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Evidence for observed projectile-z dependence in values of target parameters extracted from stopping power measurements
Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 273–280 www.elsevier.com / locate / elspec
Evidence for observed projectile-z dependence in values of target parameters extracted from stopping power measurements L.E. Porter* Radiation Safety Office, Washington State University, P.O. Box 641302, Pullman, WA 99164 -1302, USA
Abstract One version of modified Bethe–Bloch stopping power theory features two parameters characteristic of the target material, the mean excitation energy and the Barkas-effect parameter, that can be evaluated through fits to accurate stopping power measurements. In several recent investigations, the values of these parameters thus determined have revealed a dependence on projectile atomic number. Most of those studies have featured target materials possessing aggregation effects, such as solid compounds and alloys. In the current investigation, the search for systematics in projectile-z dependence has been extended to encompass other target materials, including elemental targets. Most of the results are corroborative of the previously reported trends. 2003 Elsevier Science B.V. All rights reserved. Keywords: Stopping power; Energy loss; Bethe–Bloch theory; Eight projectiles; Heavy ions; Projectile-z
1. Introduction The stopping power of matter for charged particles has long remained a topic of vital interest in numerous areas of physics, both basic and applied, since it is often necessary to ascertain the energy loss of a given projectile in a particular target. An understanding of the processes involved in energy loss can lead to a formalism for calculation based on sound theory. A formalism that has proved successful over several orders of magnitude of projectile energy is based on the modified Bethe–Bloch theory of stopping power [1]. This theory contains several target-dependent parameters, notably the shell corrections and the mean excitation energy of the basic theory and the Barkas-effect parameter associated with the second*Tel.: 11-509-335-8916; fax: 11-509-335-1615. E-mail address: [email protected] (L.E. Porter).
order Born approximation as a modification of that basic theory [2–5]. The Barkas-effect correction, initially called the projectile-z 3 correction, represents a modification of the projectile-z 2 dependence of basic Bethe–Bloch theory. When the Bloch term [6] was restored to the Bethe–Bloch formalism [7], a projectile-z 2n (n$2) correction term appeared in the Bethe–Bloch formula for stopping power as well. The author has conducted numerous analyses of stopping power measurements, commencing in 1974 [8], in order to extract values of various parameters of the formalism [3–5,9]. The focus of those studies has been on the target mean excitation energy (I) and the Barkas-effect parameter (b) in particular. A large fraction of the previous studies by the author have dealt with target materials such as compounds and alloys, which manifest aggregation effects (e.g. Refs. [10–12]). One method of treating such cases is to invoke Bragg’s rule of the (linear)
0368-2048 / 03 / $ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016 / S0368-2048(03)00078-1
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additivity of stopping effects [13]. However, the problem of assigning approximate parameter values in such cases represents added complexity in data analyses. A major portion of recent studies has been devoted to the study of observed trends in projectilez dependence of extracted values of the mean excitation energy (I) and Barkas-effect parameter (b), and the target materials studied have been compounds of low (average) atomic number and one alloy. The data revealing such trends were summarized recently [14]. The observed projectile-z dependence embraces both a value of I that decreases, and a value of b that increases, with increasing projectilez for a given target material. The objective of the current investigation was to search for such trends with elemental targets in cases where stopping power measurements of considerable accuracy had been reported. The preferable sets of measurements sought were those conducted by the same experimental group for at least two projectiles of increasing z, such as protons and alpha particles. One such case is the measurement of the stopping powers of Ti for protons and alpha particles by the Nara group [15,16]. Another is the set of definitive measurements of the stopping powers of four elemental targets (Al, Cu, Ag, and Au) for protons, alpha particles, and 7 Li ions by the Aarhus group [17]. (Only the Al and Cu data were used in this study.) A supplemental set of measurements of the stopping power of Ti for alpha particles [18], and another of the stopping power of Cu for 7 Li ions [19], were selected for comparison with the aforementioned sets of measurements of high accuracy [15–17]. Data for the stopping power of Zn for 7 Li ions, included in Ref. [19], were also analyzed. Those results were compared with some previous results for protons as projectiles [20].
