- Email: [email protected]
Stopping powers of havar for protons from 4.0 to 13.0 MeV
Nuclear
Instruments
and Methods
in Physics Research
B 117 (1996)
343-346
LOXSIB
Beam Interactions with Materials & Atoms EISEVIER
Stopping powers of havar for protons from 4.0 to 13.0 MeV N. Shiomi-Tsuda*, N. Sakamoto, H. Ogawa, M. Tanaka, T. Goto, Y. Nagata Department of Physics. Nara Women’s University, Nara 630. Japan Received
8 April
1996; revised form received 28 May
1996
Abstract Stopping powers of havar have been measured with an uncertainty of fO.35% for protons from 4.0 to 13.0 MeV. Experimental results have been compared with calculated stopping powers obtained by Bragg’s additivity rule [ W.H. Bragg and R.K. Kleeman, Philos. Mag. 10 ( 1905) 3 181 using the stopping-power values for constituent elements calculated by Andersen and Ziegler’s formula [Hydrogen Stopping Powers and Ranges in All Elements (Pergamon, New York, 1977) 1. The calculated values agree with the experimental results within the stated uncertainty. Applying the modified Bethe-Bloch formula including the Barkas term for the stopping power of compounds, the mean excitation energy, viz., the I-value, for havar has been extracted from the experimental results. In this analysis, we used the theory of Ashley et al. [Phys. Rev. B 5 ( 1972) 2393; Phys. Rev. A 8 ( 1973) 24021 to estimate the Barkas correction and Bonderup’s [K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 35 ( 1967) no.171 and Bichsel’s [UCRL Report No. 17538 ( 1967); ICRU Report 37 ( 1984)] shell corrections. With Bonderup’s shell correction, the l-value is 292.4f 6.7 eV; with Bichsel’s shell correction 296.9 f 7.5 eV. These two I-values agree well within the uncertainties.
1. Introduction
2. Experimental
Accurate data on the stopping power of havar (a cobaltbased alloy) [ 1,2] are useful as a basis for a full understanding of the stopping powers of compounds and alloys. Havar also is widely used as a window material in various experiments and knowledge of its stopping powers is indispensable for data analysis in such experiments. Duder et al. [ 31 and Rauhala and RBisgnen [ 41 respectively measured the stopping powers of havar for protons of 2.93-5.96 MeV with an accuracy of *l-2% and for those of 0.51-8.29 MeV with an accuracy of f3%. Foroughi et al. [ 1] measured energy losses of 1.26-4.43 MeV protons for 12 pm thick havar with uncertainties larger than 10%. These experimental uncertainties are too large to enable us to discuss the mean excitation energy, viz., I-value, with reasonable accuracy. Following work on Mylar [ 51 for protons from 4.0 to 11.5 MeV we have measured the stopping powers of havar with an experimental uncertainty of f0.35%. By analyzing the experimental results with the Bethe-Bloch formula, we have extracted the I-value of havar.
A proton beam from the tandem Van de Graaff accelerator at Kyoto University was used in the present measurement. The energy of the protons from the accelerator was measured with a beam analyzing magnet. The magnet was calibrated by using the elastic-scattering resonance of the protons by 160( p. p) I60 at 6.482 * 0.007 and 6.564 f 0.020 MeV [6]. Since the experimental setup and procedure are very similar to those used in previous experiments [ 5,7,8], we here will limit the description to the sample target only. The 30 pm thick havar foil, supplied by Hamilton PreciTable
author. Fax f81
742 27 5405,
e-mail
[email protected]
0168-583X/%/$15.00
Copyright
@
1996 Elsevier
Science
B.V.
All
of havar. wi(%)
denotes the weight
stituent element
Element
Wi(%)
Be
0.054
C
0.20
Si
0.18
P
0.005
S
0.003 19.73
Mo
1.68
Fe
18.598
CO
41.88
Ni
12.73
MO
2.23
W
2.7
wu.ac.jp
PfISOl68-583X(96)00343-6
I
Compositions
CK
* Corresponding
procedure
rights reserved
I
percent
of the ith con-
344
N Shio~wTsdu
er al. /Nd.
