- Email: [email protected]
Energy loss of beta particles on backscattering
I25 (I975)
N U C L E A R I N S T R U M E N T S AND METHODS
I69-I7I;
© NORTH-HOLLAND
P U B L I S H I N G CO.
ENERGY LOSS OF BETA PARTICLES ON BACKSCATTERING*
T. BALTAKMENS New Zealand Department of Health, National Radiation Laboratory, Christchurch, New Zealand Received 31 December 1974 An empirical equation for calculating the ettective maximum and average energy of beta particles after backscattering is derived from their absorption characteristics. For a beta emitter of maximum energy Em and average energy E, the effective maximum and average energies after scattering,
In the numerous practical applications of betaemitting radionuclides, their scattering characteristics are frequently of interest. One of these characteristics * Published with the authority of the Director-General of Health. 100
lOO
10
lO
1
i
1500
i
4
i
i
i
1200 i
I_
lOO
lOO
r o (D
10
1c
1
1500 Absorber
thickness
4
.1 (rag
12j00
Es and E's are given by: Es/Em = (158 +Z)/251,
Fig. 1. Absorption curves for incident (upper) and backscattered (lower) betas for four emitters with aluminium absorbers and a copper backscatterer.
169
Es/E=(123+Z)/216,
where Z is the atomic number of the scattering material.
is the energy loss which occurs when beta particles are scattered by various materials. Due to the complex nature of the energy spectra of beta emitters, theoretical calculations and direct measurements of the energy loss are difficult. This paper presents a simple empirical relationship which may be useful in determining the energy loss of beta particles on backscattering, and which can be derived from their absorption characteristics. It is a well-known empirical observation that the absorption in metals of both incident and backscattered beta particles is exponential over a large range of absorber thicknesses 1'2) and can be expressed as: C = Co e -"x
(incident),
C=
(backscattered),
Co e-Ix
where C is the counting rate produced by a beta emitter in a beta counter through an absorber of thickness x; Co is the counting rate without absorber; and n and l are the respective absorption coefficients for incident and backscattered beta particles. Fig. 1 shows typical absorption curves for the incident (upper curve) and backscattered (lower curve) beta particles from four emitters obtained with a low-background anticoincidence beta counter. The scattering material is copper of sufficient thickness to give saturation backscattering at diffuse incidence, the absorber is aluminium. It is also observed empirically that the absorption coefficient for a given emitter is related to its maximum energy, Era, by an expression of the type: n =
cr~ 2)
and;
KErn",
where K and a are constants3). A similar relationship holds also for backscattered betas, and fig. 2 shows the relationship between E m and n (line A), and E m and l
170
T. B A L T A K M E N S
(line B) for the following emitters: 2°4T1, E m = 0 . 7 6 MeV; 2]°Bi, E r a = 1.16 M e V ; 89Sr, Em = 1.45 MeV; 90y, Em = 2.27 MeV; 234mpa, E m = 2.29 MeV; la4pr, E m = 2 . 9 8 MeV; l°6Rh, E m = 3.53 MeV (68 %). These results were obtained using the same counter as above, with aluminium absorbers and copper as backscattering material. F o r each emitter, l > n , and we can regard the absorption characteristics o f backscattered betas from a given emitter as being very nearly identical to those o f incident betas from another emitter o f correspondingly lower m a x i m u m energy. Thus we can use the ratio n to l as a measure o f the loss in E m on backscattering. Consider a beta emitter with m a x i m u m energy E m and absorption coefficients n and l. Let the effective m a x i m u m energy after scattering be Es. Then, referring
Eml
to the diagram at the top o f fig. 2, it is obvious that: n
= K AEm a
I
= K a E ~ ~ = K A E ~ ~,
.'.FI/I
=
(Em/Es)
-a ,
or: E s / E m = ( n i l ) '/a.
It has been found that the relationship between n and l can be approximately given by: 1/n = C / ( D + Z),
where Z is the atomic number of the backscatterer, and C and D are constants2). Hence: =
- -
.
(1)
F r o m fig. 2, 1/a = 0.75; suitable values for the other constants are C = 200, and D - - 1 0 7 . If the function on the right-hand side of eq. (1) is plotted against Z, the resulting graphical relationship is very nearly linear (fig. 3). Thus the expression for the effective m a x i m u m energy can be simplified to:
2 I
E s / E m = (158 + Z)/251.
L~
0"5
(2)
The actual loss in m a x i m u m energy on backscattering, dEm, is given by: i
i
~
i i ~rl
i
3 10 Absorption coefficient
i
I
30
r
50
(cm2g1)
dE m = Em-Es,
and the fractional loss by: Fig. 2. Relationship between absorption coefficient and maximum energy for incident (A) and backscattered (B) betas; and between absorption coefficient and average energy for incident (A') and backscattered (B') betas. 1.o
0.8
0.6 I
o
I
I
I
t
50 z
I
I
I
I
I
1oo
Fig. 3. Ratio of maximum and average energy to effective maximum and average energy after backscattering as a function of the atomic number of the backscatterer.
d E m / E m = 1 - (Es/Em) = ( 9 3 - Z ) / 2 5 1 .
