- Email: [email protected]
E) ∼10–60%
Radiation Physics and Chemistry 119 (2016) 180–185
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Energy loss straggling in Aluminium foils for Li and C ions in fractional energy loss limits (ΔE/E) ∼10–60% P.K. Diwan a,n, Sunil Kumar a, Shyam Kumar b, V. Sharma c, S.A. Khan d, D.K. Avasthi d a
Department of Applied Science, UIET, Kurukshetra University, Kurukshetra 136119, India Department of Physics, Kurukshetra University, Kurukshetra 136119, India c Institute of Forensic Science and Criminology, Panjab University, Chandigarh 160014, India d Inter University Accelerator Centre, P.O. Box 10502, New Delhi 110 067, India b
H I G H L I G H T S
Measured energy loss straggling values are compared with the theoretical predictions. Titeica theory is recognized as a best collisional straggling theory. A fitted relation for charge-exchange straggling is developed.
art ic l e i nf o
a b s t r a c t
Article history: Received 28 January 2015 Received in revised form 3 October 2015 Accepted 15 October 2015 Available online 19 October 2015
The energy loss straggling of Li and C ions in Al foils of various thicknesses has been measured, within the fractional energy loss limit (ΔE/E) ∼ 10–60%. These measurements have been performed using the 15UD Pelletron accelerator facility available at Inter University Accelerator Centre (IUAC), New Delhi, India. The measured straggling values have been compared with the corresponding predicted values adopting popularly used collisional straggling formulations viz Bohr, Lindhard and Scharff, Bethe–Livingston, Titeica. In addition, the experimental data has been compared to the Yang et al. empirical formula and Close Form Model, recently proposed by Montanari et al. The straggling values derived by Titeica theory were found to be in better agreement with the measured values as compared to other straggling formulations. The charge-exchange straggling component has been estimated from the measured data based on Titeica’s theory. Finally, a function of the ion effective charge and the energy loss fraction within the target has been fitted to the latter straggling component. & 2015 Published by Elsevier Ltd.
Keywords: Energy loss Straggling Heavy ions Aluminum foils
1. Introduction Energy loss straggling arises mainly due to the statistical nature of ion-target's electron interaction (collisional straggling) and charge states of incident ions (charge-exchange straggling). For theoretical calculations, both collisional and charge-exchange straggling are first treated independently and then added through quadrature, in order to determine the total straggling values (Ouichaoui et al., 2000; Sigmund, 2004, 2006). Literature reveals that for collisional straggling, number of analytical formulae are available (Livingston and Bethe, 1937; Titeica, 1939; Bohr, 1948; Lindhard and Scharff, 1953; Tschalär, 1968; Bonderup and Hvelplund, 1971; Chu, 1976) but for charge-exchange straggling, limited work has been done so far (Sigmund, 1992; Narmann and n
Corresponding author. E-mail address: [email protected] (P.K. Diwan).
http://dx.doi.org/10.1016/j.radphyschem.2015.10.019 0969-806X/& 2015 Published by Elsevier Ltd.
Sigmund, 1994; Sigmund et al., 2011). Further, some of these straggling formulae are difficult to use to compute straggling values because numerous input parameters are required (Ouichaoui et al., 2000; Weick et al., 2000). So only selected theoretical formulations are available, which can be used or made to use for straggling calculations. In order to exploit such calculations for practical applications (Livingston and Bethe, 1937; Titeica, 1939; Bohr’s theory, 1948; Lindhard and Scharff, 1953; Yang et al., 1991, Montanari et al., 2007), where swift heavy ions are involved, there is a need to check the predicted values based on the straggling formulations through comparison with the measured values. These measured values may be collected from the existing literature or generated by conducting new straggling experiments. As far as literature is concerned, very limited straggling data is available, particularly for heavy ions. So, it is highly essential to generate new measurements related with energy loss straggling. In the present study, energy loss straggling for Li (22.70, 29.73, and 39.74 MeV) and C (38.96, 59.04, and 79.06 MeV) ions in
P.K. Diwan et al. / Radiation Physics and Chemistry 119 (2016) 180–185
1800 C (80 MeV) Without Foil
1600
1400
1200
1000
Counts
varying thicknesses of Al foils has been measured. Similar combination is also studied by different groups in thin materials for lower energy range (Thomas and Fallavier, 1978; Cowern et al., 1979; Hsu et al., 2005). The presently measured straggling values have been compared with the corresponding computed values based on four collisional (Livingston and Bethe, 1937; Titeica, 1939; Bohr’s theory, 1948; Lindhard and Scharff, 1953), and one empirical (Yang et al., 1991) straggling formulations. In this comparison, recently proposed Closed-Form Model (CFM) (Konac et al., 1998; Montanari et al., 2007) is also considered. Finally an attempt has been made to develop a fitted formula for charge-exchange straggling based on the presently measured values.