Instr. and Merh. in Phys. Res. B I17 (1996) 343-346
Table 2 The present stopping power data are given in keV mg- ’ cm’ and compared with the calculated values obtained by Bragg’s additivity rule using the stopping power values for constituent efements calculated by Andercen and Ziegler’s formula (calculated A-Z values). The stopping powers measured by Duder et al. [ 3] and Rauhala and RWinen 141 from commmon energy intervals also are shown together with the calculated A-Z values. The symbol A denotes the percentage difference E IMeV]
4.0 4.5 5.0 5.5 6.0 6.5 7.0 1.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 Il.5 12.0 12.5 13.0
A (%I
Stopping powers present data
calculated A-Z values
53.35 + 0.19 49.26 f 0.17 45.82 f 0.16 42.91 + 0.15 40.38 + 0. I4 38.20 zt 0. I3 36.22 f 0. I3 34.46 zt 0.12 32.92iO.12 31.49fO.ll 30.23 zt 0.1 I 29.09ztz 0.10 28.01 f 0.10 27.01 + 0.09 26.15 i 0.09 25.32 f 0.09 24.5 I + 0.09 23.76 I!C 0.08 23.08 * 0.08
53.88 49.70 46.20 43.21 40.64 38.39 36.41 34.64 33.06 3 I .64 30.35 29. I7 28.09 27.10 26.18 25.33 24.5.5 23.8 I 23 I3
E IMeVJ
Stopping powers Dnder et al.
3.94 4.45 4.95 5.45 S.96
-0.98 -0.89 -0.82 -0.69 -0.64 -0.49 -0.52 -0.52 -0.42 -0.47 -0.40 -0.27 -0.28 -0.33 -0.11
0
Et & B
a
a
AA
l
Present
c: Duder
c
a
A
et al
Rauhala
and
R&&en
1.02.
calculated A-Z values
55.0 50.0 47.2 42.9 39.8
54.44 50.08 46.52 43.49 40.83
3.87 4.33 4.37 4.82 5.33 5.82 6.31 6.81 7.30 1.79 8.29
Rauhala and R&a&en [Z]
calculated A-Z values
57.0 52.5 53.2 48.8 44.4 42.9 39.9 37.5 36.6 33.9 31.8
55.10 51.03 50.71 47.39 44.18 41.52 39.21 37.13 35.32 33.71 32.22
-O.OJ
a
1.04.
Ill
Stopping powers
-0.16 -0.21 -0.22
sion Metals Inc., was produced by the cold-rolled method. Its composition is given in Table 1 with the percent of weight (Wi( W) ). The mass thickness obtained by measuring the weight and area was 25.948 mg/cm2 ( 105.290 mg/4.0577 cm2). The data analysis procedure used is described in detail in Refs. [ 5,7,8].
s
E [MeVl
n
3. Results Table 2 shows the present results in keV/ ( mg/cm2) and compares them with calculated stopping powers obtained from Bragg’s additivity rule [9] of stopping power using the stopping power values for constituent elements calculated by Andersen and Ziegler’s formula [lo] (hereafter, “calculated A-Z values”). The experimental data of Duder et al. [3] and of Rauhala and RSisiinen [4] for the common energy intervals also are given in Table 2 together with the calculated A-Z values. To clearly show the discrepancy between data, Fig. 1 gives the stopping power ratios of the present results and the experimental data of Refs. [ 3.41 to the calculated A-Z values. The experimental uncertainty of the present results was assessed as f0.35% as estimated by the method described in Ref. [ 51.
4. Discussion 0.98 t
0
4
6
4. I. Stopping power
a
10
12
14
Proton Energy (MeV)
Fig. I. Stoppmg power ratios of the present results. those of Duder et al. [ 31 and Rauhala and RLiiiisanen 141 to the calculated A-Z values are shown as a function of proton energy.
