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Electron impact stopping powers for elemental and compound media
Vacuum 132 (2016) 123e129
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Electron impact stopping powers for elemental and compound media A.K. Fazlul Haque a, b, *, M. Atiqur R. Patoary a, M. Alfaz Uddin a, b, c, Arun K. Basak a, M. Ismail Hossain a, Mahmudul Hasan d, Bidhan C. Saha e, M. Maaza b, f, g a
Department of Physics, University of Rajshahi, Rajshahi, Bangladesh Nano Sciences African Network (NANOAFNET), iThemba LABS-National Research Foundation, Old Faure road, 7129, Somerset West, South Africa c Khwaja Yunus Ali University, Enaetpur, Chowhali, Sirajgonj, 6751, Bangladesh d Department of Physics, Pabna University of Science & Technology, Pabna 6600, Bangladesh e Department of Physics, Florida A & M University, Tallahassee, FL, USA f Council for Scientific and Industrial Research, PO Box 395, Pretoria, 0001, South Africa g UNESCO-UNISA Africa Chair in Nanosciences/Nanotechnology, College of Graduate Studies, University of South Africa (UNISA), Muckleneuk Ridge, PO Box 392, Pretoria, South Africa b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 February 2016 Received in revised form 27 July 2016 Accepted 28 July 2016 Available online 4 August 2016
A semi-analytical model is proposed for evaluating the electron stopping powers (ESPs) of material media including both elements and compounds in the energy range 1.0 eVe100 MeV. This paper reports details of our calculations using the effective atomic electron number, effective mean excitation energies of target atoms and realistic electron density distribution of the target atoms. This simple formula is capable of generating accurate cross-section for diverse systems. We present here only few simple cases. In comparison with other theoretical predictions, our calculated ESPs for elemental [C, Al, Cu, Pt, Au and Pb] and compound [H2O, C5H5N5O, GaAs, Bi2Sr2CaCu2O8, SiO2 and ZnTe] media agree reasonably well with the experimental findings. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Stopping power Material media Relativistic charge density distribution
1. Introduction Electron stopping powers (ESPs) of material media, usually defined as the average energy loss per unit distance along the electron path (in the units of energy/length), are needed in diversified areas of researches and applications, involving energy loss, including quantitative calculations of the dosages in radiation therapy for living tissues, Monte Carlo simulation study in electron microscopy for manipulating materials in both nuclear and space applications, quantitative surface analysis, the design of particle detectors [1], etc. It demands a simple-to-use formula capable of furnishing reasonably accurate ESPs of different media over a wide range of projectile energies. To date there are considerable attempts to calculate on ESPs using various formulas, methods and models [2e15]. Some of them are heavily dependent on fitting species dependent parameters, some are relied on accurate oscillator strength evaluation, some are involved in evaluating the complex dielectric response function, etc. However, for different
* Corresponding author. Department of Physics, University of Rajshahi, Rajshahi 6205, Bangladesh. E-mail address: [email protected] (A.K.F. Haque). http://dx.doi.org/10.1016/j.vacuum.2016.07.038 0042-207X/© 2016 Elsevier Ltd. All rights reserved.
energy regime, many of them are intuitively divided into slots to represent ESPs. Moreover, their application to molecular targets still poses considerable challenges. The search for a formula, with validity over a wide range of incident energies and for a large domain of species, motivated the present work. Rohrlich and Carlson [2] have reported a formula for the average rate of energy loss of positrons and electrons in passing through matter. Sugiyama [13] has reported two approaches to derive ESP formulas: (a) the Bethe-Bloch formulas [8], referred to here as Bethe formula, and (b) the Lindhard-Scharff-Schiott theory [3,14]. We follow closely the Rohrlich and Carlson [2] based on the earlier work of Bethe formula [16]. This model without the shell- and density-corrections [4,9,14] shows reasonable results for the incident electron energies ranging 50 keV to a few MeV. Sugiyama [7,13,15] introduced the effective charges [6], z* of the projectile and Z* of the target atom, and effective mean excitation energy (EMEE) I* to modify the formula of Rohrlich and Carlson [2]. Gümüs group [11,12] further modified the formulas of Rohrlich and Carlson [2] and Sugiyama [7,15] to obtain analytic expressions for Z* and I* of the target atom using the Tietz screening function [17]. For the incident energies 10 eV E 100 keV, this model yields fairly accurate ESPs. For 200 eV E 30 keV, Jablonski et al. [9] have
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reported an empirical formula with species dependent parameters that rely heavily on the available experimental data. At E 1 keV, the ESPs for elemental solids are calculated successfully and are available in Pages et al. [5] and ESTAR database [18]. From all these theoretical works, it is apparent that none of the above-mentioned formulas is able to describe the experimental ESP data at the energies E < 10 eV. Moreover, no single formula, available in the literature, goes smoothly from E ¼ 1 eV to ultrahigh energies to reproduce simultaneously the available experimental data up to about 10 MeV and the ESTAR-generated ESP data beyond. In particular, the formula of Gümüs group [11,12], which can describe the experimental data beyond 10 eV, is not applicable to the energy region dominated by Bremsstrahlung. On the other hand, ESTAR, claimed to be reliable in the high and ultra-high energies, where no experimental data are available, is found to reproduce the experimental ESP data in the light elemental targets beyond 10 keV. It is, therefore, desirable to find a simple-to-use formula capable of reproducing successfully the experimental ESPs at lower energies and the ESTAR-evaluated ESPs from the Bremsstrahlung data. The purpose of this study is to present a formula after proper modifications of the earlier formulas in Refs. [2,7,11,12] incorporating the effective charges, z* and Z* as well as the target EMEE I*. The numerical density distributions (DDs) of the target electrons are evaluated numerically using the multi-configuration code of Desclaux [19], which uses the Dirac-Hartree-Fock electron wave functions. These realistic DDs are then employed to calculate both the effective charges and mean excitation energies of the target elements and molecules considered herein. The numerical DDs are represented by the piecewise cubic spline functions to evaluate the integrals [see Eqs. (3) and (4), shown below] for the effective charge Z* and EMEE I*. Although the present semi-analytical model is capable of calculating ESPs for elemental solid targets up to Z ¼ 92, we consider here only few sample media for the energies 1 eVe100 MeV. Our results are compared with the available experimental and other theoretical findings including ESTAR [18]. To establish the suitability, the proposed formula has been tested systematically on 6 important compound media including 2 biological molecules, water (H2O) and Guanine (C5H5N5O) in nucleic acid, and 4 semiconductors, namely silica (SiO2), Gallium arsenide (GaAs), Bi-High-TC superconductor (Bi2Sr2CaCu2O8) and Zinc tellurium (ZnTe). 2. Basic theory The formula of the stopping power for electron in a medium, modified by Refs. [11,12] from the formula of [2], is;
. dE r E S ¼ ¼ 2pe4 N0 z2 Z ln þ FðtÞ=2 eV A dx I AE
(1)
with
FðtÞ ¼ 1 b2 þ
h
.
