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Evaluation of fission fragment ranges in any medium
N U C L E A R I N S T R U M E N T S AND METHODS
I27
(t975) 5 7 9 - 5 8 2 ;
© NORTH-HOLLAND
PUBLISHING
CO.
EVALUATION OF FISSION FRAGMENT RANGES IN ANY MEDIUM G. CESINI, G. L U C A R I N I
Dipartimento di Scienze Fisiche, Facoltgt di Ingegneria, Universit~ di Ancona, Italy and F. RUSTICHELLI Physics Division, Joint Research Center, E U R A T O M , Ispra, Italy, and Dipartimento di Seienze Fisiche, Facolt& di Ingegneria, Universitg~ di Ancona, Italy Received 7 February 1975 Revised manuscript received 5 May 1975 By using the set of average ranges o f 2a5U and 230pu fission fragments in all natural elements, evaluated by Rustichelli, in conjunction with the rule of additivity of atomic stopping powers of Bragg-Kleeman, ranges of 2z5U and 239pu median
light, median heavy and overall median fission fragments in some composite systems are evaluated. This is an example o f a range evaluation method in any substance.
1. Introduction
in refs. 4 and 5 and the Bragg-Kleeman rule on atomic stopping power additivity6).
Most investigations concerning fission fragment interaction with matter have been performed, for any fissile nucleus, on a few fission fragments, which were studied one by one in their interaction with few stopping medial). These investigations are very interesting concerning the physics of the energy loss of charged particles in matter. However they are far from covering the real needs for fission fragment data in science and technology and an effort is required to obtain a complete data set. This direction was followed by Hakim and Shafrirl), who measured 252Cf overall median fission fragment transmission curves in several stopping media, and by Aiello, Maracci and Rustichelli2), who determined 235U median fission fragment transmission curves in many stopping elements. Furthermore, Hakim and Shafrir 3) have developed a semi-empirical method, based on the Bragg-Kleeman additivity rule of atomic stopping power, which allows the evaluation of the overall median fission fragment absolute range in composite systems, knowing the corresponding ranges in the component atoms. More recently a method was developed that, by using the Lindhard, Scharff and Schiott theory and the limited available experimental data, allows to derive 235 U 4) and 239pu 5) median heavy, median light and overall median fission fragment absolute ranges in all natural elements. The present work shows how it is possible to evaluate median light, median heavy and overall median fission fragment ranges in any composite system in a very easy way, by using data obtained with the method reported
2. Method and results
First of all we utilize the Bragg-Kleeman rule* of additivity of atomic stopping powers to evaluate median atomic stopping power of any composite system, as a function of stopping powers of the elements in the compound
ZN
¢ = ~ wi-~i \dX/i
where N is the number of atoms per volume unit, (dE/dX) is the linear stopping power, w i is the atom fraction and subscripts i and c are related to the element and the compound, respectively. After that, by assuming that the median fission fragment transmission curve shape is independent from the stopping medium, it is possible to evaluate the range of any median fission fragment in a composite system by knowing the ranges in all constituent elements. This assumption has been proved experimentally for 23SU overall median fission fragments in the interaction with several natural elements4). With this assumption the following expression for the range, Re, in the composite system was obtained R~ =
Mc
(2)
Y~ n, (a,/n~)" i
* The conditions of validity o f this rule are discussed in ref. 7.
579
580
G. CESINI et al. TABLE 1 2asU median fission fragment ranges in some plastics.
TABLE 3 239pu median fission fragment ranges in some plastics
Absolute range (mg/cm2)
Absolute range (mg/cm2)
Compound Density (g/cruZ)
Polythene
Median Overall light median fragment fragment
Median heavy fragment
Compound Density (g/cm3)
Median light fragment
Overall median fragment
Median heavy fragment
Ziegler 0.84 Phillis 0.96
2.383 2.383
2.231 2.231
2.100 2.099
Ziegler 0.84 Phillis 0.96
2.336 2.336
2.195 2.195
2.021 2.021
Polythene
Teflon
2.15 2.20
3.214 3.216
2.975 2.976
2.704 2.706
Teflon
2.15 2.20
3.261 3.262
3.018 3.020
2.810 2.812
Polystyrene
1.06
2.557
2.394
2.198
Polystyrene
1.06
2.605
2.433
2.284
P.V.C.
1.25
3.154
2.914
2.645
PVC
1.25
3.199
2.956
2.749
TABLE 4 239pu median fission fragment ranges in some oxides.
