- Email: [email protected]
Energy loss straggling of Li, C and O ions in mylar and polycarbonate absorber foils
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 244 (2006) 289–293 www.elsevier.com/locate/nimb
Energy loss straggling of Li, C and O ions in mylar and polycarbonate absorber foils P.K. Diwan
a,*
, V. Sharma b, S. Aggarwal b, S. Kumar b, S.K. Sharma c, V.K. Mittal d, B. Sannakki e, R.D. Mathad e, S.A. Khan f, D.K. Avasthi f a
c
Department of Physics, UIET, Kurukshetra University, Kurukshetra 136 119, India b Department of Physics, Kurukshetra University, Kurukshetra 136 119, India Department of Physics, Ambala College of Engineering and Applied Research, Ambala 133 104, India d Department of Physics, Punjabi University, Patiala 147 002, India e Department of Physics, Gulbarga University, Gulbarga 585 106, India f Nuclear Science Centre, Aruna Asaf Ali Marg, P.O. Box 10502, New Delhi 110 067, India Available online 4 January 2006
Abstract Measurements of energy loss straggling for Li, C and O ions in varying thicknesses of mylar (C10H8O4) and polycarbonate (C16H14O3) absorber foils in the energy range 3.0–6.0 MeV/u have been carried out, utilizing the Pelletron accelerator facility at Nuclear Science Centre, New Delhi, India. These measured values have been compared with the calculated values in order to establish the validity of various energy loss straggling formulations viz. Bohr, Lindhard and Scharff and Bethe–Livingston. From the energy loss straggling measurements in these two polymers having same atomic constituents and almost same effective Z, the validity of Bragg’s rule has been checked. Ó 2005 Elsevier B.V. All rights reserved. PACS: 06; 07; 34; 35 Keywords: Energy loss; Straggling; Heavy ions; Polymers
1. Introduction Monoenergetic heavy ions of given Z, while passing through an absorber foil of a given thickness, lose energy, which fluctuates around a mean value. This results in finite width in the energy distribution curve of the emerging ions. The phenomenon is called energy loss straggling. Conventionally, the slowing down of a particle i.e. energy loss has been termed as first order effect while the broadening of the energy distribution curve i.e. energy loss straggling as second order effect. Considerable attention has been paid to the understanding of the energy loss phenomena from both experimental and theoretical point of view due to its wide *
Corresponding author. E-mail address: [email protected] (P.K. Diwan).
0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.11.045
range of applications in diverse disciplines [1–4]. Much less attention has been paid to the understanding of energy loss straggling considering it as a second order effect. With the increasing use of ion beam based analytical techniques, the energy loss straggling phenomena has also gained importance. As energy loss straggling is one of the main factors limiting depth resolution, thus accurate information on energy loss straggling is important for the applications of Rutherford backscattering spectrometry (RBS), elastic recoil detection analysis (ERDA), nuclear reaction analysis (NRA) etc. in material analysis [5,6]. The energy loss straggling of heavy ions in solids is basically the combined effect of collisional and charge exchange straggling. The collisional energy loss straggling is a result of the statistical fluctuations in both the number of collisions and energy transfer in each collision, when the ion
290
P.K. Diwan et al. / Nucl. Instr. and Meth. in Phys. Res. B 244 (2006) 289–293
passes through a given thickness of the absorber foil. For partially stripped ions, the statistical fluctuations of the ionic charge states, due to electron-capture and electronloss processes, lead to the charge exchange straggling. These contributions must be added in quadrature as being assumed to be independent [7,8]. Although several theoretical formulations for energy loss straggling are available [9– 11] but their validity could not be established due to the availability of only limited experimental data [7,8,12–22]. No experimental data is available for heavy ions in complex target materials. In the present study, the energy loss straggling for 7Li, 12 C and 16O ions at energies 3.0–6.0 MeV/u in mylar and polycarbonate absorber foils of varying thicknesses has been measured. The experimental measured values have been compared with those calculated using Bohr, Lindhard and Scharff and Bethe–Livingston theories. The present work is an attempt to check the reliability of these formulations in the light of the experimental measurements.
gled peaks’’. The energy loss straggling (dEexpt.) of the incident ions after passing through a given thickness of the absorber foil, has been determined experimentally, using the following relation: dEexpt. ¼ ðdE2st dE20 Þ1=2 ; where dE0 and dEst are the FWHM of the energy spectra of unstraggled and straggled peaks, respectively. The present experimental arrangement provides a unique advantage of recording unstraggled and straggled peaks simultaneously, so that they contain the same contribution due to (i) beam energy spread from the accelerator, (ii) straggling in the Au foil and (iii) detector resolution and its response, thus nullifying the error due to these factors. Adopting the above procedure, the energy loss straggling for Li (23, 30, 40 MeV), C (40, 60, 80 MeV) and O (80, 100 MeV) ions in varying thicknesses of mylar and polycarbonate foils has been measured. 3. Energy loss straggling formulations
2. Experimental details The energy loss straggling measurements were carried out utilizing the 15 MV Pelletron accelerator facility at Nuclear Science Centre, New Delhi, India. The actual experiment was performed in a low flux chamber, fixed with general purpose scattering chamber (GPSC) at an angle of 15° with respect to primary beam direction. There is a provision to introduce a target ladder in the low flux chamber, which can be moved vertically upward or downward without disturbing the vacuum in the chamber. For the present experiment, the mylar and polycarbonate foils of quoted thickness 6 and 15 lm, respectively were used. Their thickness was measured adopting a-energy loss method using 241Am source. The mean thickness values came out to be 5.81 ± 0.21 and 14.64 ± 0.28 lm, respectively for mylar and polycarbonate foils. These absorber foils in the form of a staircase were mounted on a collimator leaving its small portion blank. Finally, such collimators were mounted on a target ladder, which was then introduced in low flux chamber. The primary ion beams of 7Li, 12C and 16O from Pelletron accelerator were scattered from a thin Au foil (400 nm) in general purpose scattering chamber (GPSC). The scattered beam was allowed to pass normally through varying thicknesses of absorber foils of these two polymers by adjusting each collimator one by one in front of the beam through the vertical motion of the target ladder, before being finally detected by the silicon surface barrier detector. The detector output pulses were stored online on a computer through analogue to digital converter, which were later analyzed using a software package CANDLE. The setup thus allowed the accumulation of several peaks simultaneously: one due to ions without absorber foil, ‘‘unstraggled peak’’ and others due to ions after crossing the absorber foils of different thicknesses, the ‘‘strag-
Due to the statistical nature of the energy loss process of the incident ions while passing though an absorber foil, it is reasonable to assume that the energy loss distribution can be described by a Gaussian shape. For such a distribution (valid for energy loss fraction, DE/E 5–25%), the energy loss straggling dE relates to the straggling standard deviation X through the relation dE ¼ 2ð2 ln 2Þ
1=2
X.
