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Evaluations of proton inelastic mean free paths for 12 elemental solids over the energy range from 0.05 to 10 MeV
Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Evaluations of proton inelastic mean free paths for 12 elemental solids over the energy range from 0.05 to 10 MeV Zhenyu Tan a,⇑, Yueyuan Xia b a b
School of Electrical Engineering, Shandong University, Jinan 250061, Shandong, PR China School of Physics, Shandong University, Jinan 250100, Shandong, PR China
a r t i c l e
i n f o
Article history: Received 1 November 2010 Received in revised form 15 November 2010 Available online 20 November 2010 Keywords: Proton Mean free path Higher-order correction Dielectric theory
a b s t r a c t The systematical calculations of the inelastic mean free paths (MFPs) of 0.05–10 MeV protons in 12 elemental solids (Al, Si, Ni, Cu, Mo, Rh, Ag, W, Os, Ir, Pt, Au) have been performed. The calculations are based on the algorithm derived from Ashley’s optical-data model including the higher-order corrections to stopping power (SP) for protons. The prominence and necessity of the higher-order corrections are demonstrated by calculating the proton SPs for the 12 solids using Ashley’s optical-data model and by comparing the calculated SPs with the experimental results, the tabulated values and other corresponding theoretical evaluations. The algorithm of evaluating the proton inelastic MFP is described. In this algorithm, the Barkas-effect correction and the Bloch correction are taken into account, the minimum impact parameter from Lindhard is used in the Barkas-effect correction, and an empirical estimation of a free parameter involved in the Bloch correction to the inelastic MFP is proposed. The evaluated inelastic MFPs of 0.05–10 MeV protons for the 12 solids under two different cases, i.e. the higher-order corrections not being considered and the Barkas-effect correction and the Bloch correction being included, are presented in the tabulated form and are first results for these solids. These numerical results provide an alternative basic data for the Monte Carlo studies on low-energy proton transport in these 12 solids. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Inelastic mean free path (MFP) is an important characteristic quantity to describe the inelastic interactions of charged particle with material. Especially, the inelastic MFPs are of importance for Monte Carlo track-structure calculations of charged particles in materials, where their values should be required as the basic input data [1]. Protons are very important radiation source in many fields of application and research. Presently, there are over 25 ion therapy facilities throughout the world and most of these facilities use proton beams [2]. The advantage of using proton beams for cancer therapy is due to their depth dose curve with the characteristic Bragg peak profile, which leads to a good potential to spare healthy tissues [3,4]. In space radiation, the radiation effects due to protons require to be estimated [5]. Also, in recent years proton beam writing has been successfully applied for micromachining and the fabrication of nanostructures [6–8] because of its excellent properties of producing deeper structures with high aspect-ratios and of exhibiting low proximity effects. In the fields mentioned above, Monte Carlo track-structure calculations of protons in the related materials, such as bioorganic compounds, polymers,
⇑ Corresponding author. Tel.: +86 0531 88392806; fax: +86 0531 81696107. E-mail address: [email protected] (Z. Tan). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.11.040
semiconductor and metal materials, represent a useful tool for the investigations of corresponding radiation effects and for the simulations of the lithography in the fabrication of microelectronics devices and in micromachining. Therefore, knowledge of the values of the inelastic MFPs of protons in the above materials is very important. To this end, in our previous work [9] we reported the evaluated inelastic MFPs of 0.05–10 MeV protons in a group of important bioorganic materials. Here, we present systematical calculations of the inelastic MFPs of 0.05–10 MeV protons in 12 elemental solids (Al, Si, Ni, Cu, Mo, Rh, Ag, W, Os, Ir, Pt, Au). The calculations are based on the approach similar to that in our previous work [9], but our emphasis is on the higher-order corrections for the calculations of proton inelastic MFPs. We check the internal consistency of the experimental optical data used in calculations of proton inelastic MFPs by means of two sum rules and by calculating mean excitation energy. We demonstrate prominence and necessity of the higher-order corrections by calculating the proton SPs for the 12 solids using Ashley’s optical-data model and by comparing the calculated SPs with the experimental results [10] and the tabulated values [11,12]. We describe the algorithm of evaluating proton inelastic MFP, including the determination of the minimum impact parameter involved in the Barkas-effect correction and an empirical estimation of a free parameter in new Bloch correction to the inelastic MFP. Using the described algorithm, we evaluate the inelastic MFPs of
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
0.05–10 MeV protons for the 12 solids under two different cases, i.e. the higher-order corrections not being considered and the Barkas-effect correction and the Bloch correction being included, and present these data in the tabulated form. 2. Calculation method 2.1. Higher-order corrections in calculation of proton stopping power In this work, the method of evaluating proton inelastic MFPs originates from Ashley’s optical-data model [13] used to calculate the stopping power of the medium for protons. Ashley’s model are based on experimental optical data (the refractive index and extinction coefficients) and takes into account two higher-order corrections, i.e. the Barkas-effect correction concerned with a choice of the minimum impact parameter and the Bloch correction. Due to this, we will first perform the systematical calculations of the proton SPs in the 12 solids under consideration for the purposes of the following: examining accuracy of the optical data for the 12 solids, choosing the minimum impact parameter for the Barkas-effect correction, and showing the necessity of making the higher-order corrections to proton inelastic MFPs. Therefore, we briefly describe the principles of calculating the SPs of a medium for a proton, and then do the systematical calculations and analyses on the proton SPs in the 12 solids. Ashley [13] presented the SP for a proton, including the Barkaseffect correction and the Bloch correction, as a sum of the following three terms:
SP ¼ SPBorn þ SPBarkas þ SPBloch ;
ð1Þ
where SPBorn is referred to as basic stopping power based on the first Born approximation, and SPBarkas and SPBloch correspond to the Barkas-effect correction and the Bloch correction, respectively. According to Ashley [13], the basic stopping power is calculated by
SPBorn
1 ¼ pa0 E0
Z
E0
ðhxÞIm½1=eðxÞv ðaÞdðhxÞ;
Z eff ¼
where E0 is the electron kinetic energy with the same velocity as that of the proton, a0 is the Bohr radius, ⁄x is the energy transfer, Im[1/e(x)] is the optical energy loss function (OELF), and v(a) is
SPBarkas ¼
1 pa0 E0
4E
Peff ¼
ph X
2
Z
p
ðhxÞIm½1=eðxÞdðhxÞ;
ð7Þ
0
ðhxÞmax
ð1=hxÞIm½1=eðxÞdð hxÞ;
ð8Þ
0
ð3Þ
Material
Photon energy range (eV)
Al
0.04–10,000a 10,000–30,000b 1.0–49.59a 51.66–30,000b 0.1–50.0a 50.935–30,000b 0.13–50.0a 50.935–30,000b 0.1–50.0a 50.637–30,000b 0.1–2000a 2025.2–30,000b 0.125–50.0a 50.558–30,000b 0.05–47.0a 50.558–30,000 0.1–2000a 2025.2–30,000b 0.1–2000 a 2025.2–30,000b 0.1–2000 a 2025.2–30,000b 0.13–9919.0a 10044.0–30,000b
Si Ni
ð4Þ
Cu
with Mo
ð5Þ
Rh
where m is the mass of electron, I(n) is a tabulated function and can be approximated by a analytic function [13,14] for easy numerical evaluation. For the energy range considered in the present work, the SPBloch can be calculated by means of the following formula [15]:
SPBloch ¼
ðhxÞmax
Table 1 Optical data used in MFP calculations and the examination for these data (in this table, Error-f and Error-ps denote the errors in f-sum rule and ps-sum rule, respectively).
0
h h x L1 ðx; nÞ ¼ IðnÞ; 8ma0 E03=2
2 P
in which Peff should be equal to unit in the limit (⁄x)max ? 1.
0
ðhxÞdðhxÞIm½1=eðxÞL1 ðx; nÞ;
2
where XP = (4p ne2/m)1/2 with n the density of atoms or molecules. According to dielectric response theory, Zeff is expected to become Z, total number of electrons per atom or molecule, when (⁄x)max ? 1. Peff due to ps-sum rule is calculated by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with a = ⁄x/2E0 and s ¼ 1 2a. Following the principles described in Ashley [13] the SPBarkas is given by
Z
Z
2
ð2Þ
0
v ðaÞ ¼ ln½ð1 a þ sÞ=a;
pffiffiffi In Eq. (4), the parameter n is expressed as n ¼ 5:811ð hx= EÞa where a is a minimum impact parameter and ⁄x and E (proton kinetic energy) are in eV. There are, p inffiffiffi general, two choices for the parameter a. One is a ¼ 88:748= E given by Lindhard [16] pffiffiffiffiffiffiffi and the other is a ¼ 3:688= hx suggested by Jackson and McCarthy [17], here ⁄x and E are also in eV. After performing a detailed analyses on the effects of the two minimum impact parameters on the calculations of the proton SPs in aluminum, due to ppragmatic reasons Ashley chose the parameter a ¼ ffiffiffi 88:748= E for the calculations of the proton SPs in several other materials (C, Cu and polystyrene). pffiffiffiAs done later, we will test the rationality of using a ¼ 88:748= E in the proton SP calculations for the elemental solids under consideration. In addition, Eqs. (2) and (4) show that the calculations of both SPBorn and SPBarkas require deriving the OELF. Here, the OELFs for the 12 solids are obtained using the optical data indicated in Table 1. Commonly, the tests for the internal consistency of the optical data are made by means of the calculations of three parameters Zeff, Peff, and I. These three parameters result from f-sum rule, perfect screening (ps)-sum rule and mean excitation energy [20–22], respectively. The f-sum rule can be expressed an effective number Zeff of electrons per atom or molecule through the following formula:
Ag W Os Ir
ð h xP Þ 2 fy2 ½1:20206 y2 ð1:042 0:8549y2 þ 0:343y4 Þg; 2a0 E0 ð6Þ
where xP is the plasma angular frequency, y = a/b with a the fine structure constant and b the ratio of proton velocity to light velocity.