2. Theory The formalism of the Bethe–Bloch theory of stopping power, replete with various modifications, has been described in several recent studies [10–12]. The stopping power, S, of an elemental target of atomic number Z and atomic weight A for a projectile with atomic number z and velocity v 5 b c can
be calculated in units of keV?cm 2 / mg from the expression, 0.30706 2 Z S 5 ]]] z ]L A b2
(1)
The quantity, L, representing the (dimensionless) stopping number per target electron, can be written as a three-term sum: L 5 Lo 1 zL1 1 L2
(2)
where the basic stopping number, Lo , consists of four terms
S
D
2mc 2 b 2 Lo 5 ln ]]] 2 b 2 2 ln I 2 C /Z 12b2
(3)
with mc 2 signifying the rest mass energy of the electron, I denoting the target mean excitation energy, and C representing the sum of target shell corrections. The mean excitation energy can be evaluated through a fit to accurate stopping power measurements. The method of Bichsel [1] can be used to calculate shell corrections, wherein the Kand L-shell corrections of Walske [21,22] are employed with appropriate scaling factors inserted in the L-shell correction in order to calculate the Mand N-shell corrections: C 5 CK ( b 2 ) 1VL CL (HL b 2 ) 1VM CL (HM b 2 ) 2
1VN CL (HN b )
(4)
In this expression, CK and CL denote the K- and L-shell corrections, respectively, whereas Vi and Hi (i5L, M, N) represent the scaling factors. Two correction terms of higher order in projectilez appear in Eq. (2). The first of these terms, the (zL1 ) term, is called the Barkas-effect term, and the second term is known as the Bloch term. A brief history of the discovery and early investigations of the Barkas effect appears in Ref. [23]. Both terms have been reviewed pursuant to a random-phase approximation evaluation of the Barkas-effect term [24]. The latter term was evaluated in the present study by means of the formalism that provided the best agreement with a large set of experimental data [23], i.e. the formalism developed by Ashley et al. [3–5,9]. Hence the (zL1 ) term was calculated for the analyses of this study from the expression [3–5],
L.E. Porter / Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 273–280
F(b /x 1 / 2 ) L1 5 ]]] (5) Z 1 / 2x 3 / 2 where F signifies a function graphed in Ref. [3], x 5 (18 787)b 2 /Z, and b denotes the sole free parameter of the formalism. The Bloch term [6], which was initially included as a correction term to allow the transition from the first Born approximation to classical scattering at low projectile velocities [24], can be calculated from L2 ( y) 5 c (1) 2 Re[c (1 1 iy)] (6) where c denotes the digamma function [25] and y 5 za /b, for a signifying the fine structure constant. (In the case of highly relativistic projectiles, a density effect correction term [26] must be added to Lo and a second term [2] must be added to L1 [2,27].) A complete set of parameters for a specified target consists of the target mean excitation energy (I), the shell correction scaling parameters (Vi and Hi with i5L, M, N), and the free parameter of the Barkaseffect correction term (b). All of the above parameters used in the current study are independent of both projectile velocity and a presumably constant projectile charge (ze). However, if the projectile energies considered are so low as to permit the gain and loss of electrons by projectiles traveling at velocities comparable to those of target atomic electrons, then some sort of projectile effective charge formalism must be used to simulate the results of gain and loss of electrons. The technique used [28] is the representation of the bare projectile charge (ze) by an ‘effective charge’ defined as (g ze), where
g 5 1 2 z e 2lv r
(7)
Here vr signifies the ratio of projectile velocity in the laboratory frame of reference (v) to the Thomas– Fermi velocity (e 2 /h)z 2 / 3 , so that vr 5 b /a z 2 / 3 . The symbols, l and z, which denote the effective-charge parameters valid over the complete projectile-velocity interval considered, must be evaluated for any specified projectile–target combination [28]. 3. Method of analysis A procedural strategy for the analysis of stopping power measurements has been developed over the
275
Table 1 Target atomic number (Z), atomic weight (A), and shell correction scaling parameters (VL , HL , VM , HM ) Target
Z
A
VL
HL
VM
HM
Al Ti Cu Zn
13 22 29 30
26.98 47.90 63.55 65.38
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
0.375 1.25 2.25 2.25
12.00 11.20 6.18 5.87
past three decades [28]. Stopping power measurements are provided with a fit using modified Bethe– Bloch theory so as to establish best-fit values of I and b, with shell correction scaling parameters prescribed by the rubric of Bichsel [1]. The figure of merit used in the search procedure is the root-meansquare relative deviation calculated from measured stopping powers, s. This figure of merit is defined as ]]]]] 1 N Sm 2 Sc 2 ]]] s5 ] N i 51 DSm i
œ OS
D
(8)
for measurements at N energies, where Sc represents the calculated stopping power, and Sm and DSm denote the measured stopping power and the associated uncertainty, respectively. Thus a value of s near unity constitutes acceptable agreement between theory and experiment. Values of target parameters, including shell correction scaling parameters, assigned for each elemental target are displayed in Table 1.
4. Results and discussion The results of the analyses will be treated in two categories, those pertaining to the Ti target and those pertaining to the Al, Cu, and Zn targets.
