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Energy loss of slow protons in Au crystalline thin film
Vacuum 83 (2009) S196–S199
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Energy loss of slow protons in Au crystalline thin film Tomasz M. Gwizda11a a, *, Jerzy Czerbniak a, Roch Andrzejewski a, b, Krzysztof Gront a, Marek Moneta a a b
´dz´, Pomorska 149/153, 90-236 Ło ´dz´, Poland Department of Solid State Physics, University of Ło Centro de Micro-Analisis de Materiales, Madrid, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 June 2008 Received in revised form 22 January 2009 Accepted 30 January 2009
Energy spectra of low energy protons transmitted through w12 nm Au self-supporting single crystalline foil in channeling direction were calculated within various simulation codes and compared with published recently experimental data. It was shown that the raw energy distribution is by no means Gaussian; however, it gets this shape after accounting for the energy loss straggling and the energy resolution of the experimental setup. The most probable and the average energy loss differ significantly from experimental energy loss despite calibration of the random stopping with SRIM. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Stopping power Simulation Thin films
1. Introduction
2. Experiment and calculation models
Recently, in a series of papers the stopping of low energy ions in very thin crystalline and polycrystalline metal foils has been measured and theoretically analysed mainly on the base of dielectric function method [1,2]. It is argued that the energy distribution of 4–9 keV protons transmitted along <100> axial direction through 12 nm Au single crystal foil has perfectly Gaussian shape and the same holds for slightly thicker (24 nm) polycrystalline foils made of Bi, Al and other metals. In a similar measurement carried out w35 years ago for 62 nm Au foil it was additionally found that shape of the energy spectrum is independent of the charge of outgoing ions [3]. The intensity of neutral hydrogen atoms in outgoing beam being about 10 times greater than the intensity of protons (Hþ) has also been reported. Moreover, a threshold effect in the energy loss of slow protons channelled in Au, similar to the effect, known so far only for isolators, was suggested in Ref. [1]. It seems that in passage of low velocity ions through very thin crystalline targets in channeling direction the transmitted energy spectrum should only roughly be approximated by a Gaussian due to few scattering events. In this work we tried to analyse the experimental results on the base of binary scattering simulation codes.
The experimental details were described in Ref. [1]. Monocrystalline gold targets mounted on a 3 mm transmission electron microscope grid can be obtained from Pelco [4]. The approximately 12 nm thick evaporated gold is induced to grow in a (100) orientation. This gives lattice plane spacings of 0.204 nm for the (200) planes and 0.143 nm for the (220) planes, as shown in Fig. 1. The thickness of the foils has been determined by energy loss measurements and their normalization to the tabular stopping power values and in this context one can also consider the homogeneity of sample. The 4–9 keV energy proton beam was directed along <100> axial channeling direction of Au single crystal and registered after transmission by electrostatic analyser with the 1 acceptance angle. The precision of the energy measurement was estimated on 5%. For the description of channeling and energy loss process we used binary collisions models labeled with M1-M3. In brief, M1 comprises model presented in Refs. [5,6]. It is based on binary scattering of ions on target atoms. The scattering angle is determined by ‘‘magic-formula’’ with the Ziegler–Biersack–Littmark (ZBL) universal interaction potential [7]. The energy loss in each collision dE(b) depends on impact parameter b and can be optionally determined either from CASP [8] or from the Detman–Robinson (DR) energy loss model [9] or from the Oen–Robinson (OR) model [10] scaled for random transmission to SRIM empirical values. M2 refers to another model called TRICK [10] based on a similar binary collisions approach to projectile passage through matter. The energy loss in each collision is
* Corresponding author. E-mail address: [email protected] (T.M. Gwizda11a). 0042-207X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2009.01.062
T.M. Gwizdałła et al. / Vacuum 83 (2009) S196–S199
a
S197
100 E[keV] p+ --> Au ZBL formula JC code
scattering angle theta/2 [ °]
80
pffiffiffi 0:32 0:3b ½eV=coll dEðbÞ ¼ k E exp aTF 4paTF
s0;1
F1 ¼ F0
s1;0 s0;1
7 keV
5 keV 6 keV
8 keV 9 keV
