of the thermal spike mechanism. Acknowledgments The authors are grateful to the staff of the ISL cyclotron (HMI, Berlin) and of the CIRIL (Caen), especially to Dr. E. Balanzat for their assistance in the experiments. The partial support of the National Scientific Research Fund (OTKA, Hungary) under Contracts Nos. T031756, T043247 and T046990 is acknowledged by G. Sz. and F.P. References [1] R.L. Fleischer, R.M. Walker, P.B. Price, Nuclear Tracks in Solids, University of California Press, 1975. [2] D. Lesueur, A. Dunlop, Radiat. Eff. Defects Solids 126 (1993) 163. [3] H. Dammak, A. Dunlop, D. Lesueur, A. Brunelle, S. Della-Negra Y. LeBeyec, Phys. Rev. Lett. 74 (1995) 1135. [4] H. Dammak, A. Dunlop, D. Lesueur, Nucl. Instr. and Meth. B 107 (1996) 204. [5] F. Seitz, J.S. Koehler, Solid State Phys. 2 (1956) 305. [6] M. Toulemonde, E. Paumier, C. Dufour, Radiat. Eff. Defects Solids 126 (1993) 201. [7] A. Meftah, F. Brisard, J.M. Costantini, E. Dooryhee, M. Hage-Ali, M. Hervieu, J.P. Stoquert, F. Studer, M. Toulemonde, Phys. Rev. B 49 (1994) 12457. [8] G. Szenes, Phys. Rev. B 51 (1995) 8026. [9] G. Szenes, Phys. Rev. B 60 (1999) 3140. ´ . Pe´ter, A.I. Popov, Nucl. Instr. and Meth. B [10] G. Szenes, F. Pa´szti, A 166–167 (2000) 949. ´ . Pe´ter, D. Fink, Nucl. Instr. and Meth. B 191 [11] G. Szenes, F. Pa´szti, A (2002) 186. [12] A. Meftah, J.M. Costantini, M. Djebara, N. Khalfaoui, J.P. Stoquert, F. Studer, M. Toulemonde, Nucl. Inst. and Meth. B 122 (1997) 470. [13] B. Canut, S.M.M. Ramos, R. Brenier, P. Thevenard, J.L. Louvet, M. Toulemonde, Nucl. Instr. and Meth. B 107 (1996) 194. [14] K. Shigematsu, Y. Anzai, K. Omote, S. Kimura, J. Cryst. Growth 137 (1994) 509. [15] J.F. Ziegler, J.P. Biersack, U. Littmark, The Stopping and Range of Ions in Solids, Pergamon Press, New York, 1985. [16] G. Szenes, Nucl. Instr. and Meth. B 146 (1998) 420. [17] G. Szenes, J. Nucl. Mater. 336 (2005) 81.