4.1. Ti target Two experiments for the measurement of the stopping power of Ti, one with proton projectiles [15] and the other with alpha particle projectiles [16], have been reported very recently. Both measurements purportedly feature high accuracy, one with 0.40% [15] and the other with 0.35% [16], and both experiments had in common the two accelerators used plus the two leading reporting au-
276
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Table 2 Summary of results from analyses of Ti measurements, including the projectile energy interval of measurements (DE), number of measurements (N), mean excitation energy (I), Barkas-effect parameter (b), and figure of merit (s ) Projectile
Reference
DE (MeV)
N
I (eV)
b
s
Proton Alpha particle Alpha particle
[15] [16] [18]
0.5–13.5 2.0–20.5 5.2–12.8
39 44 9
239.660.8 229.260.6 252.0
2.2760.05 2.0460.02 1.70
1.17 1.53 0.21
thors. Hence these data were selected for analysis in the context of recently reported observed trends in the I- and b-values extracted from pairs of measurements with different projectiles [14]. Results of these detailed analyses, displayed in Table 2, were only partly consistent with expectations based on observed trends [14]. That is, the I-values decreased with increasing z as expected, but the b-values also decreased with increasing z. The uncertainties given for I- and b-values were obtained by a complicated procedure developed and described in detail previously [28,29]. The excellence of fit for these two analyses is clearly shown in Figs. 1 and 2. It is curious that both extracted b-values lay considerably higher than the value expected by two of
the architects of the Barkas-effect formalism [9] (b51.460.1), especially since the (I, b) values reported earlier by the Nara group for the proton–Ti combination was (232.364.9 eV, 1.26) [20]. Another set of measurements for the alpha particle–Ti combination, of only 3% accuracy, was reported nearly two decades ago [18]. Results of the analysis of those measurements appear in Table 1. Clearly the I-value lies well above expectation, whereas the b-value is eminently plausible.
4.2. Al, Cu, and Zn targets The measurements of the stopping powers of Al and Cu for protons, alpha particles, and 7 Li ions
Fig. 1. Comparison of calculated (open diamonds) with measured (solid circles) stopping powers [15] for protons traversing targets of Ti, using a two-parameter fit for mean excitation energy (I) and Barkas-effect parameter (b).
L.E. Porter / Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 273–280
277
Fig. 2. Comparison of calculated (open diamonds) with measured (solid circles) stopping powers [16] for alpha particles traversing targets of Ti, using a two-parameter fit for mean excitation energy (I) and Barkas-effect parameter (b).
reported by the Aarhus group [17] served as the basis for revision [9] of the initial formalism for inclusion of the projectle-z 3 effect [3–5] following the restitution of the Bloch term to the Bethe–Bloch formula and forwarding of suggestions concerning inclusion of close-collision contributions to the (then-renamed) Barkas effect [7]. Hence, these measurements were
subjected to analysis with the usual method [10–12]. Results are shown in Table 3. In the case of Al, the excellence of fits is indicated by the values of the figure of merit (s ) in Table 3. Clearly the anticipated projectile-z dependence is present in the extracted I-values, whether or not one includes an effective charge parameter for improve-
Table 3 Summary of results from analyses of Al, Cu, and Zn measurements, including the projectile energy interval of measurements (DE), number of measurements (N), mean excitation energy (I), Barkas-effect parameter (b), effective charge parameter ( l), and figure of merit (s ) Target
Projectile
Ref.
DE (MeV)
N
I (eV)
b
l
s
Al Al Al Al Cu Cu Cu Cu Cu Zn Zn Zn
Proton Alpha particle 7 Li ion 7 Li ion Proton Alpha particle 7 Li ion 7 Li ion 7 Li ion Proton 7 Li ion 7 Li ion
[17] [17] [17] [17] [17] [17] [17] [19] [19] [20] [19] [19]
0.8–6.4 3.2–22.2 8.3–23.4 8.3–23.4 1.2–7.2 5.6–19.9 11.0–19.3 3.7–16.1 3.7–16.1 6.5 3.9–8.7 3.9–8.7
29 25 12 12 31 19 7 24 24 1 7 7
166.5 165.4 156.8 160.4 329.0 328.5 324.5 311.7 327.4 331.668.2 335.5 326.8
1.37 1.36 1.79 1.42 1.30 1.41 1.58 1.78 1.55 1.2660.20 1.82 1.71
– – – 1.26 – – – – 1.46 – – 1.38
0.20 0.25 0.43 0.31 0.24 0.15 0.31 0.36 0.36 – 0.45 0.43
278
L.E. Porter / Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 273–280
ment of the fit of the 7 Li ion data. Apparently the b-value decreases slightly in proceeding from the z51 case to the z52 case, however, contrary to the anticipated behavior. In fact, the two values prior to round-off are 1.368 (z51) and 1.360 (z52), and the b-dimension of the basic cell in two-parameter space for the final search is just 0.008. Hence the (estimated) uncertainties in the two b-values [28,29] would easily provide overlap, and the anticipated behavior is approximately observed even in the z51 and z52 cases. Elsewhere, the trends are manifestly followed. Fits to the Cu data [17] are also excellent, as indicated by the values of s in Table 3. The trends are clearly followed for these three projectiles, although the I-values for the z51 and z52 cases are very close to one another. Inclusion of the additional, more recent measurements [19] further buttresses the corroborative evidence. A lower projectile energy interval was used for those measurements, so that a need for insertion of a single effective charge parameter is quite evident. Fig. 3 contains a graph showing the excellence of fit for the three-parameter option.
