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On measurements of disintegration rates of radioactive sources by coincidence methods
NUCLEAR
INSTRUMENTS
AND
ON MEASUREMENTS
METHODS
31 (1964)314-316;
OF DISINTEGRATION
0 NORTH-HOLLAND
PUBLISHING
BATES OF RADIOACTIVE
BY COINCIDENCE
co.
SOURCES
METHODS
I. S.PANASYUK I.V. Kurchatov Institute for Atomic Energy, Academy of Sciences, Moscow, USSR Received 21 May 1964
In the present paper the first steps have been made in the theoretical study of the influence of the spatial distribution of active atoms density and of the self-absorption of radiated particles
inside radioactive sources on the shape of theoretical formulas for computing absolute activities of the latter upon using the coincidence method.
In spite of rather wide use of measurements of absolute disintegration rates of radioactive sources by coincidence methods’ -‘) little attention has been paid up to the present time to the study of the influence of such characteristics, as the distribution of active atom density and to the self-absorption of radiated particles on final computation formulas. Let us consider the measurement of absolute activity of a sufficiently thick specimen by the double coincidence method and let us introduce the following variables: c(z) is the absolute specific activity of the specimen at position z (the number of disintegrations per sec. and per cm3 at z in the specimen); V is the volume of the specimen material; S is the absolute activity (disintegration rate) of the whole specimen:
taneous production of particles of all n kinds in the region of the specimen d V and in the energetic intervals
P
s=
JW)o(z)dV;
(1)
n representing the number of particles, emitted simultaneously in every event of radioactive decay; Ei is the energy of the ithparticle at the moment of its production
0’.&) N In2
N,(kl
pEimaX
JJhmill
=
+
At
a(z)Wk(E,)W,,(z,E,)dVdE,;
~(+K@,)~,~(z~E,) s V’,E,)
m2) = At
(2)
c(z) w,&) wX%) s V’,Eb%.J
.
N, = i Nkl;
(6’)
N,(l + 2) =
c
k=l
(3)
dV d&i
x w~~(z,E,)W,~(Z,E,)~E~~E,;
W,,(z,E,) representing the probability that the kth particle (k = 1,2,3.. .n) which was produced in the place of z specimen to be registered in the first detector; W,z(z,E,) is the same, but for the m’h particle (m = 1,2,3.. .n) in the second detector; c(z)dVW,(EJdE, is the frequency of production of particles of the ith kind with the energy in the interval Ei +- Ei + dEi in the region of the .specimen d Y; g(z) W,(E,) W,(E,) . . W,, (EJdVdE, . . dE,, is the frequency of cases of simul-
.
the number of registered pulses of the first detector at the expense of particles of mth kind, which got there for the time At; N,,,, is the same but with respect to the particles of mth kind, which got to the second detector; N, is the number of registered pulses of the first detector at the expense of the particles of all the n-kinds, which got to it in the time At; Nz is the same, but with respect to the second detector; N,(l + 2) is the number of coincidences of pulses in the first and in the second detectors when all possible particles with all possible energies for the time At get to it; N&k1 -F m2) is the same, but only at the expense of getting of the kth particles to the first detector and of the mth particles to the second detector. From such definitions of variables and conditions the laws of mathematical statistics give the following relations Nkl = At
due to radioactive disintegration (the energy of each of n emitted particle lies in the range from ETi” up to EF”“); Wi(Ei)dEiis the probability of production of the ith particle (i = 1,2,3 . . .n) in the energetic interval Ei f Ei + dEi on the conditions that Erin 5 Ei 5 Eyx and W,(E,)dE, = 1;
E,fE,+dE,;E~+E,+dE,...E,,iE,,+dE,,;N,,is
N, = ~ N,,; m=l
(4)
x
(5) (6)
n
N,(kl +- m2).