Several theories have been put forward to evaluate the straggling standard deviation X. 3.1. Bohr’s theory The simple Bohr’s theory [9] predicts the straggling standard deviation XB to be enegry independent and given by the following expression: X2B ¼ 4pZ 21 e4 Z 2 Nx; where e, N and x are the electronic charge, the number of absorber atoms per unit volume and the thickness of the absorber foil, respectively. Z1 and Z2 are the atomic numbers of the projectile and the absorber material, respectively. After simplification, the Bohr’s formula reduces to X2B ðMeV2 Þ ¼ 1:57 104 Z 21
Z2 x; A2
where A2 is the absorber mass number and x is expressed in mg/cm2. This theory is based on the assumptions that (i) the velocity of the projectile is much greater than the orbital velocities of the electrons of the target atoms, (ii) the energy loss is very small in comparison to the total energy of the projectile and (iii) the target atoms are randomly distributed and no channeling is involved in the penetration process.
P.K. Diwan et al. / Nucl. Instr. and Meth. in Phys. Res. B 244 (2006) 289–293
3.2. Lindhard and Scharff theory Using the dielectric electron gas picture of the atoms and a local electron charge density, Lindhard and Scharff [10] extended Bohr’s theory by applying a correction factor for low and medium energy projectiles. They developed the following expression for straggling standard deviation, XLS: 0:5LðvÞ for v < 3; X2LS ¼ 2 1 for v P 3; XB where v ¼ v2 =v20 Z 2 is a reduced energy variable with v/v0 the ratio of ion’s velocity to Bohr’s velocity and L(v) = 1.36v1/2 0.016v3/2. Lindhard and Scharff equation indicates that energy straggling approaches Bohr’s value at energy E P 75Z2 keV/u and is energy dependent only at lower energies. 3.3. Bethe and Livingston theory From the treatment of Livingston and Bethe [11], within the energy validity region of the Born approximation, the straggling standard deviation XBL is expressed by the following relation: X2BL Z 02 1 X I iZi 2me v2 ¼ þ ki ln ; 2 2 Z2 Z2 i me v Ii XB
291
where Z 02 is the effective number of absorber atomic electrons, Ii is the average excitation energy of the Zi electrons in the ith atomic orbit, v is the velocity of the incident ion and me is the rest mass of the electron. The constant ki is taken to be 4/3 for all orbits. In the summation, i extends over all orbits for which 2mev2 > Ii. Unlike Bohr’s theory, this theory is energy dependent due to involvement of the factor v2 in the above expression. The values of Ii were modified in such a way [23–26] so as to be compatible with the average ionization potential I of the target atom. Extending the Bragg’s additivity rule, the straggling standard deviation X in polymers can be calculated using the relation [27,28]: n X X2 ¼ ci X2i ; i¼1
where ci is the atomic fraction and Xi is the straggling standard deviation for the ith element in the composite material. 4. Results and discussion Tables 1 and 2 present the experimentally measured energy loss straggling (dEexpt.) values for Li, C and O ions in mylar and polycarbonate absorber foils of varying thicknesses (0.8–7.0 mg/cm2), in the energy range 3.0–6.0 MeV/u. In these tables, DE and DE/E are the
Table 1 Experimental and calculated energy loss straggling values for Li (3.24, 4.25 MeV/u), C (3.25, 4.92 MeV/u) and O (4.89, 6.14 MeV/u) ions in mylar absorber foil Einc. (MeV/u) Ion – Li 3.24 4.25 3.24 4.25 3.25 4.25 Ion – C 3.25 4.92 4.92 4.92 4.92 Ion – O 4.89 6.14 4.89 6.14 4.89 6.14 6.14 4.89 6.14 6.14
Thickness (x) (mg/cm2)
1.62 ± 0.04 2.43 ± 0.05 3.24 ± 0.06
0.81 ± 0.03 1.62 ± 0.04 2.43 ± 0.05 3.24 ± 0.06
0.81 ± 0.03 1.62 ± 0.04 2.43 ± 0.05
3.24 ± 0.06
DE (MeV)
Eav (MeV/u)
DE/E (%)
dEexpt. (keV)
Calculated energy loss straggling (keV) dEB
dELS
dEBL
1.60 1.25 2.44 1.92 3.28 2.61
3.13 4.15 3.07 4.11 3.01 4.06
7 4 11 6 14 9
99 ± 33 84 ± 37 136 ± 29 116 ± 32 158 ± 26 127 ± 30
81(18) 81(4) 99(27) 99(15) 115(27) 115(9)
81(18) 81(4) 99(27) 99(15) 115(27) 115(9)
90(9) 89(6) 110(19) 109(6) 128(19) 126(1)
2.97 2.28 4.58 6.95 9.39
3.13 4.82 4.73 4.63 4.53
8 4 8 12 16
165 ± 28 150 ± 31 228 ± 27 270 ± 26 315 ± 29
115(30) 115(23) 163(28) 199(26) 230(27)
115(30) 115(23) 163(28) 199(26) 230(27)
127(23) 124(17) 177(22) 216(20) 250(21)
3.94 3.35 7.93 6.67 12.05 10.06 10.08 16.32 13.56 13.57
4.77 6.04 4.65 5.93 4.52 5.83 5.83 4.38 5.72 5.72
5 3 10 7 15 10 10 21 14 14
210 ± 40 207 ± 37 331 ± 33 294 ± 33 427 ± 32 370 ± 31 376 ± 32 494 ± 34 451 ± 35 452 ± 35
153(27) 153(26) 217(34) 217(26) 266(38) 266(28) 266(29) 307(38) 307(32) 307(32)
153(27) 153(26) 217(34) 217(26) 266(38) 266(28) 266(29) 307(38) 307(32) 307(32)
166(21) 164(21) 236(29) 233(21) 289(32) 285(23) 285(24) 334(32) 330(27) 330(27)
Percentage differences with respect to the measured values are given in parentheses.