i. 1 þ t2 :
t2 8 ð2t þ 1Þln2
(2)
Here, t ¼ E/mc2 is the kinetic energy in units of the electronic rest mass; e, N0, r and A are, respectively, the electronic charge, Avogadro number, atomic density in gm/cm3 and atomic mass in gm. E is the electron energy in eV and b is the velocity of incident electron in the unit of the velocity of light. In Eq. (1) the factor in front of the parenthesis can be written as (ra=AEÞ eV/Å, since a ¼ 2pe4 N0 ¼ 785 [20]. The function FðtÞ [7,11,12] has dependency not only on the projectile energy but also on its kind. However, it is independent of
its atomic number Z. ESPs contain same relativistic components through t and electron density in Eq. (1). In order to determine Z* and I* of the target atoms, Gümüs and Kabadayi [12] used Thomas-Fermi theory [21] which describes an atom having Z bound electrons with radially symmetric electron density. Sugiyama [7] obtained Z* and I* from Bohr's stripping criterion for the effective charge of incident heavy ions and target atoms, and from the Lindhard and Schraff theory [22] concerning the quantities in EMEE I*. These functions in Eq. (1) are given in Refs. [7,13] as;
Z ¼
Z∞
4pr 2 nðrÞdr
(3)
rc
and
1 ln I ¼ Z
Z∞
ln gZup ðrÞ 4pr 2 nðrÞdr:
(4)
rc
here, r is the distance of the bound electron from the nucleus; n(r), the electronic charge density in target, up ðrÞ ¼ ð4pe2 nðrÞ=mÞ0:5 , the local plasma frequency and g ¼ 20:5 [7,22]. The velocity of the incident electron vi at its distance from the nucleus rc satisfying Bohr‘s criterion is equal to that vb of the bound electron. A bound electron for which r > rc moves with a velocity vb < vi and can contribute to the stopping power. The inner electrons with r < rc and vb > vi are rigidly bound leading to an insignificant energy loss of the incident electron. The quantity rc is defined by the Bohr stripping criterion [23] vi ¼ ðbZ=mÞð3p2 nðrc ÞÞ1=3 with b as the proportionality constant of the order of 1.26 [7]. The quantity rc is obtained from the potential-energy condition of Yarlagadda et al. [6]
1 2 mv þ Uðrc Þ ¼ 0; 2 i
(5)
as pointed out earlier with vi ¼ vb. Here, U(r) is the potential experienced by the incident electron. Sugiyama [7] employed the semi-empirical effective charge of the incident electron z* as given by
z ¼ 1 exp 2200b1:78 :
(6)
This conforms to the reason that z* increases with the incident energy as all processes contributing to the stopping power are enhanced with increasing velocity of the projectile electron. Our proposed formula reported in this study embodies the following modifications in addition to the use of relativistic treatment for the incident and bound electrons and the realistic DDs: in theright-hand 1. The first term within the square brackets side of Eq. (1) has to be modified from ln IE to ln 3:75E to repreI sent both the peak location and the magnitude of relative ESPs. This is linked to the fact that the energy loss in the low energy limit is proportional to the square of the effective velocity veff of the incident electron (see Fig. 3 of [6]). As the projectile electron approaches the bound electron, veff gets enhanced to increase it's effective energy Eeff, which is substantially greater than E in the low energy region. To simulate Eeff, a number of grid searches on an appropriate factor with E are found essential to provide an overall best description of the experimental data near the peak region of the elemental targets. This procedure results in the factor of 3.75 with E, which is found good also for
A.K.F. Haque et al. / Vacuum 132 (2016) 123e129
the compound targets. This factor takes care of the correlation effect [24]. 2. FðtÞ given in Eq. (2) goes to negative values beyond E ¼ 80 keV and thus cannot reproduce the ESTAR data where the main contribution to ESPs is due to Bremsstrahlung [25], which increases with E. It thus suggests to replace FðtÞ by its modified form which not only describes the ESTAR data at high energies but also supplements the Bethe contribution to reproduce the low energy behaviour of the experimental data. In the simplest form, the modified function F ðtÞ looks like
F ðtÞ ¼ 1 þ t2 :
(7)
3. To reproduce the magnitude of all the available experimental data accurately a normalization constant (x ¼ 0.529) is introduced. Incorporating all these above modifications into Eq. (1), our present formula for calculating the ESPs for elemental targets becomes;
S ¼ 2pe4 N0
. E z2 Z x ln þ FðtÞ=2 eV A I AE
r
(8)
For the compound media, the following Bragg-Kleeman rule of the linear combination SC [11,26] from the elemental ESPs is used:
SC ¼
X
n i Si :
(9)
i
here, Si represents the ESPs for the i-th element and ni denotes its molar fraction in the compound target. 3. Analysis and results All ESPs, reported here, are calculated using Eqs. (3), (4), (6) and (8)), with a generation of the electron density function from the code [19]. In Fig. 1(a) and (b), we present the effective number of electrons Z* and the mean excitation energy I* as a function of the incident energy. It is evident from the figures that our calculated values for Z* and I* approach the atomic number Z and the experimental mean excitation energies [12], respectively. For small values of Z* at lower incident energies, it is indicated that inner electrons of target atom do not contribute to the energy loss of incident electrons. The higher is the energy, the greater is the number of bound electron participating in the collision process until the saturation of Z* is reached around E < 100 keV. Similar trend is also noticed in the mean excitation energy. With increase of the incident energy, more