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On the determination of disintegration rates by the coincidence method using high efficiency detectors
m
On the ~Determlnation o f Disintegration Rates by the Coincidence Method Using High Efficiency Detectors (Received 12 J u l y 1961)
IN A recent paper, CAMPIONI1) discusses the corrections p e r t a i n i n g to the d e t e r m i n a t i o n of a disintegration rate b y the 47rfl-7 coincidence m e t h o d w h e r e h i g h efficiency detectors are employed, a n d h e gives a n e x p e r i m e n t a l verification of the validity of these corrections. T h i s c o r r e s p o n d e n t would like to m a k e a few a d d i t i o n a l observations o n the coincidence m e t h o d w h e n applied u n d e r these high-efficiency conditions. T h e b e a u t y of the coincidence m e t h o d , as it is customarily applied in situations w h e r e the detectors are of low overall efficiency, is t h a t the disintegration rate of a radioactive nuclide c a n b e expressed (where two or m o r e simultaneous radiations are emitted) in terms of directly observable quantities. O n e need know only the c o u n t i n g rates of the individual detectors a n d the true coincidence c o u n t i n g rate b e t w e e n t h e m . T h e true coincidence rate c a n be f o u n d b y s u b t r a c t i n g from the observed total coincid e n c e rate the c h a n c e or accidental coincidence rate. This a c c i d e n t a l rate c a n be d e t e r m i n e d by using two i n d e p e n d e n t a n d isolated sources or by artificially i n t r o d u c i n g a delay into one of the detector c h a n n e l s so t h a t the possibility for a true coincidence is completely eliminated. D e a d - t i m e effects do not a p p e a r explicitly in the expressions for coincidences as they are a u t o m a t i c a l l y a c c o u n t e d for w h e n the coincidence rate is expressed in terms of the singlec h a n n e l rates. For details of the m e t h o d see, for example, D U N W O R T H (2) o r P U T M A N 13). W h e n the efficiencies of the detectors are increased so t h a t h i g h e r - o r d e r terms in the efficiency c a n n o longer b e neglected, m a n y of the a b o v e a d v a n t a g e s are lost. D e a d - t i m e corrections b e c o m e i m p o r t a n t , a n d the c h a n c e coincidence rate no longer r e m a i n s a direct observable. This c o m p l i c a t i o n requires t h a t b o t h the i n d i v i d u a l c h a n n e l d e a d times a n d the coincidence resolving t i m e m u s t be d e t e r m i n e d in a d d i t i o n to the i n d i v i d u a l c h a n n e l a n d total coincidence c o u n t i n g rates. T h e question of h o w one expresses the disintegration rate in terms of these quantities is p a r t l y a philosophical one. CAMPION chooses to m a k e a separation of the total observed coincidence rate into " t r u e s " a n d " a c c i d e n t a l s " . T h e " t r u e s " c o n t a i n terms involving the single-channel d e a d time, ~ , a n d the " a c c i d e n t a l s " involving only the coincidence
resolving time, 7R. A point of interest here is t h a t CAMPION'S accidentals c o n t a i n coincidences b e t w e e n separate, b u t not necessarily u n r e l a t e d events, i.e. a n event where t h e / 3 is detected in one counter a n d no 7 in the other, w h i c h is followed b y a n o t h e r event w i t h i n the time resolution of the coincidence mixer where the 7 is counted b u t t h e / 3 is not. This latter /3 was n o t c o u n t e d because the /3-channel was d e a d from the preceeding event, b u t h a d this c h a n n e l not b e e n dead, the latter /3 either would h a v e been c o u n t e d or would