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Stopping powers of Al and Cu for protons from 3 TO 9 MeV
118
Nuclear Instruments and Methods in Physics Research B35 (1988) 118-129 North-Holland, Amsterdam
STOPPING
POWERS OF Al AND Cu FOR PROTONS
FROM 3 TO 9 MeV
R. ISHIWARI Faculty of General Education, Kinki University, Higashi-Osaka
N. SHIOMI-TSUDA Department
577, Japan
and N. SAKAMOTO
of Physics, Nara Women’s University, Nara 630, Japan
Received 19 July 1988
The stopping powers of Al and Cu for protons having energies from 3 to 9 MeV have been measured with an accuracy of +0.4%. The results have been compared with the Rise and the Aarhus data of Andersen et al. and with Tschaliir’s data for Al. The mean excitation energies have been extracted from the present results using the Bonderup shell correction, the Barkas correction and the Bloch correction as 167.4 + 0.5 eV for Al and 326.3 + 0.6 eV for Cu. A set of parameters x = 1.340 and b = 1.26 was used in these determinations. By using the mean excitation energies thus determined, the Bonderup shell corrections, the Barkas corrections and the Bloch corrections, stopping powers have been calculated up to 30 MeV for Al and up to 20 MeV for Cu. The calculated stopping powers agree very well with the Aarhus data. For Al, the agreement between the calculated stopping powers and TschaWs data is surprisingly good up to 30 MeV.
1. Introduciion
2. Experimental procedure
In our previous experiments [l-3], we have measured stopping powers of 21 kinds of metallic elements for 6.5 MeV protons. These experiments revealed that the Rise data of Andersen et al. [4-81 are too high by 0.78% on the average and the Aarhus data of Andersen et al. [9], except for Au, are in good agreement with our data. From our stopping power data of 21 metallic elements for 6.5 MeV protons [l-3], the mean excitation energies have been extracted [2,10] using the Bonderup shell correction [ll] and taking account of the Barkas correction of Ashley, Ritchie and Brandt [12-141 and the Bloch correction [15,16]. Although our mean excitation energies agree very well with the I values given by different authors [8,17-271, it is desirable to measure stopping powers at various energies to obtain more reliable mean excitation energies. In the present experiment, stopping powers of Al and Cu for protons having energies from 3 to 9 MeV have been measured and the mean excitation energies have been determined by using the Bonderup shell correction [ll] and by taking account of the Barkas correction [12-141 and the Bloch correction [15,16]. Then, by using these mean excitation energies the stopping power table of Al for protons from 3 to 30 MeV and that of Cu for protons from 3 to 20 MeV have been calculated.
The analyzed beam of protons accelerated by the tandem Van de Graaff accelerator of Kyoto University was used in the present experiment. The analyzing magnet was calibrated by the resonance of elastic scattering of protons by ‘*C at 4.808 + 0.010 MeV [28]. The experimental procedures to determine the energy loss of protons in the sample target is the same as described in detail in the previous paper on stopping powers of 16 kinds of metallic elements for 6.5 MeV protons [2]. Therefore, only a brief description of the experimental procedures will be given here. Fig. 1 shows the experimental setup. The analyzed beam of protons was collimated by a double slit system St and S,. The diameter of each slit was 2 mm and they were 40 cm apart. Then, the beam was scattered by a thin gold foil of 180 pg/cm* placed at the center of the scattering chamber. The beam scattered at an angle of 15O was used for the energy loss measurement. The scattered beam was collimated by a double slit system. The diameter of each slit was 1.5 mm and they were 5.9 cm apart. To determine the energy loss of protons in the target, we used what we call the “absorber wheel technique”. In fig. 1, the absorber wheel is a brass disk 1 mm thick and 10 cm in diameter. The disk has two windows. The sample target was fitted to one of these windows and the other window was left open. The disk was fixed to an induction motor and was rotated by the motor at 60
0168-583X/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
R. Ishiwari et al. / Stopping of Al and Cu for 3- 9 MeVp 52
51
In order to determine the absolute value of the energy loss of protons in the target from the pulse height difference shown in fig. 2, the zero point of the abscissa of fig. 2 should be determined. This was done by the use of a very high precision pulse generator (Ortec 448). During the measurement the pulses from the pulse generator with three different dial settings were recorded in addition to the proton pulses. From these three pulse heights the zero point of the abscissa of fig. 2 was determined. Let the integral pulse heights with and without the target be Ht and He. The energy loss of protons in the target is given by
Au 180 wg/cm2
53
119
2mm
Targe
Fig. 1. A schematic diagram of the experimental setup for the energy loss measurementusing the “absorber wheel technique”.
AE=Eo,,
*O-H* 0
rpm during the measurement. A surface barrier silicon
detector was placed 9.2 mm behind the target. A small armular permanent magnet of about 500 G was placed just before the silicon detector to prevent the possible energetic S-rays ejected from the target from affecting the energy loss measurement. When ac the power was turned on and the “absorbed wheel” was rotated, protons that passed through the target and protons scattered by the Au scatterer hit the silicon detector alternately. Thus, the pulse heights with, and without, the target were measured simultaneously in one exposure. By this arrangement substantially all protons that passed through the target and suffered multiple scattering were detected by the silicon detector. To avoid confusion, we call the protons that are scattered by the thin Au foil at an angle of 15 o and are used for the stopping power measurement the “incident protons”. The pulses from the detector were amplified with a low noise amplifier system. The relevant portion of the pulse height spectrum was expanded by a biased amplifier and fed into a 400 channel pulse height analyzer. Fig. 2 shows a typical pulse height spectrum.
1500
,
I
I
Al
d
.
5 a lOOO-
..
g \
.*.. . * . * . . . . .
e z 0’ 500”
0
1
200
250
x ‘A
.*
300
350
CHANNEL NUMBER
Fig. 2. A typical pulse height spectrum.
400
where E,, is the incident energy. The energy loss measurement was performed four times at each energy and the average value was obtained. Then, stopping powers were obtained by dividing the energy loss by the average path length of protons in the target. The average path length of protons in the target was obtained as t(l + (0*)/4), where t is the thickness of the target and (0’) is the mean square angular deflection of protons from the direction of the incident beam [4]. The value of (0*> was calculated by using the elementary theory of multiple scattering [29]. The energy loss divided by the average path length of protons in the target, AE/AX, corresponds, to a good approximation, to the stopping power, - dE/d X, at the “average energy” defined by i?=E,-AE/2. The incident proton energy E, was chosen in such a way that the average energy E became very near a round number energy. The stopping power was reduced to a round number energy by assuming that the stopping power is proportional to (hi u2)/v2 in a narrow velocity range. The thickness of the target was determined by the weight per area method. The weight of the target was measured with a Mettler M-5 microbalance five times and the area was measured also five times with a Tiyoda LTG bi-AI1 microscope that can read to 1 urn. The Al targets were supplied from Toyo Aluminum Co. Ltd. and three targets of 3.4828 mg/cm2, 7.5001 mg/cm2 and 11.5586 mg/cm’ were used. The Cu targets were supplied from Fukuda Metal Foil and Powder MFG Co. Ltd. and four targets of 3.8026 mg/cm2, 7.6197 mg/cm2, 11.4588 mg/cm2 and 15.2491 mg/cm’ were used. The stated purity for the Al target was 99.8% and that for the Cu target was 99.9%. No correction for the impurity of the Al target was made, because the main impurity of the Al target is Si, a nearby element.
120
R Zshiwari et al. / Stopping of Al and Cu for 3- 9 MeVp
3. Results The results are shown in tables 1 and 2 and are compared with the Rise data of Sorensen and Andersen [8] and the Aarhus data of Andersen et al. [9]. In table 1, Tschalar’s data for Al 1301 are also shown. The uncertainty of the present stopping power at each energy was assigned to be f0.4%. For Cu data at 8.5 and 9.0 MeV the uncertainty was assigned to be f0.6%. Then, the results have been expressed in the form of Bichsel’s X-variable [31]. We use Bichsel’s notation to express the Bethe-Bloch formula of stopping power 1231.
X f(P)-lnl-g+@+ZtL,-g
i
,
2
1
(3) and f(B)=ln
2muZ --p2, i l-b2 1
where Z, is the atomic number of the projectile, Z, and A the atomic number and atomic weight of the target element. The symbol I is the mean excitation energy, C/Z, the shell correction, @ the Bloch correction, Z,L, the Barkas correction and S/2 the density effect correction. p is v/c where v is the speed of the projectile and c is the speed of light. The density effect correction is quite negligible in this experiment. The values of /3’ and f( fl) have been tabulated as functions of the incident proton energy by Bichsel [23]. The stopping power is given in keV/mg cme2.
I
0
I
I
I
I
I
5
10
ENERCY%e”)
The theoretical and experimental and X=,, are defined as
X-variable,
X theo=ln Ii-C/Z,-Qi-Z,L,
X,,
(5)
and
where SexP is the experimentally determined stopping power. In fig. 3, the X_, values for Al are shown and are compared with the Rise data [8] and the Aarhus data 191. me Xexp values of Tschahar [30] are also indicated. In fig. 4, the XexP values for Cu are shown and are compared with the Rise data and the Aarhus data.
4. Analysis The mean excitation energy I was determined from the comparison of the X,, values with the XexP values taking I as an adjustable parameter. For the shell correction, we used the Bonderup shell correction [ll]. The Bonderup shell correction contains an adjustable parameter x. The parameter x is a constant which appears in the theory of stopping power using the statistical model of the atom [32] (sometimes expressed by y and taken as &). We used x = 1.340 which has been determined in our previous experiments for 21 kinds of metallic elements for 6.5 MeV protons
WI.
For the Barkas correction, we used the theory of Ashley, Ritchie and Brandt [12-141. In this theory the Barkas correction L, contains x and another parameter 6 that denotes the scaled minimum impact parameter of the collision between the projectile and electrons. The value of b has been determined as 1.26 from our previous experiments for 6.5 MeV protons [lo].
I
I
20
Fig. 3. The present results for Al expressed in the form of Bichsel’s X-variable. The present results are compared with the Rise data (solid curve), the Aarhus data (dashed curve) and TschaEr’s data (dot and dash curve).
5.81 0
I 5
I 10
ENERGY
I ;ile”
1
Fig. 4. The present results for Cu expressed in the form of Bichsel’s X-variable. The present results are compared with the Risra data (solid curve) and the Aarhus data (dashed curve).