inelastic channels contribute till saturation is reached. We present here the calculations of ESPs for the elementary species of C, Al, Cu, Pt, Au and Pb; and compound species of H2O, C5H5N5O, GaAs, Bi2Sr2CaCu2O8, SiO2 and ZnTe. Table 1 shows the densities obtained from Ref. [27] for both elemental and compound media. All reported ESPs herein are calculated using them. Sources of the experimental data are collected from Joy's database [28]. The sources comprise Luo [29] for C, Pt, and C5H5N5O; Hovington et al. [30] for C, Al, Cu, Pt, SiO2 and ZnTe; Kalil et al. [31] for Al; Fitting [32] for Al; Al-Ahmad and Watt [33] for Al, Cu and Au; Ishigure et al. [34] for Al; Garber et al. [35] for Al; Luo et al. [36] for Cu, Au, Pb, H2O, C5H5N5O and SiO2; MacPherson [37] for Al, Cu and H2O; and La Verne and Mazumder [38] for H2O, GaAs and Bi2Sr2CaCu2O8. The ESP-formula given in Eq. (8) has been tested rigorously for elemental solid media using the realistic DDs generated from the code [19]. The effective quantities Z*, I* and z*, as defined earlier, are
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calculated using (3), (4) and (6), respectively. The ESP results with Eq. (1) coupled with FðtÞ in Eq. (2) produce an underestimation of the ESP data in the lower energy region including the peak. Before discussing the role of different terms in Eq. (8), we apply it to a typical case of C to examine the influence of medium-density on ESPs. Fig. 1(c) depicts explicitly the structural dependence of ESPs leading to significant variations in values due to its different forms. For example, diamond has the highest ESP peak of around 12.14 eV/ Å at E ¼ 100 eV; at the same energy graphite yields 7.79 eV/Å and amorphous carbon produces around 6.5 eV/Å. The effect due to difference in densities is reflected dominantly in the peak region and becomes negligibly small at higher energies. The ratio of ESPs for diamond and graphite S(diamond)/S(graphite) ¼ 1.558 in the peak region is found matching with their density-ratio (see Table 1), as expected in the first order. To understand the role of individual terms in Eq. (1) we, in Fig. 2(a), show explicitly the separate contributions due to the first term in (1) denoted by BTH1 [dashed line (magenta in online)] and that due to the second term by FT1 [dotted line (cyan in online)]. BTH1 generates negative ESP values in the low energy region below 7.0 eV and FT1 produces negative values in the high energy region E > 80 keV. The sum contribution BTH1þFT1 [dash-dotted line with open triangles (violet in online)], underestimates the experimental ESP data considerably. To reproduce the ESP data in the energy region E < 2 keV, a factor of 3.75 is multiplied to E inside the square brackets in Eq. (8), as discussed earlier in Section 2. A simplified form of F ðtÞ is then searched for as a special requirement for satisfying the ESP data at higher as well as lower incident energies and this procedure produces the form of F ðtÞ shown in Eq. (7). The individual contributions are displayed in Fig. 2(a), where the first term in the right hand side of Eq. (8), denoted by BTH in short-dashed line (green in online), and the second term with F ðtÞ, depicted as FTP in thinsolid line (blue in online). FTP contributes solely to ESPs at E > 10 MeV. The shooting of ESP data at E > 10 MeV is understood as resulting from the dominant contribution of Bremsstrahlung [25], which remains proportional to the square of the incident energy E [39]. The sum BTH þ FTP, labelled ’present work’ in Fig. 2(a), governed by Eq. (8), describes the experimental data profoundly well and predicts ESPs in agreement with the ESTAR data in the higher energy domain. The predicted ESPs for all other elemental targets are compared in Fig. 2 with the experimental data and findings from other available theoretical calculations. The predicted results from the present model agree closely with the experimental data including those from MacPherson [37] and the ESTAR-derived ESPs at the higher energies beyond 30 MeV, where no experimental data are available as far as we are aware of. Formula of Shinotsuka et al. [40] works well in the energy range of 25 eVe100 MeV (see Fig. 2(c, d)) except for Al (Fig. 2(b)) where there is an overestimation of data near the peak region. The predicted ESPs from the model of ref. [12] also describe satisfactorily the data in the 10 eV < E < 100 keV range except for Al near the peak region while ESTAR works well beyond ndez-Varea et al. [41] tens of keV. The calculated results from Ferna reproduce the data well for Cu (Fig. 2(c)), reasonably well for Al (Fig. 2(b)) albeit disagreement with the data near the peak region and satisfactorily for Au (Fig. 2(e)) with an overestimation of data near the peak. Our predicted ESPs reproduce the available experimental data and ESTAR-derived data for all elemental targets in the considered range of 1 eV < E < 100 MeV nicely. The analytic form of ESPs from Eq. (8) is then applied to several compound targets. In Fig. 3 we compare our results for compound media with the experimental data and ESTAR derived data. In Fig. 3(a), our calculated ESPs for H2O in the solid phase compare well with the experimental data of Luo et al. [36] for vitreous ice, of
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Fig. 1. (a) The effective number of electrons Z* and (b) the effective mean excitation energy I* of various target atoms as a function of incident electron energy. (c) present ESP calculations for three forms of carbon: graphite, diamond and amorphous in solid, dashed and dash-dotted lines, respectively.