181
800
6.94±0.16 mg/cm2
600
2. Experimental details
13.89±0.23 mg/cm2
400
The straggling measurements for Li and C ions in varying thicknesses of Al foils were performed by utilizing 15 UD Pelletron at Inter University Accelerator Center (IUAC), New Delhi, India. For this study, Al foil with quoted thickness 6.75 mg/cm2 was procured from STREM Chemical, USA. The foil thickness and inhomogeneity in the foil was ensured with both gravimetric and energy loss method. In energy loss method, first the energy spectra of Li and C ions were recorded one by one using collimator of diameter 0.5 mm and then with the help of measured energy loss and computed values based on Ziegler et al. (1985) approach, thickness of the foil was measured. Adopting this method, different points in the active region of interest were selected for thickness measurements and mean thickness came out to be 6.94 70.16 mg/cm2. Thereafter, in order to obtain higher thicknesses of Al foil, desired sizes of foils were cut from Al sheet and laid down to form a staircase type arrangement and then mounted on a target ladder leaving its small portion blank. Finally, the target ladder was inserted in a scattering chamber. In the chamber, low vacuum was maintained which was ∼10 6–10–7 Torr. Further, Li ion beam (current intensity ∼1–3 pnA) at energy 23 MeV was generated through Pelletron accelerator. This Li beam was passed through thin Au scatterer, to reduce the flux, and then penetrated through the stack of Al foils. Finally, energies of the transmitted ions were measured with silicon surface barrier detector and energy spectra were recorded. In the same way, Li ion at two other energies (30, 40 MeV) and C ions at three energies (40, 60, and 80 MeV) were generated one by one through the accelerator and energy spectra were recorded. The energy spectra for C ions at energy 80 MeV after passing through Al foils is shown in Fig. 1. In this fig., the spectrum at higher channel number corresponds to zero thickness and other three spectra with decreasing channel number correspond to thicknesses 6.94 70.16, 13.897 0.23 and 20.83 7 0.28 mg/cm2 respectively. Finally, Full Width at Half Maxima (FWHM) of recorded spectra was determined using origin software and energy loss straggling was determined through the following relation
δE expt =
(FWHM)2with − (FWHM)2without
where subscripts in the above equation correspond to with and without foil respectively. From the standard errors in FWHMs and propagating the errors through standard method, the standard error in measured straggling values was determined (Topping, 1956). The straggling measurements with the present experimental setup are highly precise because this provides a unique advantage of recording energy spectra without and with foils simultaneously, thus nullifying any error due to beam spot fluctuations from the accelerator, straggling due to Au scatterer and detector resolution and its response.
2
20.83±0.28 mg/cm
200
0 800
1200
1600
2000
2400
2800
Channel Number
Fig. 1. Energy spectra for C ions at energy 80 MeV after passing through different thicknesses of Al metallic foils. The peak on higher channel number corresponds to absence of Al foil.
The detail of the experimental setup and procedure followed for energy loss straggling measurements also given in our previous publications (Diwan et al., 2006, 2007, 2010, 2011, Pratibha et al., 2008; Gulati et al., 2009) .
3. Results and discussion Measured energy loss straggling values for Li and C ions in Al foils, are given in Table 1 and shown in Fig. 2. The numerical values after 7 sign in Table 1 and error bars in Fig. 2 indicate the standard error in straggling measurements. The expected behavior of increase in straggling values with increase of thickness of foils has been observed from Table 1 and Fig. 2. Further, it is observed that the measured straggling decreases with increase of incident energy at particular thickness of the foil. In order to investigate the most appropriate theoretical formulation, the presently measured straggling values along with the prediction of straggling formulations (Livingston and Bethe, 1937; Titeica, 1939; Bohr’s theory, 1948; Lindhard and Scharff, 1953;
Table 1 Measured energy loss straggling of Li and C ions, at different energies, in Al foils as a function of foil thickness and fractional energy loss (E/E). E (MeV)
Thickness (mg/cm2)
(ΔE/E) %
Li
22.70 22.70 29.73 29.73 29.73 39.74 39.74
6.94 7 0.16 13.89 7 0.23 6.94 7 0.16 13.89 7 0.23 20.83 7 0.28 6.94 7 0.16 13.89 7 0.23
23 53 14 30 50 8 17
281 724 550 731 241 724 417 726 576 729 221 726 369 724
C
38.96 59.04 59.04 79.06 79.06 79.06
6.94 7 0.16 6.94 7 0.16 13.89 7 0.23 6.94 7 0.16 13.89 7 0.23 20.83 7 0.28
57 26 60 14 32 54
1072 735 691 735 1449 746 595 735 901 739 1446 737
Ion
δEexpt (keV)
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600
800
Ion - Li (22.70 MeV)
Li + Al (6.94 mg/cm ) Li + Al (13.89 mg/cm ) Li + Al (20.83 mg/cm )
Bohr Lindhard-Scharff
500
Yang et al.
