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Stopping power measurement using thick alpha sources
Nuclear Instruments and Methods in Physics Research 223 (1984) 377-381 North-Holland, Amsterdam
377
STOPPING POWER MEASUREMENT USING THICK ALPHA SOURCES M. H O S O E Department of Physic.v, Faculty of Science, Rikkvo Unit,ersitv 3. Nishi- lkebukuro, Toshima- ku, Tokyo, 171, Japan
Y. T A K A M 1 , F. S H I R A I S H I
a n d K. T O M U R A
In~titute for Atomic Energy, Rtkkvo Universi(v, Nagasaka. Yokosuka, Kanagawa, 240-01. Japan
For alpha particles emerging from a plane surface of a thick medium in which alpha activity is distributed homogeneously, the energy spectrum is inversely proportional to the stopping power of the medium. The surface barrier type SSDs made of p type silicon were used to measure the spectrum and the stopping power was calculated. Ahhough this method can be applied only to alpha emitting sources at present, it has the merit of simplicity in the experimental procedure and gives the stopping power as a continuous function of energy by measuring only one spectrum. Powder with less than 1 p.m particle size is a suitable medium to mix with alpha activity in this method. The results are compared with stopping powers calculated from the Northcliffe Schilling and Ziegler data tables.
!. d E / d X calculation from the spectrum of thick alpha sources Alpha particles emitted from a plane surface of a medium containing alpha activity can easily be measured by a silicon surface barrier detector. W h e n sources are distributed homogeneously in the medium, the energy spectrum will be:
d N / d E = 0 . 2 5 . S. D / ( d E / d R ), where d N / d E
[MeV
(l)
i] is the differential detector
10"
count, S [cm 2] the area of detector in contact with the m e d i u m and D [g t] is the alpha disintegrations per unit mass of medium. R ( E ) [ g / c m 2] is the residual range and d E / d R [ M e V / ( g / c m 2 )] is the stopping power of the medium for an alpha particle with energy E [MeV]. The stopping power can be given simply by taking the inverse of the count per channel for a given energy in the spectrum. A p-silicon surface barrier detector [1] is useful for such measurements, because its aluminium entrance window is much stronger than the gold window corn-
(b)
(a)
~200 TP ;" "1Am In AI203
AI203
I
[
2
4
J
6
L
I
I
8
10
x
I00
CHANNEL NUMBER
ALPHA ENERGY ( M e V )
Fig. 1. (a) Differential energy spectrum of alpha particles measured with Si-SSD in close contact with AI ,O~ powder, which includes 24~Am homogeneously: (b) stopping powers of AI203 powder for alpha particle, calculated from the spectrum (a). 0 1 6 7 - 5 0 8 7 / 8 4 / $ 0 3 . 0 0 f ' Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Ill. SPECTROMETRY
378
M. Hosoe et al.
Stoppmg power measurement usmg thick alpha sources
monly used for n-silicon. This m e t h o d of measuring is useful only for samples which can be mixed into the m e d i u m homogeneously and emit particles uniformly in all directions.
300
dE/dR
2. Metal oxide / For metal oxides it is easy to o b t a i n a spectrum of good quality by p r e p a r i n g powder with particle size of less than 1 F m as the source medium• Fig. 1 shows the differential energy spectrum of alpha particles from Z4~Am activity homogeneously distributed in fine a l u m i n a powder by coprecipitation, and the stopping power o b t a i n e d by applying eq. (1). Also are shown the curves calculated from the tables of Northcliffe-Schilling ( N - S ) [2] a n d Ziegler [3], where the powder is assumed to consist of pure AI20~. Experimental values are normalized to N - S values at 1 MeV. Agreement is good with the N - S curve using solid oxygen values (interpolated between solid metals in the table), but not with the curve using data of oxygen gas. which ~s not shown here. Fig. 2 shows the a l p h a spectrum of u r a n i u m oxide. In this case betas interfere with the alpha spectrum below 1.5 MeV. The u r a n i u m atoms emit alphas a n d at the same time constitute the m e d i u m molecules. Thus the D value is assured to be c o n s t a n t in atomic terms a n d this is a n ideal case for the application of eq. (1). Since 2~SU decays by a l p h a emission with a half-life of 4.468 × ].09 years, the disintegration rate per gram is
>
100 d
%,"
/
/
}
/
k\~
....
Zje~Ler Schilh nq
2
L Nortl%ljJ[e_-
o O3
U02 I
1
.