TABLE 2
235U median fission fragment ranges in some oxides. Absolute range (mg/cm2)
Absolute range (mg/cm2)
Compound
Compound Median light Overall median Median heavy fragment fragment fragment
SiO2 A1203 Fe203 CaO
3.311 3.362 4.188 3.770
3.046 3.093 3.798 3.431
2.755 2.798 3.394 3.076
SiO2 A1203 Fe203 CaO
where R i is the range in element i, ni is the n u m b e r o f a t o m s o f element i in the molecule, At is the a t o m i c weight o f element i and M c is the m o l e c u l a r weight o f the c o m p o s i t e system. By using eq. (2) it is possible to evaluate the m e d i a n light, m e d i a n heavy a n d overall m e d i a n fission fragm e n t ranges o f a n y fissile nucleus in any c o m p o s i t e system k n o w i n g the c o r r e s p o n d i n g m e d i a n fission f r a g m e n t ranges in the c o m p o n e n t atoms. T a b l e 1 a n d table 2 show the results o b t a i n e d for 23SU m e d i a n fission f r a g m e n t ranges in some plastics a n d oxides by using the d a t a o f ref. 3, a n d table 3 a n d table 4 show the c o r r e s p o n d i n g ranges for Z39pu fission fragments4). M o r e generally it is possible to derive in an easy w a y (see a p p e n d i x ) an expression to evaluate fission fragm e n t ranges in c o m p o s i t e system mixture, (M) n
Median light Overall median Median heavy fragment fragment fragment
(3)
3.352 3.404 4.220 3.803
3.088 3.136 3.844 3.475
2.864 2.908 3.531 3.200
where ( R ) a n d ( M ) are respectively the m e d i a n range a n d m e d i a n m o l e c u l a r weight o f the mixture, Re, a n d Me, are the range and m o l e c u l a r weight o f the c o m p o u n d n, and w, is the m o l a r fraction. Table 5 shows the results o b t a i n e d for 235U m e d i a n fission f r a g m e n t ranges in some mixtures o f technological interest a n d table 6 the c o r r e s p o n d i n g results for 239pu fission fragments. It is clear that these d a t a are only few examples o f a p p l i c a t i o n o f a general m e t h o d which is available for a n y s t o p p i n g m e d i u m a n d for any fissile nucleus whose m e d i a n fission f r a g m e n t ranges are k n o w n in all n a t u r a l elements.
3. Conclusion By using the B r a g g - K l e e m a n rule o f additivity o f a t o m i c s t o p p i n g p o w e r a n d the fact that evaluated d a t a are available for the 235U a n d 239pu m e d i a n light, m e d i a n heavy a n d overall m e d i a n fission f r a g m e n t ranges, it was shown that it is possible to evaluate the
581
EVALUATION OF FISSION FRAGMENT RANGES TABLE 5 235U median fission fragments ranges in three kinds of Portland cement. Absolute range (mg/cm~)
Composition (%)
I II III
SiO2
A1203
Fe203
CaO
Median light fragment
Overall median fragment
Median heavy fragment
20.0 22.0 20.0
7.0 7.0 5.5
3.0 3.0 4.5
66.0 63.0 66.0
3.644 3.632 3.656
3.327 3.317 3.338
2.991 2.982 3.000
TABLE 6
23apu median fission fragment ranges in three kinds of Portland cement. Absolute range (mg/cm 2)
Composition (%)
I II 1ll
SiO2
A1203
Fe203
CaO
Median light fragment
Overall median fragment
Median heavy fragment
20.0 22.0 20.0
7.0 7.0 5.5
3.0 3.0 4.5
66.0 63.0 66.0
3.680 3.668 3.692
3.371 3.361 3.381
3.111 3.102 3.120
ranges o f these three m e d i a n fragments in a n y c o m posite m e d i u m . T h e m e t h o d is limited by the uncertainties associated with the validity o f the B r a g g - K l e e m a n rule a n d with the available d a t a on single n a t u r a l elements. A n i m p r o v e m e n t in the accuracy o f these p r i m a r y d a t a , due to a better e v a l u a t i o n or to direct experi m e n t a l measurements, w o u l d i m p r o v e the accuracy o f the a b o v e m e t h o d , which is quite general a n d can be a p p l i e d to the e v a l u a t i o n o f the three m e d i a n fission f r a g m e n t ranges for a n y fissile nucleus, if the corres p o n d i n g ranges in the n a t u r a l elements are known.