292
P.K. Diwan et al. / Nucl. Instr. and Meth. in Phys. Res. B 244 (2006) 289–293
Table 2 Experimental and calculated energy loss straggling values for Li (3.24, 4.25, 5.68 MeV/u), C (4.92, 6.59 MeV/u) and O (4.89, 6.14 MeV/u) ions in polycarbonate absorber foil Einc. (MeV/u) Ion – Li 4.25 3.24 4.25 5.68 3.24 4.25 5.68 3.24 4.25 5.68 Ion – C 4.92 6.59 4.92 6.59 4.92 6.59 6.59 Ion – O 4.89 6.14 4.89 6.14 6.14 6.14
Thickness (x) (mg/cm2)
1.76 ± 0.03 3.52 ± 0.04
5.28 ± 0.05
7.04 ± 0.06
1.76 ± 0.03 3.52 ± 0.04 5.28 ± 0.05 7.04 ± 0.06
1.76 ± 0.03 3.52 ± 0.04 5.28 ± 0.05 7.04 ± 0.06
DE (MeV)
Eav (MeV/u)
DE/E (%)
dEexpt. (keV)
Calculated energy loss straggling (keV) dEB
dELS
BL
1.44 3.64 2.93 2.30 5.69 4.46 3.45 7.94 6.08 4.66
4.15 2.98 4.04 5.51 2.83 3.93 5.43 2.67 3.81 5.35
5 16 10 6 25 15 9 35 20 12
103 ± 31 159 ± 27 153 ± 26 149 ± 26 202 ± 24 186 ± 26 149 ± 25 240 ± 26 224 ± 28 188 ± 26
85(17) 120(24) 120(22) 120(19) 147(27) 147(21) 147(1) 170(29) 170(24) 170(10)
85(17) 120(24) 120(22) 120(19) 147(27) 147(21) 147(1) 170(29) 170(24) 170(10)
93(10) 131(18) 131(14) 129(13) 164(19) 161(13) 158(6) 190(21) 186(17) 183(3)
5.02 3.22 10.40 7.34 16.24 11.67 16.24
4.71 6.46 4.49 6.28 4.24 6.10 5.91
9 4 18 9 27 15 20
224 ± 24 209 ± 22 329 ± 28 321 ± 26 433 ± 32 390 ± 28 470 ± 31
171(24) 171(18) 240(27) 240(25) 295(32) 295(24) 341(27)
171(24) 171(18) 240(27) 240(25) 295(32) 295(24) 341(27)
184(18) 182(13) 261(21) 257(20) 320(26) 315(19) 365(22)
8.72 7.32 18.18 15.04 23.29 32.30
4.62 5.91 4.32 5.67 5.41 5.13
11 7 23 15 24 33
331 ± 35 317 ± 31 531 ± 37 457 ± 36 584 ± 38 681 ± 40
228(31) 228(28) 321(39) 321(30) 393(33) 455(33)
228(31) 228(28) 321(39) 321(30) 393(33) 455(33)
246(26) 243(23) 348(34) 344(25) 422(28) 489(28)
Percentage differences with respect to the measured values are given in parentheses.
500 EXPERIMENTAL Projectile - C Targets Polycarbonate Mylar
400
Energy loss straggling (keV)
average energy loss and the fractional energy loss, respectively, in the given thickness of the absorber foil and Eav = Einc. DE/2 with Einc. as the incident ion energy. For the calculations of energy loss straggling in polymers, first we have calculated the straggling in elemental constituents of the polymers adopting Bohr (dEB) [9], Lindhard and Scharff (dELS) [10] and Bethe–Livingston (dEBL) [11] theories and then Bragg’s additivity rule has been applied (Tables 1 and 2). On comparison between the experimental and calculated values (Tables 1 and 2), it is noticed that the Bohr’s theory underestimates the experimental values up to 30% for Li and C ions and up to 40% for O ions in the fractional energy loss (DE/E) limits 5–25%. In the energy region covered in the present study, the Lindhard and Scharff values are identical with Bohr’s values. Bethe–Livingston theory with modified values of Ii predicts a better agreement with deviations reduced to 20% for Li and C ions and 30% for O ions. Further, Bethe–Livingston theory correctly predicts slightly decreasing trend in energy loss straggling with increase of incident energy, for a particular ion, in a given thickness of the absorber foil. In order to study the dependence of energy loss straggling on target’s chemical bond structure, the energy loss straggling as a function of energy loss for C ions in mylar and polycarbonate (having almost same effective Z and
Best Fit
300
200
100
0 0
4
8
12
16
20
Energy loss (MeV) Fig. 1. Variation of experimental energy loss straggling (dEexpt.) as a function of energy loss (DE) in mylar (C10H8O4) and polycarbonate (C16H14O3) absorber foils for C projectile.