As seen in Fig. 1 and Table 2, on the whole, the present results are systematically lower than the calculated A-Z values. These results, however, agree well with the calculated A-Z values within the stated uncertainties of the calculations. In particular, agreement is excellent at the high energy side. These trends are very similar to those found in the experiment with A-l 50 tissue equivalent plastic [ 71. In
N. Shiomi-7kuda
et 01. /NW/.
lnstr. and Meth. in Phys. RPS. B 117 (1996)
343-346
contrast, the data sets of Duder et al. [ 31 and Rauhala and RIistinen [4] show a wide scatter, which reflects the large uncertainties.
t
4.2. Mean excitation energy 5051
Present
-
Bonderup : I = 292.4 eV
---
Bichsel : I = 296.9 eV
+-+$+$+
:t’ -_ The modified Bethe-Bloch formula of mass stopping power for pure elements is written as
345
-L
._
I
’
1 dE -= -; dx x
4rrZ2e4No Z2 I mu2
-x
580-
, (1)
ln~-$-ln1-~+Z~L~+@ 1
{
E
(Z2/A),
(2)
lnf =+ (Zz./A)-’ ~(MJ;Z~~/A;) lnli,
C/Z2
*
(Z2/A)-’
~Mi/~i)
(3)
(C/ZZ)~
=
(CjZ2)
(4)
and LI * (Z2/A)-’ C(W;Z2i/Ai)Lli
G (LI),
(5)
where Wi is the fraction by weight of the ith constituent element. If Bragg’s additivity rule holds, we can calculate the Ivalue of a composite material from Eq. (3). However, we may not simply adopt the I-values for the atomic constituents because the physical and chemical effects may alter the Z-values of the constituents. We therefore determined the effective I-value for havar by comparing our present data with values calculated by the Bethe-Bloch formula, modified for the description of compounds, taking the I-value as an adjustable parameter. In this comparison we used modifications of Bichsel’s X variable [ 141 to represent the stopping power. The experimental and theoretical X variables, Xexpand XtkO, are defined as
2
- 4nF;e4K,
I
(Z21A)
exp
I,,
4
6
,J
,,,
8
10
12
Proton Energy (MeV)
where p is the mass density of target material, the symbol I the mean excitation energy, C/Z? the shell correction, ZlL.1 the Barkas correction [ 11,121 and @ the Bloch correction [ 131. The other symbols have their usual meanings (e.g., Ref. [ 71). Assuming the validity of Bragg’s additivity rule [9] of stopping powers, the modified Bethe-Bloch formula (Eq. ( 1) ) can be applied to the stopping power of composite materials by making the following replacement in Eq. ( 1) : Zz/A + CwiZ2;/Ai
/
1,
_ -_ -++++ .
(6)
Fig. 2. Xexppoints andX,, curveswith the best fit I values.The solid and dashedlines, respectively,show the X,heocurves obtainedwith Bonderup’s and Bichsel’s shell corrections.
X ,hrO= In I + (C/Z=) - ZI (LI) - @.
(7)
We have used two kinds of shell corrections; Bonderup’s [IS] and Bichsel’s [16,17]. Bondemp’s shell correction has one adjustable parameter y which was introduced to consider the effect of binding forces acting on atomic electrons. We used y = 1.336, which value was determined by using the Bethe-Bloch formula which included the Barkas correction in a previous experiment [ 181. For the adjustable parameter values included in Bichsel’s shell correction, we used those adopted in ICRU Report 37 [ 171. The value of the Barkas correction, L1, was calculated by the use of the theory of Ashley et al. [ 11,121. In their theory, LI contains y (which is expressed by x in their reports) and another parameter b. We used b = 1.32, which also was obtained from Ref. [ 181. Taking the I-value in Eq. (7) as an adjustable parameter, the X,k,, values were compared with the Xexpvalues. The method of least squares was used to obtain the best fit values of I. The best-fit Ivalues are 292.4 f 6.7 eV with Bonderup’s shell correction and 296.9 f 7.5 eV with Bichsel’s shell correction. Uncertainties originate from the theoretical uncertainty of the shell correction in absolute magnitude, say, *lo%, and the experimental uncertainty of the present stopping powers, details of which are given in Ref. [5]. Fig. 2 shows the Xexppoints and XtiO curves with the best-fit l-values. 1 he solid and dashed lines respectively represent the X,k,, curves using Bonderup’s and Bichsel’s shell corrections. As seen in Fig. 2, the XhO curve obtained with Bonderup’s shell correction fits the Xexppoints well. The XLheO curve obtained with Bichsel’s shell correction has a slightly gentler slope than the Xex,points, but it too can be regarded as reproducing the experimental situation within the uncertainties. Both extracted I-values of havar, 292.4 f 6.7 and 296.9 f 7.5 eV, obtained with Bonderup’s and Bichsel’s shell corrections, respectively, are in good agreement within the uncertainties. Recently Porter et al. [ 191 have measured
346
N. Shimi-T.&a
rt 01./Nucl. Instr. and Meth. in Plys. Rex B I17 (1996) 343-346
stopping powers of havar for 0.56-2.38 MeV protons and recommended 299.3 f 3.3 eV for the I-value, which agrees with the present f-values, and b = I .33 * 0.04, which essentially agrees with the value selected in the present study.