T h e n u m b e r o f element i a t o m s in the system n which are present in a mole o f w.Nnl, where N is the A v o g a d r o number. the total n u m b e r o f a t o m s o f element i in mixture is ~w, Nni.; then n
Z Wn Nnin n.~i = n -- Z W nni.. N
A s s u m i n g for a mixture the same hypothesis we have used for a c o m p o s i t e system, i.e. t h a t the shape o f fission fragments transmission curve is i n d e p e n d e n t f r o m the s t o p p i n g m e d i u m , it is possible to show t h a t the m e a n range in the mixture ( R ) is given by an expression similar to eq. (2). In the d e r i v a t i o n o f this expression the m o l e c u l a r weight o f the c o m p o s i t e system is replaced by the m e a n m o l e c u l a r weight o f the mixture ( M ) = ~w,M. a n d the n u m b e r nl o f a t o m s
_
(M>
Z i
=
(M>
Z Z Wn Hi. (Ai/R,) i n
Hi (Ai/Ri)
(M>
(5)
Z Wn Z rlin(Ai/Ri) n
i
Eq. (2), a p p l i e d to the nth c o m p o s i t e system present in the mixture, takes the f o r m Mn
R,
~ nl, (Ai/Ri)
(6)
i
n
o f element i in the molecule o f the c o m p o s i t e system is replaced b y ~i, which represents the m e a n n u m b e r o f a t o m s o f element i in a " m e a n m o l e c u l e " o f the mixture.
(4)
A s a consequence the expression o f the m e a n range in the mixture is
(R>
Appendix
composite mixture is Therefore a mole o f
F r o m this expression we derive i
nln(AJRi) = M,/R,.
(7)
582
G. CESINI et al.
By substitution of eq. (7) in eq. (5) we o b t a i n the final expression for ( R ) ,
(M)
(R) -
L w,, ( M , / R n ) ' n
which gives the average fission fragment range in a mixture as a f u n c t i o n of the molecular weights associated to each composite system a n d of the fission fragment ranges in each composite system present in the mixture.
References 1) M. Hakim and N. H. Shafrir, Can. J. Phys. 49 (1971) 3036. 2) V. Aiello, G. Maracci and F. Rustichelli, Phys. Rev. 4B (1971) 3812. a) M. Hakim and N. H. Shafrir, Nucl. Sci. Eng. 48 (1972) 72. 4) F. Rustichelli, Z. Physik 262 (1973) 211. 5) F. Rustichelli, Nuclear Data in Science and Technology, vol. I, Proc. Symp. Paris (IAEA, Vienna, 1973) p. 559. 8) W. H. Bragg and R. Kleeman, Phil. Mag. 10 (1905) 318. 7) W. Brandt, Health Phys. I (1958) ll.
I27
(t975) 5 7 9 - 5 8 2 ;
© NORTH-HOLLAND
PUBLISHING
CO.
EVALUATION OF FISSION FRAGMENT RANGES IN ANY MEDIUM G. CESINI, G. L U C A R I N I
Dipartimento di Scienze Fisiche, Facoltgt di Ingegneria, Universit~ di Ancona, Italy and F. RUSTICHELLI Physics Division, Joint Research Center, E U R A T O M , Ispra, Italy, and Dipartimento di Seienze Fisiche, Facolt& di Ingegneria, Universitg~ di Ancona, Italy Received 7 February 1975 Revised manuscript received 5 May 1975 By using the set of average ranges o f 2a5U and 230pu fission fragments in all natural elements, evaluated by Rustichelli, in conjunction with the rule of additivity of atomic stopping powers of Bragg-Kleeman, ranges of 2z5U and 239pu median
light, median heavy and overall median fission fragments in some composite systems are evaluated. This is an example o f a range evaluation method in any substance.
1. Introduction
in refs. 4 and 5 and the Bragg-Kleeman rule on atomic stopping power additivity6).
Most investigations concerning fission fragment interaction with matter have been performed, for any fissile nucleus, on a few fission fragments, which were studied one by one in their interaction with few stopping medial). These investigations are very interesting concerning the physics of the energy loss of charged particles in matter. However they are far from covering the real needs for fission fragment data in science and technology and an effort is required to obtain a complete data set. This direction was followed by Hakim and Shafrirl), who measured 252Cf overall median fission fragment transmission curves in several stopping media, and by Aiello, Maracci and Rustichelli2), who determined 235U median fission fragment transmission curves in many stopping elements. Furthermore, Hakim and Shafrir 3) have developed a semi-empirical method, based on the Bragg-Kleeman additivity rule of atomic stopping power, which allows the evaluation of the overall median fission fragment absolute range in composite systems, knowing the corresponding ranges in the component atoms. More recently a method was developed that, by using the Lindhard, Scharff and Schiott theory and the limited available experimental data, allows to derive 235 U 4) and 239pu 5) median heavy, median light and overall median fission fragment absolute ranges in all natural elements. The present work shows how it is possible to evaluate median light, median heavy and overall median fission fragment ranges in any composite system in a very easy way, by using data obtained with the method reported
2. Method and results
First of all we utilize the Bragg-Kleeman rule* of additivity of atomic stopping powers to evaluate median atomic stopping power of any composite system, as a function of stopping powers of the elements in the compound
ZN
¢ = ~ wi-~i \dX/i
where N is the number of atoms per volume unit, (dE/dX) is the linear stopping power, w i is the atom fraction and subscripts i and c are related to the element and the compound, respectively. After that, by assuming that the median fission fragment transmission curve shape is independent from the stopping medium, it is possible to evaluate the range of any median fission fragment in a composite system by knowing the ranges in all constituent elements. This assumption has been proved experimentally for 23SU overall median fission fragments in the interaction with several natural elements4). With this assumption the following expression for the range, Re, in the composite system was obtained R~ =
Mc
(2)
Y~ n, (a,/n~)" i
* The conditions of validity o f this rule are discussed in ref. 7.
