- Email: [email protected]
Theoretical considerations for standardization of 125I by the coincidence method
NUCLEAR
INSTRUMENTS
AND
METHODS
t2 4
(I975) 585-589; ©
NORTH-HOLLAND
PUBLISHING
CO.
T H E O R E T I C A L C O N S I D E R A T I O N S FOR S T A N D A R D I Z A T I O N OF lzsI BY T H E C O I N C I D E N C E M E T H O D DONALD
L. H O R R O C K S
and P A U L R. K L E I N
Scientific Instruments Division, Beckman Instruments, Inc., [trine, CA 92644, U.S.A. Received 12 A u g u s t 1974 a n d in revised f o r m 24 D e c e m b e r 1974 T h e decay scheme o f 12aI is analyzed. T h e probabilities o f emission are calculated for each o f the several p a t h w a y s o f 125I decay. T h e equation for calculation o f the source decay rate by the coincidence m e t h o d is derived f r o m these probabilities.
1. Introduction
2. General theory
The dramatic increase in the use of 1251 as a radiotracer, especially in labeling specific antigens for radioimmunoassay (RIA), has prompted a closer look at methods of standardizing solutions of 12sI. Several excellent papers 2-4) have been published on the standardization of ~2sI by the coincidence method. Most of these papers have used the same general equation for calculation of absolute disintegration rate:
Some radionuclides decay by the emission of two coincident gamma rays. The absolute decay rate of these radionuclides can be determined by a measurement made with a single NaI(T1) crystal detector or two NaI(T1) crystal detectors in coincidence. The measurement of the ratio of the number of non-coincident (single) gamma-ray photopeak events relative to the number of coincident (sum)photopeak events can be used to calculate the absolute decay rate. Certain corrections may be required depending upon the probabilities of emission of gamma rays for the given transitions. If two gamma rays, 71 and Y2, are produced by every decay of the radionuclides, then the photopeak intensities which are measured by a counting system, can be designated: Nsl: the photopeak count rate of 71, which is not coincident with 72 ; Ns~: the photopeak count rate of 72 which is not coincident with 71 ; No: the photopeak count rate when 71 and 72 are detected coincidently; el : the photopeak counting efficiency of 71 in the counting system; e2: the photopeak counting efficiency of 72 in the counting system; S: the absolute decay rate of the radionuclide.
S -
P1 P2 (P1 q- P 2 ) 2
(Ns + 2N°)2, Nc
(1)
where S is the absolute decay rate of the radionuclide, Ns is the singles photopeak counts, No is the coincidence photopeak counts, and P1 and P2 are the probabilities of emission from the two parts of the decay process. If the value of the ratio P J P : is unity then eq. (l) reduces to: S = (Ns + 2 No) 2 4N,
(2)
Even if this ratio is as much as 20% different from unity, only about a 1% error will be made in the value of S by assuming that the ratio is unity. For P1/P2 equals 1.0, 0.8, and 1.2, we get 1/4.00, 1/4.05, and 1/4.03, respectively, for the expression: P1P2/ (P1 + P2) 2. In this paper a more rigorous, complete derivation of the equation for calculation of the value of S is developed. The same final equation as eq. (2) is obtained, but the technique for obtaining the final equation shows all the steps involved in obtaining it without any general assumptions about the value of P,/P2. This technique will be useful in development of equations for calculation of S for radionuclides by the coincidence method for which the value of P1/P2 differs markedly from unity.
585
Using these definitions, a set of equations can be written to relate the observed count rates and the absolute decay rate, S, as follows: Nsl = e a S - el e2S,
(3)
N s 2 = /~2S - - e 1 g2 S ,
(4)
No = ~l z2S ,
(5)
where first terms of eqs. (3) and (4) are the detection
586
D. L. H O R R O C K S
probability of Yl and 72, respectively. The second terms of eq. (3) and (4) are necessary for correction of the observed single rate for those gamma rays which are detected coincidently with the corresponding other gamma ray. The total singles rate is given as the sum of eqs. (3) and (4): Ns, + Ns2 = elS(1 - e 2 ) +
~;2S(1--51).