P.K. Diwan et al. / Nucl. Instr. and Meth. in Phys. Res. B 244 (2006) 289–293
same atomic constituents but different bond structures) has been plotted (Fig. 1). It is clearly depicted from the figure that experimental values of energy loss straggling lie on the same line (within experimental errors) for both these polymers, irrespective of their chemical bond structure. Similar trend has been noticed for Li and O ions. Since the energy loss straggling in polymers is independent of their chemical bond structure, therefore the validity of Bragg’s additivity rule is supported, in the energy range covered in the present study. 5. Conclusion Our’s is apparently the first study that reports the energy loss straggling of heavy ions in mylar and polycarbonate foils. Bethe–Livingston theory shows better agreement with the experimental data than Bohr’s theory. From straggling measurements in polymers, the validity of Bragg’s rule is supported. Acknowledgements The authors are grateful to the Director, Nuclear Science Centre for granting beam time and Pelletron operation staff for providing the stable beam. Authors are thankful to Mr. Sandeep Chopra and Mr. Sunil Ojha for useful technical discussion. Thanks are also due to Mr. D. Kabiraj and Mr. Abhilash for help during target preparation. One of the authors (P.K. Diwan) is thankful to the Council of Scientific and Industrial Research (CSIR), New Delhi, India, for providing financial assistance by way of Senior Research Fellowship. References [1] [2] [3] [4]
P. Sigmund, Nucl. Instr. and Meth. B 315 (1998) 1. J.F. Ziegler, J. Appl. Phys. 85 (3) (1999) 1249. S.P. Ahlen, Rev. Mod. Phys. 52 (1) (1980) 21. L.C. Feldman, J.W. Mayer, Fundamentals of Surface and Thin Film Analysis, North Holland, New York, 1986. [5] W.K. Chu, Phys. Rev. A 13 (6) (1976) 2057.
293
[6] Q. Yang, D.J. O’Connor, Z. Wang, Nucl. Instr. and Meth. B 61 (1991) 149. [7] S. Ouichaoui, E. Hourani, L. Rosier, R. Bimbot, H. Beaumevieille, B. Bouzid, S. Mammeri, Nucl. Instr. and Meth. B 164–165 (2000) 259. [8] H. Weick, H. Geissel, C. Scheidenberger, F. Attallah, D. Cortina, M. Hausmann, G. Munzenberg, T. Radon, H. Schatz, K. Schmidt, J. Stadlmann, K. Summerer, M. Winkler, Phys. Rev. Lett. 85 (13) (2000) 2725. [9] N. Bohr, Kgl. Dan. Vid. Selsk. Mat.-Fys. Medd. 18 (1948) 8. [10] J. Lindhard, M. Scharff, Kgl. Dan. Vid. Selsk. Mat.-Fys. Medd. 27 (1953) 15. [11] M.S. Livingston, H.A. Bethe, Rev. Mod. Phys. 9 (3) (1937) 245. [12] B. Efken, D. Hahn, D. Hilscher, G. Wustefeld, Nucl. Instr. and Meth. B 129 (1975) 219. [13] T. Takahashi, Y. Awaya, T. Tonuma, H. Kumagai, K. Izumo, A. Hashizume, S. Uchiyama, A. Hitachi, Nucl. Instr. and Meth. B 166 (1979) 587. [14] C.J. Sofield, N.E.B. Cowern, R.J. Petty, J.M. Freeman, J.P. Mason, Phys. Rev. A 17 (3) (1978) 859. [15] C.J. Sofield, N.E.B. Cowern, J.M. Freeman, Nucl. Instr. and Meth. B 170 (1980) 221. [16] N.E.B. Cowern, C.J. Sofield, J.M. Freeman, J.P. Mason, Phys. Rev. A 19 (1) (1979) 111. [17] N.E.B. Cowern, P.N. Trehan, C.J. Sofield, L.B. Bridwell, S.M. Ferguson, Nucl. Instr. and Meth. B 206 (1983) 595. [18] T. Schwab, H. Geissel, P. Armbruster, A. Gillibert, W. Mitting, R.E. Olson, K.B. Winterbon, H. Wollnik, G. Munzenberg, Nucl. Instr. and Meth. B 48 (1990) 69. [19] H. Ogawa, I. Katayama, Y. Haruyama, M. Saito, K. Yoshida, M. Tosaki, I. Sugai, Nucl. Instr. and Meth. B 212 (2003) 27. [20] H. Ogawa, I. Katayama, Y. Haruyama, M. Saito, K. Yoshida, M. Tosaki, I. Sugai, Nucl. Instr. and Meth. B 195 (2002) 175. [21] Q. Yang, T. Li, D.J. O’Connor, R.J. MacDonald, T.R. Ophel, H.J. Hay, D.J.H. Cockayne, A. Sikorski, Nucl. Instr. and Meth. B 83 (1993) 439. [22] L.L. Araujo, P.L. Grande, M. Behar, J.H.R. dos Santos, Nucl. Instr. and Meth. B 190 (2002) 79. [23] J.R. Comfort, J.F. Decker, E.T. Lynk, M.O. Scully, A.R. Quinton, Phys. Rev. 150 (1) (1966) 249. [24] T.A. Carlson, Photoelectrons and Auger Spectroscopy, Plenum, New York, 1975. Appendix 1. [25] R.M. Sternheimer, Phys. Rev. 103 (3) (1956) 511. [26] R.M. Sternheimer, Phys. Rev. 88 (4) (1952) 851. [27] J. Raisanen, U. Watjen, A.J.M. Plompen, F. Munnik, Nucl. Instr. and Meth. B 118 (1996) 1. [28] H. Ammi, S. mammeri, M. Allab, Nucl. Instr. and Meth. B 213 (2004) 60.