Acknowledgements We thank professor K. Imai for his kind support throughout our study. We also are grateful to Dr. K. Takimoto, Dr. M. Nakamura and the members of the tandem Van de Graaff Laboratories of Kyoto University for their kind cooperation. This research was supported by a Grant-in-Aid for Fundamental Scientific Research from the Ministry of Education, Science and Culture of Japan.
References
[ I]
F. Foroughi, B. Vuilleumier and E. Bovet, Nucl. Instr. and Meth. 159 (1979) 513. [Zj E. Rauhala and J. Rtisiinen, Nucl. Instr. and Meth. B 12 (1985) 321. 131 J.C. Dud% J.F. Glare and H. Naylor, Nucl. Instr. and Meth. 123 ( 1975) 89. [4] E. Rauhala and J. RBisBnen, Nucl. lnstr. and Merh. B 35 ( 1988) 130.
I5 1 N. Shiomi-Tsuda, N. Sakamoto and H. Ogawa. Nucl. Instr. and Meth. B 103 (1995) 255. I61 E Ajzenberg-Selove, Nucl. Phys. A 460 ( 1986) 1. 171 R. Ishiwari. N. Shiomi-Tsuda, N. Sakamoto and H. Ogawa, Nucl. Instr. and Meth. B 47 ( 1990) 1 II. I 8 1 R. Ishiwari, N. Shiomi-Tsuda and N. Sakamoto, Nucl. Instr. and Meth. B 31 (1988) 503. 191 W.H. Bragg and R.K. Kleeman, Philos. Msg. 10 (1905) 318. I IO] H.H. Andersen and J.F. Ziegler, Hydrogen Stopping Powers and Ranges in All Elements (Pergamon, New York. 1977). I I I ] J.C. Ashley, R.H. Ritchie and W. Brand& Phys. Rev. B 5 (1972) 2393. 1121 J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. A 8 (1973) 2402. 1131 F. Bloch. Ann. Phys. (Leipzig) 16 (1933) 285. [ 14) H. Bichsel, in: Studies in Penetration of Charged Particles in Matter. National Academy of Science-National Research Council, Pubi. I I33 (1964) p.17. [ 151 E. Bonderup, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 35 (1967) no.17. [ 161 H. Bichsel, UCRL Report No.17538 (1967). [ 17 1 ICRU Report 37, Stopping Powers for Electrons and Positrons (International Commission on Radiation Units and Measurements, 1984). [ 181 N. Shiomi-Tsuda. N. Sakamoto, H. Ogawa and R. Ishiwari, Radiat. Eff. and Defects in S&ids 117 (1991) 185. 1191LE. Porter, E. Rauhala and J. R&i&en, Phys. Rev. B 49 (1994) 11543.