0
5
10
15
20
impact parameter b [A]
b
100 E[keV] p+ --> Au OR model JC code
4 keV 5 keV
10
6 keV
7 keV 8 keV 9 keV
(1)
where aTF is the Thomas–Fermi screening length 2=3 2=3 aTF ¼ 0:885a0 ðZi þ Za Þ1=2 and Zi and Za are the atomic numbers. The parameter k was determined by fitting the random stopping power calculated with Eq. (1) for 10 keV protons transmitted through amorphous foil, to the SRIM2008 result at this energy. Since at 10 keV dE/dx ¼ 88.94 eV/nm, it yields DE ¼ 1067 eV of the total energy loss for the 12 nm Au foil. The proportionality of the low velocity stopping cross-section topthe ffiffiffi ion velocity was taken into account by the scaling factor E. The energy loss dE(b) shown in Fig. 2 was derived within the Oen–Robinson model and within the CASP model. OR dE(b), shown by solid dark grey line, assumes a free parameter for fitting the calculated random stopping power. The electron exchange process, optionally switched on in M1, can usually be accounted for by means of propagation equations, which in case of H-ions for charged F1 and neutral F0 fractions asymptotically takes the form:
1 s F0 ¼ 1 þ 1;0
4 keV
0
energy loss [eV/collision]
1. the scattering angle Q(b)/2 of protons colliding with Au atom, calculated with the ZBL magic-formula, shown in Fig. 2a for E ¼ 4–9 keV. 2. the energy loss dE(b) of protons in a collision with Au atom, calculated in JC code [5,6] along Eq. (1), as shown in Fig. 2b for E ¼ 4–9 keV. The dE(b) can be written in the form:
40
20
Fig. 1. (100)-oriented Au crystals [4]. Lattice spacing (200) – 0.204 nm, (220) – 0.143 nm.
determined with the OR model. No electron exchange between low velocity protons and target is assumed in this model. M3 is related to transmission of ions through random thin foils applied in SRIM08 [11]. The energy loss straggling and the energy resolution of the data acquisition system can be accounted for by means of convolution of the resultant raw spectra with an appropriate Gaussian. There are two impact parameter b dependent basic functions to be implemented in a simulation code:
60
(2)
where sj 1,j is the capture and sj þ 1,j is the electron loss crosssection by an ion of j-th initial charge. In the considered energy
1 0
1
2
3
impact parameter b [A] Fig. 2. Impact parameter b dependent scattering angle J(b) of protons scattered from Au atom (a) and the energy loss E(b), of Eq. (1) (b) calculated with JC code [5,6].
range the mean free path for equilibrating charge distribution (few A) is much shorter than the film thickness. In this case F0 w 0.9 is the completely dominating fraction in calculation of dE(b) and the stopping cross-section [3].
3. Results and discussion The angular distribution (5 range) of protons after passage through 12 nm Au along <100> axial direction calculated with TRICK is presented in Fig. 3 for protons of 4–9 keV energy. It is clear that counting rate of detector, whose acceptance angle is 1, depends on the position of the detector with respect to the incidence direction. Neither data acquisition system energy
S198
T.M. Gwizdałła et al. / Vacuum 83 (2009) S196–S199
1.4
E[keV] p+ --> 12 nm <100> Au
experiment
normalized yield
1.2
5
4
1.0
calculation JC code 7
8
9
6000
7000
8000
6
0.8
0.6
0.4
Fig. 3. Angular distributions of protons transmitted through 12 nm Au foil in <100> axial direction, as functions of initial ion energy E ¼ 4–9 keV calculated with TRICK code.
0.2
resolution nor electron loss process were incorporated in production of the raw data presented in the Fig. 3. In order to get reference random stopping for experimental conditions the energy distribution of protons registered in forward direction within 1 acceptance angle was derived from SRIM2008 for the 4–9 keV initial proton energy. As an example the spectrum for 7 keV initial proton energy and for the detector acceptance angles 1 and 45 are presented in Fig. 4 with grey lines. The acceptance angle of the detector seems to play a second order role in shaping the final distribution. The experimental transmission spectrum in <100> channeling direction is also shown in the figure with black line. Result of simulation with JC code and M1 model (with straggling and energy resolution) is shown in the figure by dark grey line, whereas a raw result of TRICK is presented with
0.0 3000
4000
5000
9000
outgoing ion energy E av [eV] Fig. 5. Experimental [1], circles and the calculated with JC code [5,6] (solid lines) energy spectra of protons after transmission through 12 nm Au foil in <100> axial direction and detected within 1, convoluted with the energy loss straggling and a 5% energy resolution of the data acquisition system, as a function of average ion energy.