.
.
.
.
.
.
.
.
5 AlDha Energy
.
.
.
.
.
5 [ MeV
Fig. 2 Stopping powers of UO, powder for alpha particle. calculated to be 2.194 x l0 4 disintegrations s 1 g ~ bv assuming that 2-a~U a n d 234U are in radioactive eqmlibrium. This enables absolute stopping power values to be deduced a n d these values agree with Ziegler's curve. The presence of both the alpha emitters does not prevent the deduction of d E ~ d R based on eq. (IL
230 Th 238u
• :%.
Z
(,_)
,.
.
i;
z "I-
~
)U 23L'U t/[ 226R =
• :. . r . ~
:~
,. '.
• ,v..
: ~k,#"
. ¥.:~...,---"..r,L..-
•
210 Po ! .,=,. •./:. 222 Rn '~,.'~. :~.:.
o
2'8p 0
;'~,
|
21z. p o
U R A N I U M ORE I
s6o
1000 CHANNEL NUMBER
Fig. 3. Differential energy spectrum of alpha particles emitted from the plane surface of powdered uramum ore.
M. tto~oe et al. / Stopping power measurement using thwk alpha sources 3.
Uranium
lOO
_
ore
Fig. 3 shows the spectrum for a uranium ore. The alpha spectrum has eight components, corresponding to the natural decay series. In spite of the multiplicity, the spectrum follov,'s eq. (1), although when closely examined, some peaking occurs at the edges of fiat parts and a relatively large drop of counts is observed for channels below 2 MeV. This is likely to be caused by a,1 excessively large sample grain size. Inspection by optical microscope showed grain sizes ranging up to 50 ffm or more. The plateau levels in the spectrum give disintegrations for each step in the decay series. By calculation assuming that eq. (I) holds, it was found that equilibrium was preserved within + 3% among alpha decays in the series.
379
E 0
S >~ 50 0
Z~egl~.'
Fe
6_ cl
." /
i
_Nort he! kff e : SGh[[li r'<2j"
o
_ _ _ _ _ [ _ _ t ~
.1
I
J
I
.5 Alpha Energy
I
[
__ __ z
L
(a)
~J
I
5
{HEY]
lOO 4.
Metal
stopping
E u
Ix)wer
For stopping power measurement of metals, powders of 300 - 700 A in grain size were used with 2mPo as the alpha source. A grain size of 0.1 # m is small enough to yield homogeneous mixed sources. This grain size did not cause serious oxidation and even with iron powder (300 A) left for 24 h in the open air, there was no difference in spectrum compared with powder freshly separated from toluene. Fig. 4 shows the stopping powers of Fe,Co and Ni. These powders were mixed into a solution polonium nitrate. About 99% of polonium adheres to the grain surfaces. After being dried and mounted, the sources were exposed to the aluminium window of a detector. To see the energy dependences better, experimental data are normalized appropriately. This is discussed in the following section. The energy dependences of the three data sets shift gradually from the Ziegler to the Northcliffe and Schilling curve in parallel with the increase in atomic number of the stopping material.
E 0
% ~ so 0 Q_
Co
a
/
/
/
/
z esLeU'
/ /
N o r t .hch f f e - S_c h d!in~/
0
i
.
_
_
_
~
k
t
L
.5
~
. . . .
i
L
_t_
(b)
_
1
Alpha Energy
[ bteV]
100
~E,j
i-1
....... I I rJ
o
5O
5. Absolute values of
dE/dR
By' integrating eq. (1), the factor S . D is expressed by an integral count N ( E ) and residual ranges of alpha particle R ( E ) ; S. D = 4. N ( E ~ ) / [ R ( E o ) - R(EI)], and
g_ E
Ni _
dE/dR = (dN/dE)
I. ,¥(EI)/[R(Eo ) -
Nor thct iffe- Sch, IIL n /
o
L
~
.c__~_,
.5 Alpha Energy
R(E,)].
,, I
_L
~
1
,
(c)
5 [ MeVJ
or
d N / N ( E,) = d R / [ R( E,,) - R( E , ) ] ,
(2)
Fig. 4. Stopping power of metal powders (300 A) for alpha particle: (a} Fe, (b) Co and (c) Ni.
where E o is the initial alpha particle energy and E] is a l o w e r threshold of integral count. If the two ranges are
given, d E / d R will be determined absolutely. For the values of the two ranges, the projected ranges would be more suitable than total path lengths. This process
eliminates experimental work to determine S and D and gives results to nearly the same accuracy as the residual ranges used. 111. SPECTROMETRY
380
M. Hosoe et al.
Stopping power measurement using ttucl~ alpha sources' By integration over surface area 3.