The measurements of the stopping power of Zn for Li ions, reported as possessing an accuracy of some 4% [19], provides plausible values of I and b, especially when an effective charge parameter is included in the fit. The I-value decreases while the b-value increases during the transition from the z51 case to the z53 case for the three-parameter fit of the 7 Li data. The excellence of fit is displayed in Fig. 4. Thus the trends are reasonably well followed for these sets of data [19,20] as well. 7
5. Summary In the course of previous analyses of the stopping powers of four low-Z solid compounds and one medium-Z alloy for protons, alpha particles, and 7 Li ions, trends in the values of mean excitation energy (I) and Barkas-effect parameter (b) extracted from the measurements were observed. The trends suggested a projectile-z dependence in the values, in that the value of I decreased, and the value of b increased, with increasing z [14]. A model was suggested in order to explain the physical basis of the
Fig. 3. Comparison of calculated (open diamonds) with measured (solid circles) stopping powers [19] for 7 Li ions traversing targets of Cu, using a three-parameter fit for mean excitation energy (I), Barkas-effect parameter (b) and effective charge parameter ( l).
L.E. Porter / Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 273–280
279
Fig. 4. Comparison of calculated (open diamonds) with measured (solid circles) stopping powers [19] for 7 Li ions traversing targets of Zn, using a three-parameter fit for mean excitation energy (I), Barkas-effect parameter (b) and effective charge parameter ( l).
observed deviation from modified Bethe–Bloch theory [14]. The present study was undertaken in order to check for the existence of the observed trends in the case of stopping power measurements of high accuracy for elemental targets and two or three low-z projectiles. In the case of a Ti target and projectiles of protons and alpha particles, the measurements by the Nara group [15,16] provided agreement with the I-value trend but behavior counter to the b-value trend. The cornerstone measurements by the Aarhus group with targets of Al and Cu, and with projectiles of protons, alpha particles, and 7 Li ions [17], provided generally good agreement with both I-value and b-value trends. Measurements of low accuracy by a member of the Helsinki group with 7 Li ion projectiles and targets of Cu and Zn [19], coupled with earlier data from the Nara group for protons on Zn [20], yielded corroborative results upon analysis. Thus the outcome of the present investigation generally confirmed the existence of the same trends for the four solid elemental targets studied. Development of further quantitative information about these trends may well enable a revision of modified Bethe–Bloch theory so as to encompass this new feature.
References [1] Stopping Power and Ranges for Protons and Alpha Particles, ICRU Report No. 49, International Commission on Radiation Units and Measurements, Bethesda, MD, 1993. [2] J.D. Jackson, R.L. McCarthy, Phys. Rev. B 6 (1972) 4131. [3] J.C. Ashley, R.H. Ritchie, W. Brandt, Phys. Rev. B 5 (1972) 2393. [4] J.C. Ashley, R.H. Ritchie, W. Brandt, Phys. Rev. A 8 (1973) 2402. [5] J.C. Ashley, Phys. Rev. B 9 (1974) 334. [6] F. Bloch, Ann. Phys. 16 (1933) 285. [7] J. Lindhard, Nucl. Instrum. Methods 132 (1976) 1. [8] L.E. Porter, C.L. Shepard, Nucl. Instrum. Methods 117 (1974) 1. [9] R.H. Ritchie, W. Brandt, Phys. Rev. A 17 (1978) 2102. [10] L.E. Porter, Phys. Rev. A 50 (1994) 2397. ¨ ¨ [11] L.E. Porter, E. Rauhala, J. Raisanen, Phys. Rev. B 49 (1994) 11543. [12] L.E. Porter, Nucl. Instrum. Methods B 149 (1999) 373. [13] W.H. Bragg, R. Kleeman, Phil. Mag. 10 (1905) 318. [14] L.E. Porter, Int. J. Quantum Chem. 90 (2002) 684. [15] N. Sakamoto, H. Ogawa, H. Tsuchida, Nucl. Instrum. Methods B 164–165 (2000) 250. [16] N. Sakamoto, H. Ogawa, N. Shiomi-Tsuda, M. Saitoh, U. Kitoba, M. Ota, Nucl. Instrum. Methods B 135 (1998) 107. [17] H.H. Andersen, J.F. Bak, H. Knudsen, B.R. Nielsen, Phys. Rev. A 16 (1977) 1929. [18] R.C. Haight, H.K. Vonach, Nucl. Instrum. Methods B 1 (1984) 9.
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¨ ¨ [19] K. Vakevainen, Nucl. Instrum. Methods B 122 (1997) 187. [20] N. Sakamoto, N. Shiomi-Tsuda, H. Ogawa, R. Ishiwari, Nucl. Instrum. Methods B 33 (1988) 158. [21] M.C. Walske, Phys. Rev. 88 (1952) 1283. [22] M.C. Walske, Phys. Rev. 101 (1956) 940. [23] L.E. Porter, H. Lin, J. Appl. Phys. 67 (1990) 6613. [24] J.M. Pitarke, R.H. Ritchie, P.M. Echenique, Phys. Rev. B 52 (1995) 13883.
[25] A. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964, p. 259. [26] R.M. Sternheimer, M.J. Berger, S.M. Seltzer, At. Data Nucl. Data Tables 30 (1984) 261. [27] L.E. Porter, Appl. Phys Lett. 61 (3) (1992) 360. [28] L.E. Porter, Radiat. Res. 110 (1987) 1. [29] L.E. Porter, S.R. Bryan, Radiat. Res. 97 (1984) 25.