Pt Au a b
Ref. [18]. Ref. [19].
Error-ps (%)
I (eV) Calculated
Expected
2
1.5
165.3
166
1
12.5
170.1
173
4.2
2.3
316.1
311
2.2
0.6
327.5
322
5.2
0.4
410.6
424
8.2
428.7
449
2.4
10.6
377.4
470
1.8
10.2
586.7
727
2.18
10.3
609.7
746
0.2
16.6
601.2
757
3.82
10.6
669.8
790
3.1
8.9
662.3
790
Error-f (%)
7
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
30
20 SPBorn SP Tabulated values [11] Binary theory [24] Denton et al. [25]
(a) Al
20
SPs (eV/Å)
SPs (eV/Å)
16
SPBorn SP Fit to Expt. data [10] Denton et al. [25]
(b) Ni
12
8
10 4
0 10
2
10
3
10
0
4
10
2
10
3
10
4
proton energy (keV)
proton energy (keV) 32
30
(c) Cu
SPBorn SP Tabulated values [11] Fit to Expt. data [10]
(d) Ag 24
SPs (eV/Å)
SPs (eV/Å)
20
SPBorn SP Tabulated values [11] Fit to Expt. data [10] Denton et al. [25]
16
10 8
0 10
2
10
3
10
0
4
10
proton energy (keV)
SPs (eV/Å)
30
2
10
3
10
4
proton energy (keV)
(e) Au
SPBorn SP Tabulated values [11] Fit to Expt. data [10]
20
10
0 10
2
10
3
10
4
proton energy (keV) Fig. 1. Comparisons between the calculated stopping powers (SPBorns and SP) and those from the fit to experimental (Expt.) data [10], the tabulated values [11], and the evaluations of both Sigmund and Schinner [24] and Denton et al. [25] as the functions of proton energy for (a) Al, (b) Ni, (c) Cu, (d) Ag, and (e) Au.
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
The mean excitation energy I is defined as
ln I ¼
R1 0
ðhxÞIm½1=eðxÞ lnðhxÞdðhxÞ R1 : ðhxÞIm½1=eðxÞdðhxÞ 0
ð9Þ
Using Eqs. (7)–(9), for each elemental solid under consideration we evaluate Zeff, Peff and I for (⁄x)max typically equal to 30 keV and compare them with those expected from Eqs. (7) and (8) when (⁄x)max ? 1 and from ICRU Report 37 [23]. The errors of the calculated values of Zeff and Peff relative to the expected ones are listed in Table 1, whereas the calculated and expected Is are also displayed this table for comparison. From Table 1, we can see that for Al, Ni, Cu, and Mo the errors in the two sum rules are small and the respective I evaluated by Eq. (9) is close to the expected one within 3%. However, for other materials there are the errors in ps-sum rule larger than 8%. In addition, for Ag, W, Os, Ir, Pt, and Au the evaluated Is differ evidently from the expected ones. The primary reason for this may be that in the integrations of Eq. (9) 30 keV used as the upper limit instead of infinity is not large enough to include fully the contributions to I from K-shell excitations for these materials. Using the algorithm described above, we have calculated the proton SPs for the 12 solids over the energy range from 0.5 to 10 MeV. According to our calculations, it is shown that for the solids with the small errors of the optical data in the two sum rules, in energy range of E > 100 keV the proton SPs calculated by taking into account pffiffiffi the higher-order corrections with the use of a ¼ 88:748= E agree well with the experiments [10] and the tabulated values [11] and those without the higher-order corrections are obviously small when compared to the experiments [10] and the tabulated values [11]. However, for the solids with the large errors of the optical data in the ps-sumpffiffiffi rule and using the higher-order corrections with a ¼ 88:748= E the calculated proton SPs are in reasonable agreement with the experiments [10] or the tabulated values [11], but present obvious large values of magnitude for proton energies from 100 keV to 1 MeV. As the examples of our calculations, Fig. 1 illustrates the calculated proton SPs as a function of energy for Al, Ni, Cu, Ag, and Au whereas the tabulated values [11] and the fits to the experimental data [10] are also plotted in this figure for comparison. It is clear from Fig. 1(a)–(c) that for Al, Ni, and Cu the calculated proton SPs show a good agreement with the experiments [10] and with the tabulated values [11] at energies above 100 keV when the
8
parameter γ
6
higher-order corrections are applied and there are evident difference between the SPBorn and the experiments [10] or the tabulated values [11]. Noticing the fact that the optical data of Al, Ni, and Cu are of small errors in the two sum rules, it is therefore demonstrated that the higher-orderpcorrections with the minimum imffiffiffi pact parameter a ¼ 88:748= E for proton SP calculations are both evident and necessary. On the other hand, from Fig. 1(d) and (e), it can be seen that for Ag and Au the proton SPs accounting for the higher-order corrections are larger than the experiments [10] or the tabulated values [11] because of their large errors of the optical data in ps-sum rule, especially for Ag, for which its error, 10.6%, in ps-sum rule leads to its obviously large SPs in contrast with the experiments [10] and the tabulated values [11] at peak energy. Recently, two groups, Sigmund and Schinner [24] and Denton et al. [25], presented the calculations of the SPs of several elemental solids for H projectiles. For these calculations Sigmund and Schinner used the binary theory based on the modifications for the Bohr’s classical theory [26] and Denton et al. applied the dielectric formalism. In the binary theory due to Sigmund and Schinner [24], the charge states of the projectile inside the target, the influence of screening of the projectiles-target interaction by accompanying electrons, the shell correction and the Barkas and Bloch corrections are included. In dielectric model of Denton et al. [25], the charge states of the projectile inside the target, the effect of the polarization of the projectile and the SP due to the electronic capture and loss events are taken into account. The SPs of Al, Ni and Cu for H projectiles calculated by these two groups are also presented in Fig. 1 for comparison. For Al, it can be seen from Fig. 1(a) that our calculated SPs are in good agreement with those from Denton et al. [25] for energies above 100 keV and that the SPs evaluated by Sigmund and Schinner [24] are larger than our calculations and those of Denton et al. [25] in the energy range below 400 keV. For Ni and Cu, from Fig. 1(b) and (c) it is clear that our calculated SPs agree well with those of Denton et al. [25] at energies higher than 200 keV. Due to similar dielectric models used in the present work and in Denton et al. [25], the differences between our calculated SPs and those of Denton et al. [25] at low energies (<100 keV) may be mainly attributed to the reasons that the charge state effects of the projectile inside the target and the SPs due to the electronic capture and loss events are not included in the present work. However, as pointed out by Denton et al., for low projectile energies other models including the binary theory [25] could be used for the SP calculations for H projectiles. Based on the above calculations and analyses, for the evaluations of the proton inelastic MFPs to be done later we will take into account the higher-order corrections and will use the minimum pffiffiffi impact parameter a ¼ 88:748= E for the Barkas-effect correction. 2.2. Calculation principle of proton inelastic MFP In a manner analogous to that expressing the constituents of proton SP calculation in Eq. (1), the inelastic inverse mean free path (IMFP) for proton impact in a medium can be expressed as
4
IMFP ¼ IMFPBorn þ IMFPBarkas þ IMFPBloch ;
ð10Þ
here IMFPBorn denotes basic inverse mean free path based on the first Born approximation, and IMFPBarkas and IMFPBloch correspond to the Barkas-effect correction and the Bloch correction, respectively.
2
0 0
30
60
90
atomic number Z Fig. 2. Estimated parameter c used in the Bloch correction to proton inelastic MFPs for 12 elemental solids.
