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Energy loss and straggling of a-particles in Ag (J^&ossMark

and Sn metallic foils

Sunil Kumar, P.K. Diwan*

Department of Applied Science, UIET, Kurufeshetra University, Kurufeshetra 136 119, India

ARTICLE INFO

ABSTRACT

Article history: Received 23 February 2015 Received in revised form 30 May 2015 Accepted 15 June 2015 Available online 26 June 2015

Keywords: Energy loss dE/dx Straggling a-Particles Metallic foils

Energy loss and straggling in Ag and Sn metallic foils for a-particles, using 241Am source, are measured. The measured energy loss values are compared with the predicted values based on Benton and Henke, Grande and Schiwietz (CasP), Ziegler et al. (SRIM) formulations and ICRU-49 report (ASTAR). Also, measured straggling values of a-particles are compared with the computed values adopting practically used four collisional (Bohr, Lindhard and Scharff, Bethe—Livingston, Titeica) formulations and one collisional plus charge exchange (Yang et al.) straggling formulation. The aim of the comparison is to identify the best energy loss and straggling formulation.

Copyright © 2015, The Egyptian Society of Radiation Sciences and Applications. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The mono-energetic charged particles, while penetrating through the target foil, slow down via number of collisions with target electrons. These collisions are statistical in nature and as a result form an energy loss distribution curve. Statistically, such energy loss distribution curve is characterized in terms of its average value and full width at half maxima (FWHM). These statistical terms provide significant information of energy loss and straggling respectively (Sigmund, 2004). This information is important to understand the fundamental interactions and serve as an input to those experiments, where energetic ions are used. Numerous experimental and theoretical groups are involved to study the energy loss, in different target-ions' energy combinations, through different experimental techniques and by adopting

different theoretical treatments (Ammi et al., 2011; Cantero et al., 2012; Diwan et al., 2003; Guesmia et al., 2014; Hubert, Bimbot, & Gauvin, 1990; Miksova, Mackova, Malinsky, Hnatowicz, & Slepicka, 2014; Moussa, Damache, & Ouichaoui, 2015; Northcliffe & Schilling, 1970; Paul & Schinner, 2001, 2002; Pratibha, Sharma, Diwan, Kumar, Khan, & Avasthi, 2008; Randhawa & Virk, 1996; Rauhala, Raisanen, Fulop, Kiss, & Hunyadi, 1992; Sharma, Kumar, Yadav, & Sharma, 1995; Wambersie, 2005; Weaver & Westphal, 2002; Zhang & Weber, 2003; Ziegler, 1999; Ziegler, Biersack, & Littmark, 1985). As far as energy loss straggling is concerned, little attention has been paid and therefore limited studies are available, which are fragmentary in nature (Behar, Fadanelli, Abril, Garcia-Molina, & Nagamine, 2011; Besenbacher, Andersen, & Bonderup, 1980; Comfort, Decker, Lynk, Scully, & Quinton, 1966; Diwan, Neetu, & Kumar, 2012; Gues-mia et al., 2015; Hsu, Liang, Yu, & Chen, 2005; Ibrahim & Al-

* Corresponding author. E-mail address: diwanpk74@gmail.com (P.K. Diwan).

Peer review under responsibility of The Egyptian Society of Radiation Sciences and Applications. http://dx.doi.org/10.1016/j.jrras.2015.06.005

1687-8507/Copyright © 2015, The Egyptian Society of Radiation Sciences and Applications. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Bedri, 2012; Kumar, Diwan, & Kumar, 2015; Miksova et al., 2014; Neetu, Sharma, Gulati, Diwan & Kumar, 2010; Ouichaoui et al., 2000). Hence, there is a need to conduct systematic studies for straggling along with energy loss, for various ions in different target materials.

In the contemporary era, a-particles emerge out as an imperative ion beam for the characterization of materials, particularly, through Rutherford backscattering (RBS) (Chu, Mayer, & Nicolet, 1978). Furthermore, a-particles play a significant role in medical applications to cure cancer via radio-nuclide therapy (Mulford, Scheinberg, & Jurcic, 2005; Sgouros, 2008; Song, Senthamizhchelvan, Hobbs, & Sgouros, 2012). For effective use of a-particles in these applications, the precise values of energy loss and straggling in different target are highly essential.