R. Ishiwari et al. / Stopping of AI and Cu for 3- 9 MeVp
121
Table 1 The present stopping power data for Al are given in keV/mg cm-* and are compared with the Rise data, the Aarhus data and Tscha&r’s data. The symbol A denotes the percentage difference. The asterisk indicates that the difference is statistically significant
Wevl
Present data
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0
82.67 + 0.33 73.60 f 0.29 66.84 + 0.27 61.70+0.25 57.23 f 0.23 52.97 + 0.21 49.59 + 0.20 46.78 L-0.19 44.33 + 0.18 41.75+0.17 40.04 f 0.16 38.26 + 0.15 36.72 f 0.15
E
Ris0
A
PI
data 83.23 & 0.25 74.55 * 0.22 47.72 + 0.20 62.19_+0.19 57.57*0.17 53.65 +0.16 50.28 f 0.15 47.34* 0.14 44.76 + 0.13 42.48 + 0.13 40.44+0.12 38.6lkO.12 36.96 f 0.11
-0.68 +oso - 1.29 * & 0.49 -1.32* +0.50 -0.79 +0.51 -0.59 +0.50 -1.28* *0.50 -1.39* *0.50 -1.20* +0.50 - 0.97 5 0.50 -1.75* +0.51 - 1.00 * f 0.50 -0.91 +0.50 -0.65 +0.51
Aarhus data
A
82.39 f 0.41 73.75 + 0.37 66.94k0.33 61.45 & 0.31 56.88 f 0.28 53.01 f 0.27 49.70 f 0.25 46.82 f 0.23
+0.34+0.64 -0.20+0.64 - 0.15 f 0.64 + 0.41+ 0.65 + 0.61* 0.63 - 0.08 f 0.65 -0.22*0.65 - 0.09 + 0.64
VI
TschalSr
A [%I
82.52 + 0.25 73.89 + 0.22 67.07 f 0.20 61.53+0.18 56.93 + 0.17 53.04kO.16 49.71+ 0.15 46.81+0.14 44.27 k 0.13 42.02 + 0.13 40.01 f 0.12 38.21+ 0.11 36.58 + 0.11
+0.18+0.50 - 0.39 + 0.49 -0.34+0.50 + 0.28 f 0.50 + 0.52 + 0.50 -0.13rto.50 - 0.24 + 0.50 - 0.06 f 0.50 + 0.14 * 0.50 -0.65rtO.51 + 0.08 f 0.50 + 0.13 f 0.49 + 0.38 & 0.51
Table 2 The present stopping power data for Cu are given in keV/mg cm-* and are compared with the F&o data and the Aarhus data. The symbol A denotes the percentage difference.. The asterisk indicates that the difference is statistically significant E WeVl
Present data
Rise data
A @I
Aarhus data
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0
61.31 f0.25 55.49 f 0.22 50.83 + 0.20 47.25 + 0.19 43.88 f 0.18 41.11+ 0.16 38.74 & 0.16 36.53 + 0.15 34.74+0.14
62.43 + 0.19 56.43 + 0.17 51.65 +0.15 47.71* 0.14 44.4O+o.13 41.58&0.12 39.14 + 0.12 37.00 + 0.11 35.11+ 0.11 33.42 + 0.10 31.91* 0.10 30.55 * 0.09 29.32 + 0.09
- 1.83* +0.51 -1.69* +0.50 -1.61* +0.49 -0.97 f0.50 -1.19* +0.51 - 1.14* * 0.49 - 1.03 + 0.52 - 1.29 * + 0.51 -1.07* +0.51
61.51 kO.31 55.64 + 0.28 50.94 + 0.25 47.08 f 0.24 43.88 + 0.22 41.16+0.21 38.79 + 0.19 36.70 + 0.18 34.83 + 0.17
30.10 t 0.18 29.03 f 0.17
- 0.33 + 0.65 - 0.27 + 0.64 -0.22kO.63 + 0.36 f 0.65 0 - 0.12 + 0.64 -0.13+0.64 - 0.47 + 0.64 -0.26*0.63
- 1.50 * + 0.67 - 1.00 + 0.66
x2
I
1
4.5
I
cu
-I
4.0-
3.5-
b
167.4
166
I VALUE
I
1 326.3
(eV)
169
Fig. 5. The least squares fitting between the X,, values and the Xexr values for Al taking the mean excitation energy Z as an adjustable parameter. The abscissa is the adjustable parameter I. The ordinate is X2 of the method of least squares.
3.01 325
326 I VALUE
327 (eV)
Fig. 6. The least squares fitting between the X,,, values and the x”s values for Cu taking the mean excitation energy I as an adjustable parameter. The abscissa is the adjustable parameter I. The ordinate is X2 of the method of least squares.
122
R. Ishiwari et al. / Stopping I
w
I
I
I
I
z P A 5
E
Fig. 7. The solid curve denotes the X,, curve for Al calculated with the best fit value of I = 167.4 eV. The solid circles are the present XaP points. The dashed curve denotes TschaWs X_, curve. Above 16 MeV the X,,, curve coincides with Tschaliir’s X_, curve within the thickness of the curves.
The best fit between the X,, values with the Xexp values was determined by the method of least squares. The mean excitation energies thus determined are 167.4 + 0.5 eV for Al and 326.3 + 0.6 eV for Cu. The attached errors are fitting errors. In figs. 5 and 6, the least squares fitting between the X,, values and the XexP values are depicted. In figs. 7 and 8 the X,, curves calculated with the best fit values of I are shown along curve was calcuwith the X_,, points. For Al the X,, lated up to 30 MeV and for Cu the X,, curve was calculated up to 20 MeV. In tables 3 and 4, the values of the Bonderup shell correction, the Bloch correction and the Barkas correction are shown along with the X,, values calculated using these values of corrections and the best fit I values for Al and Cu. The values of C/Z, and L, at 6.5 MeV are slightly different from the values given in
I
I
I
C/G
@
=1
X the0
0.16772 0.15411 0.14317 0.13306 0.12499 0.11765 0.11106 0.10511 0.10006 0.09524 0.09096 0.08710 0.08361 0.07749 0.07225 0.06747 0.06353 0.05994 0.05688 0.05406 0.05156 0.04912 0.04710 0.04528 0.04343 0.04192 0.04035 0.03891 0.03772 0.03647 0.03545 0.03436 0.03334 0.03250
- 0.00999 - 0.00857 - 0.00751 - 0.00669 - 0.00603 - 0.00549 - 0.00503 - 0.00465 - 0.00432 - 0.00404 - 0.00379 - 0.00357 - 0.00338 - 0.00304 - 0.00277 - 0.00255 - 0.00235 - 0.00219 - 0.00205 - 0.00192 - 0.00181 - 0.00171 - 0.00163 - 0.00155 - 0.00148 - 0.00141 - 0.00135 - 0.00130 - 0.00125 - 0.00120 -0.00116 - 0.00112 - 0.00108 - 0.00105
0.05603 0.04767 0.04133 0.03638 0.03240 0.02915 0.02645 0.02417 0.02222 0.02053 0.01907 0.01778 0.01664 0.01472 0.01317 0.01189 0.01082 0.00991 0.00913 0.00845 0.00786 0.00734 0.00688 0.00647 0.00611 0.00578 0.00548 0.00520 0.00495 0.00473 0.00452 0.00432 0.00414 0.00398
5.24207 5.23540 5.22874 5.22376 5.21901 5.21438 5.21003 5.20598 5.20255 5.19914 5.19607 5.19328 5.19074 5.18620 5.18224 5.17852 5.17545 5.17261 5.17019 5.16792 5.16590 5.16388 5.16224 5.16075 5.15919 5.15794 5.15661 5.15540 5.15441 5.15333 5.15248 5.15155 5.15067 5.14996
[Meal
Xtheo
I
and Cu for 3- 9 MeVp
Table 3 The Bonderup shell corrections and the Barkas corrections for Al and the Bloch corrections. The X,, values calculated by using these corrections and the best fit value of Z = 167.4 eV are also shown
I
-14
zi!J5 a
ofAl
1
1
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
ref. [2], because the values of the present paper have been calculated with finer mesh.
5. Calculation of smoothed stopping power
0
5
20
10
ENERG:
J
( MeV 1
Fig. 8. The solid curve denotes the X,&curve for Cu calculated with the best fit value of Z = 326.3 eV. The solid circles are the present Xexp points. The open circle depicts Burkig and MacKenzie’s point calculated at 20 MeV taking Al as the standard material.
Smoothed stopping power values have been calculated putting the I values determined in the present analysis and the shell corrections, and the Barkas corrections eq. (3). The results compared
are shown
with the Rise
the Bloch corrections
given in tables 3 and 4 into in tables
5 and 6 and
data [8] and the Aarhus
data
[9]. In table 5, the results for Al have also been compared with Tschalar’s data [30]. Although the present data cover the energy range from 3 to 9 MeV, stopping
R. Ishiwari etal./Stopping of Al and& for 3-9 MeVp Table 4 The Bonderup shell corrections and the Barkas corrections for Cu and the B&h corrections. The X,, values calculated by using these corrections and the best fit value of Z = 326.3 eV are also shown E
c/z,
@
-%
X the0
0.28174 0.26991 0.25843 0.24795 0.23743 0.22802 0.21937 0.21099 0.20343 0.19640 0.18967 0.18375 0.17823 0.16790 0.15857 0.15046 0.14265 0.13615 0.13034 0.12488 0.11996 0.11528 0.11104 0.10719
-0.00999 -0.00857 -0.00751 -0.00669 -0.00603 -0.00549 -0.00503 -0.00465 -0.00432 -0.00404 -0.00379 -0.00357 -0.00338 -0.00304 -0.00277 -0.00255 -0.00235 -0.00219 -0.00205 -0.00192 -0.00181 -0.00171 -0.00163 -0.00155
0.08185 0.07119 0.06284 0.05614 0.05065 0.04607 0.04221 0.03890 0.03603 0.03354 0.03134 0.02939 0.02765 0.02469 0.02227 0.02024 0.01853 0.01707 0.01580 0.01470 0.01373 0.01287 0.01211 0.01142
5.99770 5.99511 5.99092 5.98632 5.98063 5.97526 5.97001 5.96456 5.95954 5.95472 5.94994 5.94575 5.94178 5.93407 5.92689 5.92059 5.91429 5.90909 5.90441 5.89992 5.89586 5.89194 5.88838 5.88514
WV1 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10 11 12 13 14 15 16 17 18 19 20
123
values obtained by TschalL [30] are also shown. The agreement between the present and Tschallr’s data is very good. The difference between the present and Tschallir’s data is 0.008% on the average and the standard deviation is 0.32%. The above-mentioned features of the present results for Al are also shown in fig. 3 in the form of Bichsel’s X variable. Although the present X_ points are somewhat scattered, it can be clearly seen that the present results agree with the Aarhus data and also with TschaWs data but evidently lower than the Riser data. (The higher X value means the lower stopping power value.) Table 2 shows that the Riser data [8] for Cu are higher than the present results by 1.30% on the average. The difference is statistically significant. The present value at 6.5 MeV is 36.53 f 0.15 keV/mg cmW2 and is slightly lower than the value 36.71 f 0.11 keV/mg cm-* obtained in the previous experiment for 6.5 MeV [1,2]. However, the difference is 0.18 & 0.19 keV/mg cm-* and is well within the statistical uncertainty. Table 2 also shows that the present results for Cu agree well with the Aarhus data [9]. The difference between the present and the Aarhus data is 0.16% on the average and the standard deviation is 0.27%. The difference is well within the statistical uncertainty. Fig. 4 shows the above-mentioned comparisons for Cu in the form of Bichsel’s X variable. It is clearly seen that the present results agree well with the Aarhus data but are evidently lower than the Risra data. 6.2. Mean excitation energy
powers have been tentatively calculated up to 30 MeV for Al and up to 20 MeV for Cu.