(6)
If the two gamma rays are of the same or nearly equal energy, their counting efficiencies, el and e2, will be equal. If the efficiencies can be assumed to be equal, then eq. (6) becomes: Ns = 2 e S ( 1 - e ) ,
(7)
and eq. (5) becomes: Nc = e2 S.
(8)
The ratio, R, is defined as: R -- N~ _
Ns
e2S
_
2eS(1 - e )
e
(9)
2(1 - e ) '
and shows R to be independent of the S. Solving eq. (9) for e gives: =
2R - l+2R"
(IO)
Substituting eq. (10) into eq. (8) and solving for S.gives: s
=
N~ =
e2
No(I
+2R) 2
(11)
(2R) 2
Thus measuring Nc and Ns is all that is necessary for the calculation of the absolute source strength, S. Upon substitution for R, eq. (2) is obtained. 3. A p p l i c a t i o n o f t h e o r y to 1251 d e c a y
Unfortunately ~2sI does not emit two gamma or X-rays for every decay. The decay scheme 5) for 125I is: 125I (60 d)
E.C. (100%)
0.0355
taSTe (stable)
A N D P. R. K L E I N
with E.C.(L + M +...)/E.C.(K) - 0.25, meaning that 80% of the electron-capture (E.C.) events produce vacancies in the K shell of the 125Te daughter nucleus. The remaining E.C. events produce vacancies in either the L, M, or higher shell of the aZSTe daughter nucleus. Vacancies in L, M, or higher shells are treated as nonphoton-producing events even though they do produce (at least part of the time) L, M, and lower-energy X rays. However, these low-energy X rays are not able to penetrate the detector housing (0.01" A1) and thus have a zero counting efficiency. Of the E.C. events which lead to K capture only part of the vacancies will result in the production of K X rays of Te. In the remaining fraction of events the excess energy will be released by the production of a series of low-energy electrons, called Auger electrons. The probability of X ray emission from a K vacancy is called the K-fluorescence yield, coK. For Te, the value of OJK is 0.86. Thus 14% of the E.C.(K) events will not produce a Te K X ray. Overall the traction of the 125I E.C. events which produce Te K X rays is equal to: (0.80) (0.86) = 0.688. The 35.5 keV excited state of 125Te which results from the E.C. of ~2sI releases its excess energy by several pathways. Part of the time a gamma ray is emitted, but more often the energy is transferred to an orbital electron which is ejected (called conversionelectron process) leaving a vacancy in the electron shell. The ratio of electron emission to gamma-ray emission can be calculated from the nuclear dataS): eK/7 =
ll,
and: K/L
= 7.
Thus 7/8 or 0.875 of the electron conversions occur with the ejection of a K electron. Therefore, the total electron ejection probability is: (11/0.875)/(1 + 11/0.875) = 12.6/13.6 =0.93. The non-electron-conversion events result in the release of the 35.5 keV gamma ray. This occurs in 7 % of the decay events. Those events which result in conversion in other than the K shell will not be detailed because, even when the X rays are produced, there is almost zero probability that the low-energy X rays will penetrate the detector housing. Of those events which produce vacancies in the K shell the probabilities of X-ray emission are the same as after E.C. in the K shell. Thus the Te K X ray will be produced by the decay of the 35.5 keV level
STANDARDIZATION
OF 1 2 5 I B Y T H E C O I N C I D E N C E
587
METHOD
125I DECAY PATHWAYS PROBABILITY
(0.86) (0.88) (0.93) (0.86) (0.80)
ENERGY keY
-~ 0.484
55.0
0.079
27.5
~
0.077
27.5
=
0.048
63.0
=
0.008
35.5
~
0.079
27,5
--
0.013
00.0
--
0.012
00.0
--
0.141
27.5
K-X-RAY /0.86
( K 0.88 /
AUGER 0.14 ELECTRON\ 0 14
k ( " ) (0.88) (0.93) (0.86) (0.80)
/ L 0.12
I.C. 0.93
~K-X-RAY 0.86
(0.12) (0.93) (0.86) (0.80)
]/0,07 (0.07) (0.86) (0.80)
E.C. fK) 0.80
1251
/
ELECTRONS
\,
0.14
/
(0.07) (0.14) (0.80)
7 0.07 (0.86) (0.88) (0.93) (0.14) (0.80)
/ K-X-RAY 0.86 I.C. 0.93 ~-
( K< / AUGER 0"14 0.88 ELECTRONS (0.14) (0.88) (0.93) (0.14) (0.80)
L0.12 k
(0.12)(0.93)(0.14)(0.80)
E.C. 1 (L+M+ . . . . ) 0.20
(0.86) (0.88) (0.93) (0.20)
/ K-X-RAY
l ,
I.C. 0.93
(
0.86
K 0.88 AUGER 0.14 < ELECTRONS k
L 0.12 X ~
=0.023
00.0
z
0.022
00.0
=
0.014
35.5
(0.14) (0.88) (0.93) (0,20) (0.12) (0.93) (0.20)
Y 0.07
\
(0.07) (0.20)
TOTAL 1.000 Fig. 1. P a t h w a y s and probabilities for a~sI decay a s s u m i n g no detection o f L, M, or etc., X rays of Te.