Table 2 Determined parameter cs for 12 elements. Element
Al
Si
Ni
Cu
Mo
Rh
Ag
W
Os
Ir
Pt
Au
c
3.0
2.9
5.1
5.3
3.6
3.5
3.9
4.8
5.0
4.8
5.5
4.9
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
From the evaluation principle for the SPBorn due to Ashley [13], IMFPBorn can be derived and the resultant expression is written as
10
0
IMFPBarkas ¼
RSP, RMFP
Z
E0
Im½1=eðxÞv ðaÞdðhxÞ:
ð11Þ
0
1 pa0 E0
Z
4E0
dðhxÞIm½1=eðxÞL1 ðx; nÞ:
ð12Þ
0
In Eq. (12), for the minimum pffiffiffi impact parameter involved in L1(x; n) we adopt a ¼ 88:748= E. Unfortunately, there is no available theoretical expression of evaluating the IMFPBolch corresponding to the SPBloch. Therefore, in our previous work [9] and on the basis of the analyses on the SPBloch due to Bloch [27], we proposed an empirical Bloch correction to inelastic MFP. This correction takes the following form
-1
IMFPBloch ¼
RSP RMFP
10
1 2pa0 E0
Similarly, according to Ashley’s principle of describing the Barkas-effect correction for the SP calculation [13], the formula of evaluating IMFPBarkas can be obtained as
Al (a)
10
IMFPBorn ¼
Bðhx; E0 Þ ¼ 200
Z
4E0
p 2
0
ðhxP Þ2 Bðhx; E0 ÞdðhxÞ;
ð13Þ
with
-2
100
1 pa0 E0
300
400
1 cðhxÞc2 X
ð4E0 Þc
j¼1
500
1 jð4Aj E0 þ 1Þ1c ðAj hx þ 1Þ1þc
ð14Þ
;
2 2
where Aj ¼ j h =ð2Z 21 e4 mÞ and c is a free parameter.
proton energy (keV)
4000
10
4
(a) Cu
Al (b)
10
10
10
IMFP (1/μm)
IMFPs (1/μm)
3000
2
IMFPBorn IMFPBarkas − IMFPBloch
0
IMFP IMFPBorn
2000
1000
-2
0 10
2
10
3
10
10
4
2
10
3
10
4
proton energy (keV)
proton energy (keV) 4000
6000
(b) Au
IMFP IMFPBorn
4000
IMFP IMFPBorn
3000
IMFP (1/μm)
IMFPs (1/μm)
Al (c)
2000
2000
1000
0
0 10
2
10
3
10
4
proton energy (keV) Fig. 3. Comparisons between (a) the RSP and RIMFP and between (b and c) the various IMFPs as the functions of proton energy for Al.
10
2
10
3
10
4
proton energy (keV) Fig. 4. Comparisons between the IMFP and IMFPBorn as the functions of proton energy for (a) Cu and (b) Au.
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335 Table 3 MFPBorn (in unit Å) for 12 elemental solids as a function of proton energy E (in unit keV). E
Al
Si
Ni
Cu
Mo
Rh
Ag
W
Os
Ir
Pt
Au
50 60 70 80 90 100 150 200 300 400 500 600 700 800 900 1000 1500 2000 3000 4000 5000 6000 7000 8000 9000 10,000
2.472 2.532 2.649 2.789 2.939 3.095 3.882 4.583 5.848 7.042 8.193 9.313 10.41 11.48 12.53 13.57 18.56 23.31 32.35 40.96 49.28 57.38 65.31 73.10 80.77 88.33
2.920 2.935 3.041 3.182 3.340 3.506 4.374 5.226 6.704 8.061 9.370 10.65 11.89 13.12 14.32 15.50 21.19 26.61 36.91 46.74 56.23 65.47 74.52 83.40 92.14 100.80
4.416 3.935 3.667 3.534 3.484 3.485 3.763 4.092 4.850 5.634 6.415 7.187 7.947 8.696 9.434 10.16 13.66 16.97 23.22 29.17 34.91 40.49 45.94 51.28 56.54 61.72
4.356 3.940 3.757 3.685 3.670 3.683 3.938 4.313 5.117 5.932 6.748 7.554 8.349 9.132 9.904 10.66 14.33 17.80 24.35 30.59 36.60 42.44 48.15 53.75 59.26 64.68
5.077 4.182 3.756 3.493 3.354 3.257 3.220 3.592 4.459 5.326 6.166 6.965 7.731 8.473 9.196 9.906 13.30 16.53 22.67 28.52 34.16 39.63 44.99 50.23 55.40 60.48
5.197 4.239 3.640 3.351 3.231 3.163 3.076 3.311 4.039 4.789 5.524 6.242 6.940 7.612 8.263 8.900 11.93 14.81 20.28 25.48 30.50 35.38 40.15 44.82 49.42 53.95
4.914 4.367 3.979 3.718 3.539 3.424 3.300 3.470 4.163 4.915 5.655 6.378 7.086 7.776 8.445 9.098 12.21 15.15 20.74 26.06 31.20 36.19 41.07 45.85 50.55 55.18
5.790 4.268 3.701 3.353 3.126 2.999 2.894 3.151 3.790 4.430 5.048 5.648 6.232 6.804 7.364 7.915 10.56 13.07 17.83 22.36 26.71 30.94 35.06 39.11 43.09 47.01
6.591 4.928 4.026 3.61 3.372 3.222 3.015 3.17 3.735 4.332 4.916 5.485 6.039 6.581 7.113 7.636 10.14 12.52 17.04 21.34 25.47 29.48 33.39 37.23 41.00 44.71
6.252 4.799 3.874 3.447 3.209 3.060 2.851 2.997 3.542 4.119 4.685 5.234 5.766 6.287 6.798 7.300 9.705 11.99 16.32 20.45 24.41 28.26 32.01 35.69 39.31 42.88
5.531 4.441 3.724 3.374 3.208 3.120 3.084 3.318 3.982 4.661 5.316 5.948 6.562 7.159 7.744 8.318 11.070 13.60 18.63 23.34 27.87 32.27 36.57 40.78 44.91 48.99
5.414 4.665 4.079 3.687 3.441 3.299 3.176 3.397 4.064 4.769 5.452 6.112 6.753 7.378 7.988 8.585 11.45 14.17 19.32 24.22 28.94 33.52 38.00 42.38 46.69 50.93
The free parameter c in Eq. (14) needs to be determined for the solids under consideration. However, to our knowledge, no available experimental data of inelastic cross-sections of protons in the considered solids can be used for estimating the parameter c. Thus, here we propose the following empirical approach for determining the parameter c. Defining RSP = (SPBarkas + SPBloch)/SPBorn as percentage correction for SPBorn and RIMFP = (IMFPBarkas + IMFPBloch)/IMFPBorn as that for IMFPBorn, for each considered solid we determine the parameter c so that the RIMFP is approximately equal to the RSP at their respective peak energy. According to this, we determine the values of the parameter c for the 12 solids, and list these values in Table 2. Particularly, the determined c versus
atomic number Z is illustrated in Fig. 2 in order to clearly show the c dependence on atomic number Z. It can be seen from this figure that for the solids with close atomic number their values of c are close for each other. As an example of our calculations, Fig. 3 presents various proton inelastic IMFPs calculated using the principles described above and the comparison between RSPs and RIMFPs as a function of proton energy for Al. From Fig. 3(a), it can be observed that at energies below 500 keV the percentage corrections for IMFPBorn and SPBorn become obvious, bringing about a correction larger than 30% at respective peak energy, which indicates clearly the importance of the higher-order corrections at low energies. Fig. 3(b) displays the comparison between various
Table 4 MFP (in unit Å) for 12 elemental solids as a function of proton energy E (in unit keV). E
Al
Si
Ni
Cu
Mo
Rh
Ag
W
Os
Ir
Pt
Au
50 60 70 80 90 100 150 200 300 400 500 600 700 800 900 1000 1500 2000 3000 4000 5000 6000 7000 8000 9000 10,000
2.039 2.003 2.080 2.200 2.339 2.487 3.268 3.998 5.334 6.584 7.779 8.934 10.06 11.15 12.23 13.28 18.33 23.12 32.20 40.84 49.18 57.30 65.24 73.04 80.71 88.28
2.452 2.354 2.420 2.542 2.691 2.853 3.723 4.591 6.137 7.555 8.913 10.23 11.51 12.75 13.98 15.18 20.94 26.40 36.75 46.61 56.12 65.38 74.44 83.33 92.08 100.70
8.418 3.808 2.991 2.715 2.618 2.820 3.221 4.107 4.983 5.830 6.651 7.451 8.232 8.996 9.746 13.33 16.69 23.00 28.98 34.75 40.34 45.81 51.16 56.43 61.62
8.029 3.906 3.126 2.858 2.762 2.948 3.383 4.316 5.230 6.115 6.975 7.812 8.629 9.430 10.21 13.96 17.49 24.11 30.38 36.42 42.28 48.01 53.62 59.14 64.57
8.231 4.313 3.432 3.072 2.837 3.200 4.087 4.976 5.837 6.655 7.439 8.196 8.934 9.656 13.10 16.36 22.54 28.41 34.06 39.55 44.91 50.17 55.34 60.43
6.159 4.021 3.349 2.764 2.978 3.719 4.491 5.245 5.978 6.690 7.375 8.038 8.685 11.76 14.66 20.16 25.39 30.42 35.32 40.09 44.77 49.37 53.91
3.506 2.917 2.676 2.587 2.866 3.655 4.463 5.245 5.999 6.732 7.443 8.131 8.800 11.97 14.95 20.58 25.93 31.08 36.09 40.98 45.76 50.47 55.11
6.606 3.818 3.060 2.473 2.692 3.346 4.016 4.664 5.289 5.896 6.487 7.064 7.629 10.32 12.87 17.67 22.22 26.59 30.83 34.97 39.02 43.01 46.94
20.33 5.433 3.750 2.590 2.693 3.269 3.897 4.511 5.106 5.684 6.246 6.796 7.333 9.894 12.31 16.87 21.19 25.34 29.36 33.29 37.13 40.91 44.63
13.70 4.775 3.418 2.437 2.548 3.109 3.715 4.309 4.882 5.436 5.976 6.503 7.019 9.474 11.79 16.17 20.31 24.29 28.15 31.92 35.61 39.23 42.80
9.193 4.521 3.478 2.69 2.863 3.523 4.224 4.902 5.558 6.192 6.808 7.409 7.998 10.80 13.44 18.44 23.18 27.73 32.14 36.45 40.67 44.82 48.90
12.04 5.057 3.747 2.796 2.960 3.631 4.360 5.067 5.749 6.410 7.053 7.678 8.289 11.21 13.96 19.15 24.08 28.82 33.41 37.89 42.28 46.60 50.85
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
MFP (Å)
10
found elsewhere [28,29]. This approximate dispersion relation may produce the uncertainty for our calculated MFPs. Third, as shown earlier, the optical data with less accuracy used in this work result in the obvious difference in the calculated SPs compared with the tabulated values [11], and hence they also induce uncertainty in the calculations of the MFPs. Finally, the parameter c involved in the Bloch correction to proton inelastic MFPs is also a source of uncertainty due to the assumption that the RIMFP is empirically considered to be close the RSP at low energies.