Instruments
and Methods
in Physics Research
B 117 (1996)
343-346
LOXSIB
Beam Interactions with Materials & Atoms EISEVIER
Stopping powers of havar for protons from 4.0 to 13.0 MeV N. Shiomi-Tsuda*, N. Sakamoto, H. Ogawa, M. Tanaka, T. Goto, Y. Nagata Department of Physics. Nara Women’s University, Nara 630. Japan Received
8 April
1996; revised form received 28 May
1996
Abstract Stopping powers of havar have been measured with an uncertainty of fO.35% for protons from 4.0 to 13.0 MeV. Experimental results have been compared with calculated stopping powers obtained by Bragg’s additivity rule [ W.H. Bragg and R.K. Kleeman, Philos. Mag. 10 ( 1905) 3 181 using the stopping-power values for constituent elements calculated by Andersen and Ziegler’s formula [Hydrogen Stopping Powers and Ranges in All Elements (Pergamon, New York, 1977) 1. The calculated values agree with the experimental results within the stated uncertainty. Applying the modified Bethe-Bloch formula including the Barkas term for the stopping power of compounds, the mean excitation energy, viz., the I-value, for havar has been extracted from the experimental results. In this analysis, we used the theory of Ashley et al. [Phys. Rev. B 5 ( 1972) 2393; Phys. Rev. A 8 ( 1973) 24021 to estimate the Barkas correction and Bonderup’s [K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 35 ( 1967) no.171 and Bichsel’s [UCRL Report No. 17538 ( 1967); ICRU Report 37 ( 1984)] shell corrections. With Bonderup’s shell correction, the l-value is 292.4f 6.7 eV; with Bichsel’s shell correction 296.9 f 7.5 eV. These two I-values agree well within the uncertainties.
1. Introduction
2. Experimental
Accurate data on the stopping power of havar (a cobaltbased alloy) [ 1,2] are useful as a basis for a full understanding of the stopping powers of compounds and alloys. Havar also is widely used as a window material in various experiments and knowledge of its stopping powers is indispensable for data analysis in such experiments. Duder et al. [ 31 and Rauhala and RBisgnen [ 41 respectively measured the stopping powers of havar for protons of 2.93-5.96 MeV with an accuracy of *l-2% and for those of 0.51-8.29 MeV with an accuracy of f3%. Foroughi et al. [ 1] measured energy losses of 1.26-4.43 MeV protons for 12 pm thick havar with uncertainties larger than 10%. These experimental uncertainties are too large to enable us to discuss the mean excitation energy, viz., I-value, with reasonable accuracy. Following work on Mylar [ 51 for protons from 4.0 to 11.5 MeV we have measured the stopping powers of havar with an experimental uncertainty of f0.35%. By analyzing the experimental results with the Bethe-Bloch formula, we have extracted the I-value of havar.
A proton beam from the tandem Van de Graaff accelerator at Kyoto University was used in the present measurement. The energy of the protons from the accelerator was measured with a beam analyzing magnet. The magnet was calibrated by using the elastic-scattering resonance of the protons by 160( p. p) I60 at 6.482 * 0.007 and 6.564 f 0.020 MeV [6]. Since the experimental setup and procedure are very similar to those used in previous experiments [ 5,7,8], we here will limit the description to the sample target only. The 30 pm thick havar foil, supplied by Hamilton PreciTable
author. Fax f81
742 27 5405,
[email protected]
0168-583X/%/$15.00
Copyright
@
1996 Elsevier
Science
B.V.
All
of havar. wi(%)
denotes the weight
stituent element
Element
Wi(%)
Be
0.054
C
0.20
Si
0.18
P
0.005
S
0.003 19.73
Mo
1.68
Fe
18.598
CO
41.88
Ni
12.73
MO
2.23
W
2.7
wu.ac.jp
PfISOl68-583X(96)00343-6
I
Compositions
CK
* Corresponding
procedure
rights reserved
I
percent
of the ith con-
344
N Shio~wTsdu
er al. /Nd.
Instr. and Merh. in Phys. Res. B I17 (1996) 343-346
Table 2 The present stopping power data are given in keV mg- ’ cm’ and compared with the calculated values obtained by Bragg’s additivity rule using the stopping power values for constituent efements calculated by Andercen and Ziegler’s formula (calculated A-Z values). The stopping powers measured by Duder et al. [ 3] and Rauhala and RWinen 141 from commmon energy intervals also are shown together with the calculated A-Z values. The symbol A denotes the percentage difference E IMeV]
4.0 4.5 5.0 5.5 6.0 6.5 7.0 1.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 Il.5 12.0 12.5 13.0
A (%I
Stopping powers present data
calculated A-Z values
53.35 + 0.19 49.26 f 0.17 45.82 f 0.16 42.91 + 0.15 40.38 + 0. I4 38.20 zt 0. I3 36.22 f 0. I3 34.46 zt 0.12 32.92iO.12 31.49fO.ll 30.23 zt 0.1 I 29.09ztz 0.10 28.01 f 0.10 27.01 + 0.09 26.15 i 0.09 25.32 f 0.09 24.5 I + 0.09 23.76 I!C 0.08 23.08 * 0.08
53.88 49.70 46.20 43.21 40.64 38.39 36.41 34.64 33.06 3 I .64 30.35 29. I7 28.09 27.10 26.18 25.33 24.5.5 23.8 I 23 I3
E IMeVJ
Stopping powers Dnder et al.