588
D. L
H O R R O C K S A N D P. R. K L E I N TABLE 1
in the following fraction of the decays:
125I decay pathways. Relative probabilities o f the radiation of three energy groups.
(0.93) (0.875) (0.86) = 0.70. Therefore the probability of emission of either as Te K X ray or a gamma ray from the 35.5 keV excited state is (0.70 + 0.07) or 0.77. The probability of emission of a Te K X ray following E.C. is 0.688. Thus the ratio of probabilities is: P1
0.688 -- 0.894. 0.77
-
P2
Energy (keV)
Probabilities
0
0.013 0.012 0.070 0.023 0.022
27.5 or 35.5 a
0.079 0.077 0.008 0.398 0.079
(12)
While this ratio is not unity, it does not differ significantly from unity to introduce an appreciable error in the calculated values of S by eq. (2). Eq. (1) with P1/P2 = 1:
0.414 0.014 0.484" 0.048 0.532
55 or 63 a
S - (Ns;2N~)2 ; 4No Eq. (1) with
a The resolution of NaI(T1) crystals is not sufficient to be able to resolve the 27.5 and 35.5 keV radiations. Likewise, the sum peaks of 55 and 63 keV are not resolved.
Pt/P2 0.894: =
S -- ( N s + 2 N e ) 2
4.013 No Considering all the possible modes and combinations of the decay process, the probability chart of fig. 1 was constructed. Each solid line is a possible pathway for the decay process. Each part of the pathway is identified by the type of process and the probability of that process. The probability of a given pathway is the product of the probabilities of individual processes of that pathway. The total energy released is the sum of energies released in the individual processes. Again it should be pointed out that only Te K X rays and the 35.5 keV g a m m a rays are energetic enough to be detected by the NaI(T1) crystal which is inside an A1 housing. The radiation has to be energetic enough to penetrate the 0.010" thick AI window. The radiations can be put into three energy groups with their relative probabilities (see table 1). Frorn these figures it is seen that 7 % of the 125I decays gives rise to no measurable radiation, 39.8 % gives rise to a single detectable radiation (either 27.5 keV X ray or 35.5 keV g a m m a ray), and 53.2 % gives rise to two coincident radiations (either two X rays or an X ray plus the 35.5 keV y ray). Thus 46.8% of the 125I decays does not produce two coincident, measurable radiations. Assuming that the measurable strength, SM, is the sum of the single events Su and the doublet events SD gives: S M = S u -}- S D .
(13)
The number of singles measured is: Ns = 28(1--e)SD + eSu,
(14)
where the first term is a measure of those events which produce two coincident radiations but only one is measured. The coincidence count rate is: No = e2SD.
(15)
Using the definition for R [eq. (9)] gives: 1
N s _
R
N¢
2(l-e) - -
Su
+
~
(16)
eS D '
letting = S,/So,
(17)
1 _ __2(1-e) t- ~- = 2 ( 1 - e ) + c ~ R
e
~
(18)
e
Solving for e gives: (19) and substituting into eq. (15) gives:
s~ =
No = No I-(1 +2R)/R3~, (2 + .)~ JR/(1 + 2 R)3 ~ (2 +.)~
(20)
multiplying the numerator and denominator of eq. (20)
STANDARDIZATION
OF 12sI BY THE C O I N C I D E N C E M E T H O D
by 4 gives:
589
by: Nc [(1 + 2 S D ~.
g)/2R]2
[½(2 + ~)32
(21)
Thus the general eq. (2) is approximately valid because the corrective terms nearly cancel each other.