2
Al Cu Ag Au 10
1
4. Summary
10
0
10
2
10
3
10
4
proton energy (keV) Fig. 5. Mean free path (MFP) results as a function of proton energy for four typical elemental solids Al, Cu, Ag, and Au.
IMFPs and it is clear that the correction IMFPBarkas + IMFPBloch becomes negligible at high energies. In Fig. 3(c) the IMFPs with the higher-order corrections are compared particularly with the IMFPBorns, notably showing the effect of the higher-order corrections at low energies. Fig. 4 plots the IMFPs and IMFPBorns versus energy for Cu and Au, respectively. From Fig. 4, the corrections to the IMFPBorns of protons in Au are small when compared to those for Cu, and the peak energy of the IMFP for Au presents a small shift toward high energy relative to Cu.
3. Proton inelastic mean free paths in 12 elemental solids Using the method of evaluating the proton inelastic MFPs described in this work and under two different cases, i.e. the higher-order corrections not being considered, and the Barkas-effect correction and the Bloch correction being included, we have performed the systematical calculations of the proton inelastic MFPs for the 12 solids in energy range from 0.05 to 10 MeV. The resultant MFPBorns (MFPBorn = 1/IMFPBorn) are listed at selected energies in numerical form in Table 3. Under the second case, the calculated results for the MFPs (MFP = 1/IMFP) including the higher-order corrections are presented in Table 4. From Tables 3 and 4, for these 12 solids the MFP dependences on energy are very similar from solid to solid. Fig. 5 presents the calculated proton inelastic MFPs as a function of energy for typical solids Al, Cu, Ag, and Au. From Fig. 5, the MFP dependence on energy from Al to Au is clearly shown, the MFP decrease with atomic number Z at given energy, and the energy corresponding to the minimum MFP shifts toward high energy with increasing atomic number Z. It should be pointed out that at energies lower than about 100 keV, similar to that stated by Ashley [13], the two correction terms begin to dominate the behavior of the IMFP and thus the validity of this approach is questionable. Hence, in the present work all calculated proton inelastic MFPs at energies below 100 keV may be considered only as rough data. In addition, the sources of uncertainty for our calculated MFPs may arise from the following factors. First, our MFP calculations are based on the Born approximation which is not reliable at low energies. Second, in the optical-data model of Ashley [13] the dispersion relation in a simple quadratic form was adopted because of the scarcity of exact theory for solving the problem of extending the OELF into the area of momentum q > 0, on which the detailed discussions can be
In this work the evaluations of the inelastic MFPs of 0.05–10 MeV protons in 12 elemental solids (Al, Si, Ni, Cu, Mo, Rh, Ag, W, Os, Ir, Pt, Au) have been performed, based on the algorithm derived from Ashley’s optical-data model of calculating the proton SPs. The internal consistency of the optical data for the 12 solids were checked by using f-sum and ps-sum rules and by calculating the mean excitation energy, showing a higher accuracy of the optical data for Al, Ni, Cu, and Mo, and showing large errors of the optical data in ps-sum rule for other solids. By systematically calculating the proton SPs for these 12 solids, the rationality of the minimum impact parameter used in the Barkas-effect correction were examined and the influences of the used optical data on the proton SP calculations were tested, leading to the conclusion that using accurate optical data and choosing the minimum impact parameter by Lindhard [16] for the Barkas-effect correction, the higher-order corrections should be taken into account and the calculated proton SPs are in satisfactory agreement with the experiments [10] and the tabulated values [11]. Following the above, the algorithm of evaluating the proton inelastic MFPs were described. In this algorithm, the higher-order corrections are accounted for and the empirical estimation of free parameter c involved in the Bloch correction to the inelastic MFP is given. Under two different cases, i.e. the higher-order corrections not being considered, and the Barkas-effect correction and the Bloch correction being included, the evaluated inelastic MFPs of 0.05–10 MeV protons in the 12 solids were presented in numerical form. These data are first results for the 12 solids and are useful for the Monte Carlo studies on low-energy proton transport in the considered 12 solids. In addition, the experimental study of determining inelastic MFP of protons in these 12 solids should be expected in the future, so as to further complete the algorithm for the calculations of proton inelastic MFPs. Acknowledgment This work was supported partly by the Natural Science Foundation of Shandong Province of China under Grant No. ZR2009DZ006. References [1] H. Nikjoo, S. Uehara, D. Emfietzoglou, F.A. Cucinotta, Radiat. Meas. 41 (2006) 1052. [2] J. Sisterson, Nucl. Instrum. Meth. B 241 (2005) 713. [3] H. Nikjoo, S. Uehara, D. Emfietzoglou, A. Brahme, New J. Phys. 10 (2008) 075006. [4] A. Brahme, Int. J. Radiat. Oncol. Biol. Phys. 58 (2004) 603. [5] E. Grossman, I. Gouzman, Nucl. Instrum. Meth. B 208 (2003) 48. [6] V. Auzelyte, M. Elfman, P. Kristiansson, C. Nilsson, J. Pallon, N.A. Marrero, M. Wegde’n, Microelectronic Eng. 83 (2006) 2015. [7] C.N.B. Udalagama, A.A. Bettiol, F. Watt, Nucl. Instrum. Meth. B 260 (2007) 384. [8] F. Menzel, D. Spemann, J. Lenzner, W. Bölmann, G. Zimmermann, T. Butz, Nucl. Instrum. Meth. B 267 (2009) 2321. [9] Zhenyu Tan, Yueyuan Xia, Mingwen Zhao, Xiangdong Liu, Nucl. Instrum. Meth. B 268 (2010) 2337. [10] H.H. Andersen, J.F. Ziegler, Hydrogen Stopping Powers and Ranges in All Elements, Pergamon, New York, 1977. [11] International Commission on Radiation Units and Measurements, ICRU Report 49, Bethesda, MD, 1993.
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Available from:. J.C. Ashley, J. Phys.: Condens. Matter. 3 (1991) 2741. G. Maynard, C. Deutsch, J. Phys. 43 (1982) L223. H. Bichsel, L.E. Porter, Phys. Rev. A 25 (1982) 2499. J. Lindhard, Nucl. Instrum. Meth. B 2 (1976) 1. J.D. Jackson, R.L. McCarthy, Phys. Rev. B 6 (1972) 4131. E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, New York, 1985. [19] B.L. Henke, E.M. Gullikson, J.C. Dives, Atomic Data Nucl. Data Tables 54 (1993) 181. [20] S. Tanuma, C.J. Powell, D.R. Penn, Surf. Interface Anal. 11 (1988) 577.