(7)
k=l;m=l;k#m
At the deduction of formulas for the calculation of absolute activity of a specimen by the double coincidence method it is possible to use two methods of
314
MEASUREMENTS
eliminations of different kinds of probabilities* Wil(Z, Ei) and Wiz(Z, Ei): 1) by taking4) the ratio
OF DISINTEGRATION
W,(Ei),
N,,Nm,
k=i;m=l:k+m
=
dtN,(l f 2)
NklNm2
i k=l;m=l;k#m I
=
At
N&l
2
f m2)
k=l;m=l;k#m
However the analysis of expressions (I)-(7) shows, that there is a highly limited number of particular cases of specimen characteristics, when with the help of the expressions (8) and (9) it is possible to make an elimination of the above-mentioned probabilities. One of such particular cases of specimen characteristics was used in the paper4) and consists in the presentation of the probabilities I+$,(& E,) and W,,&, E,,,) as : &(z,&) Wmz(z,E,)
=
wk,
=
W,,
=
o,&&,
’ = AtN,(l
f 2)
= o,e&:ad
*(I - A) = S*(l - A),
Thus if the conditions (10) and (11) are fulfilled which means that there is practically no self-absorption of particles inside the specimens registered by their detectors in the coincidence circuits, then in this particular case there is an opportunity to use the main advantage of the coincidence method - to exclude from the computation formulas the probabilities Wil(z, Et) and l+‘iz(z,Ei). The other particular case of specimen characteristics, when in the calculation formulas the above-mentioned probabilities are also excluded, is the following one. Let us imagine, that every decay event of thick enough measured specimen would lead to the production of two particles, of which one (for example k = 1) is practically not at all absorbed in this specimen, while the second one (m = 2) is absorbed noticables (up to full absorption in the specimen, if it is producing deep enough). In this case in the expressions (3) and (5) the probability of one of the types, viz. W, ,(z,E,) and W1,(z, E,) cant be independent of z providing the possibility to represent (3), (4) and (5) as follows
(11)
N,, = At N,,
= At
J
3)
(15)
= AtS
WdE,)WdE,)dE,; J (El)
(16) (17)
o(z)W~(E,)W~~(~,E,)~V~E,;
(19
V'JW
N,(ll f 22) = At (I,,
) KW~IWI) 1
x
b(z)WZ(E2)WZ2(z,EZ)dVdEZ ; (19) O’,.W
(
W,&)W@,)d&; s (El)
n
X
A=
= AtS
Otz)Wz(E,)Wzl(z,E,)dVdE,; s V’G%)
(12)
where
2) by utilising5,6) the ratio (9):
-.
N,(kl f m2)
(10)
where ol,wZ are relative solid angles (Q/47r) formed by the effective centre of the specimen and by active parts of the detectors 1 and 2, correspondingly; e;, e: are the sensitivities of the detector 1 to particles of the k”’ kind and of detector 2 to particles of the mth kind; ai,& are the coefficients of beam attenuation of particles of the kth and mth kinds in the walls of the detectors and in other absorbers between the specimen and the detectors 1 and 2. Substituting (lo), (11) in (8) and (9) and taking into account (1) and (2), we find, that providing enough thin specimen (otherwise it is impossible to fulfil conditions (10) and (11)): 1) by utilising4) the ratio (8) is:
N,N,
2 k=l;m=l;k#m
+ 2) ’
n
S” =
”
At
2) by taking5) the ratio
c
k=,;,$l;k+,,,Nk1N (14)
s=
N,N, ” = dtN,(l
315
RATES
>
* These probabilities can hardly be calculated and measured in practice. t So that in the examined particular case Wrt(z, El) and Wrs(z, El) should not really depend on z it is necessary to secure such geometric conditions, which would permit equally probable hit of particles from any place z of specimen into the active volume both of the first and second detector.
I.
316
S. PANASYUK
where N,(kl f m2) at n = 2. Substituting (15), (18) and (19) either directly to (14) or to (12) and (13) with due account of the fact that N,=N,,; N2=NZ2; e;‘=O; NC(1t2)=NC(11f22) and e; = 0 we obtain the same result in both the cases viz: NIN, NIINZZ (21) ’ = dtN,(lli22) = dtN,(l -+ 2) ’ Substituting (15) + (18) and (20) in (14) we find that S = NIINZZ + NI,NZI
(22)
Thus, if inside the specimen, measured by the double coincidence method there is practically no absorption of one of the registered particle kinds, then even essential absorption in this specimen of particles of the second registered kind would not influence the shape of the computation formula of the type (21) and (22). Some particular cases of specimens characteristics were considered (for example the specimen having g(z) = constant, but absorbing both kinds of registered radiation and so on), however there was no cases which, as the both mentioned above, would provide the possibility to exclude the complicated coefficients Wi(Ei), Wil(z,Ej) and Wiz(Z,Ei) in the computation formulas. The author is particularly grateful to A. A. Markov for his participation in the discussions of the problems here involved. References 1)K. Geiger and A. Werner, Z. f. Phys., 21 (1924) 187. 2) J. V. Dunworth, Rev. Sci. Instr., 11 (1940) 167. 3) M. L. Wiedenbeck, Phys. Rev., 72 (1947) 974. 4) J. Bamothy, M. Forro, Rev. Sci. Instr., 22 (1957) 415. 5) I. S. Panasyuk, Pribori i Technika Experimenta, 2 (1962) 63. 6) I. S. Panasyuk, Pribori i Technika Experimenta, 2 (1963) 66. 7) D. E. Vaughan, Nucl. Instr. and Meth., 22 (1963) 186.