Substitution into eq. (13) gives: S M ~- S D - ] - ~ S D = SD(J_+OO,
s = (-~) sM :No\ (l+2R'~2(O' j 927x}\-O-..~3-d/" (27)
(22) 4. Conclusions
or;
No [(1
SM
+2R)/2R]z (1 +c 0
(233
[½(2 + ~)]2
From the flow chart of fig. 1 the values of S, = 0.398,
SD = 0.532,
give: = 0.398/0.532 = 0.748,
(24)
and: l+c~
= 0.927.
(25)
Thus: SM =
N {1 + 2 R ) 2
c \--~--~-j (0.927),
(26)
is the decay rate which involves either a single or two coincident radiations. However, only in 0.93 of the decays this does happen. The true decay rate is given
While the general equation used by previous investigations was accurate (at least within 1%) for the calculation of source strength by the coincidence method, the more detailed derivation of the equation, as shown here, seems to eliminate some reservations about the assumption of its validity. Thus the simple measurement of the relative number of counts produced in the single and sum (coincidence) photopeaks is all that is required to calculate the source strength of any radionuclide which has the proper decay characteristics. References 1) j. S. Eldridge and P. Crowther, Nucleonics 22 (6) (1964) 56. 2) H. Stephan, Isotopenpraxis 2 (1966) 161. a) j. G. V. Taylor, in Standardization o f radionuelides (IAEA, Vienna, 1967) p. 341. 4) L. Erbeszkron, L. Szokolyi, A. Szorenyi and K. Zsdanszky, Atomtech. Tajek. 11 (7) (1968) 358. 5) All data used to calculate conversion coefficients, energies, X-ray probabilities and other decay properties have been obtained from information published in C. M. Lederer, J. M. Hollander and I. Perlman, Table o f isotopes, 6th ed. (J. Wiley & Sons, New York, 1968).
INSTRUMENTS
AND
METHODS
t2 4
(I975) 585-589; ©
NORTH-HOLLAND
PUBLISHING
CO.
T H E O R E T I C A L C O N S I D E R A T I O N S FOR S T A N D A R D I Z A T I O N OF lzsI BY T H E C O I N C I D E N C E M E T H O D DONALD
L. H O R R O C K S
and P A U L R. K L E I N
Scientific Instruments Division, Beckman Instruments, Inc., [trine, CA 92644, U.S.A. Received 12 A u g u s t 1974 a n d in revised f o r m 24 D e c e m b e r 1974 T h e decay scheme o f 12aI is analyzed. T h e probabilities o f emission are calculated for each o f the several p a t h w a y s o f 125I decay. T h e equation for calculation o f the source decay rate by the coincidence m e t h o d is derived f r o m these probabilities.
1. Introduction
2. General theory
The dramatic increase in the use of 1251 as a radiotracer, especially in labeling specific antigens for radioimmunoassay (RIA), has prompted a closer look at methods of standardizing solutions of 12sI. Several excellent papers 2-4) have been published on the standardization of ~2sI by the coincidence method. Most of these papers have used the same general equation for calculation of absolute disintegration rate:
Some radionuclides decay by the emission of two coincident gamma rays. The absolute decay rate of these radionuclides can be determined by a measurement made with a single NaI(T1) crystal detector or two NaI(T1) crystal detectors in coincidence. The measurement of the ratio of the number of non-coincident (single) gamma-ray photopeak events relative to the number of coincident (sum)photopeak events can be used to calculate the absolute decay rate. Certain corrections may be required depending upon the probabilities of emission of gamma rays for the given transitions. If two gamma rays, 71 and Y2, are produced by every decay of the radionuclides, then the photopeak intensities which are measured by a counting system, can be designated: Nsl: the photopeak count rate of 71, which is not coincident with 72 ; Ns~: the photopeak count rate of 72 which is not coincident with 71 ; No: the photopeak count rate when 71 and 72 are detected coincidently; el : the photopeak counting efficiency of 71 in the counting system; e2: the photopeak counting efficiency of 72 in the counting system; S: the absolute decay rate of the radionuclide.