In the present study, energy loss and straggling of a-par-ticles in Ag and Sn metallic foils are measured and compared with the respective theoretical predictions.

2. Experimental details

For present measurements, Ag (thickness: 127 mm, purity: 3 N) and Sn (thickness: 50 mm, purity: 4 N) metallic foils were procured from STREM Chemical, USA. These metallic foils were rolled with hardened rollers machine by applying different pressure and as a result, thin foils of desired thicknesses were obtained. These metallic foils were mounted on different collimators and positioned one by one, in a chamber, between 241Am source and Silicon Surface Barrier Detector and respective energy spectra were recorded. These spectra were analyzed to obtain centroids as well as Full Width at Half Maxima (FWHM), through ORIGIN software.

With the help of observed centroids, residual energies of a-particles were determined and least square fit between thickness and residual energy was obtained. Fig. 1 shows such a curve for Ag metallic foil. Through the fitted curve, energy loss per unit length (dE/dx) of a-particles, at different energies, in Ag and Sn metallic foils was determined.

For determination of energy loss straggling of a-particles, the following relation has been used.

Meanued Value* I rlit Square Fil

Residual Energy (MeV)

Fig. 1 - Least square fit between thickness and residual energy, for a-particles in Ag metallic foil.

R(b) — I Z2 |Wb) + BZ1 (b)]

where M and Z1 are mass and charge of ions in units of proton. l(b) is the range of an ideal proton with velocity bc. An ideal proton does not capture electrons from target atoms or undergo close collision with the target's nuclei. BZ1 (b) is the extension of range, in the same material, caused by the neutralization of incident ion's charge due to charge pick-up near the end of the trajectory.

In this formulation, dE/dx values are evaluated by taking the derivative of range expression (1) with respect to energy (E). Based on this formulation, dE/dx values of heavy ions in any solid materials can be computed in the energy range 0.1-1200 MeV/n.

— \J¡0Ewith dEwithout

where dEwith and SEwithout are the FWHM of energy spectra, which are obtained with and without metallic foils.

3. Energy loss and straggling formulations

The brief description of energy loss and straggling formulations, used in the present study, are given below.

Energy loss formulations

3.1.1. Benton and Henke formulation

Benton and Henke (1969) make use of Barkas and Berger (1964) method and modified their empirical relations of range-energy data and extended to low energy region. They expressed the range of ions in matter with the following expression:

3.1.2. Grande and Schiwietz formulation (CasP code) Grande and Schiwietz (2002, 2014) have developed a CasP code to calculate the impact parameter dependent energy loss of heavy ions in different target materials. In this code, energy loss of different heavy ions can be computed either through Perturbative Convolution Approximation (PCA) or more advanced Unitary Convolution Approximation (UCA). PCA is a first order perturbation theory and assumes the straight line trajectories of incident ions while passing through the target materials. While, latter approximation (UCA) is the extension of PCA model and incorporates the non-perturbative effects in Bloch theory. For dE/dx calculation of very light and very heavy ions, UCA model is recommended.

3.1.3. ICRU 49 report (ASTAR code)

ASTAR program is based on ICRU 49 report (1993) and provides dE/dx values of a-particles, in the energy region 1 keV-1000 MeV, for 25 elemental and 48 mixtures and

compounds. In this program, dE/dx values at low energies (1 keV-2 MeV) are evaluated adopting the fitting formulas based on the existing experimental data and for high energies (>2 MeV), Bethe stopping formula with various corrections (Shell, Barkas and Bloch, density effect) (Ziegler et al., 1985) is used.

3.1.4. Ziegler, Biersack and Littmark formulation (SRIM code) SRIM code is based on Ziegler et al. approach (1985) and adopting this code, one can compute dE/dx values for incident ions with Z = 1-92 in all elemental and many complex materials, in the energy range 1 eV-2 GeV. For He ions, Ziegler et al. developed a Master dE/dx curve based on the existing measured values in different target materials and formulated the following fitted relations:

' High

(-dE/dx)l0w = At(E)A

(-dE/dx)Mgh = (As/E) ln (1 + A4/E + A5E). E is the energy of Helium ions in keV and Aa, A2, A3, A4 and A5 are the fitting coefficients. These relations are valid for energy range

1 keV-10 MeV (Ziegler, 1978).