Table 1 Densities of media in unit of gm/cm3 taken from Ziegler [25]. (a) Density of elemental media Medium Density
C (graphite) 2.253
C (diamond) 3.51
C (amorphous) 2.0
Al 2.70
Cu 8.92
Pt 21.45
Au 19.31
Pb 11.34
(b) Density of solid elements associated with the compound media Medium Density
H 0.0715
C 2.25
N 1.026
O 1.426
Si 2.3212
Verner and Mozumder for vitreous ice [38] and McPherson [37] for liquid water, and ESTAR-data for liquid phase [18]. The peak value of 3.71 eV/Å at 0.11 keV for ESPs from the proposed formula compares well with those of 3.0 eV/Å at 0.10 keV and 3.72 eV/Å at 0.08 keV from the experiments of [36] and [38], respectively. The peak value of 3.18 eV/Å at 0.10 keV predicted by Ref. [11] for water in liquid phase is close to the experiment and is less than our result. ESTAR in
Ca 1.54
Cu 8.92
Zn 7.14
Ga 5.904
As 5.727
Sr 2.6
Te 6.25
Bi 9.8
its normal workable range beyond E ¼ 30 keV is found to reproduce the sole experimental data of MacPherson [37], available for the liquid phase in the E ¼ 510 MeV range (Fig. 3(a)). Among other compound media, ESP calculations of Guanine (C5H5N5O) using the generalized oscillator strength model are provided by Akar and Gümüs [10]. Their predicted ESPs reproduce the experimental data well in the 20 eV < E < 1 MeV range. Our model reproduces all the
A.K.F. Haque et al. / Vacuum 132 (2016) 123e129
127
Fig. 2. Electron stopping powers for (a) C (graphite), (b) Al, (c) Cu, (d) Pt, (e) Au and (f) Pb. Theoretical predictions are given in lines while experimental data in symbols. The results from ESTAR [18] calculations, Gümüs and Kabadayi [12], Shinotsuka e al [40]. and Fernandez-Varea et al. [41] are shown along with the present results for comparison. Predictions marked BTH1, FT1, BTH and FTP in (a) are explained in the text.
available experimental results and ESTAR-derived data for all elemental and molecular media from 1 eV to 100 MeV successfully. The dependence of ESPs on medium density shown in Fig. 1(c) for C is also noticed in the results, displayed in Fig. 3(a), for H2O in the solid and liquid phases. The ratio of our predicted maximum
ESP in the solid phase to that of [12] S(solid Present)/S(liquid) [12] ~1.18 in the peak region conforms to the ratio rsolid =rliquid ¼ 1:22 of the corresponding densities (using r for liquid H and that for liquid O from websites https://en.wikipedia.org/wiki/Liquid_hydrogen and https://en.wikpedia.org/wiki/Liquid_oxygen, respectively). On
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A.K.F. Haque et al. / Vacuum 132 (2016) 123e129
Fig. 3. Same as in Fig. 2 for compound media e.g. (a) H2O (solid), (b) C5H5N5O, (c) GaAs, (d) Bi2Sr2CaCu2O8, (e) SiO2 and (f) ZnTe. Results from Gümüs [11] and ESTAR [18] in (a) are for liquid water.
the other hand, our predicted ESPs and those from ESTAR [18] for water in liquid phase beyond E ¼ 30 keV are very close and both agree with the experimental data of [37] in the E ¼ 525 MeV (see Fig. 3(a)). Although our predicted ESPs for water in solid phase and those for liquid water from Ref. [11] are different in the peak region
because of densities, both agree with ESTAR data [18] and experimental data [37] in the energy region beyond E ¼ 30 keV. The density effects of H2O in two phases on the ESP values thus bear a trend, similar to difference in ESPs for C in varied structures with different densities in Fig 1(c).
A.K.F. Haque et al. / Vacuum 132 (2016) 123e129
4. Conclusion The present model not only reproduces successfully the observed ESP data for 1 eV E 10 keV, below the range of ESTAR calculations, but also agrees excellently with the accurate measurements of MacPherson [37] in the range 5 E 30 MeV. Our predicted results then join smoothly to the region of dominant Bremsstrahlung effects from 500 KeV to 100 MeV associated with the ESTAR-evaluated data. The proposed model produces useful results for both elemental and compound media over wide range of the incident energies, albeit being simple in structure. This could be a lucrative model for the modeling of the electron energy loss phenomena in material and biological systems of interest. For a rapid generation of accurate ESPs for science, industries and technologies this simple-to-use model will be very useful. Acknowledgments We are thankful to Professor F. B. Malik, who passed away on 05 August 2014, for his important advice regarding the improvement of the manuscript. The authors would like to thank Professor D. G. Sarantites in Washington University at St. Louis, MO, USA for stimulating discussion. A. K. Fazlul Haque would like to thank TWAS-UNISCO associateship scheme for partial funding. References [1] M.R. Cleland, Industrial applications of electron accelerators, 2009, pp. 383e416. Published version from, CERNcds.cern.ch/record/1005393/files/ p383.pdf. [2] F. Rohrlich, B.C. Carlson, Positron-electron differences in energy loss and multiple scattering, Phys. Rev. 93 (1954) 38e44. [3] J. Lindhard, M. Scharff, H.E. Schiott, Integral equations governing radiation effects, Mat.-Fys. Medd. Dan. Vid. Selsk. 33 (1963) 1e42. [4] R.M. Sternheimer, R.F. Peierls, General expression for the density effect for the ionization loss of charged particles, Phys. Rev. B 3 (1971) 3681e3692. [5] L. Pages, E. Bertel, H. Joffre, L. Sklavenitis, Energy loss, range, and bremsstrahlung yield for 10-keV to 100-MeV electrons in various elements and chemical compounds, At. Data Tables 4 (1972) 1e127. [6] B.S. Yarlagadda, J.E. Robinson, W. Brandt, Effective-charge theory and the electronic stopping power of solids, Phys. Rev. B 17 (1978) 3473e3883. [7] H. Sugiyama, Stopping power formula for intermediate energy electrons, Phys. Med. Biol. 30 (1985) 331e335. [8] A.P. Ahlen, Theoretical and experimental aspects of the energy loss of relativistic heavily ionizing particles, Rev. Mod. Phys. 52 (1990) 121e173. [9] A. Jablonski, S. Tanuma, C.J. Powell, New universal expression for the electron stopping power for energies between 200 eV and 30 keV, J. Surf. Interface Anal. 38 (2006) 76e83. [10] Akar A, Gümüs H. Electron stopping power in biological compounds for low and intermediate energies with generalized oscillator strength (GOS) model. Radiat. Phys. Chem.; 73: 196e203. [11] H. Gümüs, New stopping power formula for intermediate energy electrons, Appl. Rad. Isot. 66 (2008) 1886e1890. [12] H. Gümüs, O. Kabadayi, Practical calculations of stopping powers for intermediate energy electrons in some elemental solids, Vacuum 85 (2010) 245e252. [13] H. Sugiyama, Electronic stopping power formula for intermediate energies, Radiat. Eff. 56 (1981) 205e209.