6. Conclusion A m e t h o d to measure the stopping power of materials is presented. The stopping powers measured for AI203,Fe,Co and Ni powder are c o m p a r e d with the N o r t h c l i f f e - S c h i l l i n g a n d Ziegler data table. The stopping power of U O 2 has been measured for an alpha particle energy range of 1.5 to 4 MeV and the results agree well with Ziegler's values. The anthers would to Dr. H. T o y o t a m a offer of the very fine oped recently in their
like to express their sincere t h a n k s and Dr. A. K o n o for their kind metal powders which were devellaboratory.
Appendix A.I. Derivation o f eq. (1)
d N - D . S f " L~(d.f2/4w ) d l
d N = ( dg2/a~r ) D d S d l . where l [g cm - 2 ] denotes the d e p t h from the surface a n d solid angle dJ~ involves a l p h a particles c o n t r i b u t i n g to d N ; d~2 = 2~r sin 0 . dO = 2w- I / r 2 d R ~ a n d r[g c m - 2 ] denotes the a l p h a path length through the medium. r( E ) = R ( E o ) - R ( E). E*dE E
I
E
~//// ,/,/
dS
dS
,, l - d l - ,
I).S/4)dR
Since d J 2 = d S . c o s O , / x 2. d V do-dx-cosO, and de0 = d o - cos O / x 2. d N = ( d S / 4 ~ r ) J2 cos O / x 2 • D . o " d o - d x cos 0 - ( d S / 4 ~ r ) f 2 , D . d R • cos 0do~. Finally for c o n s t a n t D and d R / d E . d E a n d for surface area S. using do~- sin 0- dO- d~ dN = ( D. S/4)(dR/dE)dE. measurement
It can be recognized that in m a n y cases, it would be much easier to keep D c o n s t a n t than to keep p c o n s t a n t through the medium. T h o u g h variations in 0 will reflect o n 52 a n d d x . this has no effect o n final results given by eq. (1), even for powders with high variations in mass density. O n the other hand, variations in D affect the spectrum and introduce errors. As an example, the case of metal powder with uniform gram size a n d mass # will be considered. For this purpose, a c o n s t a n t 0 formulation (1) is sufficient for an order estimation. A s s u m i n g that a fraction p of the n u m b e r of grains h a s a given activity 8 per grain a n d the rest 1 - p has no actwity: p - B - D . #. In this case. the relative error d E ~ d R will be given by the relative error in d N which is caused by fluctuation of the n u m b e r d n - d S - d l / # integrated with weight ~ = ( 1 / 2 r 2 ~8- d R ,
de" -
~{PI
A(dN )
t'f~12pd n
dN
f~pdn V fdS.
(ta}
",'(L')
(d~2/4~r}D-p.dV.
A.2. Errors in d E / d R A. 1.1. Case p = constant In fig. 5a, a mass d e m e n t d S . d l c o n t r i b u t e s a yield d N to the differential energy spectrum between E and E + dE
)t,
A. 1.2. Case 0 ~= constant tn fig. 5b. distances x and d x are m cm and surface areas d S a n d d o in cm 2 a n d p [ g c m ~] d e n o t e s the mass density of the medium, a n d residual range of a l p h a particles with energy E is d e n o t e d b y X ( E ) [cm}. A l p h a s from volume element d V emitted into a solid angle dI2 c o n t r i b u t e to the spectrum at ~ E , E + d E L when a surface Y'(P/ is defined by ,~ = X ( E 0) - X ( E ) a n d d x by p . d x - d R / d E d E Here the direction cosine between the surface normal to d o and .~ ~ denoted by cos #. dN-
Consider the energy spectrum of a l p h a particles emerging from a p l a n e surface of a medium. Nuclear scattering in the m e d i u m will be neglected in the following derivation. U n d e r this condition, eq. (1) is applicable even for a n i n h o m o g e n e o u s mass density O, especially for powder, if a l p h a activity per mass is held constant. However the case of c o n s t a n t p will be verified first for convenience.