600
Close Form Model Bethe-Livingston
400
Titeica Measured Values
300
200
200 100
0
0 600 Ion - Li (29.73 MeV)
1600 C + Al (6.94 mg/cm ) C + Al (13.89 mg/cm ) C + Al (20.83 mg/cm )
500
Energy Loss Straggling (keV)
Measured Energy Loss Straggling (keV)
400
1200
800
400
300
200
100
400
0 600
0
Ion - Li (39.74 MeV)
2
3
4
5
6
7
Energy (MeV/n)
500
Fig. 2. Variation of measured energy loss straggling with incident ion energy, for Li and C ions in Al foils.
400
Yang et al., 1991) and proposed Closed-Form Model (CFM) (Konac et al., 1998; Montanari et al., 2007) are presented in Figs. 3 and 4. Through this comparison, it is generally observed that the straggling values based on considered formulations underestimate the measured values. In case of Li ions (Fig. 3), the computed values based on Bohr’s theory underestimate ∼1.35–2.40 times the measured straggling values. Similar deviations have been observed with Lindhard– Scharff theory, within the fractional energy loss limit (ΔE/E) ∼10– 60% considered in the present study. The predicted straggling values based on Yang et al. formulation is ∼1.35–2.35 times lower. The computed values based on Close Form Model are also lower (∼1.38–2.47 times) than the measured straggling values. The results based on the Bethe–Livingston theory, using modified Ii (Sternheimer, 1952, 1956; Comfort et al. 1966; Ahlen, 1980, Lide, 2010) values (Table 2), slightly reduce the deviation (∼1.22–2.10 times) with the measured values. Further, it is observed from Fig. 3
300
200
100
0 0
15
30
45
60
75
∆E/E % Fig. 3. The comparison of measured and computed energy loss straggling as a function of fractional energy loss (ΔE/E) for Li ions at three energies (22.70, 29.73, and 39.74 MeV) in Al foils.
P.K. Diwan et al. / Radiation Physics and Chemistry 119 (2016) 180–185
1200
183
Table 2 Parameters used in the calculation of modified Ii values for Al target.
Ion - C (38.96 MeV) Bohr
1000
Lindhard-Scharff
Energy level No. of electrons per level
Oscillator strength (fj)
Binding energies (hvi) (eV)a
1s1/2 2s1/2 2p1/2 2p3/2
2/13 2/13 2/13 4/13
1559.0 117.8 72.9 72.5
Yang et al. Closed Form Model
800
Bethe-Livingston Titeica
Plasma energy (hvp) = 32. 86 eV
Measured values
600
Mean ionization potential (I)b ¼ 164 7 1 eV a b
400
Lide (2010). Ahlen (1980).
For C ions (Fig. 4), Bohr and Lindhard and Scharff predictions underestimate the measured straggling values up to ∼1.85–3.30 times. Yang et al. predicted that the straggling values are also lower (1.65–2.75 times) than the measured values. Close Form model based values underestimate the measured values up to 1.85–3.39 times. The computed values based on Bethe–Livingston theory show better results and deviation is up to ∼1.65–2.90 times the measured values. The deviation between measured and predicted values is further reduced by Titeica theory and observed deviation is up to 1.50–2.45 times the measured straggling. Above discussion indicates that the prediction of Titeica theory is better in agreement with the measured straggling values as compared to other theories. Further, the deviation between measured and computed straggling values depends upon the atomic number of the incident ion and increases with increase of ion’s atomic number as well as fractional energy loss. The enhancement in deviation may be mainly due to more fluctuations in charge state, which is only considered in Yang et al. approach. In this approach, the charge state part is purely based on the limited available experimental straggling data. Hence, it is highly essential to develop an empirical formula to evaluate charge-exchange counterpart of total straggling. Since charge-exchange component is always coupled with collisional component in the measured straggling, therefore, it cannot be measured individually. In order to exploit charge-exchange component, Titeica collisional straggling values have been subtracted in quadrature from the measured straggling data and following fitted relation for charge exchange straggling (ΩCES) has been developed.
200
0 1600 Ion - C (59.04 MeV) 1400
Energy Loss Straggling (keV)
2 2 2 4
1200
1000
800
600
400
200
0 1600 Ion - C (79.06 MeV) 1400
⎛ ΔE ⎞c ⎟ ΩCES = a (Z1* )b ⎜ ⎝ E ⎠
1200
1000
where Z1* is the ion effective atomic number,
ΔE E
is the fractional
energy loss of incident ion and a ¼3.128, b¼ 1.421 and c¼ 0.849 are the fitting coefficients. The computed results, based on the presently fitted charge-exchange component (CEC) in combination with Titeica (Tit) theory, have been compared with the measured straggling values ( Figs. 5 and 6) and observed that the computed values are in agreement with measured values, within experimental errors.
800
600
400
200
4. Conclusions
0 0
15
30
45
60
75
∆E/E % Fig. 4. Same as Fig. 3 for C ions at three energies (38.96, 59.04, and 79.06 MeV).
that the straggling values based on Titeica's theory are higher than the above theories and underestimate ∼1.10–1.75 times the measured straggling values.
