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Accurate charged particle activation analysis: calculation of the average energy in the average stopping power method
NUCLEAR I N S T R U M E N T S AND METHODS 153 ( 1 9 7 8 )
503-505 ; ©
N O R T H - H O L L A N D PUBLISHING CO.
ACCURATE CHARGED PARTICLE ACTIVATION ANALYSIS: CALCULATION OF THE AVERAGE ENERGY IN THE AVERAGE STOPPING POWER METHOD K. ISHII*, M. VALLADON, C. S. SASTRI and J. L. DEBRUN
CNRS-Groupe "Application des reactions nucleaires (* I'analyse chimique", Service du Cyclotron, 45045 Orleans, Ckdex, France Reccived 27 December 1977 In the averagc stopping power method, the average energy (Era) is normally derived from the cross-section curve. It is shown here that identical results are obtained using the average energy (Eact) derived from the thick target yield.
1. Introduction In an earlier publication'), we introduced the average stopping power m e t h o d for charged particle activation analysis. This method is very accurate and is based on the stopping power of the incident particle in the matrix, at an energy " E r a " called the " a v e r a g e e n e r g y " . Em is derived from the cross-section curve. In this work, it is shown that identical results are obtained in the average stopping power method, using either Em or an average energy (E~ ~) derived from the thick target yield. The possibility to use thick target yields is very interesting because these data are much easier to measure than the cross-section curves, and because they are also obtained with a better precision.
In the previous publicationt), it was shown that eq. (1) can be rewritten as: A(EI) = N iEl a(E) dE (
dE 1 .']E=E~ X
v,t logE/E m o log x
t-
E
,r,
dE I
--;~-~. - - - -
, (2)
E, ~r(E) dE do where the average energy, Era, is defined by:
' Ea(E) dE
]!16
Em = " ,
,
(3)
a(E) dE
2. Calculation of the average energy
from the thick target yield In charged particle activation, the saturation activity A(Ei) of a nuclide is given by:
A(Ei) = N
'
fo
a(E) - -
1
dE
'
(1)
where Ei o" N
is the energy of the incident particle, is the cross-section, is the concentration of the required element, and dE/(-pdX) is the stopping power. In eq. (1) A (Ei) is normalized to the n u m b e r of incident particles and it" is assumed that the element is uniformly distributed in the sample. * On leave from the Department of Physics, University of Tohoku, Sendai (Japan).
where I and ). are the mean ionization potential and the ratio of the mass of the incident particle to that of the electron. From eq. (2), the calibration equation for a sample " b " and a standard " a " can be written. dE
/ r=~:'~
Ab(E~) - Art' x \ ~ ] ~ '
Aa(El)
J~'ta ( dE ~=Em
(4)
\ -pdX]b and the systematic error of this equation is given by
I~ [.e~
logE/Em E 4E 4E E-m a(E) dE
log ~ ~ o log )~a log 2Ib =
(5)
j
a(E) dE 0
504
K. ISHII et al.
F r o m the definition o f the a v e r a g e e n e r g y [eq. (3)] it can be seen that it is a f u n c t i o n of E and a. In this paper, E m is redefined and the new definition o f the a v e r a g e e n e r g y E,~c~ can be obtained by rewriting eq. (3) as follows:
a(E) dE j,e,o(d) E~' =
(6)
f
*'~ 1
Eq. (8) represents the new definition and according to this, E~' is a function o f the incident energy & and o f the thick target yield A(&). For a g i v e n nuclear reaction, Em is i n d e p e n d e n t o f the s a m p l e w h e r e a s E~ t is slightly d e p e n d e n t on the s a m p l e and for this reason the suffix " o " is used in eq. (8) to indicate the s a m p l e for which the thick target yield is obtained. According to Bethe's formula for the stopping power, eq. (6) b e c o m e s :
dE
a(E)
f
v., Ea(E) dE 4E
o log
E~' =
Ie,
(9)
a(E) dE
N (aT)
4E log Moo d E
0
(7) jo
dE
F r o m eqs. (3) and (9), the difference b e t w e e n Em and E~m~ is given b y '
+ o
+['E,
1 {
"v.
\
o'(E)
,,.\
\ - pdX /l
dE
/
El
T A BLI.?
E2
dE
E m a n d E act
for the reaction
E~
~-
1 o-(E)
dE (1o)
fo ~~a(E) dE
l
Comparison of
o log 4.__E.E 2I o
(8)
El fE~ Ao(E)
1 + A-~i ) o
E m -- E act E~
160(t,
n)lS1-
/:i (McV)
Em (Mc'v)
Eact (Me',,")
Deviation (%)
3.500 3.400 3.300 3.200 3.100 3.000 2.900 2.800 2.700 2.600 2.500 2.400 2.300 2.200 2. 100 2.000
2.562 2.504 2.449 2.395 2.342 2.290 2.237 2.183 2.126 2.(/65 1.998 1.927 1.857 1.786 1.713 1.639
2.553 2.510 2.454 2.400 2.345 2.290 2.241 2.197 2.102 2.070 1.988 1.927 1.860 1.779 1.714 1.637
- 0.35 - 0.24 - 0.20 0.21 - 0.13 0.00 - 0.18 0.64 - 1.13 - 0.24 ~0.50 0.00 -0.16 • 0.39 0.[)6 0.12
O w i n g to the t e r m Iog(E/Em)(E/Em-1), this difference is very small. I t e n c e , we can use E ~ ~ instead o f Em in eq. (4). For e x a m p l e , for the reaction~60(t, n)~SF, with A1203 as the target, Em and E~~ were calculated with the data o f Revel 2) and Valladon3), respectively, for different triton energies and d e v i a t i o n s ranging from zero to a maxim u m of 1.13% were noticed (seetable 1). act N o w , rewriting eq. (2) using E• m , we obtain: •~ E i
N [ A (El)
=
/
./o
dE
a ( E ) dE .