not, d e p e n d i n g solely on the efficiency of the /3-counter w h i c h c a n be between zero a n d unity. O n l y those coincidences w h e r e the latter/3 would not h a v e b e e n c o u n t e d if t h e / / - c h a n n e l were alive c a n be considered " c h a n c e " coincidences in the statistical sense of the word, while the r e m a i n ing coincidences c a n be considered to b e true coincidences, albeit between different events. O n the other h a n d , this separation into " t r u e s " a n d " a c c i d e n t a l s " is only a n artifice, since in no case c a n the " a c c i d e n t a l s " be m e a s u r e d directly or indirectly. I t would, perhaps, be b e t t e r to start from the definition of a coincidence, true or otherwise, a n d express the disintegration rate in terms of this total coincidence rate a n d the single-channel c o u n t i n g rates. T h e single-channel rates are affected by the individual c h a n n e l d e a d time, TD. I n a m o d e r n coincidence mixer where the coincidence resolving time, ~-n, is d e t e r m i n e d b y a passive e l e m e n t in order to achieve a truly n o n - e x t e n d a b l e resolving time, ~9 is usually greater t h a n ~'R' Consider a situation w h e r e n o disintegrations per second are occurring w i t h the emission of cascade radiations 1 a n d 2 w h i c h are detected in c h a n n e l s 1 a n d 2 h a v i n g efficiencies e1 a n d e 2, respectively. T h e efficiencies are understood to include the intrinsic detection efficiency as well as t h a t d u e to the solid angle s u b t e n d e d b y each detector. I n a n interval 7~ t h e r e are n0elzD p r o b a b l e counts t h a t are lost from c h a n n e l 1 because detector 1 is dead. T h e r e are n 1 counts or intervals p e r second, so t h a t the c h a n n e l 1 true c o u n t i n g rate is noel = n 1 + nlnoelT n
(1)
giving nl
n°el 1 + noel"rD
(2)
Similarly we h a v e for c h a n n e l 2
n°e2 nz
148
1 ~- noe2"r9
(3)
Letter to the editors Likewise we can d e t e r m i n e the coincidence counting loss due to the d e a d time of the mixer circuit. This d e a d time is d e p e n d e n t on the coincidence resolving time a n d the single-channel d e a d times w h e r e for simplicity we have assumed the latter to be equal. W e also take into account the possible directional correlation which would modify the efficiency for simultaneously detecting the coincident radiations. This correlation can be expressed as a factor, f ( O ) (a) w h i c h is of the form oo
f(o)
= 1
+ ~=1:a~ (e~(cos
o)).
W e consider all possible ways t h a t a coincidence m a y be registered. T h e n , from the d e a d times involved, we find t h a t there are ncno['r~ele2f(O) + ('rz) -- "rn){e l(1 -- e2f(O) ) -t- 63(1 -- elf(O)) }] p r o b a b l e coincidences that are lost in the interval, r~, after registering an event. T h e coincidence mixer is d e a d for a n interval rD after a coincidence event has been registered, a n d is d e a d for an interval ~ ' ~ - ~'R after a single-channel event has been registered, the mixer being alive for the interval Tx immediately after the single-channel event. T h e corrected coincidence rate is thus: noele2f(O) + 2"rnno2el(1 -- e2f(O) )e2(1 -- e l f ( O ) ) =
,o + n~no[','~ele~f(0)
+ ('D -- ~'_~){el(1 -- e j ( ' O ) )
÷ e2(1 -- elf(O))}] (4) which becomes, neglecting higher-order terms in the resolving time and d e a d time,
149
w h e r e nl ha, n ~ a, a n d ncso are the b a c k g r o u n d counting rates in each of their respective channels. I f one of the detectors has uniform detection probability over all directions or if no angular correlation exists at all, t h e n f ( O ) equals unity a n d expression (5) can be rewritten in the form equivalent to CAMPION~S treatment: nc =