Evidence for observed projectile-z dependence in values of target parameters extracted from stopping power measurements L.E. Porter* Radiation Safety Office, Washington State University, P.O. Box 641302, Pullman, WA 99164 -1302, USA
Abstract One version of modified Bethe–Bloch stopping power theory features two parameters characteristic of the target material, the mean excitation energy and the Barkas-effect parameter, that can be evaluated through fits to accurate stopping power measurements. In several recent investigations, the values of these parameters thus determined have revealed a dependence on projectile atomic number. Most of those studies have featured target materials possessing aggregation effects, such as solid compounds and alloys. In the current investigation, the search for systematics in projectile-z dependence has been extended to encompass other target materials, including elemental targets. Most of the results are corroborative of the previously reported trends. 2003 Elsevier Science B.V. All rights reserved. Keywords: Stopping power; Energy loss; Bethe–Bloch theory; Eight projectiles; Heavy ions; Projectile-z
1. Introduction The stopping power of matter for charged particles has long remained a topic of vital interest in numerous areas of physics, both basic and applied, since it is often necessary to ascertain the energy loss of a given projectile in a particular target. An understanding of the processes involved in energy loss can lead to a formalism for calculation based on sound theory. A formalism that has proved successful over several orders of magnitude of projectile energy is based on the modified Bethe–Bloch theory of stopping power [1]. This theory contains several target-dependent parameters, notably the shell corrections and the mean excitation energy of the basic theory and the Barkas-effect parameter associated with the second*Tel.: 11-509-335-8916; fax: 11-509-335-1615. E-mail address: [email protected] (L.E. Porter).
order Born approximation as a modification of that basic theory [2–5]. The Barkas-effect correction, initially called the projectile-z 3 correction, represents a modification of the projectile-z 2 dependence of basic Bethe–Bloch theory. When the Bloch term [6] was restored to the Bethe–Bloch formalism [7], a projectile-z 2n (n$2) correction term appeared in the Bethe–Bloch formula for stopping power as well. The author has conducted numerous analyses of stopping power measurements, commencing in 1974 [8], in order to extract values of various parameters of the formalism [3–5,9]. The focus of those studies has been on the target mean excitation energy (I) and the Barkas-effect parameter (b) in particular. A large fraction of the previous studies by the author have dealt with target materials such as compounds and alloys, which manifest aggregation effects (e.g. Refs. [10–12]). One method of treating such cases is to invoke Bragg’s rule of the (linear)
0368-2048 / 03 / $ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016 / S0368-2048(03)00078-1
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L.E. Porter / Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 273–280
additivity of stopping effects [13]. However, the problem of assigning approximate parameter values in such cases represents added complexity in data analyses. A major portion of recent studies has been devoted to the study of observed trends in projectilez dependence of extracted values of the mean excitation energy (I) and Barkas-effect parameter (b), and the target materials studied have been compounds of low (average) atomic number and one alloy. The data revealing such trends were summarized recently [14]. The observed projectile-z dependence embraces both a value of I that decreases, and a value of b that increases, with increasing projectilez for a given target material. The objective of the current investigation was to search for such trends with elemental targets in cases where stopping power measurements of considerable accuracy had been reported. The preferable sets of measurements sought were those conducted by the same experimental group for at least two projectiles of increasing z, such as protons and alpha particles. One such case is the measurement of the stopping powers of Ti for protons and alpha particles by the Nara group [15,16]. Another is the set of definitive measurements of the stopping powers of four elemental targets (Al, Cu, Ag, and Au) for protons, alpha particles, and 7 Li ions by the Aarhus group [17]. (Only the Al and Cu data were used in this study.) A supplemental set of measurements of the stopping power of Ti for alpha particles [18], and another of the stopping power of Cu for 7 Li ions [19], were selected for comparison with the aforementioned sets of measurements of high accuracy [15–17]. Data for the stopping power of Zn for 7 Li ions, included in Ref. [19], were also analyzed. Those results were compared with some previous results for protons as projectiles [20].