.
~E=':~ " " x
\-pdXJ
o x
1-
4E log )~ I..', fo
E~ta( --
a(E) dE
)dE +
ACCURATE CHARGED PARTICLE ACTIVATION ANALYSIS
505
3. C o n c l u s i o n
t
-- 1
o log
+
a(E) d
;To e,
1) . (1
fo a(E)dE As the first and the second terms in parentheses in eq. (l l) are nearly equal to the corresponding terms in eq. (2) and si,nce the third term of eq. (11) is the same for all samples, the systematic error introduced by the use of E~ ~ in eq. (4) will be identical to the error introduced by the use of Era. This means that in the average stopping power method, E m or E~' may be used indifferently. The systematic error e, that we may rename e', can be calculated from the thick target yield:
It was shown in this paper that an average energy (E~ t) could be calculated from the thick target yield. It was also shown that the value of E~ ' is very close to the value Em derived from the cross-section curve. Although the values of Em and E~ ~ are slightly different, the systematic error when using the method of the average stopping power is exactly the same in both cases since only ratios are used. Consequently, when no suitable data are available either for the cross section or for the thick-target yield, we feel that the latter should be determined and used because it is easier to determine and is known with a better accuracy.
log la
4E i 4E i log )7~ log ;.I-~
1 [
× × E~'
~ I + (~
(E i -- E)
..o
--1) (1'
References
dE (12)
-14 El/ log )-~o /
1) K. lshii, M. Valladon and J. L. Debrun, Nucl. Instr. and Meth. 150 (1978) 213. 2) G. Revel, M. Da Cunha Belo, I. Link and L. Kraus, Rev. Phys. Appl. 12 (1977) 81. 3) M. Valladon, J. L. Debrun, J. Radioan. Chem. 37 (1977) 385.
503-505 ; ©
N O R T H - H O L L A N D PUBLISHING CO.
ACCURATE CHARGED PARTICLE ACTIVATION ANALYSIS: CALCULATION OF THE AVERAGE ENERGY IN THE AVERAGE STOPPING POWER METHOD K. ISHII*, M. VALLADON, C. S. SASTRI and J. L. DEBRUN
CNRS-Groupe "Application des reactions nucleaires (* I'analyse chimique", Service du Cyclotron, 45045 Orleans, Ckdex, France Reccived 27 December 1977 In the averagc stopping power method, the average energy (Era) is normally derived from the cross-section curve. It is shown here that identical results are obtained using the average energy (Eact) derived from the thick target yield.
1. Introduction In an earlier publication'), we introduced the average stopping power m e t h o d for charged particle activation analysis. This method is very accurate and is based on the stopping power of the incident particle in the matrix, at an energy " E r a " called the " a v e r a g e e n e r g y " . Em is derived from the cross-section curve. In this work, it is shown that identical results are obtained in the average stopping power method, using either Em or an average energy (E~ ~) derived from the thick target yield. The possibility to use thick target yields is very interesting because these data are much easier to measure than the cross-section curves, and because they are also obtained with a better precision.
In the previous publicationt), it was shown that eq. (1) can be rewritten as: A(EI) = N iEl a(E) dE (
dE 1 .']E=E~ X
v,t logE/E m o log x
t-
E
,r,
dE I
--;~-~. - - - -
, (2)
E, ~r(E) dE do where the average energy, Era, is defined by:
' Ea(E) dE
]!16
Em = " ,
,
(3)
a(E) dE
2. Calculation of the average energy
from the thick target yield In charged particle activation, the saturation activity A(Ei) of a nuclide is given by:
A(Ei) = N
'
fo
a(E) - -
1
dE
'
(1)
where Ei o" N
is the energy of the incident particle, is the cross-section, is the concentration of the required element, and dE/(-pdX) is the stopping power. In eq. (1) A (Ei) is normalized to the n u m b e r of incident particles and it" is assumed that the element is uniformly distributed in the sample. * On leave from the Department of Physics, University of Tohoku, Sendai (Japan).