INSTRUMENTS
AND
ON MEASUREMENTS
METHODS
31 (1964)314-316;
OF DISINTEGRATION
0 NORTH-HOLLAND
PUBLISHING
BATES OF RADIOACTIVE
BY COINCIDENCE
co.
SOURCES
METHODS
I. S.PANASYUK I.V. Kurchatov Institute for Atomic Energy, Academy of Sciences, Moscow, USSR Received 21 May 1964
In the present paper the first steps have been made in the theoretical study of the influence of the spatial distribution of active atoms density and of the self-absorption of radiated particles
inside radioactive sources on the shape of theoretical formulas for computing absolute activities of the latter upon using the coincidence method.
In spite of rather wide use of measurements of absolute disintegration rates of radioactive sources by coincidence methods’ -‘) little attention has been paid up to the present time to the study of the influence of such characteristics, as the distribution of active atom density and to the self-absorption of radiated particles on final computation formulas. Let us consider the measurement of absolute activity of a sufficiently thick specimen by the double coincidence method and let us introduce the following variables: c(z) is the absolute specific activity of the specimen at position z (the number of disintegrations per sec. and per cm3 at z in the specimen); V is the volume of the specimen material; S is the absolute activity (disintegration rate) of the whole specimen:
taneous production of particles of all n kinds in the region of the specimen d V and in the energetic intervals
P
s=
JW)o(z)dV;
(1)
n representing the number of particles, emitted simultaneously in every event of radioactive decay; Ei is the energy of the ithparticle at the moment of its production
0’.&) N In2
N,(kl
pEimaX
JJhmill
=
+
At
a(z)Wk(E,)W,,(z,E,)dVdE,;
~(+K@,)~,~(z~E,) s V’,E,)
m2) = At
(2)
c(z) w,&) wX%) s V’,Eb%.J
.
N, = i Nkl;
(6’)
N,(l + 2) =
c
k=l
(3)
dV d&i
x w~~(z,E,)W,~(Z,E,)~E~~E,;
W,,(z,E,) representing the probability that the kth particle (k = 1,2,3.. .n) which was produced in the place of z specimen to be registered in the first detector; W,z(z,E,) is the same, but for the m’h particle (m = 1,2,3.. .n) in the second detector; c(z)dVW,(EJdE, is the frequency of production of particles of the ith kind with the energy in the interval Ei +- Ei + dEi in the region of the .specimen d Y; g(z) W,(E,) W,(E,) . . W,, (EJdVdE, . . dE,, is the frequency of cases of simul-
.
the number of registered pulses of the first detector at the expense of particles of mth kind, which got there for the time At; N,,,, is the same but with respect to the particles of mth kind, which got to the second detector; N, is the number of registered pulses of the first detector at the expense of the particles of all the n-kinds, which got to it in the time At; Nz is the same, but with respect to the second detector; N,(l + 2) is the number of coincidences of pulses in the first and in the second detectors when all possible particles with all possible energies for the time At get to it; N&k1 -F m2) is the same, but only at the expense of getting of the kth particles to the first detector and of the mth particles to the second detector. From such definitions of variables and conditions the laws of mathematical statistics give the following relations Nkl = At
due to radioactive disintegration (the energy of each of n emitted particle lies in the range from ETi” up to EF”“); Wi(Ei)dEiis the probability of production of the ith particle (i = 1,2,3 . . .n) in the energetic interval Ei f Ei + dEi on the conditions that Erin 5 Ei 5 Eyx and W,(E,)dE, = 1;
E,fE,+dE,;E~+E,+dE,...E,,iE,,+dE,,;N,,is
N, = ~ N,,; m=l
(4)
x
(5) (6)
n
N,(kl +- m2).