The prediction of straggling theories underestimates the measured values for Li and C ions in Al foils. Although the results of Titeica theory are closer to the measured values, as compared to other theories, but still large deviations have been observed due to non consideration of the charge-exchange component. Further, the results based on the presently developed fitted formula for chargeexchange component in combination with Titeica theory are in
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P.K. Diwan et al. / Radiation Physics and Chemistry 119 (2016) 180–185
1600
700
C (38.96 MeV)
Li (22.70 MeV)
Tit + CEC
Tit + CEC
600
1400
Measured Values
Measured Values
1200
500 1000
400 800
300 600
200 400
100
200
0
0
1800
800
C (59.04MeV)
Li (29.73 MeV) 1600
Energy Loss Straggling (keV)
Energy Loss Straggling (keV)
700
600
500
400
300
200
1400 1200 1000 800 600 400
100
200
0
0 1800
800
C (79.06MeV)
Li (39.74 MeV) 1600
700
1400
600
1200 500 1000 400 800 300 600 200
400
100
200 0
0 0
15
30
45
60
75
ΔE/E% Fig. 5. The comparison of measured and computed energy loss straggling based on presently developed adhoc (Titeica þ fitted charge exchange component) method for Li ions.
0
15
30
45
ΔE/E% Fig. 6. Same as Fig. 5 for C ions.
60
75
P.K. Diwan et al. / Radiation Physics and Chemistry 119 (2016) 180–185
reasonable agreement with the measured values in ion-target combination considered in the present study. The straggling results of present study may be useful in those experiments where Li and C ions in Al foils are required. In order to generalize the charge-exchange fitted relation, the experiments related to energy loss straggling are under progress.
References Ahlen, S.P., 1980. Theoretical and experimental aspects of energy loss of relativistic heavily ionizing particles. Rev. Mod. Phys. 52, 121–173. Bohr, N., 1948. The penetration of atomic particles through matter. In: Det. Kgl. Dan. Vid. Selsk. Mat. -Fys. Medd. 18,8, pp. 1–144. Bonderup, E., Hvelplund, P., 1971. Stopping power and energy straggling for swift protons. Phys. Rev. A 4 (2), 562–569. Chu, W.K., 1976. Calculation of energy straggling for protons and helium ions. Phys. Rev. A13 (6), 2057–2060. Comfort, J.R., Decker, J.F., Lynk, E.T., Scully, M.O., Quinton, A.R., 1966. Energy loss and straggling of alpha particles in metal foils. Phys. Rev. 150 (1), 249–256. Cowern, N.E.B., Sofield, C.J., Freeman, J.M., Mason, J.P., 1979. Energy straggling of 3– 36 MeV 12C ions in aluminum. Phys. Rev. A19 (1), 111–115. Diwan, P.K., Sharma, V., Aggarwal, S., Kumar, S., Sharma, S.K., Mittal, V.K., Sannakki, B., Mathad, R.D., Khan, S.A., Avasthi, D.K., 2006. Energy loss straggling of Li, C and O ions in mylar and polycarbonate absorber foils. Nucl. Instrum. Methods Phys. Res. B244, 289–293. Diwan, P.K., Sharma, V., Kumar, S., Mittal, V.K., Khan, S.A., Avasthi, D.K., 2007. Energy loss and straggling of MeV heavy ions in polypropylene absorber foils. Nucl. Instrum. Methods Phys. Res. B258, 293–298. Diwan, P.K., Neetu, Gulati, P.K., Kumar, S., Avasthi, D.K., 2010. Statistical variations in energy loss for Li, C and O ions in thick Kapton polymeric foils. Nucl. Instrum. Methods Phys. Res. B268, 3523–3525. Diwan, P.K., Neetu, Gulati, P.K., Kumar, S., Avasthi, D.K., 2011. Energy loss straggling in thick mylar polymeric foils for swift heavy ions in fractional energy loss limits ΔE/E ∼5–80%. Nucl. Instrum. Methods Phys. Res. B269, 1786–1791. Gulati, P.K., Munjal, N., Diwan, P.K., Sharma, V., Kumar, S., Khan, S.A., Avasthi, D.K., 2009. Statistical fluctuations in energy loss for swift heavy ions in thick polymeric foils. Phys. Rev. A80, 032903 (1–5). Hsu, J.Y., Liang, J.H., Yu, Y.C., Chen, K.M., 2005. Energy straggling of He, Li, and B isotopes in aluminum and silver. Nucl. Instrum. Methods Phys. Res. B241, 160–164. Konac, G., Kalbitzer, S., Klatt, C., Niemann, D., Stoll, R., 1998. Energy loss and straggling of H and He ions of keV energies in Si and C. Nucl. Instrum. Methods Phys. Res. B136–138, 159–165. Lide, D.R., ed., Electron biding energies of the elements. In: CRC Handbook of