2. Theory The formalism of the Bethe–Bloch theory of stopping power, replete with various modifications, has been described in several recent studies [10–12]. The stopping power, S, of an elemental target of atomic number Z and atomic weight A for a projectile with atomic number z and velocity v 5 b c can
be calculated in units of keV?cm 2 / mg from the expression, 0.30706 2 Z S 5 ]]] z ]L A b2
(1)
The quantity, L, representing the (dimensionless) stopping number per target electron, can be written as a three-term sum: L 5 Lo 1 zL1 1 L2
(2)
where the basic stopping number, Lo , consists of four terms
S
D
2mc 2 b 2 Lo 5 ln ]]] 2 b 2 2 ln I 2 C /Z 12b2
(3)
with mc 2 signifying the rest mass energy of the electron, I denoting the target mean excitation energy, and C representing the sum of target shell corrections. The mean excitation energy can be evaluated through a fit to accurate stopping power measurements. The method of Bichsel [1] can be used to calculate shell corrections, wherein the Kand L-shell corrections of Walske [21,22] are employed with appropriate scaling factors inserted in the L-shell correction in order to calculate the Mand N-shell corrections: C 5 CK ( b 2 ) 1VL CL (HL b 2 ) 1VM CL (HM b 2 ) 2
1VN CL (HN b )
(4)
In this expression, CK and CL denote the K- and L-shell corrections, respectively, whereas Vi and Hi (i5L, M, N) represent the scaling factors. Two correction terms of higher order in projectilez appear in Eq. (2). The first of these terms, the (zL1 ) term, is called the Barkas-effect term, and the second term is known as the Bloch term. A brief history of the discovery and early investigations of the Barkas effect appears in Ref. [23]. Both terms have been reviewed pursuant to a random-phase approximation evaluation of the Barkas-effect term [24]. The latter term was evaluated in the present study by means of the formalism that provided the best agreement with a large set of experimental data [23], i.e. the formalism developed by Ashley et al. [3–5,9]. Hence the (zL1 ) term was calculated for the analyses of this study from the expression [3–5],
L.E. Porter / Journal of Electron Spectroscopy and Related Phenomena 129 (2003) 273–280
F(b /x 1 / 2 ) L1 5 ]]] (5) Z 1 / 2x 3 / 2 where F signifies a function graphed in Ref. [3], x 5 (18 787)b 2 /Z, and b denotes the sole free parameter of the formalism. The Bloch term [6], which was initially included as a correction term to allow the transition from the first Born approximation to classical scattering at low projectile velocities [24], can be calculated from L2 ( y) 5 c (1) 2 Re[c (1 1 iy)] (6) where c denotes the digamma function [25] and y 5 za /b, for a signifying the fine structure constant. (In the case of highly relativistic projectiles, a density effect correction term [26] must be added to Lo and a second term [2] must be added to L1 [2,27].) A complete set of parameters for a specified target consists of the target mean excitation energy (I), the shell correction scaling parameters (Vi and Hi with i5L, M, N), and the free parameter of the Barkaseffect correction term (b). All of the above parameters used in the current study are independent of both projectile velocity and a presumably constant projectile charge (ze). However, if the projectile energies considered are so low as to permit the gain and loss of electrons by projectiles traveling at velocities comparable to those of target atomic electrons, then some sort of projectile effective charge formalism must be used to simulate the results of gain and loss of electrons. The technique used [28] is the representation of the bare projectile charge (ze) by an ‘effective charge’ defined as (g ze), where
g 5 1 2 z e 2lv r
(7)
Here vr signifies the ratio of projectile velocity in the laboratory frame of reference (v) to the Thomas– Fermi velocity (e 2 /h)z 2 / 3 , so that vr 5 b /a z 2 / 3 . The symbols, l and z, which denote the effective-charge parameters valid over the complete projectile-velocity interval considered, must be evaluated for any specified projectile–target combination [28]. 3. Method of analysis A procedural strategy for the analysis of stopping power measurements has been developed over the
275
Table 1 Target atomic number (Z), atomic weight (A), and shell correction scaling parameters (VL , HL , VM , HM ) Target
Z
A
VL
HL
VM
HM
Al Ti Cu Zn
13 22 29 30
26.98 47.90 63.55 65.38
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
0.375 1.25 2.25 2.25
12.00 11.20 6.18 5.87
past three decades [28]. Stopping power measurements are provided with a fit using modified Bethe– Bloch theory so as to establish best-fit values of I and b, with shell correction scaling parameters prescribed by the rubric of Bichsel [1]. The figure of merit used in the search procedure is the root-meansquare relative deviation calculated from measured stopping powers, s. This figure of merit is defined as ]]]]] 1 N Sm 2 Sc 2 ]]] s5 ] N i 51 DSm i
œ OS
D
(8)
for measurements at N energies, where Sc represents the calculated stopping power, and Sm and DSm denote the measured stopping power and the associated uncertainty, respectively. Thus a value of s near unity constitutes acceptable agreement between theory and experiment. Values of target parameters, including shell correction scaling parameters, assigned for each elemental target are displayed in Table 1.
4. Results and discussion The results of the analyses will be treated in two categories, those pertaining to the Ti target and those pertaining to the Al, Cu, and Zn targets.