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[21] E. Shiles, T. Sasaki, M. Inokuti, D.Y. Smith, Phys. Rev. B 22 (1980) 1612. [22] S. Tanuma, C.J. Powell, D.R. Penn, J. Appl. Phys. 103 (2008) 063707. [23] International Commission on Radiation Units and Measurements, ICRU Report 37, Bethesda, MD, 1984. [24] P. Sigmund, A. Schinner, Nucl. Instrum. Meth. B 193 (2002) 49. [25] C.D. Denton, I. Abril1, J.C. Moreno-Marín, S. Heredia-Avalos, R. Garcia-Molina, Phys. Stat. Sol. (B) 245 (2008) 1498. [26] N. Bohr, Philos. Mag. 25 (1913) 10. [27] F. Bloch, Ann. Phys. (Leipzig) 16 (1933) 285. [28] D. Emfietzoglou, Radiat. Phys. Chem. 66 (2003) 373. [29] D. Emfietzoglou, H. Nikjoo, Radiat. Res. 163 (2005) 98.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Evaluations of proton inelastic mean free paths for 12 elemental solids over the energy range from 0.05 to 10 MeV Zhenyu Tan a,⇑, Yueyuan Xia b a b
School of Electrical Engineering, Shandong University, Jinan 250061, Shandong, PR China School of Physics, Shandong University, Jinan 250100, Shandong, PR China
a r t i c l e
i n f o
Article history: Received 1 November 2010 Received in revised form 15 November 2010 Available online 20 November 2010 Keywords: Proton Mean free path Higher-order correction Dielectric theory
a b s t r a c t The systematical calculations of the inelastic mean free paths (MFPs) of 0.05–10 MeV protons in 12 elemental solids (Al, Si, Ni, Cu, Mo, Rh, Ag, W, Os, Ir, Pt, Au) have been performed. The calculations are based on the algorithm derived from Ashley’s optical-data model including the higher-order corrections to stopping power (SP) for protons. The prominence and necessity of the higher-order corrections are demonstrated by calculating the proton SPs for the 12 solids using Ashley’s optical-data model and by comparing the calculated SPs with the experimental results, the tabulated values and other corresponding theoretical evaluations. The algorithm of evaluating the proton inelastic MFP is described. In this algorithm, the Barkas-effect correction and the Bloch correction are taken into account, the minimum impact parameter from Lindhard is used in the Barkas-effect correction, and an empirical estimation of a free parameter involved in the Bloch correction to the inelastic MFP is proposed. The evaluated inelastic MFPs of 0.05–10 MeV protons for the 12 solids under two different cases, i.e. the higher-order corrections not being considered and the Barkas-effect correction and the Bloch correction being included, are presented in the tabulated form and are first results for these solids. These numerical results provide an alternative basic data for the Monte Carlo studies on low-energy proton transport in these 12 solids. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Inelastic mean free path (MFP) is an important characteristic quantity to describe the inelastic interactions of charged particle with material. Especially, the inelastic MFPs are of importance for Monte Carlo track-structure calculations of charged particles in materials, where their values should be required as the basic input data [1]. Protons are very important radiation source in many fields of application and research. Presently, there are over 25 ion therapy facilities throughout the world and most of these facilities use proton beams [2]. The advantage of using proton beams for cancer therapy is due to their depth dose curve with the characteristic Bragg peak profile, which leads to a good potential to spare healthy tissues [3,4]. In space radiation, the radiation effects due to protons require to be estimated [5]. Also, in recent years proton beam writing has been successfully applied for micromachining and the fabrication of nanostructures [6–8] because of its excellent properties of producing deeper structures with high aspect-ratios and of exhibiting low proximity effects. In the fields mentioned above, Monte Carlo track-structure calculations of protons in the related materials, such as bioorganic compounds, polymers,
⇑ Corresponding author. Tel.: +86 0531 88392806; fax: +86 0531 81696107. E-mail address: [email protected] (Z. Tan). 0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2010.11.040
semiconductor and metal materials, represent a useful tool for the investigations of corresponding radiation effects and for the simulations of the lithography in the fabrication of microelectronics devices and in micromachining. Therefore, knowledge of the values of the inelastic MFPs of protons in the above materials is very important. To this end, in our previous work [9] we reported the evaluated inelastic MFPs of 0.05–10 MeV protons in a group of important bioorganic materials. Here, we present systematical calculations of the inelastic MFPs of 0.05–10 MeV protons in 12 elemental solids (Al, Si, Ni, Cu, Mo, Rh, Ag, W, Os, Ir, Pt, Au). The calculations are based on the approach similar to that in our previous work [9], but our emphasis is on the higher-order corrections for the calculations of proton inelastic MFPs. We check the internal consistency of the experimental optical data used in calculations of proton inelastic MFPs by means of two sum rules and by calculating mean excitation energy. We demonstrate prominence and necessity of the higher-order corrections by calculating the proton SPs for the 12 solids using Ashley’s optical-data model and by comparing the calculated SPs with the experimental results [10] and the tabulated values [11,12]. We describe the algorithm of evaluating proton inelastic MFP, including the determination of the minimum impact parameter involved in the Barkas-effect correction and an empirical estimation of a free parameter in new Bloch correction to the inelastic MFP. Using the described algorithm, we evaluate the inelastic MFPs of
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
0.05–10 MeV protons for the 12 solids under two different cases, i.e. the higher-order corrections not being considered and the Barkas-effect correction and the Bloch correction being included, and present these data in the tabulated form. 2. Calculation method 2.1. Higher-order corrections in calculation of proton stopping power In this work, the method of evaluating proton inelastic MFPs originates from Ashley’s optical-data model [13] used to calculate the stopping power of the medium for protons. Ashley’s model are based on experimental optical data (the refractive index and extinction coefficients) and takes into account two higher-order corrections, i.e. the Barkas-effect correction concerned with a choice of the minimum impact parameter and the Bloch correction. Due to this, we will first perform the systematical calculations of the proton SPs in the 12 solids under consideration for the purposes of the following: examining accuracy of the optical data for the 12 solids, choosing the minimum impact parameter for the Barkas-effect correction, and showing the necessity of making the higher-order corrections to proton inelastic MFPs. Therefore, we briefly describe the principles of calculating the SPs of a medium for a proton, and then do the systematical calculations and analyses on the proton SPs in the 12 solids. Ashley [13] presented the SP for a proton, including the Barkaseffect correction and the Bloch correction, as a sum of the following three terms:
SP ¼ SPBorn þ SPBarkas þ SPBloch ;
ð1Þ
where SPBorn is referred to as basic stopping power based on the first Born approximation, and SPBarkas and SPBloch correspond to the Barkas-effect correction and the Bloch correction, respectively. According to Ashley [13], the basic stopping power is calculated by
SPBorn
1 ¼ pa0 E0
Z
E0
ðhxÞIm½1=eðxÞv ðaÞdðhxÞ;
Z eff ¼
where E0 is the electron kinetic energy with the same velocity as that of the proton, a0 is the Bohr radius, ⁄x is the energy transfer, Im[1/e(x)] is the optical energy loss function (OELF), and v(a) is
SPBarkas ¼
1 pa0 E0
4E
Peff ¼
ph X
2
Z
p
ðhxÞIm½1=eðxÞdðhxÞ;
ð7Þ
0
ðhxÞmax
ð1=hxÞIm½1=eðxÞdð hxÞ;
ð8Þ
0
ð3Þ
Material
Photon energy range (eV)
Al
0.04–10,000a 10,000–30,000b 1.0–49.59a 51.66–30,000b 0.1–50.0a 50.935–30,000b 0.13–50.0a 50.935–30,000b 0.1–50.0a 50.637–30,000b 0.1–2000a 2025.2–30,000b 0.125–50.0a 50.558–30,000b 0.05–47.0a 50.558–30,000 0.1–2000a 2025.2–30,000b 0.1–2000 a 2025.2–30,000b 0.1–2000 a 2025.2–30,000b 0.13–9919.0a 10044.0–30,000b
Si Ni
ð4Þ
Cu
with Mo
ð5Þ
Rh
where m is the mass of electron, I(n) is a tabulated function and can be approximated by a analytic function [13,14] for easy numerical evaluation. For the energy range considered in the present work, the SPBloch can be calculated by means of the following formula [15]:
SPBloch ¼
ðhxÞmax
Table 1 Optical data used in MFP calculations and the examination for these data (in this table, Error-f and Error-ps denote the errors in f-sum rule and ps-sum rule, respectively).
0
h h x L1 ðx; nÞ ¼ IðnÞ; 8ma0 E03=2
2 P
in which Peff should be equal to unit in the limit (⁄x)max ? 1.