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Chemistry and Physics, 90th Edition (Internet Version 2010), CRC Press/Taylor and Francis, Boca Raton, Florida. Lindhard, J., Scharff, M., 1953. Energy loss in matter by fast particles of low charge. In: Kgl. Dan. Vid. Selsk. Mat. -Fys. Medd., 27, 15, pp. 1–32. Livingston, M.S., Bethe, H.A., 1937. Nuclear Physics-C. Nuclear dynamics, experimental. Rev. Mod. Phys. 9 (3), 245–398. Montanari, C.C., Miraglia, J.E., Heredia-Avalos, S., Garcia-Molina, R., Abril, I., 2007. Calculation of energy-loss straggling of C, Al, Si, and Cu for fast H, He, and Li ions. Phys. Rev. A75, 022903 1–11. Narmann, A., Sigmund, P., 1994. Statistics of energy loss and charge exchange of penetrating particles: higher moments and transients. Phy. Rev. A49 (6), 4709–4715. Ouichaoui, S., Hourani, E., Rosier, L., Bimbot, R., Beaumevieille, H., Bouzid, B., Mammeri, S., 2000. Energy loss straggling of swift heavy ions in metal foils at E/ A ∼ 2 MeV/u. Nucl. Instrum. Methods Phys. Res. B 164–165, 259–267. Pratibha, Sharma, V., Diwan, P.K., Kumar, S., Khan, S.A., Avasthi, D.K., 2008. Energy loss and straggling in LR-115 and Kapton polymeric foils for energetic ions. Nucl. Instrum. Methods Phys. Res. B266, 2556–2563. Sigmund, P., 1992. Statistical theory of charged - particle stopping and straggling in the presence of charge exchange. Nucl. Instrum. Methods Phys. Res. B 69, 113–122. Sigmund, P., 2004. Stopping of Heavy Ions: A Theoretical Approach. Springer, Berlin. Sigmund, P., 2006. Particle Penetration and Radiation Effects: General Aspects and Stopping of Swift Point Particles. 151. Springer-verlag, Berlin, Springer series in solid state sciences. Sigmund, P., Osmani, O., Shinner, A., 2011. Charge-exchange straggling in equilibrium. Nucl. Instrum. Methods Phys. Res. B 269, 804–809. Sternheimer, R.M., 1952. The density effect for the ionization loss in various materials. Phys. Rev. 88 (4), 851–859. Sternheimer, R.M., 1956. Density effect for the ionization loss in various materials. Phys. Rev. 103 (3), 511–515. Thomas, J.P., Fallavier, M., 1978. Lithium ion production and use for backscattering analsis in Aluminum and Aluminum oxide media. Nucl. Instrum. Methods 149, 169–175. Titeica, S., 1939. Sur les fluctuations de parcours des rayonscorpuscularies. Bull. Soc. Roum. Phys. 38, 81–100. Topping, J., 1956. Theory of Error. Unwin Brothers Limited, Great Britain. Tschalär, C., 1968. Straggling distribution of large energy losses. Nucl. Instrum. Methods 61, 141–156. Weick, H., Geissel, H., Scheidenberger, C., Attallah, F., Cortina, D., Hausmann, M., Munzenberg, G., Radon, T., Schatz, H., Schmidt, K., Stadlmann, J., Summerer, K., Winkler, M., 2000. Drastic enhancement of energy loss straggling of relativistic heavy ions due to charge-state fluctuations. Phys. Rev. Lett. 85 (13), 2725–2728. Yang, Q., O’Connor, D.J., Wang, Z., 1991. Empirical formulae for energy loss straggling of ions in matter. Nucl. Instrum. Methods Phys. Res. B61, 149–155. Ziegler, J.F., Biersack, J.P., Littmark, U., 1985. The Stopping and Ranges of Ions in Solids. 1. Pergamon, New York.
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Energy loss straggling in Aluminium foils for Li and C ions in fractional energy loss limits (ΔE/E) ∼10–60% P.K. Diwan a,n, Sunil Kumar a, Shyam Kumar b, V. Sharma c, S.A. Khan d, D.K. Avasthi d a
Department of Applied Science, UIET, Kurukshetra University, Kurukshetra 136119, India Department of Physics, Kurukshetra University, Kurukshetra 136119, India c Institute of Forensic Science and Criminology, Panjab University, Chandigarh 160014, India d Inter University Accelerator Centre, P.O. Box 10502, New Delhi 110 067, India b
H I G H L I G H T S
Measured energy loss straggling values are compared with the theoretical predictions. Titeica theory is recognized as a best collisional straggling theory. A fitted relation for charge-exchange straggling is developed.
art ic l e i nf o
a b s t r a c t
Article history: Received 28 January 2015 Received in revised form 3 October 2015 Accepted 15 October 2015 Available online 19 October 2015
The energy loss straggling of Li and C ions in Al foils of various thicknesses has been measured, within the fractional energy loss limit (ΔE/E) ∼ 10–60%. These measurements have been performed using the 15UD Pelletron accelerator facility available at Inter University Accelerator Centre (IUAC), New Delhi, India. The measured straggling values have been compared with the corresponding predicted values adopting popularly used collisional straggling formulations viz Bohr, Lindhard and Scharff, Bethe–Livingston, Titeica. In addition, the experimental data has been compared to the Yang et al. empirical formula and Close Form Model, recently proposed by Montanari et al. The straggling values derived by Titeica theory were found to be in better agreement with the measured values as compared to other straggling formulations. The charge-exchange straggling component has been estimated from the measured data based on Titeica’s theory. Finally, a function of the ion effective charge and the energy loss fraction within the target has been fitted to the latter straggling component. & 2015 Published by Elsevier Ltd.