black dots. They can be seen as narrow spectra placed in the higher energy part of the SRIM spectra with no correlation with experimental spectra. The striking feature is the pronounced difference between the average (or most probable) energy losses for these three spectra. We focused on analysis of experimental spectra within M1 model and JC code. Complete set of calculated proton spectra for all the incident energies used in experiment are presented in Fig. 5 together with
7keV p+ -> 12 nm <100>Au SRIM - random transmission 1100 E[eV] p+ --> 12 nm <100> Au TRICK experiment
1000
average energy loss [eV]
JC
counts
SRIM45°
SRIM1°
SRIM av
experiment
900
SRIM.rand
800
700 calculation JC code 600
5000
5500
6000
6500
7000
outgoing ion energy [eV] Fig. 4. Energy spectra of protons transmitted through 12 nm Au foil and detected within 1, for initial ion energy E ¼ 7 keV: black line – measured behind Au single crystal along <100> channeling direction [1], dotted line – calculated with TRICK code [10], grey solid lines – calculated with SRIM08 code [11] for amorphous sample with detectors of 1 and 45 acceptance angle, dark grey solid line – calculations with JC code [5,6], convoluted with straggling, without exchange of electrons.
500 3000
4000
5000
6000
7000
8000
9000
average ion energy E av [eV] Fig. 6. Circles – experiment [1], squares – simulationpwith JC code, down directed ffiffiffi triangles – SRIM direct results, up directed triangles – E scaled stopping at 10 keV.
T.M. Gwizdałła et al. / Vacuum 83 (2009) S196–S199
experimental spectra. The raw spectra were convoluted with the distribution accoutering for the energy loss straggling and energy resolution of the detecting system. The energy losses derived from these data are presented in Fig. 6 as functions of the average energy Eav defined as:
Eav ¼
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi2 Ein þ Eout
(3)
Despite the fact that the simulated energy loss for amorphous sample reproduces SRIM2008 results, the energy loss in channeling direction is much lower than determined experimentally. In this energy range the energy loss should be strictly proportional to the average ion velocity (Eav)n ¼ 1/2 in the film, as it results from both the transport equation and from the linear response theory in the low velocity range. However, in the experiment n w 1, whereas in the simulation n w 1/2.
4. Conclusions Energy spectra of low energy protons transmitted through w12 nm Au single crystal in <100> channeling direction and in random direction were calculated within various simulation codes and compared with recently published experimental data. A comparison of the measured spectra with those generated by SRIM differs both qualitatively and quantitatively, particularly in the
S199
width of the distributions, which can only roughly be approximated by a Gaussian. The calculated average energy loss for channeling conditions differs significantly from the experimental average energy loss, despite the fact that calculated random stopping was calibrated with SRIM. It seems that OR energy loss formula should rather not be used in this energy range. Acknowledgments Much thanks to Mrs. Zofia Fijarczyk for her help in preparing the manuscript. References [1] Figueroa EA, Arista NR, Eckardt JC, Lantschner GH. Nucl Instrum Meth B 2007;256:126. [2] Figueroa EA, Cantero DE, Eckardt JC, Lantschner GH, Valdes JE, Arista NR. Phys Rev A 2007;75. 010901(R). [3] Blume R, Eckstein W, Verbeek H. Nucl Instrum Meth 1982;194:67. [4] http://www.tedpella.com. [5] Czerbniak J, Pud1owski K, Moneta M. Radiat Rhys Chem A 2007;76:529. [6] Czerbniak J, Moneta M, Pud1owski K. Acta Phys. Pol A 2002;101:857. [7] Ziegler JF, Biersack JP, Littmark U. The stopping and ranges of ions in matter. New York: Pergamon Press; 1985. [8] Grande PL, Schiwietz G. Phys Rev A 1998;58:3796. [9] Dettman K, Robinson MT. Phys Rev B 1974;10:1. [10] Smulders PJM, Boerma DO. Nucl Instrum Meth B 1987;29:471. [11] Ziegler JF, Biersack JP. SRIM-2008 cf, http://www.srim.org.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Energy loss of slow protons in Au crystalline thin film Tomasz M. Gwizda11a a, *, Jerzy Czerbniak a, Roch Andrzejewski a, b, Krzysztof Gront a, Marek Moneta a a b
´dz´, Pomorska 149/153, 90-236 Ło ´dz´, Poland Department of Solid State Physics, University of Ło Centro de Micro-Analisis de Materiales, Madrid, Spain