S -
P1 P2 (P1 q- P 2 ) 2
(Ns + 2N°)2, Nc
(1)
where S is the absolute decay rate of the radionuclide, Ns is the singles photopeak counts, No is the coincidence photopeak counts, and P1 and P2 are the probabilities of emission from the two parts of the decay process. If the value of the ratio P J P : is unity then eq. (l) reduces to: S = (Ns + 2 No) 2 4N,
(2)
Even if this ratio is as much as 20% different from unity, only about a 1% error will be made in the value of S by assuming that the ratio is unity. For P1/P2 equals 1.0, 0.8, and 1.2, we get 1/4.00, 1/4.05, and 1/4.03, respectively, for the expression: P1P2/ (P1 + P2) 2. In this paper a more rigorous, complete derivation of the equation for calculation of the value of S is developed. The same final equation as eq. (2) is obtained, but the technique for obtaining the final equation shows all the steps involved in obtaining it without any general assumptions about the value of P,/P2. This technique will be useful in development of equations for calculation of S for radionuclides by the coincidence method for which the value of P1/P2 differs markedly from unity.
585
Using these definitions, a set of equations can be written to relate the observed count rates and the absolute decay rate, S, as follows: Nsl = e a S - el e2S,
(3)
N s 2 = /~2S - - e 1 g2 S ,
(4)
No = ~l z2S ,
(5)
where first terms of eqs. (3) and (4) are the detection
586
D. L. H O R R O C K S
probability of Yl and 72, respectively. The second terms of eq. (3) and (4) are necessary for correction of the observed single rate for those gamma rays which are detected coincidently with the corresponding other gamma ray. The total singles rate is given as the sum of eqs. (3) and (4): Ns, + Ns2 = elS(1 - e 2 ) +
~;2S(1--51).
(6)
If the two gamma rays are of the same or nearly equal energy, their counting efficiencies, el and e2, will be equal. If the efficiencies can be assumed to be equal, then eq. (6) becomes: Ns = 2 e S ( 1 - e ) ,
(7)
and eq. (5) becomes: Nc = e2 S.
(8)
The ratio, R, is defined as: R -- N~ _
Ns
e2S
_
2eS(1 - e )
e
(9)
2(1 - e ) '
and shows R to be independent of the S. Solving eq. (9) for e gives: =
2R - l+2R"
(IO)
Substituting eq. (10) into eq. (8) and solving for S.gives: s
=
N~ =
e2
No(I
+2R) 2
(11)
(2R) 2
Thus measuring Nc and Ns is all that is necessary for the calculation of the absolute source strength, S. Upon substitution for R, eq. (2) is obtained. 3. A p p l i c a t i o n o f t h e o r y to 1251 d e c a y
Unfortunately ~2sI does not emit two gamma or X-rays for every decay. The decay scheme 5) for 125I is: 125I (60 d)
E.C. (100%)
0.0355
taSTe (stable)
A N D P. R. K L E I N
with E.C.(L + M +...)/E.C.(K) - 0.25, meaning that 80% of the electron-capture (E.C.) events produce vacancies in the K shell of the 125Te daughter nucleus. The remaining E.C. events produce vacancies in either the L, M, or higher shell of the aZSTe daughter nucleus. Vacancies in L, M, or higher shells are treated as nonphoton-producing events even though they do produce (at least part of the time) L, M, and lower-energy X rays. However, these low-energy X rays are not able to penetrate the detector housing (0.01" A1) and thus have a zero counting efficiency. Of the E.C. events which lead to K capture only part of the vacancies will result in the production of K X rays of Te. In the remaining fraction of events the excess energy will be released by the production of a series of low-energy electrons, called Auger electrons. The probability of X ray emission from a K vacancy is called the K-fluorescence yield, coK. For Te, the value of OJK is 0.86. Thus 14% of the E.C.(K) events will not produce a Te K X ray. Overall the traction of the 125I E.C. events which produce Te K X rays is equal to: (0.80) (0.86) = 0.688. The 35.5 keV excited state of 125Te which results from the E.C. of ~2sI releases its excess energy by several pathways. Part of the time a gamma ray is emitted, but more often the energy is transferred to an orbital electron which is ejected (called conversionelectron process) leaving a vacancy in the electron shell. The ratio of electron emission to gamma-ray emission can be calculated from the nuclear dataS): eK/7 =
ll,
and: K/L
= 7.