Keywords: Energy loss Straggling Heavy ions Aluminum foils
1. Introduction Energy loss straggling arises mainly due to the statistical nature of ion-target's electron interaction (collisional straggling) and charge states of incident ions (charge-exchange straggling). For theoretical calculations, both collisional and charge-exchange straggling are first treated independently and then added through quadrature, in order to determine the total straggling values (Ouichaoui et al., 2000; Sigmund, 2004, 2006). Literature reveals that for collisional straggling, number of analytical formulae are available (Livingston and Bethe, 1937; Titeica, 1939; Bohr, 1948; Lindhard and Scharff, 1953; Tschalär, 1968; Bonderup and Hvelplund, 1971; Chu, 1976) but for charge-exchange straggling, limited work has been done so far (Sigmund, 1992; Narmann and n
Corresponding author. E-mail address: [email protected] (P.K. Diwan).
http://dx.doi.org/10.1016/j.radphyschem.2015.10.019 0969-806X/& 2015 Published by Elsevier Ltd.
Sigmund, 1994; Sigmund et al., 2011). Further, some of these straggling formulae are difficult to use to compute straggling values because numerous input parameters are required (Ouichaoui et al., 2000; Weick et al., 2000). So only selected theoretical formulations are available, which can be used or made to use for straggling calculations. In order to exploit such calculations for practical applications (Livingston and Bethe, 1937; Titeica, 1939; Bohr’s theory, 1948; Lindhard and Scharff, 1953; Yang et al., 1991, Montanari et al., 2007), where swift heavy ions are involved, there is a need to check the predicted values based on the straggling formulations through comparison with the measured values. These measured values may be collected from the existing literature or generated by conducting new straggling experiments. As far as literature is concerned, very limited straggling data is available, particularly for heavy ions. So, it is highly essential to generate new measurements related with energy loss straggling. In the present study, energy loss straggling for Li (22.70, 29.73, and 39.74 MeV) and C (38.96, 59.04, and 79.06 MeV) ions in
P.K. Diwan et al. / Radiation Physics and Chemistry 119 (2016) 180–185
1800 C (80 MeV) Without Foil
1600
1400
1200
1000
Counts
varying thicknesses of Al foils has been measured. Similar combination is also studied by different groups in thin materials for lower energy range (Thomas and Fallavier, 1978; Cowern et al., 1979; Hsu et al., 2005). The presently measured straggling values have been compared with the corresponding computed values based on four collisional (Livingston and Bethe, 1937; Titeica, 1939; Bohr’s theory, 1948; Lindhard and Scharff, 1953), and one empirical (Yang et al., 1991) straggling formulations. In this comparison, recently proposed Closed-Form Model (CFM) (Konac et al., 1998; Montanari et al., 2007) is also considered. Finally an attempt has been made to develop a fitted formula for charge-exchange straggling based on the presently measured values.
181
800
6.94±0.16 mg/cm2
600
2. Experimental details
13.89±0.23 mg/cm2
400
The straggling measurements for Li and C ions in varying thicknesses of Al foils were performed by utilizing 15 UD Pelletron at Inter University Accelerator Center (IUAC), New Delhi, India. For this study, Al foil with quoted thickness 6.75 mg/cm2 was procured from STREM Chemical, USA. The foil thickness and inhomogeneity in the foil was ensured with both gravimetric and energy loss method. In energy loss method, first the energy spectra of Li and C ions were recorded one by one using collimator of diameter 0.5 mm and then with the help of measured energy loss and computed values based on Ziegler et al. (1985) approach, thickness of the foil was measured. Adopting this method, different points in the active region of interest were selected for thickness measurements and mean thickness came out to be 6.94 70.16 mg/cm2. Thereafter, in order to obtain higher thicknesses of Al foil, desired sizes of foils were cut from Al sheet and laid down to form a staircase type arrangement and then mounted on a target ladder leaving its small portion blank. Finally, the target ladder was inserted in a scattering chamber. In the chamber, low vacuum was maintained which was ∼10 6–10–7 Torr. Further, Li ion beam (current intensity ∼1–3 pnA) at energy 23 MeV was generated through Pelletron accelerator. This Li beam was passed through thin Au scatterer, to reduce the flux, and then penetrated through the stack of Al foils. Finally, energies of the transmitted ions were measured with silicon surface barrier detector and energy spectra were recorded. In the same way, Li ion at two other energies (30, 40 MeV) and C ions at three energies (40, 60, and 80 MeV) were generated one by one through the accelerator and energy spectra were recorded. The energy spectra for C ions at energy 80 MeV after passing through Al foils is shown in Fig. 1. In this fig., the spectrum at higher channel number corresponds to zero thickness and other three spectra with decreasing channel number correspond to thicknesses 6.94 70.16, 13.897 0.23 and 20.83 7 0.28 mg/cm2 respectively. Finally, Full Width at Half Maxima (FWHM) of recorded spectra was determined using origin software and energy loss straggling was determined through the following relation
δE expt =
(FWHM)2with − (FWHM)2without
where subscripts in the above equation correspond to with and without foil respectively. From the standard errors in FWHMs and propagating the errors through standard method, the standard error in measured straggling values was determined (Topping, 1956). The straggling measurements with the present experimental setup are highly precise because this provides a unique advantage of recording energy spectra without and with foils simultaneously, thus nullifying any error due to beam spot fluctuations from the accelerator, straggling due to Au scatterer and detector resolution and its response.