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 June 2008 Received in revised form 22 January 2009 Accepted 30 January 2009
Energy spectra of low energy protons transmitted through w12 nm Au self-supporting single crystalline foil in channeling direction were calculated within various simulation codes and compared with published recently experimental data. It was shown that the raw energy distribution is by no means Gaussian; however, it gets this shape after accounting for the energy loss straggling and the energy resolution of the experimental setup. The most probable and the average energy loss differ significantly from experimental energy loss despite calibration of the random stopping with SRIM. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Stopping power Simulation Thin films
1. Introduction
2. Experiment and calculation models
Recently, in a series of papers the stopping of low energy ions in very thin crystalline and polycrystalline metal foils has been measured and theoretically analysed mainly on the base of dielectric function method [1,2]. It is argued that the energy distribution of 4–9 keV protons transmitted along <100> axial direction through 12 nm Au single crystal foil has perfectly Gaussian shape and the same holds for slightly thicker (24 nm) polycrystalline foils made of Bi, Al and other metals. In a similar measurement carried out w35 years ago for 62 nm Au foil it was additionally found that shape of the energy spectrum is independent of the charge of outgoing ions [3]. The intensity of neutral hydrogen atoms in outgoing beam being about 10 times greater than the intensity of protons (Hþ) has also been reported. Moreover, a threshold effect in the energy loss of slow protons channelled in Au, similar to the effect, known so far only for isolators, was suggested in Ref. [1]. It seems that in passage of low velocity ions through very thin crystalline targets in channeling direction the transmitted energy spectrum should only roughly be approximated by a Gaussian due to few scattering events. In this work we tried to analyse the experimental results on the base of binary scattering simulation codes.
The experimental details were described in Ref. [1]. Monocrystalline gold targets mounted on a 3 mm transmission electron microscope grid can be obtained from Pelco [4]. The approximately 12 nm thick evaporated gold is induced to grow in a (100) orientation. This gives lattice plane spacings of 0.204 nm for the (200) planes and 0.143 nm for the (220) planes, as shown in Fig. 1. The thickness of the foils has been determined by energy loss measurements and their normalization to the tabular stopping power values and in this context one can also consider the homogeneity of sample. The 4–9 keV energy proton beam was directed along <100> axial channeling direction of Au single crystal and registered after transmission by electrostatic analyser with the 1 acceptance angle. The precision of the energy measurement was estimated on 5%. For the description of channeling and energy loss process we used binary collisions models labeled with M1-M3. In brief, M1 comprises model presented in Refs. [5,6]. It is based on binary scattering of ions on target atoms. The scattering angle is determined by ‘‘magic-formula’’ with the Ziegler–Biersack–Littmark (ZBL) universal interaction potential [7]. The energy loss in each collision dE(b) depends on impact parameter b and can be optionally determined either from CASP [8] or from the Detman–Robinson (DR) energy loss model [9] or from the Oen–Robinson (OR) model [10] scaled for random transmission to SRIM empirical values. M2 refers to another model called TRICK [10] based on a similar binary collisions approach to projectile passage through matter. The energy loss in each collision is
* Corresponding author. E-mail address: [email protected] (T.M. Gwizda11a). 0042-207X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2009.01.062
T.M. Gwizdałła et al. / Vacuum 83 (2009) S196–S199
a
S197
100 E[keV] p+ --> Au ZBL formula JC code
scattering angle theta/2 [ °]
80
pffiffiffi 0:32 0:3b ½eV=coll dEðbÞ ¼ k E exp aTF 4paTF
s0;1
F1 ¼ F0
s1;0 s0;1
7 keV
5 keV 6 keV
8 keV 9 keV
0
5
10
15
20
impact parameter b [A]
b
100 E[keV] p+ --> Au OR model JC code
4 keV 5 keV
10
6 keV
7 keV 8 keV 9 keV
(1)