nln 2 1 + ~'nno(2 -- eI -- ez) no
(8)
1 -- .rgnoete 2
Both r 9 and r n can be evaluated in a single experiment. I n lieu of the source that is to be calibrated in the 4wfl- 7 coincidence experiment, for example, a special source is e m p l o y e d consisting of non-coincident/5- and T-radiations* w h e r e the activity is approximately equally distributed on two semicircular mounts such t h a t the total activity distribution is approximately t h a t normally used in the 4wfl c h a m b e r . These semicircular mounts need to be so constructed that one can be a d d e d or r e m o v e d without disturbing the other. No particular attention need be paid to the source thickness. T h e single-channel counting rates and coincidence rate are d e t e r m i n e d with one source in place, then with the second source added, and then, finally with the first source removed. T h e d e a d times for each c h a n n e l can b e found as in the usual two source technique (e.g. PRICE(5)) for the d e t e r m i n a t i o n of counter d e a d times. T h e coincidence resolving time can be found from the counting rate in the coincidence channel, noting t h a t in this case the coincidences are
f(0)[1 + no'rn{el(1 -- % f ( O ) ) + ee(l -- elf(0))}] + (2rnn0)(1 -- elf(0))(1 -- % f ( O ) ) nc = n0ele2 - 1 -- no'rvel%f(O ) + nor~e I + no,rDe 2
(5)
W e note t h a t to the first o r d e r nl - - n o
1 - - e 2 f ( O ) and n2 -- nc -- 1 - - e l f ( 0 ) (6) n2
nI
for which a p p r o x i m a t i o n we can express the disintegration rate in terms of physically measurable quantities; no = ~f(O) c
given by the customary expression for the accidentals (2) na == 2"rnnln 2. This test source can be used also to calibrate the energy response of the a p p a r a t u s as well as to d e t e r m i n e rD and r n in such a m a n n e r as not to
1 -}- rn(nl + n 2 -- 2ne) + (2rn/nc) (n 1 -- ne)(n 2 -- no) l
(7a)
-- net D
W h e n either detector a p p r o a c h e s 100 per cent efficiency, the angular correlation factor, f ( O ) , a p p r o a c h e s unity (4), while the genuine chance coincidences given by the last t e r m in the n u m e r ator of expressions (5) or (7a) a p p r o a c h zero. E q u a t i o n (7a) has been derived without considering b a c k g r o u n d effects. W h e n these effects are i n c o r p o r a t e d we can rewrite the expression as
require m u c h more operating time t h a n would normally be required to see that the e q u i p m e n t is in p r o p e r operating order. It m a y be instructive to see that the total coincidence rate expected from equation (5) for a typical 7-7 coincidence m e a s u r e m e n t w h e r e the overall
(n1 -- nl Ba) (n~, -- nfO)f(O) 1 + rn(n, + n 2 -- 2ne) + ( 2 m i n e ) ( n I -- ne)(n 2 -- no) nO 1 --
* A very suitable source for this application would be Cs asT.
neq" ~
(7b)
150
Letter to the editors
efficiency of each 7-counter is 0.03, ~-9 = 10 #sec, *n = l # s e c , and n o = 5 × 104 d.p.s, is about 0.3 per cent smaller than the coincidence rate that would be obtained using conventional low-efficiency expressions. This discrepancy is somewhat greater than the statistical accuracy that is usually obtained in a routine disintegration-rate determination.
National Bureau of Standards Washington 25, D.C.
R. W. HAYWARD
References
1. CAMPION P. J. Int. J. appl. Rad. Isotopes 4, 232 (1959). 2. DUNWORTHJ. V. Rev. sci. Instrum. 11, 167 (1940). 3. PUTMAN J. L. Beta and Gamma Ray Spectroscopy (SmGBAHN K. Ed.) Chap. 26, Interscience, N.Y.
(1955). 4. HAYWARDR. W., HoPPES D. D. and MANN W. B. J. Res. Nat. bur. Stand. 54, 47 (1955). 5. PRICE W. J. Nuclear Radiation Detection, M c G r a w Hill, N.Y. (1958).