Thus 7/8 or 0.875 of the electron conversions occur with the ejection of a K electron. Therefore, the total electron ejection probability is: (11/0.875)/(1 + 11/0.875) = 12.6/13.6 =0.93. The non-electron-conversion events result in the release of the 35.5 keV gamma ray. This occurs in 7 % of the decay events. Those events which result in conversion in other than the K shell will not be detailed because, even when the X rays are produced, there is almost zero probability that the low-energy X rays will penetrate the detector housing. Of those events which produce vacancies in the K shell the probabilities of X-ray emission are the same as after E.C. in the K shell. Thus the Te K X ray will be produced by the decay of the 35.5 keV level
STANDARDIZATION
OF 1 2 5 I B Y T H E C O I N C I D E N C E
587
METHOD
125I DECAY PATHWAYS PROBABILITY
(0.86) (0.88) (0.93) (0.86) (0.80)
ENERGY keY
-~ 0.484
55.0
0.079
27.5
~
0.077
27.5
=
0.048
63.0
=
0.008
35.5
~
0.079
27,5
--
0.013
00.0
--
0.012
00.0
--
0.141
27.5
K-X-RAY /0.86
( K 0.88 /
AUGER 0.14 ELECTRON\ 0 14
k ( " ) (0.88) (0.93) (0.86) (0.80)
/ L 0.12
I.C. 0.93
~K-X-RAY 0.86
(0.12) (0.93) (0.86) (0.80)
]/0,07 (0.07) (0.86) (0.80)
E.C. fK) 0.80
1251
/
ELECTRONS
\,
0.14
/
(0.07) (0.14) (0.80)
7 0.07 (0.86) (0.88) (0.93) (0.14) (0.80)
/ K-X-RAY 0.86 I.C. 0.93 ~-
( K< / AUGER 0"14 0.88 ELECTRONS (0.14) (0.88) (0.93) (0.14) (0.80)
L0.12 k
(0.12)(0.93)(0.14)(0.80)
E.C. 1 (L+M+ . . . . ) 0.20
(0.86) (0.88) (0.93) (0.20)
/ K-X-RAY
l ,
I.C. 0.93
(
0.86
K 0.88 AUGER 0.14 < ELECTRONS k
L 0.12 X ~
=0.023
00.0
z
0.022
00.0
=
0.014
35.5
(0.14) (0.88) (0.93) (0,20) (0.12) (0.93) (0.20)
Y 0.07
\
(0.07) (0.20)
TOTAL 1.000 Fig. 1. P a t h w a y s and probabilities for a~sI decay a s s u m i n g no detection o f L, M, or etc., X rays of Te.
588
D. L
H O R R O C K S A N D P. R. K L E I N TABLE 1
in the following fraction of the decays:
125I decay pathways. Relative probabilities o f the radiation of three energy groups.
(0.93) (0.875) (0.86) = 0.70. Therefore the probability of emission of either as Te K X ray or a gamma ray from the 35.5 keV excited state is (0.70 + 0.07) or 0.77. The probability of emission of a Te K X ray following E.C. is 0.688. Thus the ratio of probabilities is: P1
0.688 -- 0.894. 0.77
-
P2
Energy (keV)
Probabilities
0
0.013 0.012 0.070 0.023 0.022
27.5 or 35.5 a
0.079 0.077 0.008 0.398 0.079
(12)
While this ratio is not unity, it does not differ significantly from unity to introduce an appreciable error in the calculated values of S by eq. (2). Eq. (1) with P1/P2 = 1:
0.414 0.014 0.484" 0.048 0.532
55 or 63 a
S - (Ns;2N~)2 ; 4No Eq. (1) with
a The resolution of NaI(T1) crystals is not sufficient to be able to resolve the 27.5 and 35.5 keV radiations. Likewise, the sum peaks of 55 and 63 keV are not resolved.