6. Discussion 6. I. Experimental
stopping power
Table 1 shows that the Riser data [S] for Al are higher than the present results by 1.06% on the average. Significant differences are marked with asterisks. On the whole, it can be concluded that the differences are statistically significant over the energy range from 3.0 to 9.0 MeV. This feature agrees with that observed in the previous experiments for 6.5 MeV protons [l-3]. The present stopping power value at 6.5 MeV is 46.78 + 0.19 keV/mg cm-* and agrees very well with the value of 46.83 + 0.14 keV/mg cm-* obtained in the previous experiment for 6.5 MeV protons [1,2]. Table 1 also shows that the present results agree well with the Aarhus data [9]. The difference between the present and the Aarhus data is 0.078% on the average and the standard deviation is 0.31%. The difference is well withjn the statistical uncertainty. In table 1 the stopping power
6.2.1. Al The present 1 value for Al 167.4 + 0.5 eV agrees very well with the value 167.6 f 2.8 eV [1,2] and 167.7 + 2.8 eV [lo] obtained in the previous experiment for 6.5 MeV protons. The present value also agrees with the value 167.0 + 0.8 eV of Tschalk and Bichsel [30,33] obtained from the energy loss measurement for 31 MeV protons and with the value 165.7 f 1 eV of Shiles et al. [34] calculated from the dielectric response function. The present value also agrees with the value 166 + 2 eV given in ICRU Report 37 [26]. NCRP [18], Fano [19], Bichsel [20], Turner et al. [21,22], Andersen and Ziegler [24], Ziegler [25] and Ahlen [27] gave I values from 162 to 164 eV. Bichsel and Uehling [35] analyzed Bichsel’s range data for protons having energies from 1 to 18 MeV in Al [36] by correcting for multiple scattering and obtained 1 value as 163 eV. This I value is 4 eV lower than the result of the present analysis. It is considered that thebe are three causes of this difference, namely (a) the difference of the experimental data, (b) the difference of the shell corrections used in the both analyses, and (c) the fact that the Bloch correction and the
124
R Ishiwari et al. / Stopping of Al and Cu for 3 - 9 MeVp
Table 5 The calculated smoothed stopping power values for Al are compared with the Rise data, the Aarhus data and TschaEr’s data. The symbol S denotes the stopping power and is given in keV/mg cm- *. The symbol A denotes the percentage difference Smoothed S-value
Rise data
A VI
Aarhus data
A PI
TschaEr
A @I
WV1 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
82.26 13.75 61.00 61.51 56.94 53.06 49.74 46.85 44.31 42.06 40.05 38.24 36.60 (33.76) (31.37) (29.32) (27.56) (26.01) (24.65) (23.43) (22.34) (21.37) (20.48) (19.67) (18.93) (18.25) (17.62) (17.04) (16.50) (16.00) (15.53) (15.09) (14.68) (14.29)
83.23 74.55 67.12 62.19 57.57 53.65 50.28 47.34 44.76 42.48 40.44 38.61 36.96 34.09 31.67 29.60 27.82 26.26 24.89 23.67 22.58 21.59
-1.18 - 1.08 -1.07 - 1.11 -1.11 - 1.11 - 1.09 -1.05 -1.02 - 1.00 - 0.97 -0.97 - 0.98 - 0.98 - 0.96 -0.95 -0.94 -0.96 - 0.97 -1.02 - 1.07 - 1.03
82.39 73.75 66.94 61.45 56.88 53.01 49.70 46.82
-0.16 0 + 0.09 +0.10 + 0.11 + 0.09 + 0.08 + 0.06
82.52 73.89 67.07 61.53 56.93 53.04 49.71 46.81 44.27 42.02 40.01 38.21 36.58 33.74 31.35 29.31 27.55 26.00 24.64 23.43 22.35 21.37 20.48 19.67 18.93 18.25 17.62 17.04 16.50 16.00 15.53 15.10 14.68 14.30
-0.32 -0.19 -0.10 - 0.03 + 0.02 + 0.04 + 0.06 + 0.09 + 0.09 +0.10 f0.10 + 0.08 + 0.05 + 0.06 + 0.06 + 0.03 + 0.04 + 0.04 + 0.04 0 -0.04 0 0 0 0 0 0 0 0 0 0 - 0.07 0 - 0.07
E
Barkas correction were not considered in the analysis of Bichsel and Uehhng. In general, the larger stopping power gives the smaller I value, the smaller shell correction gives the larger I value and inclusion of the Bloch correction and the Barkas correction gives the larger I value. The experimental data of Bichsel and Uehhng as expressed in stopping powers are higher than the present results by 1.17-0.378 over the energy range from 3 to 9 MeV. Therefore, in the analysis of Bichsel and Uehhng the I value should have a smaller value. Bichsel and Uehling used WaIske’s K-shell correction [37] and the simplified L-shell correction, C, = l.S/E(MeV). Their total shell corrections are smaller than the present shell corrections by 36-12% over the energy range from 3 to 8 MeV. Thus, their I value should have somewhat a larger value. Roughly considered, the effect of the larger stopping powers nearly compensates the effect of
the smaller shell corrections. Then, the inclusion of the Bloch correction and the Barkas correction will raise their I value to about 167 eV. However, recently published ICRU Report 37 [26] recommends the Walske’s shell corrections [37,38] modified by Bichsel’s scaling method [39]. These shell corrections are larger by several percent than the present Bonderup shell corrections. Therefore, the shell corrections used by Bichsel and Uehhng in their analysis are probably too small. If we use the present Bonderup shell corrections and include the Bloch correction and the Barkas correction in the analysis of the experimental data of Bichsel and Uehhng, an I value of 163-164 eV will be obtained. The difference between this I value and the present I value of 167.4 eV arises entirely from the higher stopping power values of Bichsel and Uehhng than the present results.
R Ishiwari et al. / Stopping
ofAl
and Cu for 3- 9 MeVp
125
Table 6 The calculatedsmoothedstoppingpower valuesfor Cu are comparedwith the Rise data, the Aarhusdata and Burkigand MacKenzie value at 20 MeV takingAl as the standardmaterial.The symbol S denotes the stopping power and is given in keV/mg cmm2.The symbol A denotes the percentagedifference E
Smoothed
Rise
NV]
S-value
data
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10 11 12 13 14 15 16 17 18 19 20
61.27 55.51 50.87 47.04 43.84 41.09 38.70 36.62 34.78 33.13 31.66 30.32 29.10 (26.98) (25.18) (23.63) (22.29) (21.10) (20.05) (19.11) (18.26) (17.50) (16.81) (16.17)
62.43 56.43 51.65 47.71 44.40 41.58 39.14 37.00 35.11 33.42 31.91 30.55 29.32 27.15 25.33 23.76 22.40 21.20 20.14 19.20 18.35 17.58
A
[%I
-1.89 - 1.66 -1.53 - 1.42 - 1.28 - 1.19 - 1.14 - 1.04 - 0.95 - 0.88 -0.79 - 0.76 - 0.76 - 0.63 - 0.60 - 0.55 - 0.49 - 0.47 - 0.45 - 0.47 - 0.49 - 0.46
Recently, McGuire [40] made a drastic suggestion that the Z value for Al should be in the range 145-150 eV rather than 162-167 eV. McGuire argued that the high energy experiment of Mather and Segre [41] indicates that the Z value for Al is 145-150 eV. The only stopping power data below 20 MeV which suggests the Z value of 163-169 eV are those of Bichsel and Uehhng [35]. He argued that below 20 MeV the Z value depends critically on the experimental data and the shell correction used in the analysis. If he applies his own shell corrections [42] to the Rise data, an Z value of 145-150 eV is obtained. He also argued that the calculation of the Z value of Shiles et al. [34] from the dielectric response function is subject to errors arising from experimental errors of optical experiments and oxygen contamination. McGuire’s suggestion became a subject of controversy [43,44]. Bichsel, Inokuti and Smith [43] argued against McGuire the following points, (a) there is an elementary error in McGuire’s analysis, i.e. McGuire used the atomic mass unit in place of the proton mass. When corrected for this error, McGuire’s suggestion becomes 150-155 eV, (b) in the discussion of the stopping power, McGuire puts unwarranted emphasis on the Rise data and ignored the Aarhus data, Tschahir’s
Aarhus data
A [%I
61.51 55.64 50.94 47.08 43.88 41.16 38.79 36.70 34.83
-0.39 - 0.23 - 0.14 - 0.09 - 0.09 -0.17 - 0.23 - 0.22 -0.14
Burkigand MacKenzie
A [%I
16.146
+0.15
data and the present data. The latter three sets of data agree with each other and are decisively lower than the Rise data. These three sets of data raise the Z value by about 8 eV, (c) although in principle McGuire’s method in obtaining the shell correction is correct, the shape of McGuire’s shell correction is unnatural as a function of the energy and the magnitude of his shell correction is not accurate enough numerically. The use of McGuire’s shell correction accounts for the additional 8 eV difference in the Z value, (d) the high energy experiment of Mather and Segre [41] is not accurate enough and their Z value is too low, because they did not take into account nuclear reactions, (e) the analysis of Bichsel and Uehling indicates the Z value - 163 eV, and (f) the two sources of error in the analysis of Shiles et al., which McGuire pointed out would increase not decrease the Z value. It appears that McGuire’s arguments indicating the Z value of 145-150 eV are almost entirely refuted. The most important points regarding the present experimental results are that McGuire placed a special emphasis on the l&o data and McGuire used his own shell corrections [44]. If we apply the present Bonderup shell corrections, the Bloch corrections and the Barkas corrections to the Risa data, an Z value of about 159 eV
126
R. Ishiwari et al. / Stopping of Al and Cu for 3- 9 MeVp
will be obtained. McGuire’s shell corrections are larger than the present Bonderup shell corrections by 111-27% over the energy range from 3 to 8 MeV. If we apply McGuire’s shell corrections to the Rise data, an I value of 150-155 eV will be obtained. However, as pointed out by Bichsel et al. [43], McGuire’s shell correction is not a smooth function of the energy. Thus, McGuire’s shell correction is considered to contain considerable numerical errors. McGuire argues that his shell corrections are in good agreement with the experimental shell corrections derived from the Aarhus data. However, this claim has almost no physical meaning. The Aarhus experimental shell correction is obtained by the following equation [9]: (C/Z, >,
= xeyp -In
I+*+ZiL,,
(7)
where Andersen et al. used experimentally determined Cp and L, and for the value of 1 they used the value 162 eV which is much larger than the value 150-155 eV suggested by McGuire. Therefore, the Aarhus experimental shell correction never accords with McGuire’s claim that the I value for Al should be 150-155 eV. R&ently, Oddershede and Sabin 1451 have combined Sigmund’s kinetic theory of stopping power of a medium with internal motion [46] and experimental Compton profile data 1471. Taking the I value as an adjustable parameter, they fitted the calculated stopping power values with the Aarhus experimental data [9]. The best fit was obtained with I = 163 eV. This evaluation of the I value utilizes a quite independent experimental data of the stopping power data, i.e. Compton profile data, and is in support of the I value of 162-16’7 eV. The I value extracted from the stopping power data depends on the choice of the shell correction. As mentioned above, ICRU Report 37 recommends Bichsel’s scaled she11 corrections [39] which are several percent larger than the present Bonderup shell corrections. If we apply shell corrections recommended by ICRU Report 37 to the present data, an I value of about 166 eV will be obtained. Although NCRP [18], Fano [19], Bichsel [20], Turner et al. (21,221, Andersen and Ziegler [24], Ziegler [ZS], and Ahlen 1271 gave I values from 162 to 164 eV, it appears that the I value for Al is more probably 166-167 eV. The uncertainty of the I value kO.5 eV given in this paper is only the fitting error of the method of least squares. If we take the ~~rt~nty of the Bonderup she11 correction to be +lO%, the uncertainty of the I value becomes 42.8 eV. 6.2.2. cu The present result of 326.3 f 0.6 eV is larger than the previous results of 323.5 rt: 7.9 eV [2] by 2.8 eV and 323.4 f 7.9 eV [lo] by 2.9 eV. The previous results for I were obtained from the stopping power measurement at one proton energy, i.e. at 6.5 MeV. The present I value