0
ðhxÞdðhxÞIm½1=eðxÞL1 ðx; nÞ;
2
where XP = (4p ne2/m)1/2 with n the density of atoms or molecules. According to dielectric response theory, Zeff is expected to become Z, total number of electrons per atom or molecule, when (⁄x)max ? 1. Peff due to ps-sum rule is calculated by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with a = ⁄x/2E0 and s ¼ 1 2a. Following the principles described in Ashley [13] the SPBarkas is given by
Z
Z
2
ð2Þ
0
v ðaÞ ¼ ln½ð1 a þ sÞ=a;
pffiffiffi In Eq. (4), the parameter n is expressed as n ¼ 5:811ð hx= EÞa where a is a minimum impact parameter and ⁄x and E (proton kinetic energy) are in eV. There are, p inffiffiffi general, two choices for the parameter a. One is a ¼ 88:748= E given by Lindhard [16] pffiffiffiffiffiffiffi and the other is a ¼ 3:688= hx suggested by Jackson and McCarthy [17], here ⁄x and E are also in eV. After performing a detailed analyses on the effects of the two minimum impact parameters on the calculations of the proton SPs in aluminum, due to ppragmatic reasons Ashley chose the parameter a ¼ ffiffiffi 88:748= E for the calculations of the proton SPs in several other materials (C, Cu and polystyrene). pffiffiffiAs done later, we will test the rationality of using a ¼ 88:748= E in the proton SP calculations for the elemental solids under consideration. In addition, Eqs. (2) and (4) show that the calculations of both SPBorn and SPBarkas require deriving the OELF. Here, the OELFs for the 12 solids are obtained using the optical data indicated in Table 1. Commonly, the tests for the internal consistency of the optical data are made by means of the calculations of three parameters Zeff, Peff, and I. These three parameters result from f-sum rule, perfect screening (ps)-sum rule and mean excitation energy [20–22], respectively. The f-sum rule can be expressed an effective number Zeff of electrons per atom or molecule through the following formula:
Ag W Os Ir
ð h xP Þ 2 fy2 ½1:20206 y2 ð1:042 0:8549y2 þ 0:343y4 Þg; 2a0 E0 ð6Þ
where xP is the plasma angular frequency, y = a/b with a the fine structure constant and b the ratio of proton velocity to light velocity.
Pt Au a b
Ref. [18]. Ref. [19].
Error-ps (%)
I (eV) Calculated
Expected
2
1.5
165.3
166
1
12.5
170.1
173
4.2
2.3
316.1
311
2.2
0.6
327.5
322
5.2
0.4
410.6
424
8.2
428.7
449
2.4
10.6
377.4
470
1.8
10.2
586.7
727
2.18
10.3
609.7
746
0.2
16.6
601.2
757
3.82
10.6
669.8
790
3.1
8.9
662.3
790
Error-f (%)
7
330
Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
30
20 SPBorn SP Tabulated values [11] Binary theory [24] Denton et al. [25]
(a) Al
20
SPs (eV/Å)
SPs (eV/Å)
16
SPBorn SP Fit to Expt. data [10] Denton et al. [25]
(b) Ni
12
8
10 4
0 10
2
10
3
10
0
4
10
2
10
3
10
4
proton energy (keV)
proton energy (keV) 32
30
(c) Cu
SPBorn SP Tabulated values [11] Fit to Expt. data [10]
(d) Ag 24
SPs (eV/Å)
SPs (eV/Å)
20
SPBorn SP Tabulated values [11] Fit to Expt. data [10] Denton et al. [25]
16
10 8
0 10
2
10
3
10
0
4
10
proton energy (keV)
SPs (eV/Å)
30
2
10
3
10
4
proton energy (keV)
(e) Au
SPBorn SP Tabulated values [11] Fit to Expt. data [10]
20
10
0 10
2
10
3
10
4
proton energy (keV) Fig. 1. Comparisons between the calculated stopping powers (SPBorns and SP) and those from the fit to experimental (Expt.) data [10], the tabulated values [11], and the evaluations of both Sigmund and Schinner [24] and Denton et al. [25] as the functions of proton energy for (a) Al, (b) Ni, (c) Cu, (d) Ag, and (e) Au.
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
The mean excitation energy I is defined as
ln I ¼
R1 0
ðhxÞIm½1=eðxÞ lnðhxÞdðhxÞ R1 : ðhxÞIm½1=eðxÞdðhxÞ 0
ð9Þ
Using Eqs. (7)–(9), for each elemental solid under consideration we evaluate Zeff, Peff and I for (⁄x)max typically equal to 30 keV and compare them with those expected from Eqs. (7) and (8) when (⁄x)max ? 1 and from ICRU Report 37 [23]. The errors of the calculated values of Zeff and Peff relative to the expected ones are listed in Table 1, whereas the calculated and expected Is are also displayed this table for comparison. From Table 1, we can see that for Al, Ni, Cu, and Mo the errors in the two sum rules are small and the respective I evaluated by Eq. (9) is close to the expected one within 3%. However, for other materials there are the errors in ps-sum rule larger than 8%. In addition, for Ag, W, Os, Ir, Pt, and Au the evaluated Is differ evidently from the expected ones. The primary reason for this may be that in the integrations of Eq. (9) 30 keV used as the upper limit instead of infinity is not large enough to include fully the contributions to I from K-shell excitations for these materials. Using the algorithm described above, we have calculated the proton SPs for the 12 solids over the energy range from 0.5 to 10 MeV. According to our calculations, it is shown that for the solids with the small errors of the optical data in the two sum rules, in energy range of E > 100 keV the proton SPs calculated by taking into account pffiffiffi the higher-order corrections with the use of a ¼ 88:748= E agree well with the experiments [10] and the tabulated values [11] and those without the higher-order corrections are obviously small when compared to the experiments [10] and the tabulated values [11]. However, for the solids with the large errors of the optical data in the ps-sumpffiffiffi rule and using the higher-order corrections with a ¼ 88:748= E the calculated proton SPs are in reasonable agreement with the experiments [10] or the tabulated values [11], but present obvious large values of magnitude for proton energies from 100 keV to 1 MeV. As the examples of our calculations, Fig. 1 illustrates the calculated proton SPs as a function of energy for Al, Ni, Cu, Ag, and Au whereas the tabulated values [11] and the fits to the experimental data [10] are also plotted in this figure for comparison. It is clear from Fig. 1(a)–(c) that for Al, Ni, and Cu the calculated proton SPs show a good agreement with the experiments [10] and with the tabulated values [11] at energies above 100 keV when the
8
parameter γ
6
higher-order corrections are applied and there are evident difference between the SPBorn and the experiments [10] or the tabulated values [11]. Noticing the fact that the optical data of Al, Ni, and Cu are of small errors in the two sum rules, it is therefore demonstrated that the higher-orderpcorrections with the minimum imffiffiffi pact parameter a ¼ 88:748= E for proton SP calculations are both evident and necessary. On the other hand, from Fig. 1(d) and (e), it can be seen that for Ag and Au the proton SPs accounting for the higher-order corrections are larger than the experiments [10] or the tabulated values [11] because of their large errors of the optical data in ps-sum rule, especially for Ag, for which its error, 10.6%, in ps-sum rule leads to its obviously large SPs in contrast with the experiments [10] and the tabulated values [11] at peak energy. Recently, two groups, Sigmund and Schinner [24] and Denton et al. [25], presented the calculations of the SPs of several elemental solids for H projectiles. For these calculations Sigmund and Schinner used the binary theory based on the modifications for the Bohr’s classical theory [26] and Denton et al. applied the dielectric formalism. In the binary theory due to Sigmund and Schinner [24], the charge states of the projectile inside the target, the influence of screening of the projectiles-target interaction by accompanying electrons, the shell correction and the Barkas and Bloch corrections are included. In dielectric model of Denton et al. [25], the charge states of the projectile inside the target, the effect of the polarization of the projectile and the SP due to the electronic capture and loss events are taken into account. The SPs of Al, Ni and Cu for H projectiles calculated by these two groups are also presented in Fig. 1 for comparison. For Al, it can be seen from Fig. 1(a) that our calculated SPs are in good agreement with those from Denton et al. [25] for energies above 100 keV and that the SPs evaluated by Sigmund and Schinner [24] are larger than our calculations and those of Denton et al. [25] in the energy range below 400 keV. For Ni and Cu, from Fig. 1(b) and (c) it is clear that our calculated SPs agree well with those of Denton et al. [25] at energies higher than 200 keV. Due to similar dielectric models used in the present work and in Denton et al. [25], the differences between our calculated SPs and those of Denton et al. [25] at low energies (<100 keV) may be mainly attributed to the reasons that the charge state effects of the projectile inside the target and the SPs due to the electronic capture and loss events are not included in the present work. However, as pointed out by Denton et al., for low projectile energies other models including the binary theory [25] could be used for the SP calculations for H projectiles. Based on the above calculations and analyses, for the evaluations of the proton inelastic MFPs to be done later we will take into account the higher-order corrections and will use the minimum pffiffiffi impact parameter a ¼ 88:748= E for the Barkas-effect correction. 2.2. Calculation principle of proton inelastic MFP In a manner analogous to that expressing the constituents of proton SP calculation in Eq. (1), the inelastic inverse mean free path (IMFP) for proton impact in a medium can be expressed as
4
IMFP ¼ IMFPBorn þ IMFPBarkas þ IMFPBloch ;
ð10Þ
here IMFPBorn denotes basic inverse mean free path based on the first Born approximation, and IMFPBarkas and IMFPBloch correspond to the Barkas-effect correction and the Bloch correction, respectively.