On the ~Determlnation o f Disintegration Rates by the Coincidence Method Using High Efficiency Detectors (Received 12 J u l y 1961)
IN A recent paper, CAMPIONI1) discusses the corrections p e r t a i n i n g to the d e t e r m i n a t i o n of a disintegration rate b y the 47rfl-7 coincidence m e t h o d w h e r e h i g h efficiency detectors are employed, a n d h e gives a n e x p e r i m e n t a l verification of the validity of these corrections. T h i s c o r r e s p o n d e n t would like to m a k e a few a d d i t i o n a l observations o n the coincidence m e t h o d w h e n applied u n d e r these high-efficiency conditions. T h e b e a u t y of the coincidence m e t h o d , as it is customarily applied in situations w h e r e the detectors are of low overall efficiency, is t h a t the disintegration rate of a radioactive nuclide c a n b e expressed (where two or m o r e simultaneous radiations are emitted) in terms of directly observable quantities. O n e need know only the c o u n t i n g rates of the individual detectors a n d the true coincidence c o u n t i n g rate b e t w e e n t h e m . T h e true coincidence rate c a n be f o u n d b y s u b t r a c t i n g from the observed total coincid e n c e rate the c h a n c e or accidental coincidence rate. This a c c i d e n t a l rate c a n be d e t e r m i n e d by using two i n d e p e n d e n t a n d isolated sources or by artificially i n t r o d u c i n g a delay into one of the detector c h a n n e l s so t h a t the possibility for a true coincidence is completely eliminated. D e a d - t i m e effects do not a p p e a r explicitly in the expressions for coincidences as they are a u t o m a t i c a l l y a c c o u n t e d for w h e n the coincidence rate is expressed in terms of the singlec h a n n e l rates. For details of the m e t h o d see, for example, D U N W O R T H (2) o r P U T M A N 13). W h e n the efficiencies of the detectors are increased so t h a t h i g h e r - o r d e r terms in the efficiency c a n n o longer b e neglected, m a n y of the a b o v e a d v a n t a g e s are lost. D e a d - t i m e corrections b e c o m e i m p o r t a n t , a n d the c h a n c e coincidence rate no longer r e m a i n s a direct observable. This c o m p l i c a t i o n requires t h a t b o t h the i n d i v i d u a l c h a n n e l d e a d times a n d the coincidence resolving t i m e m u s t be d e t e r m i n e d in a d d i t i o n to the i n d i v i d u a l c h a n n e l a n d total coincidence c o u n t i n g rates. T h e question of h o w one expresses the disintegration rate in terms of these quantities is p a r t l y a philosophical one. CAMPION chooses to m a k e a separation of the total observed coincidence rate into " t r u e s " a n d " a c c i d e n t a l s " . T h e " t r u e s " c o n t a i n terms involving the single-channel d e a d time, ~ , a n d the " a c c i d e n t a l s " involving only the coincidence
resolving time, 7R. A point of interest here is t h a t CAMPION'S accidentals c o n t a i n coincidences b e t w e e n separate, b u t not necessarily u n r e l a t e d events, i.e. a n event where t h e / 3 is detected in one counter a n d no 7 in the other, w h i c h is followed b y a n o t h e r event w i t h i n the time resolution of the coincidence mixer where the 7 is counted b u t t h e / 3 is not. This latter /3 was n o t c o u n t e d because the /3-channel was d e a d from the preceeding event, b u t h a d this c h a n n e l not b e e n dead, the latter /3 either would h a v e been c o u n t e d or would not, d e p e n d i n g solely on the efficiency of the /3-counter w h i c h c a n be between zero a n d unity. O n l y those coincidences w h e r e the latter/3 would not h a v e b e e n c o u n t e d if t h e / / - c h a n n e l were alive c a n be considered " c h a n c e " coincidences in the statistical sense of the word, while the r e m a i n ing coincidences c a n be considered to b e true coincidences, albeit between different events. O n the other h a n d , this separation into " t r u e s " a n d " a c c i d e n t a l s " is only a n artifice, since in no case c a n the " a c c i d e n t a l s " be m e a s u r e d directly or indirectly. I t would, perhaps, be b e t t e r to start from the definition of a coincidence, true or otherwise, a n d express the disintegration rate in terms of this total coincidence rate a n d the single-channel c o u n t i n g rates. T h e single-channel rates are affected by the individual c h a n n e l d e a d time, TD. I n a m o d e r n coincidence mixer where the coincidence resolving time, ~-n, is d e t e r m i n e d b y a passive e l e m e n t in order to achieve a truly n o n - e x t e n d a b l e resolving time, ~9 is usually greater t h a n ~'R' Consider a situation w h e r e n o disintegrations per second are occurring w i t h the emission of cascade radiations 1 a n d 2 w h i c h are detected in c h a n n e l s 1 a n d 2 h a v i n g efficiencies e1 a n d e 2, respectively. T h e efficiencies are understood to include the intrinsic detection efficiency as well as t h a t d u e to the solid angle s u b t e n d e d b y each detector. I n a n interval 7~ t h e r e are n0elzD p r o b a b l e counts t h a t are lost from c h a n n e l 1 because detector 1 is dead. T h e r e are n 1 counts or intervals p e r second, so t h a t the c h a n n e l 1 true c o u n t i n g rate is noel = n 1 + nlnoelT n
(1)
giving nl
n°el 1 + noel"rD
(2)
Similarly we h a v e for c h a n n e l 2
n°e2 nz
148
1 ~- noe2"r9
(3)
Letter to the editors Likewise we can d e t e r m i n e the coincidence counting loss due to the d e a d time of the mixer circuit. This d e a d time is d e p e n d e n t on the coincidence resolving time a n d the single-channel d e a d times w h e r e for simplicity we have assumed the latter to be equal. W e also take into account the possible directional correlation which would modify the efficiency for simultaneously detecting the coincident radiations. This correlation can be expressed as a factor, f ( O ) (a) w h i c h is of the form oo
f(o)
= 1
+ ~=1:a~ (e~(cos
o)).