Nuclear Instruments and Methods in Physics Research B 244 (2006) 289–293 www.elsevier.com/locate/nimb
Energy loss straggling of Li, C and O ions in mylar and polycarbonate absorber foils P.K. Diwan
a,*
, V. Sharma b, S. Aggarwal b, S. Kumar b, S.K. Sharma c, V.K. Mittal d, B. Sannakki e, R.D. Mathad e, S.A. Khan f, D.K. Avasthi f a
c
Department of Physics, UIET, Kurukshetra University, Kurukshetra 136 119, India b Department of Physics, Kurukshetra University, Kurukshetra 136 119, India Department of Physics, Ambala College of Engineering and Applied Research, Ambala 133 104, India d Department of Physics, Punjabi University, Patiala 147 002, India e Department of Physics, Gulbarga University, Gulbarga 585 106, India f Nuclear Science Centre, Aruna Asaf Ali Marg, P.O. Box 10502, New Delhi 110 067, India Available online 4 January 2006
Abstract Measurements of energy loss straggling for Li, C and O ions in varying thicknesses of mylar (C10H8O4) and polycarbonate (C16H14O3) absorber foils in the energy range 3.0–6.0 MeV/u have been carried out, utilizing the Pelletron accelerator facility at Nuclear Science Centre, New Delhi, India. These measured values have been compared with the calculated values in order to establish the validity of various energy loss straggling formulations viz. Bohr, Lindhard and Scharff and Bethe–Livingston. From the energy loss straggling measurements in these two polymers having same atomic constituents and almost same effective Z, the validity of Bragg’s rule has been checked. Ó 2005 Elsevier B.V. All rights reserved. PACS: 06; 07; 34; 35 Keywords: Energy loss; Straggling; Heavy ions; Polymers
1. Introduction Monoenergetic heavy ions of given Z, while passing through an absorber foil of a given thickness, lose energy, which fluctuates around a mean value. This results in finite width in the energy distribution curve of the emerging ions. The phenomenon is called energy loss straggling. Conventionally, the slowing down of a particle i.e. energy loss has been termed as first order effect while the broadening of the energy distribution curve i.e. energy loss straggling as second order effect. Considerable attention has been paid to the understanding of the energy loss phenomena from both experimental and theoretical point of view due to its wide *
Corresponding author. E-mail address: [email protected] (P.K. Diwan).
0168-583X/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.11.045
range of applications in diverse disciplines [1–4]. Much less attention has been paid to the understanding of energy loss straggling considering it as a second order effect. With the increasing use of ion beam based analytical techniques, the energy loss straggling phenomena has also gained importance. As energy loss straggling is one of the main factors limiting depth resolution, thus accurate information on energy loss straggling is important for the applications of Rutherford backscattering spectrometry (RBS), elastic recoil detection analysis (ERDA), nuclear reaction analysis (NRA) etc. in material analysis [5,6]. The energy loss straggling of heavy ions in solids is basically the combined effect of collisional and charge exchange straggling. The collisional energy loss straggling is a result of the statistical fluctuations in both the number of collisions and energy transfer in each collision, when the ion
290
P.K. Diwan et al. / Nucl. Instr. and Meth. in Phys. Res. B 244 (2006) 289–293
passes through a given thickness of the absorber foil. For partially stripped ions, the statistical fluctuations of the ionic charge states, due to electron-capture and electronloss processes, lead to the charge exchange straggling. These contributions must be added in quadrature as being assumed to be independent [7,8]. Although several theoretical formulations for energy loss straggling are available [9– 11] but their validity could not be established due to the availability of only limited experimental data [7,8,12–22]. No experimental data is available for heavy ions in complex target materials. In the present study, the energy loss straggling for 7Li, 12 C and 16O ions at energies 3.0–6.0 MeV/u in mylar and polycarbonate absorber foils of varying thicknesses has been measured. The experimental measured values have been compared with those calculated using Bohr, Lindhard and Scharff and Bethe–Livingston theories. The present work is an attempt to check the reliability of these formulations in the light of the experimental measurements.
gled peaks’’. The energy loss straggling (dEexpt.) of the incident ions after passing through a given thickness of the absorber foil, has been determined experimentally, using the following relation: dEexpt. ¼ ðdE2st dE20 Þ1=2 ; where dE0 and dEst are the FWHM of the energy spectra of unstraggled and straggled peaks, respectively. The present experimental arrangement provides a unique advantage of recording unstraggled and straggled peaks simultaneously, so that they contain the same contribution due to (i) beam energy spread from the accelerator, (ii) straggling in the Au foil and (iii) detector resolution and its response, thus nullifying the error due to these factors. Adopting the above procedure, the energy loss straggling for Li (23, 30, 40 MeV), C (40, 60, 80 MeV) and O (80, 100 MeV) ions in varying thicknesses of mylar and polycarbonate foils has been measured. 3. Energy loss straggling formulations
2. Experimental details The energy loss straggling measurements were carried out utilizing the 15 MV Pelletron accelerator facility at Nuclear Science Centre, New Delhi, India. The actual experiment was performed in a low flux chamber, fixed with general purpose scattering chamber (GPSC) at an angle of 15° with respect to primary beam direction. There is a provision to introduce a target ladder in the low flux chamber, which can be moved vertically upward or downward without disturbing the vacuum in the chamber. For the present experiment, the mylar and polycarbonate foils of quoted thickness 6 and 15 lm, respectively were used. Their thickness was measured adopting a-energy loss method using 241Am source. The mean thickness values came out to be 5.81 ± 0.21 and 14.64 ± 0.28 lm, respectively for mylar and polycarbonate foils. These absorber foils in the form of a staircase were mounted on a collimator leaving its small portion blank. Finally, such collimators were mounted on a target ladder, which was then introduced in low flux chamber. The primary ion beams of 7Li, 12C and 16O from Pelletron accelerator were scattered from a thin Au foil (400 nm) in general purpose scattering chamber (GPSC). The scattered beam was allowed to pass normally through varying thicknesses of absorber foils of these two polymers by adjusting each collimator one by one in front of the beam through the vertical motion of the target ladder, before being finally detected by the silicon surface barrier detector. The detector output pulses were stored online on a computer through analogue to digital converter, which were later analyzed using a software package CANDLE. The setup thus allowed the accumulation of several peaks simultaneously: one due to ions without absorber foil, ‘‘unstraggled peak’’ and others due to ions after crossing the absorber foils of different thicknesses, the ‘‘strag-
Due to the statistical nature of the energy loss process of the incident ions while passing though an absorber foil, it is reasonable to assume that the energy loss distribution can be described by a Gaussian shape. For such a distribution (valid for energy loss fraction, DE/E 5–25%), the energy loss straggling dE relates to the straggling standard deviation X through the relation dE ¼ 2ð2 ln 2Þ
1=2
X.