has been determined from the fitting with 11 points with different energies in the energy range from 3 to 9 MeV. Therefore, as far as the Bonderup shell corrections are used, the present I value of 326.3 eV is more reliable than the previous values of 323.5 eV or 323.4 eV. As mentioned earlier, the present attached error of the 1 value is the fitting error of the method of least squares. If we take the uncertainty of the Bonderup shell correction to be *lo%, the uncertainty of the I value becomes 18 eV. The difference between the present and the previous I values is well within the statistical uncertainty. Sorensen and Andersen obtained and I value of 319.8 f 3.2 eV from the Rise data. Their method of extracting the I value from the stopping power data depends neither on the choice of the type of the shell correction nor the disregard of the Bloch correction and the Barkas correction, but directly depends on the absolute value of the experimental stopping power. Since their stopping power data are higher than the present data by 1.30% on the average, and I value about 5% lower than the present result, i.e. 310-315 eV, is expected from the Riser data. Actually, however, their I value is somewhat higher than this expectation. The origin of this deviation is not understood. For the 1 value for Cu, NCRP [IS], Fano [19], Turner et al. 121,221, Bichsel 1231, and Ahlen [27] gave I values from 306 to 319 eV. Bichsel [20] and Andersen and Ziegler [24] gave 322 eV and Ziegler [25] gave 330 eV. ICRU Report 37 1261 recommends an I value of 322 zt:10 eV. Although Ahlen gave a very small uncertainty, _+2 eV, for his I value 317 eV, the above-mentioned I values agree with the present result within the statistical uncertainty, provided that the uncertainty of the present result is taken as + 8 eV. ICRU Report 37 recommends Bichsel’s scaled shell corrections [39] for Cu which are about 10% smaller than the present Bonderup shell corrections in the energy range from 2 to 6.5 MeV and nearly equal to the present shell corrections at energies from 10 to 12 MeV. If we apply these shell corrections to the present experimental data, an I value 333-336 eV will be obtained for the data from 3 to 6.5 MeV and an I value - 327 eV will be obtained at 9 MeV. Although at present there is no experimental evidence whether the present Bonderup shell corrections or Bichsel’s scaled shell corrections recommended in ICRU Report 37 represent the experimental situation better, it appears that the present experimental data indicates an I value equal to or larger than 326 eV for Cu. As mentioned earlier, the Aarhus stopping power data for Cu are larger than the present results by 0.16% on the average. Therefore, it is expected that the Aarhus data will give a slightly smaller I value than the present I value. If we apply the present Bonderup shell corrections, the Bloch corrections and the Barkas corrections
R. Ishiwari et al. /Stopping
to the Aarhus data, an I value of 323 - 325 eV is obtained. Burkig and MacKenzie [48] performed very accurate relative stopping power measurements for 19.8 MeV protons. Using the present Z value for Al 167.4 eV and the Bonderup shell correction, the Bloch correction and the Barkas correction, the stopping power of Al for 20 MeV protons is calculated as 19.666 keV/mg cm-*. The stopping power of Cu for 20 MeV protons is obtained by multiplying the relative stopping power of Cu of Burkig and Mackenzie, 0.821, to 19.666 keV/mg cm-* as 16.146 keV/mg cm-*. Applying the present Bonderup shell correction, the Bloch correction and the Barkas correction to this value, the Z value for Cu is obtained as 328.6 eV. ICRU Report 37 gives the shell correction which is larger by 4.30% than the present Bonderup shell correction at 19.8 MeV. If we apply the shell correction that is 1.0430 times the present Bonderup shell correction, the Bloch correction and the Barkas correction to the stopping power value 16.146 keV/mg cm-* of Burkig and McKenzie at 20 MeV, an I value of 327.1 eV is obtained. On the whole, it appears that the Z value for Cu is more probably equal to 326 eV or somewhat larger rather than the 322 eV that is recommended in ICRU Report 37. 6.3. Cakxdated smoothed stopping power Table 5 shows the calculated smoothed stopping powers for Al. Because the present experiment covers only the energy range from 3 to 9 MeV, the calculated values above 10 MeV are given in parentheses. The calculated smoothed stopping powers are decisively lower than the Riser data. The difference is 1.03% on the average. The agreement between the calculated values and the Aarhus data is better than that between the experimental stopping powers and the Aarhus data shown in table 1. The difference is 0.046% on the average and the standard deviation is 0.096%. The agreement between the calculated values and TschalL’s data [30] is surprisingly good up to 30 MeV. The difference is only 0.0024% on the average and the standard deviation is 0.0812%. The calculated stopping power at 6.5 MeV is 46.85 keV/mg cm-* and agrees very well with our previous value of 46.83 keV/mg cm-* [1,2]. curve calculated with the present Z value The X,, for Al 167.4 eV and the Bonderup shell corrections, the Bloch corrections and the Barkas corrections given in table 3 is shown in fig. 7. As expected from table 5, above 16 MeV the present X,,,, curve coincides with TschalL’s experimental XexP curve within the thickness of the curves. Table 6 shows the calculated smoothed stopping powers for Cu. The values above 10 MeV are also given in parentheses. The calculated smoothed stopping
of Al and Cu for 3- 9 MeVp
127
powers are lower than the Rise data by 0.90% on the average. The difference between the calculated stopping powers and the Rise data decreases as the energy increases. The difference in the range from 3.0 to 9.0 MeV is 1.18% on the average. The calculated values are lower than the Aarhus data by 0.19% on the average. However the differences between the calculated values and the Aarhus data are well within the statistical uncertainties. In table 6 Burkig and MacKenzie’s value obtained by taking the Z value for Al as 167.4 eV is also shown. The calculated value agrees with the value of Burkig and MacKenzie very well. Therefore, the calculated stopping powers from 10 to 20 MeV are considered to be fairly reliable. The X&n curve calculated with the present Z value for Cu 326.3 eV and the Bonderup shell corrections, the Bloch corrections and the Barkas corrections given in table 4 is shown in fig. 8. As expected from table 6, Burkig and MacKenzie’s point falls very near the Xtheo curve. 6.4. Shell correction 6.4.1. AI As shown in fig. 7, the Bonderup shell corrections for Al can reproduce very well not only the,present X_, points but also TschaEr’s X_, values [30] up to 30 MeV. As mentioned earlier, the shell corrections for Al recommended by ICRU Report 37 [26], which are Bichsel’s scaled shell corrections [39] based on Walske’s K- and L-shell corrections [37,38], are larger by several percent than the present Bonderup shell corrections in the energy range from 3 to 9 MeV. These shell corrections give an I value of about 166 eV when applied to the present experimental data taking account of the Bloch corrections and the Barkas corrections and can reproduce Tschahar’s X_, values up to 30 MeV. Therefore, the shell corrections for Al recommended by ICRU Report 37 are in fundamental agreement with the present Bonderup shell corrections within the uncertainties accompanying theoretical calculations. Recently, Sabin and Oddershede [49] have calculated the shell corrections for Al shell by shell using Sigmund’s kinetic theory of stopping power [46] and the Hartree-Slater calculations of oscillator strength moments given by Dehmer et al. [50] and Inokuti et al. [51,52]. The shell corrections of Sabin and Oddershede are larger by 42-40% than the present Bonderup shell corrections over the energy range from 3 to 9 MeV. If we apply the shell corrections of Sabin and Oddershede to the present calculated stopping powers using the Bloch corrections and the Barkas corrections, the Z values are obtained as 156.03 eV for 3 MeV, 159.87 eV for 6 MeV and 161-97 eV for 9 MeV. Thus, the shell corrections of Sabin and Oddershede do not give a
128
R. Ishiwari et al. / Stopping of Al and Cu for 3- 9 MeVp I
y
m
I
I
I
5.3
t
---_ .$&in
&j-------
Oddershede 0
I
I
5
10
I
ENERG’)5( MeV1
I
20
Fig. 9. The solid curve denotes the present Xtt,_, curve. The dashed curve denotes the X,,,_, curve calculated with Sabin and Odderschede’s shell corrections, I value of 159.87 eV and the present Bloch and Barkas corrections. The solid circles indicate the present Xexp points.
I value. This means that the shape of Sabin and Oddershede’s shell correction as a function of the energy does not agree with the present Xexp points. In fig. 9, the X,, curve calculated with Sabin and Oddershede’s shell corrections, I value of 159.87 eV, the present Bloch and Barkas corrections is shown and is compared with the present X,, curve. As already mentioned, the X,, curve using the present Bonderup shell correction agrees very well with Tschallr’s X_, curve up to 30 MeV. As is clearly seen from the figure, Sabin and Oddershede’s Xtheo curve deviates from the X thee curve using the present Bonderup shell correction. Since the change of the I value only shifts the X,, curve up and down without changing its shape, we can conclude that Sabin and Oddershede’s shell correction does not reproduce the Xex,, point in a fundamental way. Andersen et al. [9] state that their experimental shell corrections, eq. (7), derived from the Aarhus data do not agree with the Bonderup shell corrections when they use the I value of 162 eV. However, if they use the I value of 169 eV, much better agreement is obtained. This fact indicates that the Bonderup shell correction can reproduce fundamentally the energy dependence of the Xexp curve of the Aarhus data over the energy range from 0.8 to 6.4 MeV. The difference between their I value of 169 eV and the present I value of 167.4 eV comes mainly from the fact that the Aarhus empirical L, values are larger than the present estimates of L, by 24-282 over the energy range from 3 to 6 MeV. constant
6.4.2. Cu At present two types of the shell correction are usually used in analyzing experimental data, i.e. the Bonderup shell correction [ll] and Bichsel’s scaled shell correction based on Walske’s K- and L-shell correction
WI.
As already mentioned, if we apply Bichsel’s scaled shell corrections recommended in ICRU 37 to the present experimental data, I values of 333-336 eV are obtained in the energy range from 3 to 6.5 MeV. Although at present there exists no criterion which type of the shell correction represents the experimental situation of Cu better, Z values of 333-336 eV appear to be slightly too large as compared with the I values given by other authors, i.e. 306-322 eV [18-24,26,27]. Another accurate stopping power measurement up to 30 MeV is strongly desired. The authors would like to thank Prof. S. Kobayashi for his kind support throughout this experiment. Thanks are also due to Drs. K. Takimoto, M, Nakamura and the members of the Van de Graaff Laboratory of Kyoto University for their kind cooperation.
References 111 R. Ishiwari, N. Shiomi and N. Sakamoto, Nucl. Instr. and Meth. 194 (1982) 61.
PI R. Ishiwari, N. Shiomi and N. Sakamoto, Nucl. Instr. and
Meth. B31 (1988) 503. [31 R. Ishiwari, N. Shiomi and N. Sakamoto, Nucl. Instr. and Meth B2 (1984) 195. t41 H.H. Andersen, A.F. Garfinkel. C.C. Hanke and H. Sorensen, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 35 (1966) no. 4. [51 H.H. Andersen, C.C. Hanke, H. Sorensen and P. Vajda, Phys. Rev. 153 (1967) 338. WI H.H. Andersen, C.C. Hanke, H. Simonsen, H. Sorensen and P. Vajda, Phys. Rev. 175 (1968) 389. 171 H.H. Andersen, H. Simonsen, H. Sorensen and P. Vajda, Phys. Rev. 186 (1969) 372. PI H. Sorensen and H.H. Andersen, Phys. Rev. B8 (1973) 1854. t91 H.H. Andersen, J.F. Bak, H. Knudsen and B.R. Nielsen, Phys. Rev. Al6 (1977) 1929. WI N. Sakamoto, N. Shiomi-Tsuda, H. Ogawa and R. Ishiwari, Nucl. Instr. and Meth. B33 (1988) 158. VI E. Bonderup, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 35 (1967) no. 17. WI J.C. Ashley, R.H. Ritchie and W. Brandt, Phys. Rev. B5 (1972) 2393. [I31 J.C. Ashley, R.H. Ritchie and W. Brand& Phys. Rev. A8 (1973) 2402. t141 R.H. Ritchie and W. Brandt, Phys. Rev. Al7 (1978) 2102. P51 F. Bloch, Ann. Phys. (Leipzig) 16 (1933) 285. WI J. Lmdbard, Nucl. Instr. and Meth. 132 (1976) 1. t171 H.H. Andersen, H. Sorensen and P. Vajda, Phys. Rev. 180 (1969) 373. WI NCRP, National Committee on Radiation Protection, Stopping Powers for Use with Cavity Chambers (National Bureau of Standards, Washington DC, 1961). P91 U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. WI H. Bichsel, Charged Particle Interactions in Radiation Dosimetry, vol. 1, eds. F.H. Attix and WC. Roesch (Academic Press, New York, 1968) p. 157.