2
20.83±0.28 mg/cm
200
0 800
1200
1600
2000
2400
2800
Channel Number
Fig. 1. Energy spectra for C ions at energy 80 MeV after passing through different thicknesses of Al metallic foils. The peak on higher channel number corresponds to absence of Al foil.
The detail of the experimental setup and procedure followed for energy loss straggling measurements also given in our previous publications (Diwan et al., 2006, 2007, 2010, 2011, Pratibha et al., 2008; Gulati et al., 2009) .
3. Results and discussion Measured energy loss straggling values for Li and C ions in Al foils, are given in Table 1 and shown in Fig. 2. The numerical values after 7 sign in Table 1 and error bars in Fig. 2 indicate the standard error in straggling measurements. The expected behavior of increase in straggling values with increase of thickness of foils has been observed from Table 1 and Fig. 2. Further, it is observed that the measured straggling decreases with increase of incident energy at particular thickness of the foil. In order to investigate the most appropriate theoretical formulation, the presently measured straggling values along with the prediction of straggling formulations (Livingston and Bethe, 1937; Titeica, 1939; Bohr’s theory, 1948; Lindhard and Scharff, 1953;
Table 1 Measured energy loss straggling of Li and C ions, at different energies, in Al foils as a function of foil thickness and fractional energy loss (E/E). E (MeV)
Thickness (mg/cm2)
(ΔE/E) %
Li
22.70 22.70 29.73 29.73 29.73 39.74 39.74
6.94 7 0.16 13.89 7 0.23 6.94 7 0.16 13.89 7 0.23 20.83 7 0.28 6.94 7 0.16 13.89 7 0.23
23 53 14 30 50 8 17
281 724 550 731 241 724 417 726 576 729 221 726 369 724
C
38.96 59.04 59.04 79.06 79.06 79.06
6.94 7 0.16 6.94 7 0.16 13.89 7 0.23 6.94 7 0.16 13.89 7 0.23 20.83 7 0.28
57 26 60 14 32 54
1072 735 691 735 1449 746 595 735 901 739 1446 737
Ion
δEexpt (keV)
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600
800
Ion - Li (22.70 MeV)
Li + Al (6.94 mg/cm ) Li + Al (13.89 mg/cm ) Li + Al (20.83 mg/cm )
Bohr Lindhard-Scharff
500
Yang et al.
600
Close Form Model Bethe-Livingston
400
Titeica Measured Values
300
200
200 100
0
0 600 Ion - Li (29.73 MeV)
1600 C + Al (6.94 mg/cm ) C + Al (13.89 mg/cm ) C + Al (20.83 mg/cm )
500
Energy Loss Straggling (keV)
Measured Energy Loss Straggling (keV)
400
1200
800
400
300
200
100
400
0 600
0
Ion - Li (39.74 MeV)
2
3
4
5
6
7
Energy (MeV/n)
500
Fig. 2. Variation of measured energy loss straggling with incident ion energy, for Li and C ions in Al foils.
400
Yang et al., 1991) and proposed Closed-Form Model (CFM) (Konac et al., 1998; Montanari et al., 2007) are presented in Figs. 3 and 4. Through this comparison, it is generally observed that the straggling values based on considered formulations underestimate the measured values. In case of Li ions (Fig. 3), the computed values based on Bohr’s theory underestimate ∼1.35–2.40 times the measured straggling values. Similar deviations have been observed with Lindhard– Scharff theory, within the fractional energy loss limit (ΔE/E) ∼10– 60% considered in the present study. The predicted straggling values based on Yang et al. formulation is ∼1.35–2.35 times lower. The computed values based on Close Form Model are also lower (∼1.38–2.47 times) than the measured straggling values. The results based on the Bethe–Livingston theory, using modified Ii (Sternheimer, 1952, 1956; Comfort et al. 1966; Ahlen, 1980, Lide, 2010) values (Table 2), slightly reduce the deviation (∼1.22–2.10 times) with the measured values. Further, it is observed from Fig. 3
300
200
100
0 0
15
30
45
60
75
∆E/E % Fig. 3. The comparison of measured and computed energy loss straggling as a function of fractional energy loss (ΔE/E) for Li ions at three energies (22.70, 29.73, and 39.74 MeV) in Al foils.