Several theories have been put forward to evaluate the straggling standard deviation X. 3.1. Bohr’s theory The simple Bohr’s theory [9] predicts the straggling standard deviation XB to be enegry independent and given by the following expression: X2B ¼ 4pZ 21 e4 Z 2 Nx; where e, N and x are the electronic charge, the number of absorber atoms per unit volume and the thickness of the absorber foil, respectively. Z1 and Z2 are the atomic numbers of the projectile and the absorber material, respectively. After simplification, the Bohr’s formula reduces to X2B ðMeV2 Þ ¼ 1:57 104 Z 21
Z2 x; A2
where A2 is the absorber mass number and x is expressed in mg/cm2. This theory is based on the assumptions that (i) the velocity of the projectile is much greater than the orbital velocities of the electrons of the target atoms, (ii) the energy loss is very small in comparison to the total energy of the projectile and (iii) the target atoms are randomly distributed and no channeling is involved in the penetration process.
P.K. Diwan et al. / Nucl. Instr. and Meth. in Phys. Res. B 244 (2006) 289–293
3.2. Lindhard and Scharff theory Using the dielectric electron gas picture of the atoms and a local electron charge density, Lindhard and Scharff [10] extended Bohr’s theory by applying a correction factor for low and medium energy projectiles. They developed the following expression for straggling standard deviation, XLS: 0:5LðvÞ for v < 3; X2LS ¼ 2 1 for v P 3; XB where v ¼ v2 =v20 Z 2 is a reduced energy variable with v/v0 the ratio of ion’s velocity to Bohr’s velocity and L(v) = 1.36v1/2 0.016v3/2. Lindhard and Scharff equation indicates that energy straggling approaches Bohr’s value at energy E P 75Z2 keV/u and is energy dependent only at lower energies. 3.3. Bethe and Livingston theory From the treatment of Livingston and Bethe [11], within the energy validity region of the Born approximation, the straggling standard deviation XBL is expressed by the following relation: X2BL Z 02 1 X I iZi 2me v2 ¼ þ ki ln ; 2 2 Z2 Z2 i me v Ii XB
291
where Z 02 is the effective number of absorber atomic electrons, Ii is the average excitation energy of the Zi electrons in the ith atomic orbit, v is the velocity of the incident ion and me is the rest mass of the electron. The constant ki is taken to be 4/3 for all orbits. In the summation, i extends over all orbits for which 2mev2 > Ii. Unlike Bohr’s theory, this theory is energy dependent due to involvement of the factor v2 in the above expression. The values of Ii were modified in such a way [23–26] so as to be compatible with the average ionization potential I of the target atom. Extending the Bragg’s additivity rule, the straggling standard deviation X in polymers can be calculated using the relation [27,28]: n X X2 ¼ ci X2i ; i¼1
where ci is the atomic fraction and Xi is the straggling standard deviation for the ith element in the composite material. 4. Results and discussion Tables 1 and 2 present the experimentally measured energy loss straggling (dEexpt.) values for Li, C and O ions in mylar and polycarbonate absorber foils of varying thicknesses (0.8–7.0 mg/cm2), in the energy range 3.0–6.0 MeV/u. In these tables, DE and DE/E are the
Table 1 Experimental and calculated energy loss straggling values for Li (3.24, 4.25 MeV/u), C (3.25, 4.92 MeV/u) and O (4.89, 6.14 MeV/u) ions in mylar absorber foil Einc. (MeV/u) Ion – Li 3.24 4.25 3.24 4.25 3.25 4.25 Ion – C 3.25 4.92 4.92 4.92 4.92 Ion – O 4.89 6.14 4.89 6.14 4.89 6.14 6.14 4.89 6.14 6.14
Thickness (x) (mg/cm2)
1.62 ± 0.04 2.43 ± 0.05 3.24 ± 0.06
0.81 ± 0.03 1.62 ± 0.04 2.43 ± 0.05 3.24 ± 0.06
0.81 ± 0.03 1.62 ± 0.04 2.43 ± 0.05
3.24 ± 0.06
DE (MeV)
Eav (MeV/u)
DE/E (%)
dEexpt. (keV)
Calculated energy loss straggling (keV) dEB
dELS
dEBL
1.60 1.25 2.44 1.92 3.28 2.61
3.13 4.15 3.07 4.11 3.01 4.06
7 4 11 6 14 9
99 ± 33 84 ± 37 136 ± 29 116 ± 32 158 ± 26 127 ± 30
81(18) 81(4) 99(27) 99(15) 115(27) 115(9)
81(18) 81(4) 99(27) 99(15) 115(27) 115(9)
90(9) 89(6) 110(19) 109(6) 128(19) 126(1)
2.97 2.28 4.58 6.95 9.39
3.13 4.82 4.73 4.63 4.53
8 4 8 12 16
165 ± 28 150 ± 31 228 ± 27 270 ± 26 315 ± 29
115(30) 115(23) 163(28) 199(26) 230(27)
115(30) 115(23) 163(28) 199(26) 230(27)
127(23) 124(17) 177(22) 216(20) 250(21)
3.94 3.35 7.93 6.67 12.05 10.06 10.08 16.32 13.56 13.57
4.77 6.04 4.65 5.93 4.52 5.83 5.83 4.38 5.72 5.72
5 3 10 7 15 10 10 21 14 14
210 ± 40 207 ± 37 331 ± 33 294 ± 33 427 ± 32 370 ± 31 376 ± 32 494 ± 34 451 ± 35 452 ± 35
153(27) 153(26) 217(34) 217(26) 266(38) 266(28) 266(29) 307(38) 307(32) 307(32)
153(27) 153(26) 217(34) 217(26) 266(38) 266(28) 266(29) 307(38) 307(32) 307(32)
166(21) 164(21) 236(29) 233(21) 289(32) 285(23) 285(24) 334(32) 330(27) 330(27)
Percentage differences with respect to the measured values are given in parentheses.