R. Zshiwari et al. / Stopping of Al and Cu for 3- 9 MeVp [21] P. Dalton and J.E. Turner, ORNL-TM-1777, Oak Ridge National Laboratory Report (1967) unpublished. [22] J.E. Turner, P.D. RoeckIein and RB. Vora, Health Phys. 18 (1970) 159. [23] H. Bichsel, Passage of Charged Particles Through Matter, in: American Institute of Physics Handbook, ed. D.E. Gray (McGraw-Hill, New York, 1972) p. 8-142. 1241 H.H. Andersen and J.F. Ziegler, Hydrogen Stopping Powers and Ranges in All Elements (Pergamon, New York, 1977). [25] J.F. Ziegler, Nucl. Instr. and Meth. 168 (1980) 17. [26] ICRU Report 37, Stopping Powers for Electrons and Positrons (International Commission on Radiation Units and Measurements, 1984). [27] S.P. Ahlen, Rev. Mod. Phys. 52 (1980) 121. [28] C.W. Reich, G.C. Philips, and J.L. Russell, Jr., Phys. Rev. 104 (1956) 143. [29] H.A. Bethe and J. Ashkin, Passage of Radiations Through Matter, in: Experimental Nuclear Physics, ed. E. Segre, vol. 1 (Wiley, New York, 1953) p. 166. [30] C. Tscha&, Thesis, University of Southern California (1967) unpublished. [31] H. Bichsel, A Critical Review of Experimental Stopping Power and Range Data, in National Academy of ScienceNational Research Council, Publ. 1133, Studies in Penetration of Charged Particles in Matter (1964) p. 17. [32] J. Lindhard and M. Scharff, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 35 (1967) no. 15. [33] C. Tsch&r and H. Bichsel, Phys. Rev. 175 (1968) 476.
129
[34] E. Shiles, T. Sasaki, M. Inokuti, and D.Y. Smith, Phys. Rev. B22 (1980) 1612. [35] H. Bichsel and E.A. Uehling, Phys. Rev. 119 (1960) 1670. [36] H. Bichsel, Phys. Rev. 112 (1958) 1089. [37] M.C. WaIske, Phys. Rev. 88 (1952) 1283. [38] M.C. WaIske, Phys. Rev. 101 (1956) 940. [39] H. Bichsel, UCRL-17538 Lawrence Radiation Laboratory, University of California (1967) unpublished. [40] E.J. McGuire, Phys. Rev. A28 (1983) 53. [41] R. Mather and E. Segre, Phys. Rev. 84 (1951) 191. [42] E.J. McGuire, Phys. Rev. A28 (1983) 49. [43] H. Bichsel, M. Inokuti and D.Y. Smith, Phys. Rev. A33 (1986) 3567. [44] E.J. McGuire, Phys. Rev. A33 (1986) 3572. [45] J. Oddershede and J.R. Sabin, Phys. Rev. A35 (1987) 3283. [46] P. Sigmund, Phys. Rev. A26 (1982) 2497. [47] S. Manninen, T. Paakkari and K. Kajantie, Philos. Mag. 29 (1974) 167. [48] V.C. Burkig and K.R. MacKenzie, Phys. Rev. 106 (1957) 848. [49] J.R. Sabin and J. Oddershede, Phys. Rev. A26 (1982) 3209. [50] J.L. Dehmer, M. Inokuti, and R.P. Saxon, Phys. Rev. Al2 (1975) 102. [51] M. Inokuti, T. Baer and J.L. Dehmer, Phys. Rev. Al7 (1978) 1229. [52] M. Inokuti, J.L. Dehmer, T. Baer and H. Hanson, Phys. Rev. A23 (1981) 95.
Nuclear Instruments and Methods in Physics Research B35 (1988) 118-129 North-Holland, Amsterdam
STOPPING
POWERS OF Al AND Cu FOR PROTONS
FROM 3 TO 9 MeV
R. ISHIWARI Faculty of General Education, Kinki University, Higashi-Osaka
N. SHIOMI-TSUDA Department
577, Japan
and N. SAKAMOTO
of Physics, Nara Women’s University, Nara 630, Japan
Received 19 July 1988
The stopping powers of Al and Cu for protons having energies from 3 to 9 MeV have been measured with an accuracy of +0.4%. The results have been compared with the Rise and the Aarhus data of Andersen et al. and with Tschaliir’s data for Al. The mean excitation energies have been extracted from the present results using the Bonderup shell correction, the Barkas correction and the Bloch correction as 167.4 + 0.5 eV for Al and 326.3 + 0.6 eV for Cu. A set of parameters x = 1.340 and b = 1.26 was used in these determinations. By using the mean excitation energies thus determined, the Bonderup shell corrections, the Barkas corrections and the Bloch corrections, stopping powers have been calculated up to 30 MeV for Al and up to 20 MeV for Cu. The calculated stopping powers agree very well with the Aarhus data. For Al, the agreement between the calculated stopping powers and TschaWs data is surprisingly good up to 30 MeV.
1. Introduciion
2. Experimental procedure
In our previous experiments [l-3], we have measured stopping powers of 21 kinds of metallic elements for 6.5 MeV protons. These experiments revealed that the Rise data of Andersen et al. [4-81 are too high by 0.78% on the average and the Aarhus data of Andersen et al. [9], except for Au, are in good agreement with our data. From our stopping power data of 21 metallic elements for 6.5 MeV protons [l-3], the mean excitation energies have been extracted [2,10] using the Bonderup shell correction [ll] and taking account of the Barkas correction of Ashley, Ritchie and Brandt [12-141 and the Bloch correction [15,16]. Although our mean excitation energies agree very well with the I values given by different authors [8,17-271, it is desirable to measure stopping powers at various energies to obtain more reliable mean excitation energies. In the present experiment, stopping powers of Al and Cu for protons having energies from 3 to 9 MeV have been measured and the mean excitation energies have been determined by using the Bonderup shell correction [ll] and by taking account of the Barkas correction [12-141 and the Bloch correction [15,16]. Then, by using these mean excitation energies the stopping power table of Al for protons from 3 to 30 MeV and that of Cu for protons from 3 to 20 MeV have been calculated.
The analyzed beam of protons accelerated by the tandem Van de Graaff accelerator of Kyoto University was used in the present experiment. The analyzing magnet was calibrated by the resonance of elastic scattering of protons by ‘*C at 4.808 + 0.010 MeV [28]. The experimental procedures to determine the energy loss of protons in the sample target is the same as described in detail in the previous paper on stopping powers of 16 kinds of metallic elements for 6.5 MeV protons [2]. Therefore, only a brief description of the experimental procedures will be given here. Fig. 1 shows the experimental setup. The analyzed beam of protons was collimated by a double slit system St and S,. The diameter of each slit was 2 mm and they were 40 cm apart. Then, the beam was scattered by a thin gold foil of 180 pg/cm* placed at the center of the scattering chamber. The beam scattered at an angle of 15O was used for the energy loss measurement. The scattered beam was collimated by a double slit system. The diameter of each slit was 1.5 mm and they were 5.9 cm apart. To determine the energy loss of protons in the target, we used what we call the “absorber wheel technique”. In fig. 1, the absorber wheel is a brass disk 1 mm thick and 10 cm in diameter. The disk has two windows. The sample target was fitted to one of these windows and the other window was left open. The disk was fixed to an induction motor and was rotated by the motor at 60
0168-583X/88/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
R. Ishiwari et al. / Stopping of Al and Cu for 3- 9 MeVp 52
51
In order to determine the absolute value of the energy loss of protons in the target from the pulse height difference shown in fig. 2, the zero point of the abscissa of fig. 2 should be determined. This was done by the use of a very high precision pulse generator (Ortec 448). During the measurement the pulses from the pulse generator with three different dial settings were recorded in addition to the proton pulses. From these three pulse heights the zero point of the abscissa of fig. 2 was determined. Let the integral pulse heights with and without the target be Ht and He. The energy loss of protons in the target is given by
Au 180 wg/cm2
53
119
2mm
Targe
Fig. 1. A schematic diagram of the experimental setup for the energy loss measurementusing the “absorber wheel technique”.
AE=Eo,,
*O-H* 0
rpm during the measurement. A surface barrier silicon
detector was placed 9.2 mm behind the target. A small armular permanent magnet of about 500 G was placed just before the silicon detector to prevent the possible energetic S-rays ejected from the target from affecting the energy loss measurement. When ac the power was turned on and the “absorbed wheel” was rotated, protons that passed through the target and protons scattered by the Au scatterer hit the silicon detector alternately. Thus, the pulse heights with, and without, the target were measured simultaneously in one exposure. By this arrangement substantially all protons that passed through the target and suffered multiple scattering were detected by the silicon detector. To avoid confusion, we call the protons that are scattered by the thin Au foil at an angle of 15 o and are used for the stopping power measurement the “incident protons”. The pulses from the detector were amplified with a low noise amplifier system. The relevant portion of the pulse height spectrum was expanded by a biased amplifier and fed into a 400 channel pulse height analyzer. Fig. 2 shows a typical pulse height spectrum.
1500
,
I
I
Al
d
.
5 a lOOO-
..
g \
.*.. . * . * . . . . .
e z 0’ 500”
0
1
200
250
x ‘A
.*
300
350
CHANNEL NUMBER
Fig. 2. A typical pulse height spectrum.
400
where E,, is the incident energy. The energy loss measurement was performed four times at each energy and the average value was obtained. Then, stopping powers were obtained by dividing the energy loss by the average path length of protons in the target. The average path length of protons in the target was obtained as t(l + (0*)/4), where t is the thickness of the target and (0’) is the mean square angular deflection of protons from the direction of the incident beam [4]. The value of (0*> was calculated by using the elementary theory of multiple scattering [29]. The energy loss divided by the average path length of protons in the target, AE/AX, corresponds, to a good approximation, to the stopping power, - dE/d X, at the “average energy” defined by i?=E,-AE/2. The incident proton energy E, was chosen in such a way that the average energy E became very near a round number energy. The stopping power was reduced to a round number energy by assuming that the stopping power is proportional to (hi u2)/v2 in a narrow velocity range. The thickness of the target was determined by the weight per area method. The weight of the target was measured with a Mettler M-5 microbalance five times and the area was measured also five times with a Tiyoda LTG bi-AI1 microscope that can read to 1 urn. The Al targets were supplied from Toyo Aluminum Co. Ltd. and three targets of 3.4828 mg/cm2, 7.5001 mg/cm2 and 11.5586 mg/cm’ were used. The Cu targets were supplied from Fukuda Metal Foil and Powder MFG Co. Ltd. and four targets of 3.8026 mg/cm2, 7.6197 mg/cm2, 11.4588 mg/cm2 and 15.2491 mg/cm’ were used. The stated purity for the Al target was 99.8% and that for the Cu target was 99.9%. No correction for the impurity of the Al target was made, because the main impurity of the Al target is Si, a nearby element.
120
R Zshiwari et al. / Stopping of Al and Cu for 3- 9 MeVp
3. Results The results are shown in tables 1 and 2 and are compared with the Rise data of Sorensen and Andersen [8] and the Aarhus data of Andersen et al. [9]. In table 1, Tschalar’s data for Al 1301 are also shown. The uncertainty of the present stopping power at each energy was assigned to be f0.4%. For Cu data at 8.5 and 9.0 MeV the uncertainty was assigned to be f0.6%. Then, the results have been expressed in the form of Bichsel’s X-variable [31]. We use Bichsel’s notation to express the Bethe-Bloch formula of stopping power 1231.
X f(P)-lnl-g+@+ZtL,-g
i
,
2
1
(3) and f(B)=ln
2muZ --p2, i l-b2 1
where Z, is the atomic number of the projectile, Z, and A the atomic number and atomic weight of the target element. The symbol I is the mean excitation energy, C/Z, the shell correction, @ the Bloch correction, Z,L, the Barkas correction and S/2 the density effect correction. p is v/c where v is the speed of the projectile and c is the speed of light. The density effect correction is quite negligible in this experiment. The values of /3’ and f( fl) have been tabulated as functions of the incident proton energy by Bichsel [23]. The stopping power is given in keV/mg cme2.
I
0
I
I
I
I
I
5
10
ENERCY%e”)
The theoretical and experimental and X=,, are defined as
X-variable,
X theo=ln Ii-C/Z,-Qi-Z,L,
X,,
(5)
and
where SexP is the experimentally determined stopping power. In fig. 3, the X_, values for Al are shown and are compared with the Rise data [8] and the Aarhus data 191. me Xexp values of Tschahar [30] are also indicated. In fig. 4, the XexP values for Cu are shown and are compared with the Rise data and the Aarhus data.