2
0 0
30
60
90
atomic number Z Fig. 2. Estimated parameter c used in the Bloch correction to proton inelastic MFPs for 12 elemental solids.
Table 2 Determined parameter cs for 12 elements. Element
Al
Si
Ni
Cu
Mo
Rh
Ag
W
Os
Ir
Pt
Au
c
3.0
2.9
5.1
5.3
3.6
3.5
3.9
4.8
5.0
4.8
5.5
4.9
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
From the evaluation principle for the SPBorn due to Ashley [13], IMFPBorn can be derived and the resultant expression is written as
10
0
IMFPBarkas ¼
RSP, RMFP
Z
E0
Im½1=eðxÞv ðaÞdðhxÞ:
ð11Þ
0
1 pa0 E0
Z
4E0
dðhxÞIm½1=eðxÞL1 ðx; nÞ:
ð12Þ
0
In Eq. (12), for the minimum pffiffiffi impact parameter involved in L1(x; n) we adopt a ¼ 88:748= E. Unfortunately, there is no available theoretical expression of evaluating the IMFPBolch corresponding to the SPBloch. Therefore, in our previous work [9] and on the basis of the analyses on the SPBloch due to Bloch [27], we proposed an empirical Bloch correction to inelastic MFP. This correction takes the following form
-1
IMFPBloch ¼
RSP RMFP
10
1 2pa0 E0
Similarly, according to Ashley’s principle of describing the Barkas-effect correction for the SP calculation [13], the formula of evaluating IMFPBarkas can be obtained as
Al (a)
10
IMFPBorn ¼
Bðhx; E0 Þ ¼ 200
Z
4E0
p 2
0
ðhxP Þ2 Bðhx; E0 ÞdðhxÞ;
ð13Þ
with
-2
100
1 pa0 E0
300
400
1 cðhxÞc2 X
ð4E0 Þc
j¼1
500
1 jð4Aj E0 þ 1Þ1c ðAj hx þ 1Þ1þc
ð14Þ
;
2 2
where Aj ¼ j h =ð2Z 21 e4 mÞ and c is a free parameter.
proton energy (keV)
4000
10
4
(a) Cu
Al (b)
10
10
10
IMFP (1/μm)
IMFPs (1/μm)
3000
2
IMFPBorn IMFPBarkas − IMFPBloch
0
IMFP IMFPBorn
2000
1000
-2
0 10
2
10
3
10
10
4
2
10
3
10
4
proton energy (keV)
proton energy (keV) 4000
6000
(b) Au
IMFP IMFPBorn
4000
IMFP IMFPBorn
3000
IMFP (1/μm)
IMFPs (1/μm)
Al (c)
2000
2000
1000
0
0 10
2
10
3
10
4
proton energy (keV) Fig. 3. Comparisons between (a) the RSP and RIMFP and between (b and c) the various IMFPs as the functions of proton energy for Al.
10
2
10
3
10
4
proton energy (keV) Fig. 4. Comparisons between the IMFP and IMFPBorn as the functions of proton energy for (a) Cu and (b) Au.
333
Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335 Table 3 MFPBorn (in unit Å) for 12 elemental solids as a function of proton energy E (in unit keV). E
Al
Si
Ni
Cu
Mo
Rh
Ag
W
Os
Ir
Pt
Au
50 60 70 80 90 100 150 200 300 400 500 600 700 800 900 1000 1500 2000 3000 4000 5000 6000 7000 8000 9000 10,000
2.472 2.532 2.649 2.789 2.939 3.095 3.882 4.583 5.848 7.042 8.193 9.313 10.41 11.48 12.53 13.57 18.56 23.31 32.35 40.96 49.28 57.38 65.31 73.10 80.77 88.33
2.920 2.935 3.041 3.182 3.340 3.506 4.374 5.226 6.704 8.061 9.370 10.65 11.89 13.12 14.32 15.50 21.19 26.61 36.91 46.74 56.23 65.47 74.52 83.40 92.14 100.80
4.416 3.935 3.667 3.534 3.484 3.485 3.763 4.092 4.850 5.634 6.415 7.187 7.947 8.696 9.434 10.16 13.66 16.97 23.22 29.17 34.91 40.49 45.94 51.28 56.54 61.72
4.356 3.940 3.757 3.685 3.670 3.683 3.938 4.313 5.117 5.932 6.748 7.554 8.349 9.132 9.904 10.66 14.33 17.80 24.35 30.59 36.60 42.44 48.15 53.75 59.26 64.68
5.077 4.182 3.756 3.493 3.354 3.257 3.220 3.592 4.459 5.326 6.166 6.965 7.731 8.473 9.196 9.906 13.30 16.53 22.67 28.52 34.16 39.63 44.99 50.23 55.40 60.48
5.197 4.239 3.640 3.351 3.231 3.163 3.076 3.311 4.039 4.789 5.524 6.242 6.940 7.612 8.263 8.900 11.93 14.81 20.28 25.48 30.50 35.38 40.15 44.82 49.42 53.95
4.914 4.367 3.979 3.718 3.539 3.424 3.300 3.470 4.163 4.915 5.655 6.378 7.086 7.776 8.445 9.098 12.21 15.15 20.74 26.06 31.20 36.19 41.07 45.85 50.55 55.18
5.790 4.268 3.701 3.353 3.126 2.999 2.894 3.151 3.790 4.430 5.048 5.648 6.232 6.804 7.364 7.915 10.56 13.07 17.83 22.36 26.71 30.94 35.06 39.11 43.09 47.01
6.591 4.928 4.026 3.61 3.372 3.222 3.015 3.17 3.735 4.332 4.916 5.485 6.039 6.581 7.113 7.636 10.14 12.52 17.04 21.34 25.47 29.48 33.39 37.23 41.00 44.71
6.252 4.799 3.874 3.447 3.209 3.060 2.851 2.997 3.542 4.119 4.685 5.234 5.766 6.287 6.798 7.300 9.705 11.99 16.32 20.45 24.41 28.26 32.01 35.69 39.31 42.88
5.531 4.441 3.724 3.374 3.208 3.120 3.084 3.318 3.982 4.661 5.316 5.948 6.562 7.159 7.744 8.318 11.070 13.60 18.63 23.34 27.87 32.27 36.57 40.78 44.91 48.99
5.414 4.665 4.079 3.687 3.441 3.299 3.176 3.397 4.064 4.769 5.452 6.112 6.753 7.378 7.988 8.585 11.45 14.17 19.32 24.22 28.94 33.52 38.00 42.38 46.69 50.93
The free parameter c in Eq. (14) needs to be determined for the solids under consideration. However, to our knowledge, no available experimental data of inelastic cross-sections of protons in the considered solids can be used for estimating the parameter c. Thus, here we propose the following empirical approach for determining the parameter c. Defining RSP = (SPBarkas + SPBloch)/SPBorn as percentage correction for SPBorn and RIMFP = (IMFPBarkas + IMFPBloch)/IMFPBorn as that for IMFPBorn, for each considered solid we determine the parameter c so that the RIMFP is approximately equal to the RSP at their respective peak energy. According to this, we determine the values of the parameter c for the 12 solids, and list these values in Table 2. Particularly, the determined c versus
atomic number Z is illustrated in Fig. 2 in order to clearly show the c dependence on atomic number Z. It can be seen from this figure that for the solids with close atomic number their values of c are close for each other. As an example of our calculations, Fig. 3 presents various proton inelastic IMFPs calculated using the principles described above and the comparison between RSPs and RIMFPs as a function of proton energy for Al. From Fig. 3(a), it can be observed that at energies below 500 keV the percentage corrections for IMFPBorn and SPBorn become obvious, bringing about a correction larger than 30% at respective peak energy, which indicates clearly the importance of the higher-order corrections at low energies. Fig. 3(b) displays the comparison between various
Table 4 MFP (in unit Å) for 12 elemental solids as a function of proton energy E (in unit keV). E
Al
Si
Ni
Cu
Mo
Rh
Ag
W
Os
Ir
Pt
Au
50 60 70 80 90 100 150 200 300 400 500 600 700 800 900 1000 1500 2000 3000 4000 5000 6000 7000 8000 9000 10,000
2.039 2.003 2.080 2.200 2.339 2.487 3.268 3.998 5.334 6.584 7.779 8.934 10.06 11.15 12.23 13.28 18.33 23.12 32.20 40.84 49.18 57.30 65.24 73.04 80.71 88.28
2.452 2.354 2.420 2.542 2.691 2.853 3.723 4.591 6.137 7.555 8.913 10.23 11.51 12.75 13.98 15.18 20.94 26.40 36.75 46.61 56.12 65.38 74.44 83.33 92.08 100.70
8.418 3.808 2.991 2.715 2.618 2.820 3.221 4.107 4.983 5.830 6.651 7.451 8.232 8.996 9.746 13.33 16.69 23.00 28.98 34.75 40.34 45.81 51.16 56.43 61.62
8.029 3.906 3.126 2.858 2.762 2.948 3.383 4.316 5.230 6.115 6.975 7.812 8.629 9.430 10.21 13.96 17.49 24.11 30.38 36.42 42.28 48.01 53.62 59.14 64.57
8.231 4.313 3.432 3.072 2.837 3.200 4.087 4.976 5.837 6.655 7.439 8.196 8.934 9.656 13.10 16.36 22.54 28.41 34.06 39.55 44.91 50.17 55.34 60.43
6.159 4.021 3.349 2.764 2.978 3.719 4.491 5.245 5.978 6.690 7.375 8.038 8.685 11.76 14.66 20.16 25.39 30.42 35.32 40.09 44.77 49.37 53.91
3.506 2.917 2.676 2.587 2.866 3.655 4.463 5.245 5.999 6.732 7.443 8.131 8.800 11.97 14.95 20.58 25.93 31.08 36.09 40.98 45.76 50.47 55.11
6.606 3.818 3.060 2.473 2.692 3.346 4.016 4.664 5.289 5.896 6.487 7.064 7.629 10.32 12.87 17.67 22.22 26.59 30.83 34.97 39.02 43.01 46.94