4.1. Ti target Two experiments for the measurement of the stopping power of Ti, one with proton projectiles [15] and the other with alpha particle projectiles [16], have been reported very recently. Both measurements purportedly feature high accuracy, one with 0.40% [15] and the other with 0.35% [16], and both experiments had in common the two accelerators used plus the two leading reporting au-
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Table 2 Summary of results from analyses of Ti measurements, including the projectile energy interval of measurements (DE), number of measurements (N), mean excitation energy (I), Barkas-effect parameter (b), and figure of merit (s ) Projectile
Reference
DE (MeV)
N
I (eV)
b
s
Proton Alpha particle Alpha particle
[15] [16] [18]
0.5–13.5 2.0–20.5 5.2–12.8
39 44 9
239.660.8 229.260.6 252.0
2.2760.05 2.0460.02 1.70
1.17 1.53 0.21
thors. Hence these data were selected for analysis in the context of recently reported observed trends in the I- and b-values extracted from pairs of measurements with different projectiles [14]. Results of these detailed analyses, displayed in Table 2, were only partly consistent with expectations based on observed trends [14]. That is, the I-values decreased with increasing z as expected, but the b-values also decreased with increasing z. The uncertainties given for I- and b-values were obtained by a complicated procedure developed and described in detail previously [28,29]. The excellence of fit for these two analyses is clearly shown in Figs. 1 and 2. It is curious that both extracted b-values lay considerably higher than the value expected by two of
the architects of the Barkas-effect formalism [9] (b51.460.1), especially since the (I, b) values reported earlier by the Nara group for the proton–Ti combination was (232.364.9 eV, 1.26) [20]. Another set of measurements for the alpha particle–Ti combination, of only 3% accuracy, was reported nearly two decades ago [18]. Results of the analysis of those measurements appear in Table 1. Clearly the I-value lies well above expectation, whereas the b-value is eminently plausible.
4.2. Al, Cu, and Zn targets The measurements of the stopping powers of Al and Cu for protons, alpha particles, and 7 Li ions
Fig. 1. Comparison of calculated (open diamonds) with measured (solid circles) stopping powers [15] for protons traversing targets of Ti, using a two-parameter fit for mean excitation energy (I) and Barkas-effect parameter (b).
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277
Fig. 2. Comparison of calculated (open diamonds) with measured (solid circles) stopping powers [16] for alpha particles traversing targets of Ti, using a two-parameter fit for mean excitation energy (I) and Barkas-effect parameter (b).
reported by the Aarhus group [17] served as the basis for revision [9] of the initial formalism for inclusion of the projectle-z 3 effect [3–5] following the restitution of the Bloch term to the Bethe–Bloch formula and forwarding of suggestions concerning inclusion of close-collision contributions to the (then-renamed) Barkas effect [7]. Hence, these measurements were
subjected to analysis with the usual method [10–12]. Results are shown in Table 3. In the case of Al, the excellence of fits is indicated by the values of the figure of merit (s ) in Table 3. Clearly the anticipated projectile-z dependence is present in the extracted I-values, whether or not one includes an effective charge parameter for improve-
Table 3 Summary of results from analyses of Al, Cu, and Zn measurements, including the projectile energy interval of measurements (DE), number of measurements (N), mean excitation energy (I), Barkas-effect parameter (b), effective charge parameter ( l), and figure of merit (s ) Target
Projectile
Ref.
DE (MeV)
N
I (eV)
b
l
s
Al Al Al Al Cu Cu Cu Cu Cu Zn Zn Zn
Proton Alpha particle 7 Li ion 7 Li ion Proton Alpha particle 7 Li ion 7 Li ion 7 Li ion Proton 7 Li ion 7 Li ion
[17] [17] [17] [17] [17] [17] [17] [19] [19] [20] [19] [19]
0.8–6.4 3.2–22.2 8.3–23.4 8.3–23.4 1.2–7.2 5.6–19.9 11.0–19.3 3.7–16.1 3.7–16.1 6.5 3.9–8.7 3.9–8.7
29 25 12 12 31 19 7 24 24 1 7 7
166.5 165.4 156.8 160.4 329.0 328.5 324.5 311.7 327.4 331.668.2 335.5 326.8
1.37 1.36 1.79 1.42 1.30 1.41 1.58 1.78 1.55 1.2660.20 1.82 1.71
– – – 1.26 – – – – 1.46 – – 1.38
0.20 0.25 0.43 0.31 0.24 0.15 0.31 0.36 0.36 – 0.45 0.43
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ment of the fit of the 7 Li ion data. Apparently the b-value decreases slightly in proceeding from the z51 case to the z52 case, however, contrary to the anticipated behavior. In fact, the two values prior to round-off are 1.368 (z51) and 1.360 (z52), and the b-dimension of the basic cell in two-parameter space for the final search is just 0.008. Hence the (estimated) uncertainties in the two b-values [28,29] would easily provide overlap, and the anticipated behavior is approximately observed even in the z51 and z52 cases. Elsewhere, the trends are manifestly followed. Fits to the Cu data [17] are also excellent, as indicated by the values of s in Table 3. The trends are clearly followed for these three projectiles, although the I-values for the z51 and z52 cases are very close to one another. Inclusion of the additional, more recent measurements [19] further buttresses the corroborative evidence. A lower projectile energy interval was used for those measurements, so that a need for insertion of a single effective charge parameter is quite evident. Fig. 3 contains a graph showing the excellence of fit for the three-parameter option.