4. Analysis The mean excitation energy I was determined from the comparison of the X,, values with the XexP values taking I as an adjustable parameter. For the shell correction, we used the Bonderup shell correction [ll]. The Bonderup shell correction contains an adjustable parameter x. The parameter x is a constant which appears in the theory of stopping power using the statistical model of the atom [32] (sometimes expressed by y and taken as &). We used x = 1.340 which has been determined in our previous experiments for 21 kinds of metallic elements for 6.5 MeV protons
WI.
For the Barkas correction, we used the theory of Ashley, Ritchie and Brandt [12-141. In this theory the Barkas correction L, contains x and another parameter 6 that denotes the scaled minimum impact parameter of the collision between the projectile and electrons. The value of b has been determined as 1.26 from our previous experiments for 6.5 MeV protons [lo].
I
I
20
Fig. 3. The present results for Al expressed in the form of Bichsel’s X-variable. The present results are compared with the Rise data (solid curve), the Aarhus data (dashed curve) and TschaEr’s data (dot and dash curve).
5.81 0
I 5
I 10
ENERGY
I ;ile”
1
Fig. 4. The present results for Cu expressed in the form of Bichsel’s X-variable. The present results are compared with the Risra data (solid curve) and the Aarhus data (dashed curve).
R. Ishiwari et al. / Stopping of AI and Cu for 3- 9 MeVp
121
Table 1 The present stopping power data for Al are given in keV/mg cm-* and are compared with the Rise data, the Aarhus data and Tscha&r’s data. The symbol A denotes the percentage difference. The asterisk indicates that the difference is statistically significant
Wevl
Present data
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0
82.67 + 0.33 73.60 f 0.29 66.84 + 0.27 61.70+0.25 57.23 f 0.23 52.97 + 0.21 49.59 + 0.20 46.78 L-0.19 44.33 + 0.18 41.75+0.17 40.04 f 0.16 38.26 + 0.15 36.72 f 0.15
E
Ris0
A
PI
data 83.23 & 0.25 74.55 * 0.22 47.72 + 0.20 62.19_+0.19 57.57*0.17 53.65 +0.16 50.28 f 0.15 47.34* 0.14 44.76 + 0.13 42.48 + 0.13 40.44+0.12 38.6lkO.12 36.96 f 0.11
-0.68 +oso - 1.29 * & 0.49 -1.32* +0.50 -0.79 +0.51 -0.59 +0.50 -1.28* *0.50 -1.39* *0.50 -1.20* +0.50 - 0.97 5 0.50 -1.75* +0.51 - 1.00 * f 0.50 -0.91 +0.50 -0.65 +0.51
Aarhus data
A
82.39 f 0.41 73.75 + 0.37 66.94k0.33 61.45 & 0.31 56.88 f 0.28 53.01 f 0.27 49.70 f 0.25 46.82 f 0.23
+0.34+0.64 -0.20+0.64 - 0.15 f 0.64 + 0.41+ 0.65 + 0.61* 0.63 - 0.08 f 0.65 -0.22*0.65 - 0.09 + 0.64
VI
TschalSr
A [%I
82.52 + 0.25 73.89 + 0.22 67.07 f 0.20 61.53+0.18 56.93 + 0.17 53.04kO.16 49.71+ 0.15 46.81+0.14 44.27 k 0.13 42.02 + 0.13 40.01 f 0.12 38.21+ 0.11 36.58 + 0.11
+0.18+0.50 - 0.39 + 0.49 -0.34+0.50 + 0.28 f 0.50 + 0.52 + 0.50 -0.13rto.50 - 0.24 + 0.50 - 0.06 f 0.50 + 0.14 * 0.50 -0.65rtO.51 + 0.08 f 0.50 + 0.13 f 0.49 + 0.38 & 0.51
Table 2 The present stopping power data for Cu are given in keV/mg cm-* and are compared with the F&o data and the Aarhus data. The symbol A denotes the percentage difference.. The asterisk indicates that the difference is statistically significant E WeVl
Present data
Rise data
A @I
Aarhus data
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0
61.31 f0.25 55.49 f 0.22 50.83 + 0.20 47.25 + 0.19 43.88 f 0.18 41.11+ 0.16 38.74 & 0.16 36.53 + 0.15 34.74+0.14
62.43 + 0.19 56.43 + 0.17 51.65 +0.15 47.71* 0.14 44.4O+o.13 41.58&0.12 39.14 + 0.12 37.00 + 0.11 35.11+ 0.11 33.42 + 0.10 31.91* 0.10 30.55 * 0.09 29.32 + 0.09
- 1.83* +0.51 -1.69* +0.50 -1.61* +0.49 -0.97 f0.50 -1.19* +0.51 - 1.14* * 0.49 - 1.03 + 0.52 - 1.29 * + 0.51 -1.07* +0.51
61.51 kO.31 55.64 + 0.28 50.94 + 0.25 47.08 f 0.24 43.88 + 0.22 41.16+0.21 38.79 + 0.19 36.70 + 0.18 34.83 + 0.17
30.10 t 0.18 29.03 f 0.17
- 0.33 + 0.65 - 0.27 + 0.64 -0.22kO.63 + 0.36 f 0.65 0 - 0.12 + 0.64 -0.13+0.64 - 0.47 + 0.64 -0.26*0.63
- 1.50 * + 0.67 - 1.00 + 0.66
x2
I
1
4.5
I
cu
-I
4.0-
3.5-
b
167.4
166
I VALUE
I
1 326.3
(eV)
169
Fig. 5. The least squares fitting between the X,, values and the Xexr values for Al taking the mean excitation energy Z as an adjustable parameter. The abscissa is the adjustable parameter I. The ordinate is X2 of the method of least squares.
3.01 325
326 I VALUE
327 (eV)
Fig. 6. The least squares fitting between the X,,, values and the x”s values for Cu taking the mean excitation energy I as an adjustable parameter. The abscissa is the adjustable parameter I. The ordinate is X2 of the method of least squares.
122
R. Ishiwari et al. / Stopping I
w
I
I
I
I
z P A 5
E
Fig. 7. The solid curve denotes the X,, curve for Al calculated with the best fit value of I = 167.4 eV. The solid circles are the present XaP points. The dashed curve denotes TschaWs X_, curve. Above 16 MeV the X,,, curve coincides with Tschaliir’s X_, curve within the thickness of the curves.
The best fit between the X,, values with the Xexp values was determined by the method of least squares. The mean excitation energies thus determined are 167.4 + 0.5 eV for Al and 326.3 + 0.6 eV for Cu. The attached errors are fitting errors. In figs. 5 and 6, the least squares fitting between the X,, values and the XexP values are depicted. In figs. 7 and 8 the X,, curves calculated with the best fit values of I are shown along curve was calcuwith the X_,, points. For Al the X,, lated up to 30 MeV and for Cu the X,, curve was calculated up to 20 MeV. In tables 3 and 4, the values of the Bonderup shell correction, the Bloch correction and the Barkas correction are shown along with the X,, values calculated using these values of corrections and the best fit I values for Al and Cu. The values of C/Z, and L, at 6.5 MeV are slightly different from the values given in
I
I
I
C/G
@
=1
X the0
0.16772 0.15411 0.14317 0.13306 0.12499 0.11765 0.11106 0.10511 0.10006 0.09524 0.09096 0.08710 0.08361 0.07749 0.07225 0.06747 0.06353 0.05994 0.05688 0.05406 0.05156 0.04912 0.04710 0.04528 0.04343 0.04192 0.04035 0.03891 0.03772 0.03647 0.03545 0.03436 0.03334 0.03250
- 0.00999 - 0.00857 - 0.00751 - 0.00669 - 0.00603 - 0.00549 - 0.00503 - 0.00465 - 0.00432 - 0.00404 - 0.00379 - 0.00357 - 0.00338 - 0.00304 - 0.00277 - 0.00255 - 0.00235 - 0.00219 - 0.00205 - 0.00192 - 0.00181 - 0.00171 - 0.00163 - 0.00155 - 0.00148 - 0.00141 - 0.00135 - 0.00130 - 0.00125 - 0.00120 -0.00116 - 0.00112 - 0.00108 - 0.00105
0.05603 0.04767 0.04133 0.03638 0.03240 0.02915 0.02645 0.02417 0.02222 0.02053 0.01907 0.01778 0.01664 0.01472 0.01317 0.01189 0.01082 0.00991 0.00913 0.00845 0.00786 0.00734 0.00688 0.00647 0.00611 0.00578 0.00548 0.00520 0.00495 0.00473 0.00452 0.00432 0.00414 0.00398
5.24207 5.23540 5.22874 5.22376 5.21901 5.21438 5.21003 5.20598 5.20255 5.19914 5.19607 5.19328 5.19074 5.18620 5.18224 5.17852 5.17545 5.17261 5.17019 5.16792 5.16590 5.16388 5.16224 5.16075 5.15919 5.15794 5.15661 5.15540 5.15441 5.15333 5.15248 5.15155 5.15067 5.14996
[Meal
Xtheo
I
and Cu for 3- 9 MeVp
Table 3 The Bonderup shell corrections and the Barkas corrections for Al and the Bloch corrections. The X,, values calculated by using these corrections and the best fit value of Z = 167.4 eV are also shown
I
-14
zi!J5 a
ofAl
1
1
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
ref. [2], because the values of the present paper have been calculated with finer mesh.
5. Calculation of smoothed stopping power
0
5
20
10
ENERG:
J
( MeV 1
Fig. 8. The solid curve denotes the X,&curve for Cu calculated with the best fit value of Z = 326.3 eV. The solid circles are the present Xexp points. The open circle depicts Burkig and MacKenzie’s point calculated at 20 MeV taking Al as the standard material.
Smoothed stopping power values have been calculated putting the I values determined in the present analysis and the shell corrections, and the Barkas corrections eq. (3). The results compared
are shown
with the Rise
the Bloch corrections
given in tables 3 and 4 into in tables
5 and 6 and
data [8] and the Aarhus
data
[9]. In table 5, the results for Al have also been compared with Tschalar’s data [30]. Although the present data cover the energy range from 3 to 9 MeV, stopping
R. Ishiwari etal./Stopping of Al and& for 3-9 MeVp Table 4 The Bonderup shell corrections and the Barkas corrections for Cu and the B&h corrections. The X,, values calculated by using these corrections and the best fit value of Z = 326.3 eV are also shown E
c/z,
@
-%
X the0
0.28174 0.26991 0.25843 0.24795 0.23743 0.22802 0.21937 0.21099 0.20343 0.19640 0.18967 0.18375 0.17823 0.16790 0.15857 0.15046 0.14265 0.13615 0.13034 0.12488 0.11996 0.11528 0.11104 0.10719
-0.00999 -0.00857 -0.00751 -0.00669 -0.00603 -0.00549 -0.00503 -0.00465 -0.00432 -0.00404 -0.00379 -0.00357 -0.00338 -0.00304 -0.00277 -0.00255 -0.00235 -0.00219 -0.00205 -0.00192 -0.00181 -0.00171 -0.00163 -0.00155
0.08185 0.07119 0.06284 0.05614 0.05065 0.04607 0.04221 0.03890 0.03603 0.03354 0.03134 0.02939 0.02765 0.02469 0.02227 0.02024 0.01853 0.01707 0.01580 0.01470 0.01373 0.01287 0.01211 0.01142
5.99770 5.99511 5.99092 5.98632 5.98063 5.97526 5.97001 5.96456 5.95954 5.95472 5.94994 5.94575 5.94178 5.93407 5.92689 5.92059 5.91429 5.90909 5.90441 5.89992 5.89586 5.89194 5.88838 5.88514
WV1 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10 11 12 13 14 15 16 17 18 19 20
123
values obtained by TschalL [30] are also shown. The agreement between the present and Tschallr’s data is very good. The difference between the present and Tschallir’s data is 0.008% on the average and the standard deviation is 0.32%. The above-mentioned features of the present results for Al are also shown in fig. 3 in the form of Bichsel’s X variable. Although the present X_ points are somewhat scattered, it can be clearly seen that the present results agree with the Aarhus data and also with TschaWs data but evidently lower than the Riser data. (The higher X value means the lower stopping power value.) Table 2 shows that the Riser data [8] for Cu are higher than the present results by 1.30% on the average. The difference is statistically significant. The present value at 6.5 MeV is 36.53 f 0.15 keV/mg cmW2 and is slightly lower than the value 36.71 f 0.11 keV/mg cm-* obtained in the previous experiment for 6.5 MeV [1,2]. However, the difference is 0.18 & 0.19 keV/mg cm-* and is well within the statistical uncertainty. Table 2 also shows that the present results for Cu agree well with the Aarhus data [9]. The difference between the present and the Aarhus data is 0.16% on the average and the standard deviation is 0.27%. The difference is well within the statistical uncertainty. Fig. 4 shows the above-mentioned comparisons for Cu in the form of Bichsel’s X variable. It is clearly seen that the present results agree well with the Aarhus data but are evidently lower than the Risra data. 6.2. Mean excitation energy
powers have been tentatively calculated up to 30 MeV for Al and up to 20 MeV for Cu.