P.K. Diwan et al. / Radiation Physics and Chemistry 119 (2016) 180–185
1200
183
Table 2 Parameters used in the calculation of modified Ii values for Al target.
Ion - C (38.96 MeV) Bohr
1000
Lindhard-Scharff
Energy level No. of electrons per level
Oscillator strength (fj)
Binding energies (hvi) (eV)a
1s1/2 2s1/2 2p1/2 2p3/2
2/13 2/13 2/13 4/13
1559.0 117.8 72.9 72.5
Yang et al. Closed Form Model
800
Bethe-Livingston Titeica
Plasma energy (hvp) = 32. 86 eV
Measured values
600
Mean ionization potential (I)b ¼ 164 7 1 eV a b
400
Lide (2010). Ahlen (1980).
For C ions (Fig. 4), Bohr and Lindhard and Scharff predictions underestimate the measured straggling values up to ∼1.85–3.30 times. Yang et al. predicted that the straggling values are also lower (1.65–2.75 times) than the measured values. Close Form model based values underestimate the measured values up to 1.85–3.39 times. The computed values based on Bethe–Livingston theory show better results and deviation is up to ∼1.65–2.90 times the measured values. The deviation between measured and predicted values is further reduced by Titeica theory and observed deviation is up to 1.50–2.45 times the measured straggling. Above discussion indicates that the prediction of Titeica theory is better in agreement with the measured straggling values as compared to other theories. Further, the deviation between measured and computed straggling values depends upon the atomic number of the incident ion and increases with increase of ion’s atomic number as well as fractional energy loss. The enhancement in deviation may be mainly due to more fluctuations in charge state, which is only considered in Yang et al. approach. In this approach, the charge state part is purely based on the limited available experimental straggling data. Hence, it is highly essential to develop an empirical formula to evaluate charge-exchange counterpart of total straggling. Since charge-exchange component is always coupled with collisional component in the measured straggling, therefore, it cannot be measured individually. In order to exploit charge-exchange component, Titeica collisional straggling values have been subtracted in quadrature from the measured straggling data and following fitted relation for charge exchange straggling (ΩCES) has been developed.
200
0 1600 Ion - C (59.04 MeV) 1400
Energy Loss Straggling (keV)
2 2 2 4
1200
1000
800
600
400
200
0 1600 Ion - C (79.06 MeV) 1400
⎛ ΔE ⎞c ⎟ ΩCES = a (Z1* )b ⎜ ⎝ E ⎠
1200
1000
where Z1* is the ion effective atomic number,
ΔE E
is the fractional
energy loss of incident ion and a ¼3.128, b¼ 1.421 and c¼ 0.849 are the fitting coefficients. The computed results, based on the presently fitted charge-exchange component (CEC) in combination with Titeica (Tit) theory, have been compared with the measured straggling values ( Figs. 5 and 6) and observed that the computed values are in agreement with measured values, within experimental errors.
800
600
400
200
4. Conclusions
0 0
15
30
45
60
75
∆E/E % Fig. 4. Same as Fig. 3 for C ions at three energies (38.96, 59.04, and 79.06 MeV).
that the straggling values based on Titeica's theory are higher than the above theories and underestimate ∼1.10–1.75 times the measured straggling values.
The prediction of straggling theories underestimates the measured values for Li and C ions in Al foils. Although the results of Titeica theory are closer to the measured values, as compared to other theories, but still large deviations have been observed due to non consideration of the charge-exchange component. Further, the results based on the presently developed fitted formula for chargeexchange component in combination with Titeica theory are in
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1600
700
C (38.96 MeV)
Li (22.70 MeV)
Tit + CEC
Tit + CEC
600
1400
Measured Values
Measured Values
1200
500 1000
400 800
300 600
200 400
100
200
0
0
1800
800
C (59.04MeV)
Li (29.73 MeV) 1600
Energy Loss Straggling (keV)
Energy Loss Straggling (keV)
700
600
500
400
300
200
1400 1200 1000 800 600 400
100
200
0
0 1800
800
C (79.06MeV)
Li (39.74 MeV) 1600
700
1400
600
1200 500 1000 400 800 300 600 200
400
100
200 0
0 0
15
30
45
60
75
ΔE/E% Fig. 5. The comparison of measured and computed energy loss straggling based on presently developed adhoc (Titeica þ fitted charge exchange component) method for Li ions.
0
15
30
45
ΔE/E% Fig. 6. Same as Fig. 5 for C ions.
60
75
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reasonable agreement with the measured values in ion-target combination considered in the present study. The straggling results of present study may be useful in those experiments where Li and C ions in Al foils are required. In order to generalize the charge-exchange fitted relation, the experiments related to energy loss straggling are under progress.
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