292
P.K. Diwan et al. / Nucl. Instr. and Meth. in Phys. Res. B 244 (2006) 289–293
Table 2 Experimental and calculated energy loss straggling values for Li (3.24, 4.25, 5.68 MeV/u), C (4.92, 6.59 MeV/u) and O (4.89, 6.14 MeV/u) ions in polycarbonate absorber foil Einc. (MeV/u) Ion – Li 4.25 3.24 4.25 5.68 3.24 4.25 5.68 3.24 4.25 5.68 Ion – C 4.92 6.59 4.92 6.59 4.92 6.59 6.59 Ion – O 4.89 6.14 4.89 6.14 6.14 6.14
Thickness (x) (mg/cm2)
1.76 ± 0.03 3.52 ± 0.04
5.28 ± 0.05
7.04 ± 0.06
1.76 ± 0.03 3.52 ± 0.04 5.28 ± 0.05 7.04 ± 0.06
1.76 ± 0.03 3.52 ± 0.04 5.28 ± 0.05 7.04 ± 0.06
DE (MeV)
Eav (MeV/u)
DE/E (%)
dEexpt. (keV)
Calculated energy loss straggling (keV) dEB
dELS
BL
1.44 3.64 2.93 2.30 5.69 4.46 3.45 7.94 6.08 4.66
4.15 2.98 4.04 5.51 2.83 3.93 5.43 2.67 3.81 5.35
5 16 10 6 25 15 9 35 20 12
103 ± 31 159 ± 27 153 ± 26 149 ± 26 202 ± 24 186 ± 26 149 ± 25 240 ± 26 224 ± 28 188 ± 26
85(17) 120(24) 120(22) 120(19) 147(27) 147(21) 147(1) 170(29) 170(24) 170(10)
85(17) 120(24) 120(22) 120(19) 147(27) 147(21) 147(1) 170(29) 170(24) 170(10)
93(10) 131(18) 131(14) 129(13) 164(19) 161(13) 158(6) 190(21) 186(17) 183(3)
5.02 3.22 10.40 7.34 16.24 11.67 16.24
4.71 6.46 4.49 6.28 4.24 6.10 5.91
9 4 18 9 27 15 20
224 ± 24 209 ± 22 329 ± 28 321 ± 26 433 ± 32 390 ± 28 470 ± 31
171(24) 171(18) 240(27) 240(25) 295(32) 295(24) 341(27)
171(24) 171(18) 240(27) 240(25) 295(32) 295(24) 341(27)
184(18) 182(13) 261(21) 257(20) 320(26) 315(19) 365(22)
8.72 7.32 18.18 15.04 23.29 32.30
4.62 5.91 4.32 5.67 5.41 5.13
11 7 23 15 24 33
331 ± 35 317 ± 31 531 ± 37 457 ± 36 584 ± 38 681 ± 40
228(31) 228(28) 321(39) 321(30) 393(33) 455(33)
228(31) 228(28) 321(39) 321(30) 393(33) 455(33)
246(26) 243(23) 348(34) 344(25) 422(28) 489(28)
Percentage differences with respect to the measured values are given in parentheses.
500 EXPERIMENTAL Projectile - C Targets Polycarbonate Mylar
400
Energy loss straggling (keV)
average energy loss and the fractional energy loss, respectively, in the given thickness of the absorber foil and Eav = Einc. DE/2 with Einc. as the incident ion energy. For the calculations of energy loss straggling in polymers, first we have calculated the straggling in elemental constituents of the polymers adopting Bohr (dEB) [9], Lindhard and Scharff (dELS) [10] and Bethe–Livingston (dEBL) [11] theories and then Bragg’s additivity rule has been applied (Tables 1 and 2). On comparison between the experimental and calculated values (Tables 1 and 2), it is noticed that the Bohr’s theory underestimates the experimental values up to 30% for Li and C ions and up to 40% for O ions in the fractional energy loss (DE/E) limits 5–25%. In the energy region covered in the present study, the Lindhard and Scharff values are identical with Bohr’s values. Bethe–Livingston theory with modified values of Ii predicts a better agreement with deviations reduced to 20% for Li and C ions and 30% for O ions. Further, Bethe–Livingston theory correctly predicts slightly decreasing trend in energy loss straggling with increase of incident energy, for a particular ion, in a given thickness of the absorber foil. In order to study the dependence of energy loss straggling on target’s chemical bond structure, the energy loss straggling as a function of energy loss for C ions in mylar and polycarbonate (having almost same effective Z and
Best Fit
300
200
100
0 0
4
8
12
16
20
Energy loss (MeV) Fig. 1. Variation of experimental energy loss straggling (dEexpt.) as a function of energy loss (DE) in mylar (C10H8O4) and polycarbonate (C16H14O3) absorber foils for C projectile.