2
=O.S.dR(1/2r
Ib}
Fig. 5. Schematic diagram of the geometry used for derivation of eq. (1).
dl(1/2r2)2(p"
fdS . dl{1/2r2 - ~/ 3 p r - d S
32~#)dR 2
)( p " 8 ~ # ) d R
M. Hosoe et al. / Stopping power measurement using thick alpha source.s Thus for surface area S, the relative error is determined by the statistics of the n u m b e r of active grains c o n t a i n e d in the mass rS. As an example, even for a very severe case of S = 0.1 cm 2, r = ,o. 2.42/~m (Fe, E = 4, E , = 5 . 3 MeV), ~ = o ' l /~m3, p = 0 . 0 1 , the relative error in d E / d R will be 0.2%. The powder made by coprecipitation may be a molecular mixture, to which eq. (1) and the above error estimation can be applied, regardless of grain size. For alpha sources adhering to the grain surface of metal powder particles, some careful preparation of the mixture would be required. At the same time a closer examination of the error estimation for small r, or for near m a x i m u m energy E o should be made. In the preceding discussion, 6 and d R = d R ~ d E , d E have played no part. These quantities determine the c o u n t i n g statistics of the spectrum, which will not be discussed here. The energy E at the surface defined by R ( E o ) R ( E ) = r is an average value. There is a spread a r o u n d this value. This energy spreading at the surface results
381
in a rounding effect on the spectrum. Because of the very smooth dependence of d E / d R on energy, the relative error is expected not to exceed 1%. except at very low energy a r o u n d 0.1 MeV and at the m a x i m u m energy E o. At low energy, an asymmetrical energy distribution should be considered and also the nuclear scattering should not be neglected.
References [1] F. Shiraishi, Y. Takami, M. Hosoe, Y. Ohsawa and H. Sato, Proc. Material Research Society Annual Meeting, Boston 1982, vol. 16, Nuclear Radiation Detector Materials eds., E.E. Hailer, H.W. Kramer and W.A. Higinbotham (1983): F. Shiraishi, Y. Takami and M. Hosoe, Third Eur. Syrup. on Semiconductor Detectors, Munich, 1983: Nucl. Instr. and Meth., to be published. [2] L.C. Northcliffe and R.F. Schilling, Nucl. Data Tables A7 (1970) 233. [3] J.F. Ziegler, Helium stopping powers and ranges in all material (Pergamon, New York, 1977).
Ill. SPECTROMETRY
377
STOPPING POWER MEASUREMENT USING THICK ALPHA SOURCES M. H O S O E Department of Physic.v, Faculty of Science, Rikkvo Unit,ersitv 3. Nishi- lkebukuro, Toshima- ku, Tokyo, 171, Japan
Y. T A K A M 1 , F. S H I R A I S H I
a n d K. T O M U R A
In~titute for Atomic Energy, Rtkkvo Universi(v, Nagasaka. Yokosuka, Kanagawa, 240-01. Japan
For alpha particles emerging from a plane surface of a thick medium in which alpha activity is distributed homogeneously, the energy spectrum is inversely proportional to the stopping power of the medium. The surface barrier type SSDs made of p type silicon were used to measure the spectrum and the stopping power was calculated. Ahhough this method can be applied only to alpha emitting sources at present, it has the merit of simplicity in the experimental procedure and gives the stopping power as a continuous function of energy by measuring only one spectrum. Powder with less than 1 p.m particle size is a suitable medium to mix with alpha activity in this method. The results are compared with stopping powers calculated from the Northcliffe Schilling and Ziegler data tables.
!. d E / d X calculation from the spectrum of thick alpha sources Alpha particles emitted from a plane surface of a medium containing alpha activity can easily be measured by a silicon surface barrier detector. W h e n sources are distributed homogeneously in the medium, the energy spectrum will be:
d N / d E = 0 . 2 5 . S. D / ( d E / d R ), where d N / d E
[MeV
(l)
i] is the differential detector
10"
count, S [cm 2] the area of detector in contact with the m e d i u m and D [g t] is the alpha disintegrations per unit mass of medium. R ( E ) [ g / c m 2] is the residual range and d E / d R [ M e V / ( g / c m 2 )] is the stopping power of the medium for an alpha particle with energy E [MeV]. The stopping power can be given simply by taking the inverse of the count per channel for a given energy in the spectrum. A p-silicon surface barrier detector [1] is useful for such measurements, because its aluminium entrance window is much stronger than the gold window corn-
(b)
(a)
~200 TP ;" "1Am In AI203
AI203
I
[
2
4
J
6
L
I
I
8
10
x
I00
CHANNEL NUMBER
ALPHA ENERGY ( M e V )
Fig. 1. (a) Differential energy spectrum of alpha particles measured with Si-SSD in close contact with AI ,O~ powder, which includes 24~Am homogeneously: (b) stopping powers of AI203 powder for alpha particle, calculated from the spectrum (a). 0 1 6 7 - 5 0 8 7 / 8 4 / $ 0 3 . 0 0 f ' Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