Pt/P2 0.894: =
S -- ( N s + 2 N e ) 2
4.013 No Considering all the possible modes and combinations of the decay process, the probability chart of fig. 1 was constructed. Each solid line is a possible pathway for the decay process. Each part of the pathway is identified by the type of process and the probability of that process. The probability of a given pathway is the product of the probabilities of individual processes of that pathway. The total energy released is the sum of energies released in the individual processes. Again it should be pointed out that only Te K X rays and the 35.5 keV g a m m a rays are energetic enough to be detected by the NaI(T1) crystal which is inside an A1 housing. The radiation has to be energetic enough to penetrate the 0.010" thick AI window. The radiations can be put into three energy groups with their relative probabilities (see table 1). Frorn these figures it is seen that 7 % of the 125I decays gives rise to no measurable radiation, 39.8 % gives rise to a single detectable radiation (either 27.5 keV X ray or 35.5 keV g a m m a ray), and 53.2 % gives rise to two coincident radiations (either two X rays or an X ray plus the 35.5 keV y ray). Thus 46.8% of the 125I decays does not produce two coincident, measurable radiations. Assuming that the measurable strength, SM, is the sum of the single events Su and the doublet events SD gives: S M = S u -}- S D .
(13)
The number of singles measured is: Ns = 28(1--e)SD + eSu,
(14)
where the first term is a measure of those events which produce two coincident radiations but only one is measured. The coincidence count rate is: No = e2SD.
(15)
Using the definition for R [eq. (9)] gives: 1
N s _
R
N¢
2(l-e) - -
Su
+
~
(16)
eS D '
letting = S,/So,
(17)
1 _ __2(1-e) t- ~- = 2 ( 1 - e ) + c ~ R
e
~
(18)
e
Solving for e gives: (19) and substituting into eq. (15) gives:
s~ =
No = No I-(1 +2R)/R3~, (2 + .)~ JR/(1 + 2 R)3 ~ (2 +.)~
(20)
multiplying the numerator and denominator of eq. (20)
STANDARDIZATION
OF 12sI BY THE C O I N C I D E N C E M E T H O D
by 4 gives:
589
by: Nc [(1 + 2 S D ~.
g)/2R]2
[½(2 + ~)32
(21)
Thus the general eq. (2) is approximately valid because the corrective terms nearly cancel each other.
Substitution into eq. (13) gives: S M ~- S D - ] - ~ S D = SD(J_+OO,
s = (-~) sM :No\ (l+2R'~2(O' j 927x}\-O-..~3-d/" (27)
(22) 4. Conclusions
or;
No [(1
SM
+2R)/2R]z (1 +c 0
(233
[½(2 + ~)]2
From the flow chart of fig. 1 the values of S, = 0.398,
SD = 0.532,
give: = 0.398/0.532 = 0.748,
(24)
and: l+c~
= 0.927.
(25)
Thus: SM =
N {1 + 2 R ) 2
c \--~--~-j (0.927),
(26)
is the decay rate which involves either a single or two coincident radiations. However, only in 0.93 of the decays this does happen. The true decay rate is given
While the general equation used by previous investigations was accurate (at least within 1%) for the calculation of source strength by the coincidence method, the more detailed derivation of the equation, as shown here, seems to eliminate some reservations about the assumption of its validity. Thus the simple measurement of the relative number of counts produced in the single and sum (coincidence) photopeaks is all that is required to calculate the source strength of any radionuclide which has the proper decay characteristics. References 1) j. S. Eldridge and P. Crowther, Nucleonics 22 (6) (1964) 56. 2) H. Stephan, Isotopenpraxis 2 (1966) 161. a) j. G. V. Taylor, in Standardization o f radionuelides (IAEA, Vienna, 1967) p. 341. 4) L. Erbeszkron, L. Szokolyi, A. Szorenyi and K. Zsdanszky, Atomtech. Tajek. 11 (7) (1968) 358. 5) All data used to calculate conversion coefficients, energies, X-ray probabilities and other decay properties have been obtained from information published in C. M. Lederer, J. M. Hollander and I. Perlman, Table o f isotopes, 6th ed. (J. Wiley & Sons, New York, 1968).