P.K. Diwan et al. / Nucl. Instr. and Meth. in Phys. Res. B 244 (2006) 289–293
same atomic constituents but different bond structures) has been plotted (Fig. 1). It is clearly depicted from the figure that experimental values of energy loss straggling lie on the same line (within experimental errors) for both these polymers, irrespective of their chemical bond structure. Similar trend has been noticed for Li and O ions. Since the energy loss straggling in polymers is independent of their chemical bond structure, therefore the validity of Bragg’s additivity rule is supported, in the energy range covered in the present study. 5. Conclusion Our’s is apparently the first study that reports the energy loss straggling of heavy ions in mylar and polycarbonate foils. Bethe–Livingston theory shows better agreement with the experimental data than Bohr’s theory. From straggling measurements in polymers, the validity of Bragg’s rule is supported. Acknowledgements The authors are grateful to the Director, Nuclear Science Centre for granting beam time and Pelletron operation staff for providing the stable beam. Authors are thankful to Mr. Sandeep Chopra and Mr. Sunil Ojha for useful technical discussion. Thanks are also due to Mr. D. Kabiraj and Mr. Abhilash for help during target preparation. One of the authors (P.K. Diwan) is thankful to the Council of Scientific and Industrial Research (CSIR), New Delhi, India, for providing financial assistance by way of Senior Research Fellowship. References [1] [2] [3] [4]
P. Sigmund, Nucl. Instr. and Meth. B 315 (1998) 1. J.F. Ziegler, J. Appl. Phys. 85 (3) (1999) 1249. S.P. Ahlen, Rev. Mod. Phys. 52 (1) (1980) 21. L.C. Feldman, J.W. Mayer, Fundamentals of Surface and Thin Film Analysis, North Holland, New York, 1986. [5] W.K. Chu, Phys. Rev. A 13 (6) (1976) 2057.
293
[6] Q. Yang, D.J. O’Connor, Z. Wang, Nucl. Instr. and Meth. B 61 (1991) 149. [7] S. Ouichaoui, E. Hourani, L. Rosier, R. Bimbot, H. Beaumevieille, B. Bouzid, S. Mammeri, Nucl. Instr. and Meth. B 164–165 (2000) 259. [8] H. Weick, H. Geissel, C. Scheidenberger, F. Attallah, D. Cortina, M. Hausmann, G. Munzenberg, T. Radon, H. Schatz, K. Schmidt, J. Stadlmann, K. Summerer, M. Winkler, Phys. Rev. Lett. 85 (13) (2000) 2725. [9] N. Bohr, Kgl. Dan. Vid. Selsk. Mat.-Fys. Medd. 18 (1948) 8. [10] J. Lindhard, M. Scharff, Kgl. Dan. Vid. Selsk. Mat.-Fys. Medd. 27 (1953) 15. [11] M.S. Livingston, H.A. Bethe, Rev. Mod. Phys. 9 (3) (1937) 245. [12] B. Efken, D. Hahn, D. Hilscher, G. Wustefeld, Nucl. Instr. and Meth. B 129 (1975) 219. [13] T. Takahashi, Y. Awaya, T. Tonuma, H. Kumagai, K. Izumo, A. Hashizume, S. Uchiyama, A. Hitachi, Nucl. Instr. and Meth. B 166 (1979) 587. [14] C.J. Sofield, N.E.B. Cowern, R.J. Petty, J.M. Freeman, J.P. Mason, Phys. Rev. A 17 (3) (1978) 859. [15] C.J. Sofield, N.E.B. Cowern, J.M. Freeman, Nucl. Instr. and Meth. B 170 (1980) 221. [16] N.E.B. Cowern, C.J. Sofield, J.M. Freeman, J.P. Mason, Phys. Rev. A 19 (1) (1979) 111. [17] N.E.B. Cowern, P.N. Trehan, C.J. Sofield, L.B. Bridwell, S.M. Ferguson, Nucl. Instr. and Meth. B 206 (1983) 595. [18] T. Schwab, H. Geissel, P. Armbruster, A. Gillibert, W. Mitting, R.E. Olson, K.B. Winterbon, H. Wollnik, G. Munzenberg, Nucl. Instr. and Meth. B 48 (1990) 69. [19] H. Ogawa, I. Katayama, Y. Haruyama, M. Saito, K. Yoshida, M. Tosaki, I. Sugai, Nucl. Instr. and Meth. B 212 (2003) 27. [20] H. Ogawa, I. Katayama, Y. Haruyama, M. Saito, K. Yoshida, M. Tosaki, I. Sugai, Nucl. Instr. and Meth. B 195 (2002) 175. [21] Q. Yang, T. Li, D.J. O’Connor, R.J. MacDonald, T.R. Ophel, H.J. Hay, D.J.H. Cockayne, A. Sikorski, Nucl. Instr. and Meth. B 83 (1993) 439. [22] L.L. Araujo, P.L. Grande, M. Behar, J.H.R. dos Santos, Nucl. Instr. and Meth. B 190 (2002) 79. [23] J.R. Comfort, J.F. Decker, E.T. Lynk, M.O. Scully, A.R. Quinton, Phys. Rev. 150 (1) (1966) 249. [24] T.A. Carlson, Photoelectrons and Auger Spectroscopy, Plenum, New York, 1975. Appendix 1. [25] R.M. Sternheimer, Phys. Rev. 103 (3) (1956) 511. [26] R.M. Sternheimer, Phys. Rev. 88 (4) (1952) 851. [27] J. Raisanen, U. Watjen, A.J.M. Plompen, F. Munnik, Nucl. Instr. and Meth. B 118 (1996) 1. [28] H. Ammi, S. mammeri, M. Allab, Nucl. Instr. and Meth. B 213 (2004) 60.