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Electron impact stopping powers for elemental and compound media A.K. Fazlul Haque a, b, *, M. Atiqur R. Patoary a, M. Alfaz Uddin a, b, c, Arun K. Basak a, M. Ismail Hossain a, Mahmudul Hasan d, Bidhan C. Saha e, M. Maaza b, f, g a
Department of Physics, University of Rajshahi, Rajshahi, Bangladesh Nano Sciences African Network (NANOAFNET), iThemba LABS-National Research Foundation, Old Faure road, 7129, Somerset West, South Africa c Khwaja Yunus Ali University, Enaetpur, Chowhali, Sirajgonj, 6751, Bangladesh d Department of Physics, Pabna University of Science & Technology, Pabna 6600, Bangladesh e Department of Physics, Florida A & M University, Tallahassee, FL, USA f Council for Scientific and Industrial Research, PO Box 395, Pretoria, 0001, South Africa g UNESCO-UNISA Africa Chair in Nanosciences/Nanotechnology, College of Graduate Studies, University of South Africa (UNISA), Muckleneuk Ridge, PO Box 392, Pretoria, South Africa b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 21 February 2016 Received in revised form 27 July 2016 Accepted 28 July 2016 Available online 4 August 2016
A semi-analytical model is proposed for evaluating the electron stopping powers (ESPs) of material media including both elements and compounds in the energy range 1.0 eVe100 MeV. This paper reports details of our calculations using the effective atomic electron number, effective mean excitation energies of target atoms and realistic electron density distribution of the target atoms. This simple formula is capable of generating accurate cross-section for diverse systems. We present here only few simple cases. In comparison with other theoretical predictions, our calculated ESPs for elemental [C, Al, Cu, Pt, Au and Pb] and compound [H2O, C5H5N5O, GaAs, Bi2Sr2CaCu2O8, SiO2 and ZnTe] media agree reasonably well with the experimental findings. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Stopping power Material media Relativistic charge density distribution
1. Introduction Electron stopping powers (ESPs) of material media, usually defined as the average energy loss per unit distance along the electron path (in the units of energy/length), are needed in diversified areas of researches and applications, involving energy loss, including quantitative calculations of the dosages in radiation therapy for living tissues, Monte Carlo simulation study in electron microscopy for manipulating materials in both nuclear and space applications, quantitative surface analysis, the design of particle detectors [1], etc. It demands a simple-to-use formula capable of furnishing reasonably accurate ESPs of different media over a wide range of projectile energies. To date there are considerable attempts to calculate on ESPs using various formulas, methods and models [2e15]. Some of them are heavily dependent on fitting species dependent parameters, some are relied on accurate oscillator strength evaluation, some are involved in evaluating the complex dielectric response function, etc. However, for different
* Corresponding author. Department of Physics, University of Rajshahi, Rajshahi 6205, Bangladesh. E-mail address: [email protected] (A.K.F. Haque). http://dx.doi.org/10.1016/j.vacuum.2016.07.038 0042-207X/© 2016 Elsevier Ltd. All rights reserved.
energy regime, many of them are intuitively divided into slots to represent ESPs. Moreover, their application to molecular targets still poses considerable challenges. The search for a formula, with validity over a wide range of incident energies and for a large domain of species, motivated the present work. Rohrlich and Carlson [2] have reported a formula for the average rate of energy loss of positrons and electrons in passing through matter. Sugiyama [13] has reported two approaches to derive ESP formulas: (a) the Bethe-Bloch formulas [8], referred to here as Bethe formula, and (b) the Lindhard-Scharff-Schiott theory [3,14]. We follow closely the Rohrlich and Carlson [2] based on the earlier work of Bethe formula [16]. This model without the shell- and density-corrections [4,9,14] shows reasonable results for the incident electron energies ranging 50 keV to a few MeV. Sugiyama [7,13,15] introduced the effective charges [6], z* of the projectile and Z* of the target atom, and effective mean excitation energy (EMEE) I* to modify the formula of Rohrlich and Carlson [2]. Gümüs group [11,12] further modified the formulas of Rohrlich and Carlson [2] and Sugiyama [7,15] to obtain analytic expressions for Z* and I* of the target atom using the Tietz screening function [17]. For the incident energies 10 eV E 100 keV, this model yields fairly accurate ESPs. For 200 eV E 30 keV, Jablonski et al. [9] have
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reported an empirical formula with species dependent parameters that rely heavily on the available experimental data. At E 1 keV, the ESPs for elemental solids are calculated successfully and are available in Pages et al. [5] and ESTAR database [18]. From all these theoretical works, it is apparent that none of the above-mentioned formulas is able to describe the experimental ESP data at the energies E < 10 eV. Moreover, no single formula, available in the literature, goes smoothly from E ¼ 1 eV to ultrahigh energies to reproduce simultaneously the available experimental data up to about 10 MeV and the ESTAR-generated ESP data beyond. In particular, the formula of Gümüs group [11,12], which can describe the experimental data beyond 10 eV, is not applicable to the energy region dominated by Bremsstrahlung. On the other hand, ESTAR, claimed to be reliable in the high and ultra-high energies, where no experimental data are available, is found to reproduce the experimental ESP data in the light elemental targets beyond 10 keV. It is, therefore, desirable to find a simple-to-use formula capable of reproducing successfully the experimental ESPs at lower energies and the ESTAR-evaluated ESPs from the Bremsstrahlung data. The purpose of this study is to present a formula after proper modifications of the earlier formulas in Refs. [2,7,11,12] incorporating the effective charges, z* and Z* as well as the target EMEE I*. The numerical density distributions (DDs) of the target electrons are evaluated numerically using the multi-configuration code of Desclaux [19], which uses the Dirac-Hartree-Fock electron wave functions. These realistic DDs are then employed to calculate both the effective charges and mean excitation energies of the target elements and molecules considered herein. The numerical DDs are represented by the piecewise cubic spline functions to evaluate the integrals [see Eqs. (3) and (4), shown below] for the effective charge Z* and EMEE I*. Although the present semi-analytical model is capable of calculating ESPs for elemental solid targets up to Z ¼ 92, we consider here only few sample media for the energies 1 eVe100 MeV. Our results are compared with the available experimental and other theoretical findings including ESTAR [18]. To establish the suitability, the proposed formula has been tested systematically on 6 important compound media including 2 biological molecules, water (H2O) and Guanine (C5H5N5O) in nucleic acid, and 4 semiconductors, namely silica (SiO2), Gallium arsenide (GaAs), Bi-High-TC superconductor (Bi2Sr2CaCu2O8) and Zinc tellurium (ZnTe). 2. Basic theory The formula of the stopping power for electron in a medium, modified by Refs. [11,12] from the formula of [2], is;
. dE r E S ¼ ¼ 2pe4 N0 z2 Z ln þ FðtÞ=2 eV A dx I AE
(1)
with
FðtÞ ¼ 1 b2 þ
h
.