where aTF is the Thomas–Fermi screening length 2=3 2=3 aTF ¼ 0:885a0 ðZi þ Za Þ1=2 and Zi and Za are the atomic numbers. The parameter k was determined by fitting the random stopping power calculated with Eq. (1) for 10 keV protons transmitted through amorphous foil, to the SRIM2008 result at this energy. Since at 10 keV dE/dx ¼ 88.94 eV/nm, it yields DE ¼ 1067 eV of the total energy loss for the 12 nm Au foil. The proportionality of the low velocity stopping cross-section topthe ffiffiffi ion velocity was taken into account by the scaling factor E. The energy loss dE(b) shown in Fig. 2 was derived within the Oen–Robinson model and within the CASP model. OR dE(b), shown by solid dark grey line, assumes a free parameter for fitting the calculated random stopping power. The electron exchange process, optionally switched on in M1, can usually be accounted for by means of propagation equations, which in case of H-ions for charged F1 and neutral F0 fractions asymptotically takes the form:
1 s F0 ¼ 1 þ 1;0
4 keV
0
energy loss [eV/collision]
1. the scattering angle Q(b)/2 of protons colliding with Au atom, calculated with the ZBL magic-formula, shown in Fig. 2a for E ¼ 4–9 keV. 2. the energy loss dE(b) of protons in a collision with Au atom, calculated in JC code [5,6] along Eq. (1), as shown in Fig. 2b for E ¼ 4–9 keV. The dE(b) can be written in the form:
40
20
Fig. 1. (100)-oriented Au crystals [4]. Lattice spacing (200) – 0.204 nm, (220) – 0.143 nm.
determined with the OR model. No electron exchange between low velocity protons and target is assumed in this model. M3 is related to transmission of ions through random thin foils applied in SRIM08 [11]. The energy loss straggling and the energy resolution of the data acquisition system can be accounted for by means of convolution of the resultant raw spectra with an appropriate Gaussian. There are two impact parameter b dependent basic functions to be implemented in a simulation code:
60
(2)
where sj 1,j is the capture and sj þ 1,j is the electron loss crosssection by an ion of j-th initial charge. In the considered energy
1 0
1
2
3
impact parameter b [A] Fig. 2. Impact parameter b dependent scattering angle J(b) of protons scattered from Au atom (a) and the energy loss E(b), of Eq. (1) (b) calculated with JC code [5,6].
range the mean free path for equilibrating charge distribution (few A) is much shorter than the film thickness. In this case F0 w 0.9 is the completely dominating fraction in calculation of dE(b) and the stopping cross-section [3].
3. Results and discussion The angular distribution (5 range) of protons after passage through 12 nm Au along <100> axial direction calculated with TRICK is presented in Fig. 3 for protons of 4–9 keV energy. It is clear that counting rate of detector, whose acceptance angle is 1, depends on the position of the detector with respect to the incidence direction. Neither data acquisition system energy
S198
T.M. Gwizdałła et al. / Vacuum 83 (2009) S196–S199
1.4
E[keV] p+ --> 12 nm <100> Au
experiment
normalized yield
1.2
5
4
1.0
calculation JC code 7
8
9
6000
7000
8000
6
0.8
0.6
0.4
Fig. 3. Angular distributions of protons transmitted through 12 nm Au foil in <100> axial direction, as functions of initial ion energy E ¼ 4–9 keV calculated with TRICK code.
0.2
resolution nor electron loss process were incorporated in production of the raw data presented in the Fig. 3. In order to get reference random stopping for experimental conditions the energy distribution of protons registered in forward direction within 1 acceptance angle was derived from SRIM2008 for the 4–9 keV initial proton energy. As an example the spectrum for 7 keV initial proton energy and for the detector acceptance angles 1 and 45 are presented in Fig. 4 with grey lines. The acceptance angle of the detector seems to play a second order role in shaping the final distribution. The experimental transmission spectrum in <100> channeling direction is also shown in the figure with black line. Result of simulation with JC code and M1 model (with straggling and energy resolution) is shown in the figure by dark grey line, whereas a raw result of TRICK is presented with
0.0 3000
4000
5000
9000
outgoing ion energy E av [eV] Fig. 5. Experimental [1], circles and the calculated with JC code [5,6] (solid lines) energy spectra of protons after transmission through 12 nm Au foil in <100> axial direction and detected within 1, convoluted with the energy loss straggling and a 5% energy resolution of the data acquisition system, as a function of average ion energy.