3.94 4.45 4.95 5.45 S.96
-0.98 -0.89 -0.82 -0.69 -0.64 -0.49 -0.52 -0.52 -0.42 -0.47 -0.40 -0.27 -0.28 -0.33 -0.11
0
Et & B
a
a
AA
l
Present
c: Duder
c
a
A
et al
Rauhala
and
R&&en
1.02.
calculated A-Z values
55.0 50.0 47.2 42.9 39.8
54.44 50.08 46.52 43.49 40.83
3.87 4.33 4.37 4.82 5.33 5.82 6.31 6.81 7.30 1.79 8.29
Rauhala and R&a&en [Z]
calculated A-Z values
57.0 52.5 53.2 48.8 44.4 42.9 39.9 37.5 36.6 33.9 31.8
55.10 51.03 50.71 47.39 44.18 41.52 39.21 37.13 35.32 33.71 32.22
-O.OJ
a
1.04.
Ill
Stopping powers
-0.16 -0.21 -0.22
sion Metals Inc., was produced by the cold-rolled method. Its composition is given in Table 1 with the percent of weight (Wi( W) ). The mass thickness obtained by measuring the weight and area was 25.948 mg/cm2 ( 105.290 mg/4.0577 cm2). The data analysis procedure used is described in detail in Refs. [ 5,7,8].
s
E [MeVl
n
3. Results Table 2 shows the present results in keV/ ( mg/cm2) and compares them with calculated stopping powers obtained from Bragg’s additivity rule [9] of stopping power using the stopping power values for constituent elements calculated by Andersen and Ziegler’s formula [lo] (hereafter, “calculated A-Z values”). The experimental data of Duder et al. [3] and of Rauhala and RSisiinen [4] for the common energy intervals also are given in Table 2 together with the calculated A-Z values. To clearly show the discrepancy between data, Fig. 1 gives the stopping power ratios of the present results and the experimental data of Refs. [ 3.41 to the calculated A-Z values. The experimental uncertainty of the present results was assessed as f0.35% as estimated by the method described in Ref. [ 51.
4. Discussion 0.98 t
0
4
6
4. I. Stopping power
a
10
12
14
Proton Energy (MeV)
Fig. I. Stoppmg power ratios of the present results. those of Duder et al. [ 31 and Rauhala and RLiiiisanen 141 to the calculated A-Z values are shown as a function of proton energy.
As seen in Fig. 1 and Table 2, on the whole, the present results are systematically lower than the calculated A-Z values. These results, however, agree well with the calculated A-Z values within the stated uncertainties of the calculations. In particular, agreement is excellent at the high energy side. These trends are very similar to those found in the experiment with A-l 50 tissue equivalent plastic [ 71. In
N. Shiomi-7kuda
et 01. /NW/.
lnstr. and Meth. in Phys. RPS. B 117 (1996)
343-346
contrast, the data sets of Duder et al. [ 31 and Rauhala and RIistinen [4] show a wide scatter, which reflects the large uncertainties.