6. Discussion 6. I. Experimental
stopping power
Table 1 shows that the Riser data [S] for Al are higher than the present results by 1.06% on the average. Significant differences are marked with asterisks. On the whole, it can be concluded that the differences are statistically significant over the energy range from 3.0 to 9.0 MeV. This feature agrees with that observed in the previous experiments for 6.5 MeV protons [l-3]. The present stopping power value at 6.5 MeV is 46.78 + 0.19 keV/mg cm-* and agrees very well with the value of 46.83 + 0.14 keV/mg cm-* obtained in the previous experiment for 6.5 MeV protons [1,2]. Table 1 also shows that the present results agree well with the Aarhus data [9]. The difference between the present and the Aarhus data is 0.078% on the average and the standard deviation is 0.31%. The difference is well withjn the statistical uncertainty. In table 1 the stopping power
6.2.1. Al The present 1 value for Al 167.4 + 0.5 eV agrees very well with the value 167.6 f 2.8 eV [1,2] and 167.7 + 2.8 eV [lo] obtained in the previous experiment for 6.5 MeV protons. The present value also agrees with the value 167.0 + 0.8 eV of Tschalk and Bichsel [30,33] obtained from the energy loss measurement for 31 MeV protons and with the value 165.7 f 1 eV of Shiles et al. [34] calculated from the dielectric response function. The present value also agrees with the value 166 + 2 eV given in ICRU Report 37 [26]. NCRP [18], Fano [19], Bichsel [20], Turner et al. [21,22], Andersen and Ziegler [24], Ziegler [25] and Ahlen [27] gave I values from 162 to 164 eV. Bichsel and Uehling [35] analyzed Bichsel’s range data for protons having energies from 1 to 18 MeV in Al [36] by correcting for multiple scattering and obtained 1 value as 163 eV. This I value is 4 eV lower than the result of the present analysis. It is considered that thebe are three causes of this difference, namely (a) the difference of the experimental data, (b) the difference of the shell corrections used in the both analyses, and (c) the fact that the Bloch correction and the
124
R Ishiwari et al. / Stopping of Al and Cu for 3 - 9 MeVp
Table 5 The calculated smoothed stopping power values for Al are compared with the Rise data, the Aarhus data and TschaEr’s data. The symbol S denotes the stopping power and is given in keV/mg cm- *. The symbol A denotes the percentage difference Smoothed S-value
Rise data
A VI
Aarhus data
A PI
TschaEr
A @I
WV1 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
82.26 13.75 61.00 61.51 56.94 53.06 49.74 46.85 44.31 42.06 40.05 38.24 36.60 (33.76) (31.37) (29.32) (27.56) (26.01) (24.65) (23.43) (22.34) (21.37) (20.48) (19.67) (18.93) (18.25) (17.62) (17.04) (16.50) (16.00) (15.53) (15.09) (14.68) (14.29)
83.23 74.55 67.12 62.19 57.57 53.65 50.28 47.34 44.76 42.48 40.44 38.61 36.96 34.09 31.67 29.60 27.82 26.26 24.89 23.67 22.58 21.59
-1.18 - 1.08 -1.07 - 1.11 -1.11 - 1.11 - 1.09 -1.05 -1.02 - 1.00 - 0.97 -0.97 - 0.98 - 0.98 - 0.96 -0.95 -0.94 -0.96 - 0.97 -1.02 - 1.07 - 1.03
82.39 73.75 66.94 61.45 56.88 53.01 49.70 46.82
-0.16 0 + 0.09 +0.10 + 0.11 + 0.09 + 0.08 + 0.06
82.52 73.89 67.07 61.53 56.93 53.04 49.71 46.81 44.27 42.02 40.01 38.21 36.58 33.74 31.35 29.31 27.55 26.00 24.64 23.43 22.35 21.37 20.48 19.67 18.93 18.25 17.62 17.04 16.50 16.00 15.53 15.10 14.68 14.30
-0.32 -0.19 -0.10 - 0.03 + 0.02 + 0.04 + 0.06 + 0.09 + 0.09 +0.10 f0.10 + 0.08 + 0.05 + 0.06 + 0.06 + 0.03 + 0.04 + 0.04 + 0.04 0 -0.04 0 0 0 0 0 0 0 0 0 0 - 0.07 0 - 0.07
E
Barkas correction were not considered in the analysis of Bichsel and Uehhng. In general, the larger stopping power gives the smaller I value, the smaller shell correction gives the larger I value and inclusion of the Bloch correction and the Barkas correction gives the larger I value. The experimental data of Bichsel and Uehhng as expressed in stopping powers are higher than the present results by 1.17-0.378 over the energy range from 3 to 9 MeV. Therefore, in the analysis of Bichsel and Uehhng the I value should have a smaller value. Bichsel and Uehling used WaIske’s K-shell correction [37] and the simplified L-shell correction, C, = l.S/E(MeV). Their total shell corrections are smaller than the present shell corrections by 36-12% over the energy range from 3 to 8 MeV. Thus, their I value should have somewhat a larger value. Roughly considered, the effect of the larger stopping powers nearly compensates the effect of
the smaller shell corrections. Then, the inclusion of the Bloch correction and the Barkas correction will raise their I value to about 167 eV. However, recently published ICRU Report 37 [26] recommends the Walske’s shell corrections [37,38] modified by Bichsel’s scaling method [39]. These shell corrections are larger by several percent than the present Bonderup shell corrections. Therefore, the shell corrections used by Bichsel and Uehhng in their analysis are probably too small. If we use the present Bonderup shell corrections and include the Bloch correction and the Barkas correction in the analysis of the experimental data of Bichsel and Uehhng, an I value of 163-164 eV will be obtained. The difference between this I value and the present I value of 167.4 eV arises entirely from the higher stopping power values of Bichsel and Uehhng than the present results.
R Ishiwari et al. / Stopping
ofAl
and Cu for 3- 9 MeVp
125
Table 6 The calculatedsmoothedstoppingpower valuesfor Cu are comparedwith the Rise data, the Aarhusdata and Burkigand MacKenzie value at 20 MeV takingAl as the standardmaterial.The symbol S denotes the stopping power and is given in keV/mg cmm2.The symbol A denotes the percentagedifference E
Smoothed
Rise
NV]
S-value
data
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10 11 12 13 14 15 16 17 18 19 20
61.27 55.51 50.87 47.04 43.84 41.09 38.70 36.62 34.78 33.13 31.66 30.32 29.10 (26.98) (25.18) (23.63) (22.29) (21.10) (20.05) (19.11) (18.26) (17.50) (16.81) (16.17)
62.43 56.43 51.65 47.71 44.40 41.58 39.14 37.00 35.11 33.42 31.91 30.55 29.32 27.15 25.33 23.76 22.40 21.20 20.14 19.20 18.35 17.58
A
[%I
-1.89 - 1.66 -1.53 - 1.42 - 1.28 - 1.19 - 1.14 - 1.04 - 0.95 - 0.88 -0.79 - 0.76 - 0.76 - 0.63 - 0.60 - 0.55 - 0.49 - 0.47 - 0.45 - 0.47 - 0.49 - 0.46
Recently, McGuire [40] made a drastic suggestion that the Z value for Al should be in the range 145-150 eV rather than 162-167 eV. McGuire argued that the high energy experiment of Mather and Segre [41] indicates that the Z value for Al is 145-150 eV. The only stopping power data below 20 MeV which suggests the Z value of 163-169 eV are those of Bichsel and Uehhng [35]. He argued that below 20 MeV the Z value depends critically on the experimental data and the shell correction used in the analysis. If he applies his own shell corrections [42] to the Rise data, an Z value of 145-150 eV is obtained. He also argued that the calculation of the Z value of Shiles et al. [34] from the dielectric response function is subject to errors arising from experimental errors of optical experiments and oxygen contamination. McGuire’s suggestion became a subject of controversy [43,44]. Bichsel, Inokuti and Smith [43] argued against McGuire the following points, (a) there is an elementary error in McGuire’s analysis, i.e. McGuire used the atomic mass unit in place of the proton mass. When corrected for this error, McGuire’s suggestion becomes 150-155 eV, (b) in the discussion of the stopping power, McGuire puts unwarranted emphasis on the Rise data and ignored the Aarhus data, Tschahir’s
Aarhus data
A [%I
61.51 55.64 50.94 47.08 43.88 41.16 38.79 36.70 34.83
-0.39 - 0.23 - 0.14 - 0.09 - 0.09 -0.17 - 0.23 - 0.22 -0.14
Burkigand MacKenzie
A [%I
16.146
+0.15
data and the present data. The latter three sets of data agree with each other and are decisively lower than the Rise data. These three sets of data raise the Z value by about 8 eV, (c) although in principle McGuire’s method in obtaining the shell correction is correct, the shape of McGuire’s shell correction is unnatural as a function of the energy and the magnitude of his shell correction is not accurate enough numerically. The use of McGuire’s shell correction accounts for the additional 8 eV difference in the Z value, (d) the high energy experiment of Mather and Segre [41] is not accurate enough and their Z value is too low, because they did not take into account nuclear reactions, (e) the analysis of Bichsel and Uehling indicates the Z value - 163 eV, and (f) the two sources of error in the analysis of Shiles et al., which McGuire pointed out would increase not decrease the Z value. It appears that McGuire’s arguments indicating the Z value of 145-150 eV are almost entirely refuted. The most important points regarding the present experimental results are that McGuire placed a special emphasis on the l&o data and McGuire used his own shell corrections [44]. If we apply the present Bonderup shell corrections, the Bloch corrections and the Barkas corrections to the Risa data, an Z value of about 159 eV
126
R. Ishiwari et al. / Stopping of Al and Cu for 3- 9 MeVp
will be obtained. McGuire’s shell corrections are larger than the present Bonderup shell corrections by 111-27% over the energy range from 3 to 8 MeV. If we apply McGuire’s shell corrections to the Rise data, an I value of 150-155 eV will be obtained. However, as pointed out by Bichsel et al. [43], McGuire’s shell correction is not a smooth function of the energy. Thus, McGuire’s shell correction is considered to contain considerable numerical errors. McGuire argues that his shell corrections are in good agreement with the experimental shell corrections derived from the Aarhus data. However, this claim has almost no physical meaning. The Aarhus experimental shell correction is obtained by the following equation [9]: (C/Z, >,
= xeyp -In
I+*+ZiL,,
(7)
where Andersen et al. used experimentally determined Cp and L, and for the value of 1 they used the value 162 eV which is much larger than the value 150-155 eV suggested by McGuire. Therefore, the Aarhus experimental shell correction never accords with McGuire’s claim that the I value for Al should be 150-155 eV. R&ently, Oddershede and Sabin 1451 have combined Sigmund’s kinetic theory of stopping power of a medium with internal motion [46] and experimental Compton profile data 1471. Taking the I value as an adjustable parameter, they fitted the calculated stopping power values with the Aarhus experimental data [9]. The best fit was obtained with I = 163 eV. This evaluation of the I value utilizes a quite independent experimental data of the stopping power data, i.e. Compton profile data, and is in support of the I value of 162-16’7 eV. The I value extracted from the stopping power data depends on the choice of the shell correction. As mentioned above, ICRU Report 37 recommends Bichsel’s scaled she11 corrections [39] which are several percent larger than the present Bonderup shell corrections. If we apply shell corrections recommended by ICRU Report 37 to the present data, an I value of about 166 eV will be obtained. Although NCRP [18], Fano [19], Bichsel [20], Turner et al. (21,221, Andersen and Ziegler [24], Ziegler [ZS], and Ahlen 1271 gave I values from 162 to 164 eV, it appears that the I value for Al is more probably 166-167 eV. The uncertainty of the I value kO.5 eV given in this paper is only the fitting error of the method of least squares. If we take the ~~rt~nty of the Bonderup she11 correction to be +lO%, the uncertainty of the I value becomes 42.8 eV. 6.2.2. cu The present result of 326.3 f 0.6 eV is larger than the previous results of 323.5 rt: 7.9 eV [2] by 2.8 eV and 323.4 f 7.9 eV [lo] by 2.9 eV. The previous results for I were obtained from the stopping power measurement at one proton energy, i.e. at 6.5 MeV. The present I value