i. 1 þ t2 :
t2 8 ð2t þ 1Þln2
(2)
Here, t ¼ E/mc2 is the kinetic energy in units of the electronic rest mass; e, N0, r and A are, respectively, the electronic charge, Avogadro number, atomic density in gm/cm3 and atomic mass in gm. E is the electron energy in eV and b is the velocity of incident electron in the unit of the velocity of light. In Eq. (1) the factor in front of the parenthesis can be written as (ra=AEÞ eV/Å, since a ¼ 2pe4 N0 ¼ 785 [20]. The function FðtÞ [7,11,12] has dependency not only on the projectile energy but also on its kind. However, it is independent of
its atomic number Z. ESPs contain same relativistic components through t and electron density in Eq. (1). In order to determine Z* and I* of the target atoms, Gümüs and Kabadayi [12] used Thomas-Fermi theory [21] which describes an atom having Z bound electrons with radially symmetric electron density. Sugiyama [7] obtained Z* and I* from Bohr's stripping criterion for the effective charge of incident heavy ions and target atoms, and from the Lindhard and Schraff theory [22] concerning the quantities in EMEE I*. These functions in Eq. (1) are given in Refs. [7,13] as;
Z ¼
Z∞
4pr 2 nðrÞdr
(3)
rc
and
1 ln I ¼ Z
Z∞
ln gZup ðrÞ 4pr 2 nðrÞdr:
(4)
rc
here, r is the distance of the bound electron from the nucleus; n(r), the electronic charge density in target, up ðrÞ ¼ ð4pe2 nðrÞ=mÞ0:5 , the local plasma frequency and g ¼ 20:5 [7,22]. The velocity of the incident electron vi at its distance from the nucleus rc satisfying Bohr‘s criterion is equal to that vb of the bound electron. A bound electron for which r > rc moves with a velocity vb < vi and can contribute to the stopping power. The inner electrons with r < rc and vb > vi are rigidly bound leading to an insignificant energy loss of the incident electron. The quantity rc is defined by the Bohr stripping criterion [23] vi ¼ ðbZ=mÞð3p2 nðrc ÞÞ1=3 with b as the proportionality constant of the order of 1.26 [7]. The quantity rc is obtained from the potential-energy condition of Yarlagadda et al. [6]
1 2 mv þ Uðrc Þ ¼ 0; 2 i
(5)
as pointed out earlier with vi ¼ vb. Here, U(r) is the potential experienced by the incident electron. Sugiyama [7] employed the semi-empirical effective charge of the incident electron z* as given by
z ¼ 1 exp 2200b1:78 :
(6)
This conforms to the reason that z* increases with the incident energy as all processes contributing to the stopping power are enhanced with increasing velocity of the projectile electron. Our proposed formula reported in this study embodies the following modifications in addition to the use of relativistic treatment for the incident and bound electrons and the realistic DDs: in theright-hand 1. The first term within the square brackets side of Eq. (1) has to be modified from ln IE to ln 3:75E to repreI sent both the peak location and the magnitude of relative ESPs. This is linked to the fact that the energy loss in the low energy limit is proportional to the square of the effective velocity veff of the incident electron (see Fig. 3 of [6]). As the projectile electron approaches the bound electron, veff gets enhanced to increase it's effective energy Eeff, which is substantially greater than E in the low energy region. To simulate Eeff, a number of grid searches on an appropriate factor with E are found essential to provide an overall best description of the experimental data near the peak region of the elemental targets. This procedure results in the factor of 3.75 with E, which is found good also for
A.K.F. Haque et al. / Vacuum 132 (2016) 123e129
the compound targets. This factor takes care of the correlation effect [24]. 2. FðtÞ given in Eq. (2) goes to negative values beyond E ¼ 80 keV and thus cannot reproduce the ESTAR data where the main contribution to ESPs is due to Bremsstrahlung [25], which increases with E. It thus suggests to replace FðtÞ by its modified form which not only describes the ESTAR data at high energies but also supplements the Bethe contribution to reproduce the low energy behaviour of the experimental data. In the simplest form, the modified function F ðtÞ looks like
F ðtÞ ¼ 1 þ t2 :
(7)
3. To reproduce the magnitude of all the available experimental data accurately a normalization constant (x ¼ 0.529) is introduced. Incorporating all these above modifications into Eq. (1), our present formula for calculating the ESPs for elemental targets becomes;
S ¼ 2pe4 N0
. E z2 Z x ln þ FðtÞ=2 eV A I AE
r
(8)
For the compound media, the following Bragg-Kleeman rule of the linear combination SC [11,26] from the elemental ESPs is used:
SC ¼
X
n i Si :
(9)
i
here, Si represents the ESPs for the i-th element and ni denotes its molar fraction in the compound target. 3. Analysis and results All ESPs, reported here, are calculated using Eqs. (3), (4), (6) and (8)), with a generation of the electron density function from the code [19]. In Fig. 1(a) and (b), we present the effective number of electrons Z* and the mean excitation energy I* as a function of the incident energy. It is evident from the figures that our calculated values for Z* and I* approach the atomic number Z and the experimental mean excitation energies [12], respectively. For small values of Z* at lower incident energies, it is indicated that inner electrons of target atom do not contribute to the energy loss of incident electrons. The higher is the energy, the greater is the number of bound electron participating in the collision process until the saturation of Z* is reached around E < 100 keV. Similar trend is also noticed in the mean excitation energy. With increase of