t
4.2. Mean excitation energy 5051
Present
-
Bonderup : I = 292.4 eV
---
Bichsel : I = 296.9 eV
+-+$+$+
:t’ -_ The modified Bethe-Bloch formula of mass stopping power for pure elements is written as
345
-L
._
I
’
1 dE -= -; dx x
4rrZ2e4No Z2 I mu2
-x
580-
, (1)
ln~-$-ln1-~+Z~L~+@ 1
{
E
(Z2/A),
(2)
lnf =+ (Zz./A)-’ ~(MJ;Z~~/A;) lnli,
C/Z2
*
(Z2/A)-’
~Mi/~i)
(3)
(C/ZZ)~
=
(CjZ2)
(4)
and LI * (Z2/A)-’ C(W;Z2i/Ai)Lli
G (LI),
(5)
where Wi is the fraction by weight of the ith constituent element. If Bragg’s additivity rule holds, we can calculate the Ivalue of a composite material from Eq. (3). However, we may not simply adopt the I-values for the atomic constituents because the physical and chemical effects may alter the Z-values of the constituents. We therefore determined the effective I-value for havar by comparing our present data with values calculated by the Bethe-Bloch formula, modified for the description of compounds, taking the I-value as an adjustable parameter. In this comparison we used modifications of Bichsel’s X variable [ 141 to represent the stopping power. The experimental and theoretical X variables, Xexpand XtkO, are defined as
2
- 4nF;e4K,
I
(Z21A)
exp
I,,
4
6
,J
,,,
8
10
12
Proton Energy (MeV)
where p is the mass density of target material, the symbol I the mean excitation energy, C/Z? the shell correction, ZlL.1 the Barkas correction [ 11,121 and @ the Bloch correction [ 131. The other symbols have their usual meanings (e.g., Ref. [ 71). Assuming the validity of Bragg’s additivity rule [9] of stopping powers, the modified Bethe-Bloch formula (Eq. ( 1) ) can be applied to the stopping power of composite materials by making the following replacement in Eq. ( 1) : Zz/A + CwiZ2;/Ai
/
1,
_ -_ -++++ .
(6)
Fig. 2. Xexppoints andX,, curveswith the best fit I values.The solid and dashedlines, respectively,show the X,heocurves obtainedwith Bonderup’s and Bichsel’s shell corrections.
X ,hrO= In I + (C/Z=) - ZI (LI) - @.
(7)
We have used two kinds of shell corrections; Bonderup’s [IS] and Bichsel’s [16,17]. Bondemp’s shell correction has one adjustable parameter y which was introduced to consider the effect of binding forces acting on atomic electrons. We used y = 1.336, which value was determined by using the Bethe-Bloch formula which included the Barkas correction in a previous experiment [ 181. For the adjustable parameter values included in Bichsel’s shell correction, we used those adopted in ICRU Report 37 [ 171. The value of the Barkas correction, L1, was calculated by the use of the theory of Ashley et al. [ 11,121. In their theory, LI contains y (which is expressed by x in their reports) and another parameter b. We used b = 1.32, which also was obtained from Ref. [ 181. Taking the I-value in Eq. (7) as an adjustable parameter, the X,k,, values were compared with the Xexpvalues. The method of least squares was used to obtain the best fit values of I. The best-fit Ivalues are 292.4 f 6.7 eV with Bonderup’s shell correction and 296.9 f 7.5 eV with Bichsel’s shell correction. Uncertainties originate from the theoretical uncertainty of the shell correction in absolute magnitude, say, *lo%, and the experimental uncertainty of the present stopping powers, details of which are given in Ref. [5]. Fig. 2 shows the Xexppoints and XtiO curves with the best-fit l-values. 1 he solid and dashed lines respectively represent the X,k,, curves using Bonderup’s and Bichsel’s shell corrections. As seen in Fig. 2, the XhO curve obtained with Bonderup’s shell correction fits the Xexppoints well. The XLheO curve obtained with Bichsel’s shell correction has a slightly gentler slope than the Xex,points, but it too can be regarded as reproducing the experimental situation within the uncertainties. Both extracted I-values of havar, 292.4 f 6.7 and 296.9 f 7.5 eV, obtained with Bonderup’s and Bichsel’s shell corrections, respectively, are in good agreement within the uncertainties. Recently Porter et al. [ 191 have measured
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stopping powers of havar for 0.56-2.38 MeV protons and recommended 299.3 f 3.3 eV for the I-value, which agrees with the present f-values, and b = I .33 * 0.04, which essentially agrees with the value selected in the present study.
Acknowledgements We thank professor K. Imai for his kind support throughout our study. We also are grateful to Dr. K. Takimoto, Dr. M. Nakamura and the members of the tandem Van de Graaff Laboratories of Kyoto University for their kind cooperation. This research was supported by a Grant-in-Aid for Fundamental Scientific Research from the Ministry of Education, Science and Culture of Japan.
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