579
580
G. CESINI et al. TABLE 1 2asU median fission fragment ranges in some plastics.
TABLE 3 239pu median fission fragment ranges in some plastics
Absolute range (mg/cm2)
Absolute range (mg/cm2)
Compound Density (g/cruZ)
Polythene
Median Overall light median fragment fragment
Median heavy fragment
Compound Density (g/cm3)
Median light fragment
Overall median fragment
Median heavy fragment
Ziegler 0.84 Phillis 0.96
2.383 2.383
2.231 2.231
2.100 2.099
Ziegler 0.84 Phillis 0.96
2.336 2.336
2.195 2.195
2.021 2.021
Polythene
Teflon
2.15 2.20
3.214 3.216
2.975 2.976
2.704 2.706
Teflon
2.15 2.20
3.261 3.262
3.018 3.020
2.810 2.812
Polystyrene
1.06
2.557
2.394
2.198
Polystyrene
1.06
2.605
2.433
2.284
P.V.C.
1.25
3.154
2.914
2.645
PVC
1.25
3.199
2.956
2.749
TABLE 4 239pu median fission fragment ranges in some oxides.
TABLE 2
235U median fission fragment ranges in some oxides. Absolute range (mg/cm2)
Absolute range (mg/cm2)
Compound
Compound Median light Overall median Median heavy fragment fragment fragment
SiO2 A1203 Fe203 CaO
3.311 3.362 4.188 3.770
3.046 3.093 3.798 3.431
2.755 2.798 3.394 3.076
SiO2 A1203 Fe203 CaO
where R i is the range in element i, ni is the n u m b e r o f a t o m s o f element i in the molecule, At is the a t o m i c weight o f element i and M c is the m o l e c u l a r weight o f the c o m p o s i t e system. By using eq. (2) it is possible to evaluate the m e d i a n light, m e d i a n heavy a n d overall m e d i a n fission fragm e n t ranges o f a n y fissile nucleus in any c o m p o s i t e system k n o w i n g the c o r r e s p o n d i n g m e d i a n fission f r a g m e n t ranges in the c o m p o n e n t atoms. T a b l e 1 a n d table 2 show the results o b t a i n e d for 23SU m e d i a n fission f r a g m e n t ranges in some plastics a n d oxides by using the d a t a o f ref. 3, a n d table 3 a n d table 4 show the c o r r e s p o n d i n g ranges for Z39pu fission fragments4). M o r e generally it is possible to derive in an easy w a y (see a p p e n d i x ) an expression to evaluate fission fragm e n t ranges in c o m p o s i t e system mixture, (M) n
Median light Overall median Median heavy fragment fragment fragment
(3)
3.352 3.404 4.220 3.803
3.088 3.136 3.844 3.475
2.864 2.908 3.531 3.200
where ( R ) a n d ( M ) are respectively the m e d i a n range a n d m e d i a n m o l e c u l a r weight o f the mixture, Re, a n d Me, are the range and m o l e c u l a r weight o f the c o m p o u n d n, and w, is the m o l a r fraction. Table 5 shows the results o b t a i n e d for 235U m e d i a n fission f r a g m e n t ranges in some mixtures o f technological interest a n d table 6 the c o r r e s p o n d i n g results for 239pu fission fragments. It is clear that these d a t a are only few examples o f a p p l i c a t i o n o f a general m e t h o d which is available for a n y s t o p p i n g m e d i u m a n d for any fissile nucleus whose m e d i a n fission f r a g m e n t ranges are k n o w n in all n a t u r a l elements.