the incident energy, more inelastic channels contribute till saturation is reached. We present here the calculations of ESPs for the elementary species of C, Al, Cu, Pt, Au and Pb; and compound species of H2O, C5H5N5O, GaAs, Bi2Sr2CaCu2O8, SiO2 and ZnTe. Table 1 shows the densities obtained from Ref. [27] for both elemental and compound media. All reported ESPs herein are calculated using them. Sources of the experimental data are collected from Joy's database [28]. The sources comprise Luo [29] for C, Pt, and C5H5N5O; Hovington et al. [30] for C, Al, Cu, Pt, SiO2 and ZnTe; Kalil et al. [31] for Al; Fitting [32] for Al; Al-Ahmad and Watt [33] for Al, Cu and Au; Ishigure et al. [34] for Al; Garber et al. [35] for Al; Luo et al. [36] for Cu, Au, Pb, H2O, C5H5N5O and SiO2; MacPherson [37] for Al, Cu and H2O; and La Verne and Mazumder [38] for H2O, GaAs and Bi2Sr2CaCu2O8. The ESP-formula given in Eq. (8) has been tested rigorously for elemental solid media using the realistic DDs generated from the code [19]. The effective quantities Z*, I* and z*, as defined earlier, are
125
calculated using (3), (4) and (6), respectively. The ESP results with Eq. (1) coupled with FðtÞ in Eq. (2) produce an underestimation of the ESP data in the lower energy region including the peak. Before discussing the role of different terms in Eq. (8), we apply it to a typical case of C to examine the influence of medium-density on ESPs. Fig. 1(c) depicts explicitly the structural dependence of ESPs leading to significant variations in values due to its different forms. For example, diamond has the highest ESP peak of around 12.14 eV/ Å at E ¼ 100 eV; at the same energy graphite yields 7.79 eV/Å and amorphous carbon produces around 6.5 eV/Å. The effect due to difference in densities is reflected dominantly in the peak region and becomes negligibly small at higher energies. The ratio of ESPs for diamond and graphite S(diamond)/S(graphite) ¼ 1.558 in the peak region is found matching with their density-ratio (see Table 1), as expected in the first order. To understand the role of individual terms in Eq. (1) we, in Fig. 2(a), show explicitly the separate contributions due to the first term in (1) denoted by BTH1 [dashed line (magenta in online)] and that due to the second term by FT1 [dotted line (cyan in online)]. BTH1 generates negative ESP values in the low energy region below 7.0 eV and FT1 produces negative values in the high energy region E > 80 keV. The sum contribution BTH1þFT1 [dash-dotted line with open triangles (violet in online)], underestimates the experimental ESP data considerably. To reproduce the ESP data in the energy region E < 2 keV, a factor of 3.75 is multiplied to E inside the square brackets in Eq. (8), as discussed earlier in Section 2. A simplified form of F ðtÞ is then searched for as a special requirement for satisfying the ESP data at higher as well as lower incident energies and this procedure produces the form of F ðtÞ shown in Eq. (7). The individual contributions are displayed in Fig. 2(a), where the first term in the right hand side of Eq. (8), denoted by BTH in short-dashed line (green in online), and the second term with F ðtÞ, depicted as FTP in thinsolid line (blue in online). FTP contributes solely to ESPs at E > 10 MeV. The shooting of ESP data at E > 10 MeV is understood as resulting from the dominant contribution of Bremsstrahlung [25], which remains proportional to the square of the incident energy E [39]. The sum BTH þ FTP, labelled ’present work’ in Fig. 2(a), governed by Eq. (8), describes the experimental data profoundly well and predicts ESPs in agreement with the ESTAR data in the higher energy domain. The predicted ESPs for all other elemental targets are compared in Fig. 2 with the experimental data and findings from other available theoretical calculations. The predicted results from the present model agree closely with the experimental data including those from MacPherson [37] and the ESTAR-derived ESPs at the higher energies beyond 30 MeV, where no experimental data are available as far as we are aware of. Formula of Shinotsuka et al. [40] works well in the energy range of 25 eVe100 MeV (see Fig. 2(c, d)) except for Al (Fig. 2(b)) where there is an overestimation of data near the peak region. The predicted ESPs from the model of ref. [12] also describe satisfactorily the data in the 10 eV < E < 100 keV range except for Al near the peak region while ESTAR works well beyond ndez-Varea et al. [41] tens of keV. The calculated results from Ferna reproduce the data well for Cu (Fig. 2(c)), reasonably well for Al (Fig. 2(b)) albeit disagreement with the data near the peak region and satisfactorily for Au (Fig. 2(e)) with an overestimation of data near the peak. Our predicted ESPs reproduce the available experimental data and ESTAR-derived data for all elemental targets in the considered range of 1 eV < E < 100 MeV nicely. The analytic form of ESPs from Eq. (8) is then applied to several compound targets. In Fig. 3 we compare our results for compound media with the experimental data and ESTAR derived data. In Fig. 3(a), our calculated ESPs for H2O in the solid phase compare well with the experimental data of Luo et al. [36] for vitreous ice, of
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Fig. 1. (a) The effective number of electrons Z* and (b) the effective mean excitation energy I* of various target atoms as a function of incident electron energy. (c) present ESP calculations for three forms of carbon: graphite, diamond and amorphous in solid, dashed and dash-dotted lines, respectively.