3.2. Straggling theories

3.2.1. Bohr theory

Based on the assumptions that (i) atoms are randomly distributed in the target materials, (ii) energy of the incident ion is very high as compared to the energy of target electrons and (iii) energy transfer per collision is much smaller than the energy of the incident ions, Bohr (1948) derived the following expression for energy loss straggling:

— 4pZ2e4Z2Nx

where Z1 is the ion atomic number, Z2 is the target atomic number, e is the charge on the electron, N is the atomic density and x is the thickness of target material. According to expression (3), the ion is fully stripped. So, the theory is valid for high energy region only.

3.2.2. Lindhard and Scharff theory

Lindhard and Scharff (1953) introduced some correction factors in Bohr's straggling expression, in order to extend for low and medium energy region. They gave the following expression:

= 0.5Qlohr [1.36y12 - 0.016y

where y = v2/v0Z2, v is the velocity of incident ions and v0 is the Bohr's velocity. The theory is applicable for y < 3.

3.2.3. Bethe—Livingston theory

By employing Born approximation, Livingston and Bethe (BL) (1937) derived the following expression for energy loss straggling:

Z2 + Z2

4 IiZi l 2mev2

^ 3 mv n

where Z' is the effective number of target electrons, Ii is the mean excitation energy of Zi electrons in ith shell of target

atom and me is the mass of electron. According to this theory, only those target electrons, which satisfy the condition, 2mev2 > Ii, are involved in the interaction process. In the present calculations, Ii values are modified adopting Comfort et al. (1966) approach.

3.2.4. Titeica theory

Incorporating Bloch correction (1933) and average kinetic energy (EKin) per electron of target material, Titeica (1939) developed following straggling expression:

1 4 EKin

3 mev2

. 2mev2

ln—+ j(1)-

Reji 1 + i

where I is mean excitation energy of target material, j is logarithmic derivative of gamma function and Rej is its real part of j. For calculations of Bloch correction, Bichsel's (1990) empirical parameterization is used.

3.2.5. Yang et al. formulation

By introducing effective charge (g) in Chu's (1976) straggling (Uchu) and adding the contribution (DU2on) due to correlation effects, Yang, O'Connor, and Wang (1991) derived the following expression for straggling calculations:

Q2 - Q2

U Yang _ QBohr

T2(Z!, Z2, V)

Q2 QChu Q2

The empirical formulas and related constants required in the expression are available (Yang et al., 1991).

Results and discussion

4.1. Measured dE/dx values and their comparison with theoretical formulations

The measured dE/dx values of a-particles, in energy range ~1.5—5.0 MeV, in Ag and Sn metallic foils are given in Table 1 and presented in Fig. 2. In order to compare these measured values with the predicted values of theoretical formulations

Table 1 - Measured dE/dx values of Sn metallic foils. a-particles in Ag and

E (MeV) (dE/dx)Ag (MeV mg-1 cm2) (dE/dx)sn (MeV mg-1 cm2)

1.50 0.579 0.562

1.75 0.553 0.536

2.00 0.529 0.511

2.25 0.507 0.489

2.50 0.486 0.469

2.75 0.468 0.45

3.00 0.450 0.433

3.25 0.434 0.417

3.50 0.419 0.402

3.75 0.405 0.388

4.00 0.392 0.375

4.25 0.380 0.363

4.50 0.368 0.352

4.75 0.358 0.341

5.00 0.347 0.331

Q2 Q2 QTit — QBohr

Q2 Q2 QBl — QBohr

Table 2 - Measured energy loss straggling values of a-particles as a function of fractional energy loss (DE/E), in Ag and Sn metallic foils.

a 0.00

- 0.80

Ф Measured Vaines

----- BH

---CmP

- - A8TAR ..........SHIM

•V.N

Measured Values

Energy (MeV)

Fig. 2 - Comparison of measured and predicted dE/dx values, for a-particles in Ag and Sn metallic foils, as a function of energy.