Ill. SPECTROMETRY
378
M. Hosoe et al.
Stoppmg power measurement usmg thick alpha sources
monly used for n-silicon. This m e t h o d of measuring is useful only for samples which can be mixed into the m e d i u m homogeneously and emit particles uniformly in all directions.
300
dE/dR
2. Metal oxide / For metal oxides it is easy to o b t a i n a spectrum of good quality by p r e p a r i n g powder with particle size of less than 1 F m as the source medium• Fig. 1 shows the differential energy spectrum of alpha particles from Z4~Am activity homogeneously distributed in fine a l u m i n a powder by coprecipitation, and the stopping power o b t a i n e d by applying eq. (1). Also are shown the curves calculated from the tables of Northcliffe-Schilling ( N - S ) [2] a n d Ziegler [3], where the powder is assumed to consist of pure AI20~. Experimental values are normalized to N - S values at 1 MeV. Agreement is good with the N - S curve using solid oxygen values (interpolated between solid metals in the table), but not with the curve using data of oxygen gas. which ~s not shown here. Fig. 2 shows the a l p h a spectrum of u r a n i u m oxide. In this case betas interfere with the alpha spectrum below 1.5 MeV. The u r a n i u m atoms emit alphas a n d at the same time constitute the m e d i u m molecules. Thus the D value is assured to be c o n s t a n t in atomic terms a n d this is a n ideal case for the application of eq. (1). Since 2~SU decays by a l p h a emission with a half-life of 4.468 × ].09 years, the disintegration rate per gram is
>
100 d
%,"
/
/
}
/
k\~
....
Zje~Ler Schilh nq
2
L Nortl%ljJ[e_-
o O3
U02 I
1
.
.
.
.
.
.
.
.
.
5 AlDha Energy
.
.
.
.
.
5 [ MeV
Fig. 2 Stopping powers of UO, powder for alpha particle. calculated to be 2.194 x l0 4 disintegrations s 1 g ~ bv assuming that 2-a~U a n d 234U are in radioactive eqmlibrium. This enables absolute stopping power values to be deduced a n d these values agree with Ziegler's curve. The presence of both the alpha emitters does not prevent the deduction of d E ~ d R based on eq. (IL
230 Th 238u
• :%.
Z
(,_)
,.
.
i;
z "I-
~
)U 23L'U t/[ 226R =
• :. . r . ~
:~
,. '.
• ,v..
: ~k,#"
. ¥.:~...,---"..r,L..-
•
210 Po ! .,=,. •./:. 222 Rn '~,.'~. :~.:.
o
2'8p 0
;'~,
|
21z. p o
U R A N I U M ORE I
s6o
1000 CHANNEL NUMBER
Fig. 3. Differential energy spectrum of alpha particles emitted from the plane surface of powdered uramum ore.
M. tto~oe et al. / Stopping power measurement using thwk alpha sources 3.
Uranium
lOO
_
ore
Fig. 3 shows the spectrum for a uranium ore. The alpha spectrum has eight components, corresponding to the natural decay series. In spite of the multiplicity, the spectrum follov,'s eq. (1), although when closely examined, some peaking occurs at the edges of fiat parts and a relatively large drop of counts is observed for channels below 2 MeV. This is likely to be caused by a,1 excessively large sample grain size. Inspection by optical microscope showed grain sizes ranging up to 50 ffm or more. The plateau levels in the spectrum give disintegrations for each step in the decay series. By calculation assuming that eq. (I) holds, it was found that equilibrium was preserved within + 3% among alpha decays in the series.
379
E 0
S >~ 50 0
Z~egl~.'
Fe
6_ cl
." /
i
_Nort he! kff e : SGh[[li r'<2j"
o
_ _ _ _ _ [ _ _ t ~
.1
I
J
I
.5 Alpha Energy
I
[
__ __ z
L
(a)
~J
I
5
{HEY]
lOO 4.