Table 1 Densities of media in unit of gm/cm3 taken from Ziegler [25]. (a) Density of elemental media Medium Density
C (graphite) 2.253
C (diamond) 3.51
C (amorphous) 2.0
Al 2.70
Cu 8.92
Pt 21.45
Au 19.31
Pb 11.34
(b) Density of solid elements associated with the compound media Medium Density
H 0.0715
C 2.25
N 1.026
O 1.426
Si 2.3212
Verner and Mozumder for vitreous ice [38] and McPherson [37] for liquid water, and ESTAR-data for liquid phase [18]. The peak value of 3.71 eV/Å at 0.11 keV for ESPs from the proposed formula compares well with those of 3.0 eV/Å at 0.10 keV and 3.72 eV/Å at 0.08 keV from the experiments of [36] and [38], respectively. The peak value of 3.18 eV/Å at 0.10 keV predicted by Ref. [11] for water in liquid phase is close to the experiment and is less than our result. ESTAR in
Ca 1.54
Cu 8.92
Zn 7.14
Ga 5.904
As 5.727
Sr 2.6
Te 6.25
Bi 9.8
its normal workable range beyond E ¼ 30 keV is found to reproduce the sole experimental data of MacPherson [37], available for the liquid phase in the E ¼ 510 MeV range (Fig. 3(a)). Among other compound media, ESP calculations of Guanine (C5H5N5O) using the generalized oscillator strength model are provided by Akar and Gümüs [10]. Their predicted ESPs reproduce the experimental data well in the 20 eV < E < 1 MeV range. Our model reproduces all the
A.K.F. Haque et al. / Vacuum 132 (2016) 123e129
127
Fig. 2. Electron stopping powers for (a) C (graphite), (b) Al, (c) Cu, (d) Pt, (e) Au and (f) Pb. Theoretical predictions are given in lines while experimental data in symbols. The results from ESTAR [18] calculations, Gümüs and Kabadayi [12], Shinotsuka e al [40]. and Fernandez-Varea et al. [41] are shown along with the present results for comparison. Predictions marked BTH1, FT1, BTH and FTP in (a) are explained in the text.
available experimental results and ESTAR-derived data for all elemental and molecular media from 1 eV to 100 MeV successfully. The dependence of ESPs on medium density shown in Fig. 1(c) for C is also noticed in the results, displayed in Fig. 3(a), for H2O in the solid and liquid phases. The ratio of our predicted maximum
ESP in the solid phase to that of [12] S(solid Present)/S(liquid) [12] ~1.18 in the peak region conforms to the ratio rsolid =rliquid ¼ 1:22 of the corresponding densities (using r for liquid H and that for liquid O from websites https://en.wikipedia.org/wiki/Liquid_hydrogen and https://en.wikpedia.org/wiki/Liquid_oxygen, respectively). On
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A.K.F. Haque et al. / Vacuum 132 (2016) 123e129
Fig. 3. Same as in Fig. 2 for compound media e.g. (a) H2O (solid), (b) C5H5N5O, (c) GaAs, (d) Bi2Sr2CaCu2O8, (e) SiO2 and (f) ZnTe. Results from Gümüs [11] and ESTAR [18] in (a) are for liquid water.
the other hand, our predicted ESPs and those from ESTAR [18] for water in liquid phase beyond E ¼ 30 keV are very close and both agree with the experimental data of [37] in the E ¼ 525 MeV (see Fig. 3(a)). Although our predicted ESPs for water in solid phase and those for liquid water from Ref. [11] are different in the peak region
because of densities, both agree with ESTAR data [18] and experimental data [37] in the energy region beyond E ¼ 30 keV. The density effects of H2O in two phases on the ESP values thus bear a trend, similar to difference in ESPs for C in varied structures with different densities in Fig 1(c).
A.K.F. Haque et al. / Vacuum 132 (2016) 123e129
4. Conclusion The present model not only reproduces successfully the observed ESP data for 1 eV E 10 keV, below the range of ESTAR calculations, but also agrees excellently with the accurate measurements of MacPherson [37] in the range 5 E 30 MeV. Our predicted results then join smoothly to the region of dominant Bremsstrahlung effects from 500 KeV to 100 MeV associated with the ESTAR-evaluated data. The proposed model produces useful results for both elemental and compound media over wide range of the incident energies, albeit being simple in structure. This could be a lucrative model for the modeling of the electron energy loss phenomena in material and biological systems of interest. For a rapid generation of accurate ESPs for science, industries and technologies this simple-to-use model will be very useful. Acknowledgments We are thankful to Professor F. B. Malik, who passed away on 05 August 2014, for his important advice regarding the improvement of the manuscript. The authors would like to thank Professor D. G. Sarantites in Washington University at St. Louis, MO, USA for stimulating discussion. A. K. Fazlul Haque would like to thank TWAS-UNISCO associateship scheme for partial funding. References [1] M.R. Cleland, Industrial applications of electron accelerators, 2009, pp. 383e416. Published version from, CERNcds.cern.ch/record/1005393/files/ p383.pdf. [2] F. Rohrlich, B.C. Carlson, Positron-electron differences in energy loss and multiple scattering, Phys. Rev. 93 (1954) 38e44. [3] J. Lindhard, M. Scharff, H.E. Schiott, Integral equations governing radiation effects, Mat.-Fys. Medd. Dan. Vid. Selsk. 33 (1963) 1e42. [4] R.M. Sternheimer, R.F. Peierls, General expression for the density effect for the ionization loss of charged particles, Phys. Rev. B 3 (1971) 3681e3692. [5] L. Pages, E. Bertel, H. Joffre, L. Sklavenitis, Energy loss, range, and bremsstrahlung yield for 10-keV to 100-MeV electrons in various elements and chemical compounds, At. Data Tables 4 (1972) 1e127. [6] B.S. Yarlagadda, J.E. Robinson, W. Brandt, Effective-charge theory and the electronic stopping power of solids, Phys. Rev. B 17 (1978) 3473e3883. [7] H. Sugiyama, Stopping power formula for intermediate energy electrons, Phys. Med. Biol. 30 (1985) 331e335. [8] A.P. Ahlen, Theoretical and experimental aspects of the energy loss of relativistic heavily ionizing particles, Rev. Mod. Phys. 52 (1990) 121e173. [9] A. Jablonski, S. Tanuma, C.J. Powell, New universal expression for the electron stopping power for energies between 200 eV and 30 keV, J. Surf. Interface Anal. 38 (2006) 76e83. [10] Akar A, Gümüs H. Electron stopping power in biological compounds for low and intermediate energies with generalized oscillator strength (GOS) model. Radiat. Phys. Chem.; 73: 196e203. [11] H. Gümüs, New stopping power formula for intermediate energy electrons, Appl. Rad. Isot. 66 (2008) 1886e1890. [12] H. Gümüs, O. Kabadayi, Practical calculations of stopping powers for intermediate energy electrons in some elemental solids, Vacuum 85 (2010) 245e252. [13] H. Sugiyama, Electronic stopping power formula for intermediate energies, Radiat. Eff. 56 (1981) 205e209.
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