has been determined from the fitting with 11 points with different energies in the energy range from 3 to 9 MeV. Therefore, as far as the Bonderup shell corrections are used, the present I value of 326.3 eV is more reliable than the previous values of 323.5 eV or 323.4 eV. As mentioned earlier, the present attached error of the 1 value is the fitting error of the method of least squares. If we take the uncertainty of the Bonderup shell correction to be *lo%, the uncertainty of the I value becomes 18 eV. The difference between the present and the previous I values is well within the statistical uncertainty. Sorensen and Andersen obtained and I value of 319.8 f 3.2 eV from the Rise data. Their method of extracting the I value from the stopping power data depends neither on the choice of the type of the shell correction nor the disregard of the Bloch correction and the Barkas correction, but directly depends on the absolute value of the experimental stopping power. Since their stopping power data are higher than the present data by 1.30% on the average, and I value about 5% lower than the present result, i.e. 310-315 eV, is expected from the Riser data. Actually, however, their I value is somewhat higher than this expectation. The origin of this deviation is not understood. For the 1 value for Cu, NCRP [IS], Fano [19], Turner et al. 121,221, Bichsel 1231, and Ahlen [27] gave I values from 306 to 319 eV. Bichsel [20] and Andersen and Ziegler [24] gave 322 eV and Ziegler [25] gave 330 eV. ICRU Report 37 1261 recommends an I value of 322 zt:10 eV. Although Ahlen gave a very small uncertainty, _+2 eV, for his I value 317 eV, the above-mentioned I values agree with the present result within the statistical uncertainty, provided that the uncertainty of the present result is taken as + 8 eV. ICRU Report 37 recommends Bichsel’s scaled shell corrections [39] for Cu which are about 10% smaller than the present Bonderup shell corrections in the energy range from 2 to 6.5 MeV and nearly equal to the present shell corrections at energies from 10 to 12 MeV. If we apply these shell corrections to the present experimental data, an I value 333-336 eV will be obtained for the data from 3 to 6.5 MeV and an I value - 327 eV will be obtained at 9 MeV. Although at present there is no experimental evidence whether the present Bonderup shell corrections or Bichsel’s scaled shell corrections recommended in ICRU Report 37 represent the experimental situation better, it appears that the present experimental data indicates an I value equal to or larger than 326 eV for Cu. As mentioned earlier, the Aarhus stopping power data for Cu are larger than the present results by 0.16% on the average. Therefore, it is expected that the Aarhus data will give a slightly smaller I value than the present I value. If we apply the present Bonderup shell corrections, the Bloch corrections and the Barkas corrections
R. Ishiwari et al. /Stopping
to the Aarhus data, an I value of 323 - 325 eV is obtained. Burkig and MacKenzie [48] performed very accurate relative stopping power measurements for 19.8 MeV protons. Using the present Z value for Al 167.4 eV and the Bonderup shell correction, the Bloch correction and the Barkas correction, the stopping power of Al for 20 MeV protons is calculated as 19.666 keV/mg cm-*. The stopping power of Cu for 20 MeV protons is obtained by multiplying the relative stopping power of Cu of Burkig and Mackenzie, 0.821, to 19.666 keV/mg cm-* as 16.146 keV/mg cm-*. Applying the present Bonderup shell correction, the Bloch correction and the Barkas correction to this value, the Z value for Cu is obtained as 328.6 eV. ICRU Report 37 gives the shell correction which is larger by 4.30% than the present Bonderup shell correction at 19.8 MeV. If we apply the shell correction that is 1.0430 times the present Bonderup shell correction, the Bloch correction and the Barkas correction to the stopping power value 16.146 keV/mg cm-* of Burkig and McKenzie at 20 MeV, an I value of 327.1 eV is obtained. On the whole, it appears that the Z value for Cu is more probably equal to 326 eV or somewhat larger rather than the 322 eV that is recommended in ICRU Report 37. 6.3. Cakxdated smoothed stopping power Table 5 shows the calculated smoothed stopping powers for Al. Because the present experiment covers only the energy range from 3 to 9 MeV, the calculated values above 10 MeV are given in parentheses. The calculated smoothed stopping powers are decisively lower than the Riser data. The difference is 1.03% on the average. The agreement between the calculated values and the Aarhus data is better than that between the experimental stopping powers and the Aarhus data shown in table 1. The difference is 0.046% on the average and the standard deviation is 0.096%. The agreement between the calculated values and TschalL’s data [30] is surprisingly good up to 30 MeV. The difference is only 0.0024% on the average and the standard deviation is 0.0812%. The calculated stopping power at 6.5 MeV is 46.85 keV/mg cm-* and agrees very well with our previous value of 46.83 keV/mg cm-* [1,2]. curve calculated with the present Z value The X,, for Al 167.4 eV and the Bonderup shell corrections, the Bloch corrections and the Barkas corrections given in table 3 is shown in fig. 7. As expected from table 5, above 16 MeV the present X,,,, curve coincides with TschalL’s experimental XexP curve within the thickness of the curves. Table 6 shows the calculated smoothed stopping powers for Cu. The values above 10 MeV are also given in parentheses. The calculated smoothed stopping
of Al and Cu for 3- 9 MeVp
127
powers are lower than the Rise data by 0.90% on the average. The difference between the calculated stopping powers and the Rise data decreases as the energy increases. The difference in the range from 3.0 to 9.0 MeV is 1.18% on the average. The calculated values are lower than the Aarhus data by 0.19% on the average. However the differences between the calculated values and the Aarhus data are well within the statistical uncertainties. In table 6 Burkig and MacKenzie’s value obtained by taking the Z value for Al as 167.4 eV is also shown. The calculated value agrees with the value of Burkig and MacKenzie very well. Therefore, the calculated stopping powers from 10 to 20 MeV are considered to be fairly reliable. The X&n curve calculated with the present Z value for Cu 326.3 eV and the Bonderup shell corrections, the Bloch corrections and the Barkas corrections given in table 4 is shown in fig. 8. As expected from table 6, Burkig and MacKenzie’s point falls very near the Xtheo curve. 6.4. Shell correction 6.4.1. AI As shown in fig. 7, the Bonderup shell corrections for Al can reproduce very well not only the,present X_, points but also TschaEr’s X_, values [30] up to 30 MeV. As mentioned earlier, the shell corrections for Al recommended by ICRU Report 37 [26], which are Bichsel’s scaled shell corrections [39] based on Walske’s K- and L-shell corrections [37,38], are larger by several percent than the present Bonderup shell corrections in the energy range from 3 to 9 MeV. These shell corrections give an I value of about 166 eV when applied to the present experimental data taking account of the Bloch corrections and the Barkas corrections and can reproduce Tschahar’s X_, values up to 30 MeV. Therefore, the shell corrections for Al recommended by ICRU Report 37 are in fundamental agreement with the present Bonderup shell corrections within the uncertainties accompanying theoretical calculations. Recently, Sabin and Oddershede [49] have calculated the shell corrections for Al shell by shell using Sigmund’s kinetic theory of stopping power [46] and the Hartree-Slater calculations of oscillator strength moments given by Dehmer et al. [50] and Inokuti et al. [51,52]. The shell corrections of Sabin and Oddershede are larger by 42-40% than the present Bonderup shell corrections over the energy range from 3 to 9 MeV. If we apply the shell corrections of Sabin and Oddershede to the present calculated stopping powers using the Bloch corrections and the Barkas corrections, the Z values are obtained as 156.03 eV for 3 MeV, 159.87 eV for 6 MeV and 161-97 eV for 9 MeV. Thus, the shell corrections of Sabin and Oddershede do not give a
128
R. Ishiwari et al. / Stopping of Al and Cu for 3- 9 MeVp I
y
m
I
I
I
5.3
t
---_ .$&in
&j-------
Oddershede 0
I
I
5
10
I
ENERG’)5( MeV1
I
20
Fig. 9. The solid curve denotes the present Xtt,_, curve. The dashed curve denotes the X,,,_, curve calculated with Sabin and Odderschede’s shell corrections, I value of 159.87 eV and the present Bloch and Barkas corrections. The solid circles indicate the present Xexp points.
I value. This means that the shape of Sabin and Oddershede’s shell correction as a function of the energy does not agree with the present Xexp points. In fig. 9, the X,, curve calculated with Sabin and Oddershede’s shell corrections, I value of 159.87 eV, the present Bloch and Barkas corrections is shown and is compared with the present X,, curve. As already mentioned, the X,, curve using the present Bonderup shell correction agrees very well with Tschallr’s X_, curve up to 30 MeV. As is clearly seen from the figure, Sabin and Oddershede’s Xtheo curve deviates from the X thee curve using the present Bonderup shell correction. Since the change of the I value only shifts the X,, curve up and down without changing its shape, we can conclude that Sabin and Oddershede’s shell correction does not reproduce the Xex,, point in a fundamental way. Andersen et al. [9] state that their experimental shell corrections, eq. (7), derived from the Aarhus data do not agree with the Bonderup shell corrections when they use the I value of 162 eV. However, if they use the I value of 169 eV, much better agreement is obtained. This fact indicates that the Bonderup shell correction can reproduce fundamentally the energy dependence of the Xexp curve of the Aarhus data over the energy range from 0.8 to 6.4 MeV. The difference between their I value of 169 eV and the present I value of 167.4 eV comes mainly from the fact that the Aarhus empirical L, values are larger than the present estimates of L, by 24-282 over the energy range from 3 to 6 MeV. constant
6.4.2. Cu At present two types of the shell correction are usually used in analyzing experimental data, i.e. the Bonderup shell correction [ll] and Bichsel’s scaled shell correction based on Walske’s K- and L-shell correction
WI.
As already mentioned, if we apply Bichsel’s scaled shell corrections recommended in ICRU 37 to the present experimental data, I values of 333-336 eV are obtained in the energy range from 3 to 6.5 MeV. Although at present there exists no criterion which type of the shell correction represents the experimental situation of Cu better, Z values of 333-336 eV appear to be slightly too large as compared with the I values given by other authors, i.e. 306-322 eV [18-24,26,27]. Another accurate stopping power measurement up to 30 MeV is strongly desired. The authors would like to thank Prof. S. Kobayashi for his kind support throughout this experiment. Thanks are also due to Drs. K. Takimoto, M, Nakamura and the members of the Van de Graaff Laboratory of Kyoto University for their kind cooperation.
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