(Benton and Henke (BH), Grande and Schiwietz (CasP-5.2), ICRU-49 report (ASTAR), Ziegler et al. (SRIM-2008.04)), the corresponding computed values are also appended in Fig. 2. Through this comparison, it is clearly observed that the predicted values based on presently considered formulations show good agreement with the measured values except for very few exceptions.

4.2. Measured straggling values and their comparison with theoretical formulations

Ag Sn

(ДБ/Б) % (dE)Measured (keV) (ДБ/Б) % (dE)Measured (keV)

11 164 ± 12 24 236 ± 16

18 197 ± 14 26 314 ± 19

24 347 ± 20 43 621 ± 27

38 389 ± 19 52 701 ± 27

51 700 ± 27 73 858 ± 32

73 972 ± 39 81 896 ± 34

I 1000

^ Measured Values Bohr

-----LS

------------ BL

Ы 800

▲ Measured Values

"T" 20

(ДЕ/Е) %

"T" 60

~~1 80

Fig. 3 - Comparison of measured and predicted energy loss straggling, for a-particles in Ag and Sn metallic foils, as a function of fractional energy loss.

Energy loss straggling for 5.486 MeV a-particles, as a function of fractional energy loss limits DE/E ~10-80%, in Ag and Sn metallic foils is given in Table 2 and presented in Fig. 3. It is observed that there is direct relationship between straggling

values and fractional energy loss. With the increase in fractional energy loss, straggling values also increase.

These measured values are compared with the corresponding computed values adopting Bohr, Lindhard & Scharff

1200 —1

£ aa aa

400 —

♦ Ag

▲ Sn

Best Fit

Energy Loss (MeV)

Fig. 4 - A linear relationship between measured energy loss straggling and energy loss of a-particles in Ag and Sn metallic foils.

(LS), Bethe-Livingston (BL), Yang et al. (Yang) and Titeica (Tit) formulations (Fig. 3) and it is noticed that computed straggling values, generally, underestimate the measured values.

In case of Ag metallic foil (Fig. 3), Bohr's predicted values are ~3.0-8.0 times lower than the measured values. The computed values based on Lindhard-Scharff theory show slightly lesser values as compared to Bohr's prediction and large deviations (~3.5-9.5 times) with the measured values. Bethe-Livingston theory (with presently modified Ii values) based computed values are slightly reducing the deviation and the predicted values are ~2.75-7.25 times lower than the measured values. The predicted values based on Titeica theory show better agreement with the measured values, as compared to other theories, and underestimate ~1.20-3.10 times.

Almost similar behavior has been observed when predicted values, based on different collisional straggling theories, have been compared with measured values in case of Sn metallic foil (Fig. 3). Here, predicted values depict the lower trend of ~3.15-7.15 times for Bohr theory, ~3.75-8.50 times for Lind-hard-Scharff theory, ~2.80-6.40 times for Bethe-Livingston theory and ~1.20-2.70 times for Titeica's theory.

Such large deviations, between predicted and measured straggling values, may be due to the reason that these theories only consider the collisional part of straggling. However, for thicker target, the ion is partially stripped, So, fluctuations in charge state of incident ions during interactions also contribute. Till now, only one semi-empirical straggling formulation is available (Yang et al.), which considers both collisional as well charge-exchange component of total straggling. The computed values based on this approach are also appended in Fig. 3 and through the comparison with measured values, the large deviation is also observed.

4.3. Relation between energy loss and straggling

Since energy loss and straggling are inter-related, so when we plot energy loss straggling (<5E) as a function of energy loss (DE) (Fig. 4), a linear relation is observed, through the following fitted expression:

<5E(keV) = 224.542DE(MeV)

Similar linear trends have also been noticed for a-particles and heavier ions in elemental and complex materials in our earlier publications (Diwan et al., 2007, 2012; Kumar et al., 2015).

Conclusions

Energy loss and straggling of a-particles, in Ag and Sn metallic foils are measured and compared with the predictions of theoretical formulations. As far as energy loss formulations are concerned, all the formulations considered in the present study show good agreement with the measured values while the prediction of straggling formulations underestimate the measured values. Further, Titeica theory's prediction show better agreement with the measured values, as compared to other considered theories, in both Ag and Sn metallic foils. The present study can be utilized as an input to modify the existing theoretical formulations and for ion beam based techniques to analyze the given materials.

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