where I and ). are the mean ionization potential and the ratio of the mass of the incident particle to that of the electron. From eq. (2), the calibration equation for a sample " b " and a standard " a " can be written. dE
/ r=~:'~
Ab(E~) - Art' x \ ~ ] ~ '
Aa(El)
J~'ta ( dE ~=Em
(4)
\ -pdX]b and the systematic error of this equation is given by
I~ [.e~
logE/Em E 4E 4E E-m a(E) dE
log ~ ~ o log )~a log 2Ib =
(5)
j
a(E) dE 0
504
K. ISHII et al.
F r o m the definition o f the a v e r a g e e n e r g y [eq. (3)] it can be seen that it is a f u n c t i o n of E and a. In this paper, E m is redefined and the new definition o f the a v e r a g e e n e r g y E,~c~ can be obtained by rewriting eq. (3) as follows:
a(E) dE j,e,o(d) E~' =
(6)
f
*'~ 1
Eq. (8) represents the new definition and according to this, E~' is a function o f the incident energy & and o f the thick target yield A(&). For a g i v e n nuclear reaction, Em is i n d e p e n d e n t o f the s a m p l e w h e r e a s E~ t is slightly d e p e n d e n t on the s a m p l e and for this reason the suffix " o " is used in eq. (8) to indicate the s a m p l e for which the thick target yield is obtained. According to Bethe's formula for the stopping power, eq. (6) b e c o m e s :
dE
a(E)
f
v., Ea(E) dE 4E
o log
E~' =
Ie,
(9)
a(E) dE
N (aT)
4E log Moo d E
0
(7) jo
dE
F r o m eqs. (3) and (9), the difference b e t w e e n Em and E~m~ is given b y '
+ o
+['E,
1 {
"v.
\
o'(E)
,,.\
\ - pdX /l
dE
/
El
T A BLI.?
E2
dE
E m a n d E act
for the reaction
E~
~-
1 o-(E)
dE (1o)
fo ~~a(E) dE
l
Comparison of
o log 4.__E.E 2I o
(8)
El fE~ Ao(E)
1 + A-~i ) o
E m -- E act E~
160(t,
n)lS1-
/:i (McV)
Em (Mc'v)
Eact (Me',,")
Deviation (%)
3.500 3.400 3.300 3.200 3.100 3.000 2.900 2.800 2.700 2.600 2.500 2.400 2.300 2.200 2. 100 2.000
2.562 2.504 2.449 2.395 2.342 2.290 2.237 2.183 2.126 2.(/65 1.998 1.927 1.857 1.786 1.713 1.639
2.553 2.510 2.454 2.400 2.345 2.290 2.241 2.197 2.102 2.070 1.988 1.927 1.860 1.779 1.714 1.637
- 0.35 - 0.24 - 0.20 0.21 - 0.13 0.00 - 0.18 0.64 - 1.13 - 0.24 ~0.50 0.00 -0.16 • 0.39 0.[)6 0.12
O w i n g to the t e r m Iog(E/Em)(E/Em-1), this difference is very small. I t e n c e , we can use E ~ ~ instead o f Em in eq. (4). For e x a m p l e , for the reaction~60(t, n)~SF, with A1203 as the target, Em and E~~ were calculated with the data o f Revel 2) and Valladon3), respectively, for different triton energies and d e v i a t i o n s ranging from zero to a maxim u m of 1.13% were noticed (seetable 1). act N o w , rewriting eq. (2) using E• m , we obtain: •~ E i
N [ A (El)
=
/
./o
dE
a ( E ) dE .
.
~E=':~ " " x
\-pdXJ
o x
1-
4E log )~ I..', fo
E~ta( --
a(E) dE
)dE +
ACCURATE CHARGED PARTICLE ACTIVATION ANALYSIS
505
3. C o n c l u s i o n
t
-- 1
o log
+
a(E) d
;To e,
1) . (1
fo a(E)dE As the first and the second terms in parentheses in eq. (l l) are nearly equal to the corresponding terms in eq. (2) and si,nce the third term of eq. (11) is the same for all samples, the systematic error introduced by the use of E~ ~ in eq. (4) will be identical to the error introduced by the use of Era. This means that in the average stopping power method, E m or E~' may be used indifferently. The systematic error e, that we may rename e', can be calculated from the thick target yield:
It was shown in this paper that an average energy (E~ t) could be calculated from the thick target yield. It was also shown that the value of E~ ' is very close to the value Em derived from the cross-section curve. Although the values of Em and E~ ~ are slightly different, the systematic error when using the method of the average stopping power is exactly the same in both cases since only ratios are used. Consequently, when no suitable data are available either for the cross section or for the thick-target yield, we feel that the latter should be determined and used because it is easier to determine and is known with a better accuracy.
log la
4E i 4E i log )7~ log ;.I-~
1 [
× × E~'
~ I + (~
(E i -- E)
..o
--1) (1'
References
dE (12)
-14 El/ log )-~o /
1) K. lshii, M. Valladon and J. L. Debrun, Nucl. Instr. and Meth. 150 (1978) 213. 2) G. Revel, M. Da Cunha Belo, I. Link and L. Kraus, Rev. Phys. Appl. 12 (1977) 81. 3) M. Valladon, J. L. Debrun, J. Radioan. Chem. 37 (1977) 385.