It is interesting to note that d E m / E m = 0 for Z = 93, which implies that the energy loss tends to zero if the scatterer has a value o f Z near the end o f the periodic table. The loss in the average energy per beta disintegration, E (as distinct from the loss in m a x i m u m energy, Era), can be calculated in a similar way. In general, the ratio E / E m for a given beta emitter depends on Era, on the type o f transition, and on the atomic n u m b e r o f the emitter4). W h e n beta particles are scattered, there is not only a loss in energy, but also a change in the shape o f the energy spectrum, which tends to be shifted towards the low-energy end with a corresponding decrease in the proportion of high-energy betas in the spectrum s.6). Thus we m a y expect that the fractional loss in E on scattering is not the same as that in E m. The relationship between E and (n, l) is similar to that between Em
171
E N E R G Y LOSS OF BETA P A R T I C L E S
and (n,/): lines A' and B' in fig. 2. A similar derivation as for Em gives:
for values of Z from 1 to 93. Also:
dE/E = 1.16dEm/Em, with 1/b = 0.88 from fig. 2. Again, using fig. 3, the above expression simplifies to:
£s/E = (123 + Z)/216,
i.e., for any emitter the fractional loss in the average energy is greater than that in the maximum energy for the above values of Z.
(3)
hence:
References
dE/E = (93-Z)/216. Eqs. (2) and (3) give two simple linear relationships which can be used to calculate the effective maximum and average energies after backscattering for any combination of emitter and scattering material. In general the energy loss decreases with increase in Z, and: Es/E m >
£s/E,
1) K. Siegbahn, ed., Alpha-, beta- and gamma-ray spectroscopy (North-Holland Publishing Company, Amsterdam, 1965) p. 21-24. 2) C. W. Tittle, Technical Bulletin no. 8 (1960) (Nuclear Chicago Corporation, Des Plaines, Illinois). 3) T. Baltakmens, Nucl. Instr. and Meth. 82 (1970) 264. 4) L. T. Dillman, Health Phys. 19, no. 3 (1970) 385. 5) G. C. Snyman and C. G. Clayton, Intern. J. Appl. Radiation Isotopes 14, no. 4 (1963) 186. 6) B. Owen, Phys. Med. Biol. 18, no. 3 (1973) 355.
N U C L E A R I N S T R U M E N T S AND METHODS
I69-I7I;
© NORTH-HOLLAND
P U B L I S H I N G CO.
ENERGY LOSS OF BETA PARTICLES ON BACKSCATTERING*
T. BALTAKMENS New Zealand Department of Health, National Radiation Laboratory, Christchurch, New Zealand Received 31 December 1974 An empirical equation for calculating the ettective maximum and average energy of beta particles after backscattering is derived from their absorption characteristics. For a beta emitter of maximum energy Em and average energy E, the effective maximum and average energies after scattering,
In the numerous practical applications of betaemitting radionuclides, their scattering characteristics are frequently of interest. One of these characteristics * Published with the authority of the Director-General of Health. 100
lOO
10
lO
1
i
1500
i
4
i
i
i
1200 i
I_
lOO
lOO
r o (D
10
1c
1
1500 Absorber
thickness
4
.1 (rag
12j00
Es and E's are given by: Es/Em = (158 +Z)/251,
Fig. 1. Absorption curves for incident (upper) and backscattered (lower) betas for four emitters with aluminium absorbers and a copper backscatterer.
169
Es/E=(123+Z)/216,
where Z is the atomic number of the scattering material.