Metal
stopping
E u
Ix)wer
For stopping power measurement of metals, powders of 300 - 700 A in grain size were used with 2mPo as the alpha source. A grain size of 0.1 # m is small enough to yield homogeneous mixed sources. This grain size did not cause serious oxidation and even with iron powder (300 A) left for 24 h in the open air, there was no difference in spectrum compared with powder freshly separated from toluene. Fig. 4 shows the stopping powers of Fe,Co and Ni. These powders were mixed into a solution polonium nitrate. About 99% of polonium adheres to the grain surfaces. After being dried and mounted, the sources were exposed to the aluminium window of a detector. To see the energy dependences better, experimental data are normalized appropriately. This is discussed in the following section. The energy dependences of the three data sets shift gradually from the Ziegler to the Northcliffe and Schilling curve in parallel with the increase in atomic number of the stopping material.
E 0
% ~ so 0 Q_
Co
a
/
/
/
/
z esLeU'
/ /
N o r t .hch f f e - S_c h d!in~/
0
i
.
_
_
_
~
k
t
L
.5
~
. . . .
i
L
_t_
(b)
_
1
Alpha Energy
[ bteV]
100
~E,j
i-1
....... I I rJ
o
5O
5. Absolute values of
dE/dR
By' integrating eq. (1), the factor S . D is expressed by an integral count N ( E ) and residual ranges of alpha particle R ( E ) ; S. D = 4. N ( E ~ ) / [ R ( E o ) - R(EI)], and
g_ E
Ni _
dE/dR = (dN/dE)
I. ,¥(EI)/[R(Eo ) -
Nor thct iffe- Sch, IIL n /
o
L
~
.c__~_,
.5 Alpha Energy
R(E,)].
,, I
_L
~
1
,
(c)
5 [ MeVJ
or
d N / N ( E,) = d R / [ R( E,,) - R( E , ) ] ,
(2)
Fig. 4. Stopping power of metal powders (300 A) for alpha particle: (a} Fe, (b) Co and (c) Ni.
where E o is the initial alpha particle energy and E] is a l o w e r threshold of integral count. If the two ranges are
given, d E / d R will be determined absolutely. For the values of the two ranges, the projected ranges would be more suitable than total path lengths. This process
eliminates experimental work to determine S and D and gives results to nearly the same accuracy as the residual ranges used. 111. SPECTROMETRY
380
M. Hosoe et al.
Stopping power measurement using ttucl~ alpha sources' By integration over surface area 3.
6. Conclusion A m e t h o d to measure the stopping power of materials is presented. The stopping powers measured for AI203,Fe,Co and Ni powder are c o m p a r e d with the N o r t h c l i f f e - S c h i l l i n g a n d Ziegler data table. The stopping power of U O 2 has been measured for an alpha particle energy range of 1.5 to 4 MeV and the results agree well with Ziegler's values. The anthers would to Dr. H. T o y o t a m a offer of the very fine oped recently in their
like to express their sincere t h a n k s and Dr. A. K o n o for their kind metal powders which were devellaboratory.
Appendix A.I. Derivation o f eq. (1)
d N - D . S f " L~(d.f2/4w ) d l
d N = ( dg2/a~r ) D d S d l . where l [g cm - 2 ] denotes the d e p t h from the surface a n d solid angle dJ~ involves a l p h a particles c o n t r i b u t i n g to d N ; d~2 = 2~r sin 0 . dO = 2w- I / r 2 d R ~ a n d r[g c m - 2 ] denotes the a l p h a path length through the medium. r( E ) = R ( E o ) - R ( E). E*dE E
I
E
~//// ,/,/
dS
dS
,, l - d l - ,
I).S/4)dR
Since d J 2 = d S . c o s O , / x 2. d V do-dx-cosO, and de0 = d o - cos O / x 2. d N = ( d S / 4 ~ r ) J2 cos O / x 2 • D . o " d o - d x cos 0 - ( d S / 4 ~ r ) f 2 , D . d R • cos 0do~. Finally for c o n s t a n t D and d R / d E . d E a n d for surface area S. using do~- sin 0- dO- d~ dN = ( D. S/4)(dR/dE)dE. measurement
It can be recognized that in m a n y cases, it would be much easier to keep D c o n s t a n t than to keep p c o n s t a n t through the medium. T h o u g h variations in 0 will reflect o n 52 a n d d x . this has no effect o n final results given by eq. (1), even for powders with high variations in mass density. O n the other hand, variations in D affect the spectrum and introduce errors. As an example, the case of metal powder with uniform gram size a n d mass # will be considered. For this purpose, a c o n s t a n t 0 formulation (1) is sufficient for an order estimation. A s s u m i n g that a fraction p of the n u m b e r of grains h a s a given activity 8 per grain a n d the rest 1 - p has no actwity: p - B - D . #. In this case. the relative error d E ~ d R will be given by the relative error in d N which is caused by fluctuation of the n u m b e r d n - d S - d l / # integrated with weight ~ = ( 1 / 2 r 2 ~8- d R ,
de" -
~{PI
A(dN )
t'f~12pd n
dN
f~pdn V fdS.