black dots. They can be seen as narrow spectra placed in the higher energy part of the SRIM spectra with no correlation with experimental spectra. The striking feature is the pronounced difference between the average (or most probable) energy losses for these three spectra. We focused on analysis of experimental spectra within M1 model and JC code. Complete set of calculated proton spectra for all the incident energies used in experiment are presented in Fig. 5 together with
7keV p+ -> 12 nm <100>Au SRIM - random transmission 1100 E[eV] p+ --> 12 nm <100> Au TRICK experiment
1000
average energy loss [eV]
JC
counts
SRIM45°
SRIM1°
SRIM av
experiment
900
SRIM.rand
800
700 calculation JC code 600
5000
5500
6000
6500
7000
outgoing ion energy [eV] Fig. 4. Energy spectra of protons transmitted through 12 nm Au foil and detected within 1, for initial ion energy E ¼ 7 keV: black line – measured behind Au single crystal along <100> channeling direction [1], dotted line – calculated with TRICK code [10], grey solid lines – calculated with SRIM08 code [11] for amorphous sample with detectors of 1 and 45 acceptance angle, dark grey solid line – calculations with JC code [5,6], convoluted with straggling, without exchange of electrons.
500 3000
4000
5000
6000
7000
8000
9000
average ion energy E av [eV] Fig. 6. Circles – experiment [1], squares – simulationpwith JC code, down directed ffiffiffi triangles – SRIM direct results, up directed triangles – E scaled stopping at 10 keV.
T.M. Gwizdałła et al. / Vacuum 83 (2009) S196–S199
experimental spectra. The raw spectra were convoluted with the distribution accoutering for the energy loss straggling and energy resolution of the detecting system. The energy losses derived from these data are presented in Fig. 6 as functions of the average energy Eav defined as:
Eav ¼
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi2 Ein þ Eout
(3)
Despite the fact that the simulated energy loss for amorphous sample reproduces SRIM2008 results, the energy loss in channeling direction is much lower than determined experimentally. In this energy range the energy loss should be strictly proportional to the average ion velocity (Eav)n ¼ 1/2 in the film, as it results from both the transport equation and from the linear response theory in the low velocity range. However, in the experiment n w 1, whereas in the simulation n w 1/2.
4. Conclusions Energy spectra of low energy protons transmitted through w12 nm Au single crystal in <100> channeling direction and in random direction were calculated within various simulation codes and compared with recently published experimental data. A comparison of the measured spectra with those generated by SRIM differs both qualitatively and quantitatively, particularly in the
S199
width of the distributions, which can only roughly be approximated by a Gaussian. The calculated average energy loss for channeling conditions differs significantly from the experimental average energy loss, despite the fact that calculated random stopping was calibrated with SRIM. It seems that OR energy loss formula should rather not be used in this energy range. Acknowledgments Much thanks to Mrs. Zofia Fijarczyk for her help in preparing the manuscript. References [1] Figueroa EA, Arista NR, Eckardt JC, Lantschner GH. Nucl Instrum Meth B 2007;256:126. [2] Figueroa EA, Cantero DE, Eckardt JC, Lantschner GH, Valdes JE, Arista NR. Phys Rev A 2007;75. 010901(R). [3] Blume R, Eckstein W, Verbeek H. Nucl Instrum Meth 1982;194:67. [4] http://www.tedpella.com. [5] Czerbniak J, Pud1owski K, Moneta M. Radiat Rhys Chem A 2007;76:529. [6] Czerbniak J, Moneta M, Pud1owski K. Acta Phys. Pol A 2002;101:857. [7] Ziegler JF, Biersack JP, Littmark U. The stopping and ranges of ions in matter. New York: Pergamon Press; 1985. [8] Grande PL, Schiwietz G. Phys Rev A 1998;58:3796. [9] Dettman K, Robinson MT. Phys Rev B 1974;10:1. [10] Smulders PJM, Boerma DO. Nucl Instrum Meth B 1987;29:471. [11] Ziegler JF, Biersack JP. SRIM-2008 cf, http://www.srim.org.