is the energy loss which occurs when beta particles are scattered by various materials. Due to the complex nature of the energy spectra of beta emitters, theoretical calculations and direct measurements of the energy loss are difficult. This paper presents a simple empirical relationship which may be useful in determining the energy loss of beta particles on backscattering, and which can be derived from their absorption characteristics. It is a well-known empirical observation that the absorption in metals of both incident and backscattered beta particles is exponential over a large range of absorber thicknesses 1'2) and can be expressed as: C = Co e -"x
(incident),
C=
(backscattered),
Co e-Ix
where C is the counting rate produced by a beta emitter in a beta counter through an absorber of thickness x; Co is the counting rate without absorber; and n and l are the respective absorption coefficients for incident and backscattered beta particles. Fig. 1 shows typical absorption curves for the incident (upper curve) and backscattered (lower curve) beta particles from four emitters obtained with a low-background anticoincidence beta counter. The scattering material is copper of sufficient thickness to give saturation backscattering at diffuse incidence, the absorber is aluminium. It is also observed empirically that the absorption coefficient for a given emitter is related to its maximum energy, Era, by an expression of the type: n =
cr~ 2)
and;
KErn",
where K and a are constants3). A similar relationship holds also for backscattered betas, and fig. 2 shows the relationship between E m and n (line A), and E m and l
170
T. B A L T A K M E N S
(line B) for the following emitters: 2°4T1, E m = 0 . 7 6 MeV; 2]°Bi, E r a = 1.16 M e V ; 89Sr, Em = 1.45 MeV; 90y, Em = 2.27 MeV; 234mpa, E m = 2.29 MeV; la4pr, E m = 2 . 9 8 MeV; l°6Rh, E m = 3.53 MeV (68 %). These results were obtained using the same counter as above, with aluminium absorbers and copper as backscattering material. F o r each emitter, l > n , and we can regard the absorption characteristics o f backscattered betas from a given emitter as being very nearly identical to those o f incident betas from another emitter o f correspondingly lower m a x i m u m energy. Thus we can use the ratio n to l as a measure o f the loss in E m on backscattering. Consider a beta emitter with m a x i m u m energy E m and absorption coefficients n and l. Let the effective m a x i m u m energy after scattering be Es. Then, referring
Eml
to the diagram at the top o f fig. 2, it is obvious that: n
= K AEm a
I
= K a E ~ ~ = K A E ~ ~,
.'.FI/I
=
(Em/Es)
-a ,
or: E s / E m = ( n i l ) '/a.
It has been found that the relationship between n and l can be approximately given by: 1/n = C / ( D + Z),
where Z is the atomic number of the backscatterer, and C and D are constants2). Hence: =
- -
.
(1)
F r o m fig. 2, 1/a = 0.75; suitable values for the other constants are C = 200, and D - - 1 0 7 . If the function on the right-hand side of eq. (1) is plotted against Z, the resulting graphical relationship is very nearly linear (fig. 3). Thus the expression for the effective m a x i m u m energy can be simplified to:
2 I
E s / E m = (158 + Z)/251.
L~
0"5
(2)
The actual loss in m a x i m u m energy on backscattering, dEm, is given by: i
i
~
i i ~rl
i
3 10 Absorption coefficient
i
I
30
r
50
(cm2g1)
dE m = Em-Es,
and the fractional loss by: Fig. 2. Relationship between absorption coefficient and maximum energy for incident (A) and backscattered (B) betas; and between absorption coefficient and average energy for incident (A') and backscattered (B') betas. 1.o
0.8
0.6 I
o
I
I
I
t
50 z
I
I
I
I
I
1oo
Fig. 3. Ratio of maximum and average energy to effective maximum and average energy after backscattering as a function of the atomic number of the backscatterer.
d E m / E m = 1 - (Es/Em) = ( 9 3 - Z ) / 2 5 1 .
It is interesting to note that d E m / E m = 0 for Z = 93, which implies that the energy loss tends to zero if the scatterer has a value o f Z near the end o f the periodic table. The loss in the average energy per beta disintegration, E (as distinct from the loss in m a x i m u m energy, Era), can be calculated in a similar way. In general, the ratio E / E m for a given beta emitter depends on Era, on the type o f transition, and on the atomic n u m b e r o f the emitter4). W h e n beta particles are scattered, there is not only a loss in energy, but also a change in the shape o f the energy spectrum, which tends to be shifted towards the low-energy end with a corresponding decrease in the proportion of high-energy betas in the spectrum s.6). Thus we m a y expect that the fractional loss in E on scattering is not the same as that in E m. The relationship between E and (n, l) is similar to that between Em
171
E N E R G Y LOSS OF BETA P A R T I C L E S
and (n,/): lines A' and B' in fig. 2. A similar derivation as for Em gives:
for values of Z from 1 to 93. Also:
dE/E = 1.16dEm/Em, with 1/b = 0.88 from fig. 2. Again, using fig. 3, the above expression simplifies to:
£s/E = (123 + Z)/216,
i.e., for any emitter the fractional loss in the average energy is greater than that in the maximum energy for the above values of Z.
(3)
hence:
References
dE/E = (93-Z)/216. Eqs. (2) and (3) give two simple linear relationships which can be used to calculate the effective maximum and average energies after backscattering for any combination of emitter and scattering material. In general the energy loss decreases with increase in Z, and: Es/E m >
£s/E,
1) K. Siegbahn, ed., Alpha-, beta- and gamma-ray spectroscopy (North-Holland Publishing Company, Amsterdam, 1965) p. 21-24. 2) C. W. Tittle, Technical Bulletin no. 8 (1960) (Nuclear Chicago Corporation, Des Plaines, Illinois). 3) T. Baltakmens, Nucl. Instr. and Meth. 82 (1970) 264. 4) L. T. Dillman, Health Phys. 19, no. 3 (1970) 385. 5) G. C. Snyman and C. G. Clayton, Intern. J. Appl. Radiation Isotopes 14, no. 4 (1963) 186. 6) B. Owen, Phys. Med. Biol. 18, no. 3 (1973) 355.