3. Conclusion By using the B r a g g - K l e e m a n rule o f additivity o f a t o m i c s t o p p i n g p o w e r a n d the fact that evaluated d a t a are available for the 235U a n d 239pu m e d i a n light, m e d i a n heavy a n d overall m e d i a n fission f r a g m e n t ranges, it was shown that it is possible to evaluate the
581
EVALUATION OF FISSION FRAGMENT RANGES TABLE 5 235U median fission fragments ranges in three kinds of Portland cement. Absolute range (mg/cm~)
Composition (%)
I II III
SiO2
A1203
Fe203
CaO
Median light fragment
Overall median fragment
Median heavy fragment
20.0 22.0 20.0
7.0 7.0 5.5
3.0 3.0 4.5
66.0 63.0 66.0
3.644 3.632 3.656
3.327 3.317 3.338
2.991 2.982 3.000
TABLE 6
23apu median fission fragment ranges in three kinds of Portland cement. Absolute range (mg/cm 2)
Composition (%)
I II 1ll
SiO2
A1203
Fe203
CaO
Median light fragment
Overall median fragment
Median heavy fragment
20.0 22.0 20.0
7.0 7.0 5.5
3.0 3.0 4.5
66.0 63.0 66.0
3.680 3.668 3.692
3.371 3.361 3.381
3.111 3.102 3.120
ranges o f these three m e d i a n fragments in a n y c o m posite m e d i u m . T h e m e t h o d is limited by the uncertainties associated with the validity o f the B r a g g - K l e e m a n rule a n d with the available d a t a on single n a t u r a l elements. A n i m p r o v e m e n t in the accuracy o f these p r i m a r y d a t a , due to a better e v a l u a t i o n or to direct experi m e n t a l measurements, w o u l d i m p r o v e the accuracy o f the a b o v e m e t h o d , which is quite general a n d can be a p p l i e d to the e v a l u a t i o n o f the three m e d i a n fission f r a g m e n t ranges for a n y fissile nucleus, if the corres p o n d i n g ranges in the n a t u r a l elements are known.
T h e n u m b e r o f element i a t o m s in the system n which are present in a mole o f w.Nnl, where N is the A v o g a d r o number. the total n u m b e r o f a t o m s o f element i in mixture is ~w, Nni.; then n
Z Wn Nnin n.~i = n -- Z W nni.. N
A s s u m i n g for a mixture the same hypothesis we have used for a c o m p o s i t e system, i.e. t h a t the shape o f fission fragments transmission curve is i n d e p e n d e n t f r o m the s t o p p i n g m e d i u m , it is possible to show t h a t the m e a n range in the mixture ( R ) is given by an expression similar to eq. (2). In the d e r i v a t i o n o f this expression the m o l e c u l a r weight o f the c o m p o s i t e system is replaced by the m e a n m o l e c u l a r weight o f the mixture ( M ) = ~w,M. a n d the n u m b e r nl o f a t o m s
_
(M>
Z i
=
(M>
Z Z Wn Hi. (Ai/R,) i n
Hi (Ai/Ri)
(M>
(5)
Z Wn Z rlin(Ai/Ri) n
i
Eq. (2), a p p l i e d to the nth c o m p o s i t e system present in the mixture, takes the f o r m Mn
R,
~ nl, (Ai/Ri)
(6)
i
n
o f element i in the molecule o f the c o m p o s i t e system is replaced b y ~i, which represents the m e a n n u m b e r o f a t o m s o f element i in a " m e a n m o l e c u l e " o f the mixture.
(4)
A s a consequence the expression o f the m e a n range in the mixture is
(R>
Appendix
composite mixture is Therefore a mole o f
F r o m this expression we derive i
nln(AJRi) = M,/R,.
(7)
582
G. CESINI et al.
By substitution of eq. (7) in eq. (5) we o b t a i n the final expression for ( R ) ,
(M)
(R) -
L w,, ( M , / R n ) ' n
which gives the average fission fragment range in a mixture as a f u n c t i o n of the molecular weights associated to each composite system a n d of the fission fragment ranges in each composite system present in the mixture.
References 1) M. Hakim and N. H. Shafrir, Can. J. Phys. 49 (1971) 3036. 2) V. Aiello, G. Maracci and F. Rustichelli, Phys. Rev. 4B (1971) 3812. a) M. Hakim and N. H. Shafrir, Nucl. Sci. Eng. 48 (1972) 72. 4) F. Rustichelli, Z. Physik 262 (1973) 211. 5) F. Rustichelli, Nuclear Data in Science and Technology, vol. I, Proc. Symp. Paris (IAEA, Vienna, 1973) p. 559. 8) W. H. Bragg and R. Kleeman, Phil. Mag. 10 (1905) 318. 7) W. Brandt, Health Phys. I (1958) ll.