20.33 5.433 3.750 2.590 2.693 3.269 3.897 4.511 5.106 5.684 6.246 6.796 7.333 9.894 12.31 16.87 21.19 25.34 29.36 33.29 37.13 40.91 44.63
13.70 4.775 3.418 2.437 2.548 3.109 3.715 4.309 4.882 5.436 5.976 6.503 7.019 9.474 11.79 16.17 20.31 24.29 28.15 31.92 35.61 39.23 42.80
9.193 4.521 3.478 2.69 2.863 3.523 4.224 4.902 5.558 6.192 6.808 7.409 7.998 10.80 13.44 18.44 23.18 27.73 32.14 36.45 40.67 44.82 48.90
12.04 5.057 3.747 2.796 2.960 3.631 4.360 5.067 5.749 6.410 7.053 7.678 8.289 11.21 13.96 19.15 24.08 28.82 33.41 37.89 42.28 46.60 50.85
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Z. Tan, Y. Xia / Nuclear Instruments and Methods in Physics Research B 269 (2011) 328–335
MFP (Å)
10
found elsewhere [28,29]. This approximate dispersion relation may produce the uncertainty for our calculated MFPs. Third, as shown earlier, the optical data with less accuracy used in this work result in the obvious difference in the calculated SPs compared with the tabulated values [11], and hence they also induce uncertainty in the calculations of the MFPs. Finally, the parameter c involved in the Bloch correction to proton inelastic MFPs is also a source of uncertainty due to the assumption that the RIMFP is empirically considered to be close the RSP at low energies.
2
Al Cu Ag Au 10
1
4. Summary
10
0
10
2
10
3
10
4
proton energy (keV) Fig. 5. Mean free path (MFP) results as a function of proton energy for four typical elemental solids Al, Cu, Ag, and Au.
IMFPs and it is clear that the correction IMFPBarkas + IMFPBloch becomes negligible at high energies. In Fig. 3(c) the IMFPs with the higher-order corrections are compared particularly with the IMFPBorns, notably showing the effect of the higher-order corrections at low energies. Fig. 4 plots the IMFPs and IMFPBorns versus energy for Cu and Au, respectively. From Fig. 4, the corrections to the IMFPBorns of protons in Au are small when compared to those for Cu, and the peak energy of the IMFP for Au presents a small shift toward high energy relative to Cu.
3. Proton inelastic mean free paths in 12 elemental solids Using the method of evaluating the proton inelastic MFPs described in this work and under two different cases, i.e. the higher-order corrections not being considered, and the Barkas-effect correction and the Bloch correction being included, we have performed the systematical calculations of the proton inelastic MFPs for the 12 solids in energy range from 0.05 to 10 MeV. The resultant MFPBorns (MFPBorn = 1/IMFPBorn) are listed at selected energies in numerical form in Table 3. Under the second case, the calculated results for the MFPs (MFP = 1/IMFP) including the higher-order corrections are presented in Table 4. From Tables 3 and 4, for these 12 solids the MFP dependences on energy are very similar from solid to solid. Fig. 5 presents the calculated proton inelastic MFPs as a function of energy for typical solids Al, Cu, Ag, and Au. From Fig. 5, the MFP dependence on energy from Al to Au is clearly shown, the MFP decrease with atomic number Z at given energy, and the energy corresponding to the minimum MFP shifts toward high energy with increasing atomic number Z. It should be pointed out that at energies lower than about 100 keV, similar to that stated by Ashley [13], the two correction terms begin to dominate the behavior of the IMFP and thus the validity of this approach is questionable. Hence, in the present work all calculated proton inelastic MFPs at energies below 100 keV may be considered only as rough data. In addition, the sources of uncertainty for our calculated MFPs may arise from the following factors. First, our MFP calculations are based on the Born approximation which is not reliable at low energies. Second, in the optical-data model of Ashley [13] the dispersion relation in a simple quadratic form was adopted because of the scarcity of exact theory for solving the problem of extending the OELF into the area of momentum q > 0, on which the detailed discussions can be
In this work the evaluations of the inelastic MFPs of 0.05–10 MeV protons in 12 elemental solids (Al, Si, Ni, Cu, Mo, Rh, Ag, W, Os, Ir, Pt, Au) have been performed, based on the algorithm derived from Ashley’s optical-data model of calculating the proton SPs. The internal consistency of the optical data for the 12 solids were checked by using f-sum and ps-sum rules and by calculating the mean excitation energy, showing a higher accuracy of the optical data for Al, Ni, Cu, and Mo, and showing large errors of the optical data in ps-sum rule for other solids. By systematically calculating the proton SPs for these 12 solids, the rationality of the minimum impact parameter used in the Barkas-effect correction were examined and the influences of the used optical data on the proton SP calculations were tested, leading to the conclusion that using accurate optical data and choosing the minimum impact parameter by Lindhard [16] for the Barkas-effect correction, the higher-order corrections should be taken into account and the calculated proton SPs are in satisfactory agreement with the experiments [10] and the tabulated values [11]. Following the above, the algorithm of evaluating the proton inelastic MFPs were described. In this algorithm, the higher-order corrections are accounted for and the empirical estimation of free parameter c involved in the Bloch correction to the inelastic MFP is given. Under two different cases, i.e. the higher-order corrections not being considered, and the Barkas-effect correction and the Bloch correction being included, the evaluated inelastic MFPs of 0.05–10 MeV protons in the 12 solids were presented in numerical form. These data are first results for the 12 solids and are useful for the Monte Carlo studies on low-energy proton transport in the considered 12 solids. In addition, the experimental study of determining inelastic MFP of protons in these 12 solids should be expected in the future, so as to further complete the algorithm for the calculations of proton inelastic MFPs. Acknowledgment This work was supported partly by the Natural Science Foundation of Shandong Province of China under Grant No. ZR2009DZ006. References [1] H. Nikjoo, S. Uehara, D. Emfietzoglou, F.A. Cucinotta, Radiat. Meas. 41 (2006) 1052. [2] J. Sisterson, Nucl. Instrum. Meth. B 241 (2005) 713. [3] H. Nikjoo, S. Uehara, D. Emfietzoglou, A. Brahme, New J. Phys. 10 (2008) 075006. [4] A. Brahme, Int. J. Radiat. Oncol. Biol. Phys. 58 (2004) 603. [5] E. Grossman, I. Gouzman, Nucl. Instrum. Meth. B 208 (2003) 48. [6] V. Auzelyte, M. Elfman, P. Kristiansson, C. Nilsson, J. Pallon, N.A. Marrero, M. Wegde’n, Microelectronic Eng. 83 (2006) 2015. [7] C.N.B. Udalagama, A.A. Bettiol, F. Watt, Nucl. Instrum. Meth. B 260 (2007) 384. [8] F. Menzel, D. Spemann, J. Lenzner, W. Bölmann, G. Zimmermann, T. Butz, Nucl. Instrum. Meth. B 267 (2009) 2321. [9] Zhenyu Tan, Yueyuan Xia, Mingwen Zhao, Xiangdong Liu, Nucl. Instrum. Meth. B 268 (2010) 2337. [10] H.H. Andersen, J.F. Ziegler, Hydrogen Stopping Powers and Ranges in All Elements, Pergamon, New York, 1977. [11] International Commission on Radiation Units and Measurements, ICRU Report 49, Bethesda, MD, 1993.
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