(7)
k=l;m=l;k#m
At the deduction of formulas for the calculation of absolute activity of a specimen by the double coincidence method it is possible to use two methods of
314
MEASUREMENTS
eliminations of different kinds of probabilities* Wil(Z, Ei) and Wiz(Z, Ei): 1) by taking4) the ratio
OF DISINTEGRATION
W,(Ei),
N,,Nm,
k=i;m=l:k+m
=
dtN,(l f 2)
NklNm2
i k=l;m=l;k#m I
=
At
N&l
2
f m2)
k=l;m=l;k#m
However the analysis of expressions (I)-(7) shows, that there is a highly limited number of particular cases of specimen characteristics, when with the help of the expressions (8) and (9) it is possible to make an elimination of the above-mentioned probabilities. One of such particular cases of specimen characteristics was used in the paper4) and consists in the presentation of the probabilities I+$,(& E,) and W,,&, E,,,) as : &(z,&) Wmz(z,E,)
=
wk,
=
W,,
=
o,&&,
’ = AtN,(l
f 2)
= o,e&:ad
*(I - A) = S*(l - A),
Thus if the conditions (10) and (11) are fulfilled which means that there is practically no self-absorption of particles inside the specimens registered by their detectors in the coincidence circuits, then in this particular case there is an opportunity to use the main advantage of the coincidence method - to exclude from the computation formulas the probabilities Wil(z, Et) and l+‘iz(z,Ei). The other particular case of specimen characteristics, when in the calculation formulas the above-mentioned probabilities are also excluded, is the following one. Let us imagine, that every decay event of thick enough measured specimen would lead to the production of two particles, of which one (for example k = 1) is practically not at all absorbed in this specimen, while the second one (m = 2) is absorbed noticables (up to full absorption in the specimen, if it is producing deep enough). In this case in the expressions (3) and (5) the probability of one of the types, viz. W, ,(z,E,) and W1,(z, E,) cant be independent of z providing the possibility to represent (3), (4) and (5) as follows
(11)
N,, = At N,,
= At
J
3)
(15)
= AtS
WdE,)WdE,)dE,; J (El)
(16) (17)
o(z)W~(E,)W~~(~,E,)~V~E,;
(19
V'JW
N,(ll f 22) = At (I,,
) KW~IWI) 1
x
b(z)WZ(E2)WZ2(z,EZ)dVdEZ ; (19) O’,.W
(
W,&)W@,)d&; s (El)
n
X
A=
= AtS
Otz)Wz(E,)Wzl(z,E,)dVdE,; s V’G%)
(12)
where
2) by utilising5,6) the ratio (9):
-.
N,(kl f m2)
(10)
where ol,wZ are relative solid angles (Q/47r) formed by the effective centre of the specimen and by active parts of the detectors 1 and 2, correspondingly; e;, e: are the sensitivities of the detector 1 to particles of the k”’ kind and of detector 2 to particles of the mth kind; ai,& are the coefficients of beam attenuation of particles of the kth and mth kinds in the walls of the detectors and in other absorbers between the specimen and the detectors 1 and 2. Substituting (lo), (11) in (8) and (9) and taking into account (1) and (2), we find, that providing enough thin specimen (otherwise it is impossible to fulfil conditions (10) and (11)): 1) by utilising4) the ratio (8) is:
N,N,
2 k=l;m=l;k#m
+ 2) ’
n
S” =
”
At
2) by taking5) the ratio
c
k=,;,$l;k+,,,Nk1N (14)
s=
N,N, ” = dtN,(l
315
RATES
>
* These probabilities can hardly be calculated and measured in practice. t So that in the examined particular case Wrt(z, El) and Wrs(z, El) should not really depend on z it is necessary to secure such geometric conditions, which would permit equally probable hit of particles from any place z of specimen into the active volume both of the first and second detector.
I.
316
S. PANASYUK
where N,(kl f m2) at n = 2. Substituting (15), (18) and (19) either directly to (14) or to (12) and (13) with due account of the fact that N,=N,,; N2=NZ2; e;‘=O; NC(1t2)=NC(11f22) and e; = 0 we obtain the same result in both the cases viz: NIN, NIINZZ (21) ’ = dtN,(lli22) = dtN,(l -+ 2) ’ Substituting (15) + (18) and (20) in (14) we find that S = NIINZZ + NI,NZI
(22)
Thus, if inside the specimen, measured by the double coincidence method there is practically no absorption of one of the registered particle kinds, then even essential absorption in this specimen of particles of the second registered kind would not influence the shape of the computation formula of the type (21) and (22). Some particular cases of specimens characteristics were considered (for example the specimen having g(z) = constant, but absorbing both kinds of registered radiation and so on), however there was no cases which, as the both mentioned above, would provide the possibility to exclude the complicated coefficients Wi(Ei), Wil(z,Ej) and Wiz(Z,Ei) in the computation formulas. The author is particularly grateful to A. A. Markov for his participation in the discussions of the problems here involved. References 1)K. Geiger and A. Werner, Z. f. Phys., 21 (1924) 187. 2) J. V. Dunworth, Rev. Sci. Instr., 11 (1940) 167. 3) M. L. Wiedenbeck, Phys. Rev., 72 (1947) 974. 4) J. Bamothy, M. Forro, Rev. Sci. Instr., 22 (1957) 415. 5) I. S. Panasyuk, Pribori i Technika Experimenta, 2 (1962) 63. 6) I. S. Panasyuk, Pribori i Technika Experimenta, 2 (1963) 66. 7) D. E. Vaughan, Nucl. Instr. and Meth., 22 (1963) 186.