(ta}
",'(L')
(d~2/4~r}D-p.dV.
A.2. Errors in d E / d R A. 1.1. Case p = constant In fig. 5a, a mass d e m e n t d S . d l c o n t r i b u t e s a yield d N to the differential energy spectrum between E and E + dE
)t,
A. 1.2. Case 0 ~= constant tn fig. 5b. distances x and d x are m cm and surface areas d S a n d d o in cm 2 a n d p [ g c m ~] d e n o t e s the mass density of the medium, a n d residual range of a l p h a particles with energy E is d e n o t e d b y X ( E ) [cm}. A l p h a s from volume element d V emitted into a solid angle dI2 c o n t r i b u t e to the spectrum at ~ E , E + d E L when a surface Y'(P/ is defined by ,~ = X ( E 0) - X ( E ) a n d d x by p . d x - d R / d E d E Here the direction cosine between the surface normal to d o and .~ ~ denoted by cos #. dN-
Consider the energy spectrum of a l p h a particles emerging from a p l a n e surface of a medium. Nuclear scattering in the m e d i u m will be neglected in the following derivation. U n d e r this condition, eq. (1) is applicable even for a n i n h o m o g e n e o u s mass density O, especially for powder, if a l p h a activity per mass is held constant. However the case of c o n s t a n t p will be verified first for convenience.
2
=O.S.dR(1/2r
Ib}
Fig. 5. Schematic diagram of the geometry used for derivation of eq. (1).
dl(1/2r2)2(p"
fdS . dl{1/2r2 - ~/ 3 p r - d S
32~#)dR 2
)( p " 8 ~ # ) d R
M. Hosoe et al. / Stopping power measurement using thick alpha source.s Thus for surface area S, the relative error is determined by the statistics of the n u m b e r of active grains c o n t a i n e d in the mass rS. As an example, even for a very severe case of S = 0.1 cm 2, r = ,o. 2.42/~m (Fe, E = 4, E , = 5 . 3 MeV), ~ = o ' l /~m3, p = 0 . 0 1 , the relative error in d E / d R will be 0.2%. The powder made by coprecipitation may be a molecular mixture, to which eq. (1) and the above error estimation can be applied, regardless of grain size. For alpha sources adhering to the grain surface of metal powder particles, some careful preparation of the mixture would be required. At the same time a closer examination of the error estimation for small r, or for near m a x i m u m energy E o should be made. In the preceding discussion, 6 and d R = d R ~ d E , d E have played no part. These quantities determine the c o u n t i n g statistics of the spectrum, which will not be discussed here. The energy E at the surface defined by R ( E o ) R ( E ) = r is an average value. There is a spread a r o u n d this value. This energy spreading at the surface results
381
in a rounding effect on the spectrum. Because of the very smooth dependence of d E / d R on energy, the relative error is expected not to exceed 1%. except at very low energy a r o u n d 0.1 MeV and at the m a x i m u m energy E o. At low energy, an asymmetrical energy distribution should be considered and also the nuclear scattering should not be neglected.
References [1] F. Shiraishi, Y. Takami, M. Hosoe, Y. Ohsawa and H. Sato, Proc. Material Research Society Annual Meeting, Boston 1982, vol. 16, Nuclear Radiation Detector Materials eds., E.E. Hailer, H.W. Kramer and W.A. Higinbotham (1983): F. Shiraishi, Y. Takami and M. Hosoe, Third Eur. Syrup. on Semiconductor Detectors, Munich, 1983: Nucl. Instr. and Meth., to be published. [2] L.C. Northcliffe and R.F. Schilling, Nucl. Data Tables A7 (1970) 233. [3] J.F. Ziegler, Helium stopping powers and ranges in all material (Pergamon, New York, 1977).
Ill. SPECTROMETRY