The measurements of the stopping power of Zn for Li ions, reported as possessing an accuracy of some 4% [19], provides plausible values of I and b, especially when an effective charge parameter is included in the fit. The I-value decreases while the b-value increases during the transition from the z51 case to the z53 case for the three-parameter fit of the 7 Li data. The excellence of fit is displayed in Fig. 4. Thus the trends are reasonably well followed for these sets of data [19,20] as well. 7
5. Summary In the course of previous analyses of the stopping powers of four low-Z solid compounds and one medium-Z alloy for protons, alpha particles, and 7 Li ions, trends in the values of mean excitation energy (I) and Barkas-effect parameter (b) extracted from the measurements were observed. The trends suggested a projectile-z dependence in the values, in that the value of I decreased, and the value of b increased, with increasing z [14]. A model was suggested in order to explain the physical basis of the
Fig. 3. Comparison of calculated (open diamonds) with measured (solid circles) stopping powers [19] for 7 Li ions traversing targets of Cu, using a three-parameter fit for mean excitation energy (I), Barkas-effect parameter (b) and effective charge parameter ( l).
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279
Fig. 4. Comparison of calculated (open diamonds) with measured (solid circles) stopping powers [19] for 7 Li ions traversing targets of Zn, using a three-parameter fit for mean excitation energy (I), Barkas-effect parameter (b) and effective charge parameter ( l).
observed deviation from modified Bethe–Bloch theory [14]. The present study was undertaken in order to check for the existence of the observed trends in the case of stopping power measurements of high accuracy for elemental targets and two or three low-z projectiles. In the case of a Ti target and projectiles of protons and alpha particles, the measurements by the Nara group [15,16] provided agreement with the I-value trend but behavior counter to the b-value trend. The cornerstone measurements by the Aarhus group with targets of Al and Cu, and with projectiles of protons, alpha particles, and 7 Li ions [17], provided generally good agreement with both I-value and b-value trends. Measurements of low accuracy by a member of the Helsinki group with 7 Li ion projectiles and targets of Cu and Zn [19], coupled with earlier data from the Nara group for protons on Zn [20], yielded corroborative results upon analysis. Thus the outcome of the present investigation generally confirmed the existence of the same trends for the four solid elemental targets studied. Development of further quantitative information about these trends may well enable a revision of modified Bethe–Bloch theory so as to encompass this new feature.
References [1] Stopping Power and Ranges for Protons and Alpha Particles, ICRU Report No. 49, International Commission on Radiation Units and Measurements, Bethesda, MD, 1993. [2] J.D. Jackson, R.L. McCarthy, Phys. Rev. B 6 (1972) 4131. [3] J.C. Ashley, R.H. Ritchie, W. Brandt, Phys. Rev. B 5 (1972) 2393. [4] J.C. Ashley, R.H. Ritchie, W. Brandt, Phys. Rev. A 8 (1973) 2402. [5] J.C. Ashley, Phys. Rev. B 9 (1974) 334. [6] F. Bloch, Ann. Phys. 16 (1933) 285. [7] J. Lindhard, Nucl. Instrum. Methods 132 (1976) 1. [8] L.E. Porter, C.L. Shepard, Nucl. Instrum. Methods 117 (1974) 1. [9] R.H. Ritchie, W. Brandt, Phys. Rev. A 17 (1978) 2102. [10] L.E. Porter, Phys. Rev. A 50 (1994) 2397. ¨ ¨ [11] L.E. Porter, E. Rauhala, J. Raisanen, Phys. Rev. B 49 (1994) 11543. [12] L.E. Porter, Nucl. Instrum. Methods B 149 (1999) 373. [13] W.H. Bragg, R. Kleeman, Phil. Mag. 10 (1905) 318. [14] L.E. Porter, Int. J. Quantum Chem. 90 (2002) 684. [15] N. Sakamoto, H. Ogawa, H. Tsuchida, Nucl. Instrum. Methods B 164–165 (2000) 250. [16] N. Sakamoto, H. Ogawa, N. Shiomi-Tsuda, M. Saitoh, U. Kitoba, M. Ota, Nucl. Instrum. Methods B 135 (1998) 107. [17] H.H. Andersen, J.F. Bak, H. Knudsen, B.R. Nielsen, Phys. Rev. A 16 (1977) 1929. [18] R.C. Haight, H.K. Vonach, Nucl. Instrum. Methods B 1 (1984) 9.
280
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¨ ¨ [19] K. Vakevainen, Nucl. Instrum. Methods B 122 (1997) 187. [20] N. Sakamoto, N. Shiomi-Tsuda, H. Ogawa, R. Ishiwari, Nucl. Instrum. Methods B 33 (1988) 158. [21] M.C. Walske, Phys. Rev. 88 (1952) 1283. [22] M.C. Walske, Phys. Rev. 101 (1956) 940. [23] L.E. Porter, H. Lin, J. Appl. Phys. 67 (1990) 6613. [24] J.M. Pitarke, R.H. Ritchie, P.M. Echenique, Phys. Rev. B 52 (1995) 13883.
[25] A. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964, p. 259. [26] R.M. Sternheimer, M.J. Berger, S.M. Seltzer, At. Data Nucl. Data Tables 30 (1984) 261. [27] L.E. Porter, Appl. Phys Lett. 61 (3) (1992) 360. [28] L.E. Porter, Radiat. Res. 110 (1987) 1. [29] L.E. Porter, S.R. Bryan, Radiat. Res. 97 (1984) 25.