W e consider all possible ways t h a t a coincidence m a y be registered. T h e n , from the d e a d times involved, we find t h a t there are ncno['r~ele2f(O) + ('rz) -- "rn){e l(1 -- e2f(O) ) -t- 63(1 -- elf(O)) }] p r o b a b l e coincidences that are lost in the interval, r~, after registering an event. T h e coincidence mixer is d e a d for a n interval rD after a coincidence event has been registered, a n d is d e a d for an interval ~ ' ~ - ~'R after a single-channel event has been registered, the mixer being alive for the interval Tx immediately after the single-channel event. T h e corrected coincidence rate is thus: noele2f(O) + 2"rnno2el(1 -- e2f(O) )e2(1 -- e l f ( O ) ) =
,o + n~no[','~ele~f(0)
+ ('D -- ~'_~){el(1 -- e j ( ' O ) )
÷ e2(1 -- elf(O))}] (4) which becomes, neglecting higher-order terms in the resolving time and d e a d time,
149
w h e r e nl ha, n ~ a, a n d ncso are the b a c k g r o u n d counting rates in each of their respective channels. I f one of the detectors has uniform detection probability over all directions or if no angular correlation exists at all, t h e n f ( O ) equals unity a n d expression (5) can be rewritten in the form equivalent to CAMPION~S treatment: nc =
nln 2 1 + ~'nno(2 -- eI -- ez) no
(8)
1 -- .rgnoete 2
Both r 9 and r n can be evaluated in a single experiment. I n lieu of the source that is to be calibrated in the 4wfl- 7 coincidence experiment, for example, a special source is e m p l o y e d consisting of non-coincident/5- and T-radiations* w h e r e the activity is approximately equally distributed on two semicircular mounts such t h a t the total activity distribution is approximately t h a t normally used in the 4wfl c h a m b e r . These semicircular mounts need to be so constructed that one can be a d d e d or r e m o v e d without disturbing the other. No particular attention need be paid to the source thickness. T h e single-channel counting rates and coincidence rate are d e t e r m i n e d with one source in place, then with the second source added, and then, finally with the first source removed. T h e d e a d times for each c h a n n e l can b e found as in the usual two source technique (e.g. PRICE(5)) for the d e t e r m i n a t i o n of counter d e a d times. T h e coincidence resolving time can be found from the counting rate in the coincidence channel, noting t h a t in this case the coincidences are
f(0)[1 + no'rn{el(1 -- % f ( O ) ) + ee(l -- elf(0))}] + (2rnn0)(1 -- elf(0))(1 -- % f ( O ) ) nc = n0ele2 - 1 -- no'rvel%f(O ) + nor~e I + no,rDe 2
(5)
W e note t h a t to the first o r d e r nl - - n o
1 - - e 2 f ( O ) and n2 -- nc -- 1 - - e l f ( 0 ) (6) n2
nI
for which a p p r o x i m a t i o n we can express the disintegration rate in terms of physically measurable quantities; no = ~f(O) c
given by the customary expression for the accidentals (2) na == 2"rnnln 2. This test source can be used also to calibrate the energy response of the a p p a r a t u s as well as to d e t e r m i n e rD and r n in such a m a n n e r as not to
1 -}- rn(nl + n 2 -- 2ne) + (2rn/nc) (n 1 -- ne)(n 2 -- no) l
(7a)
-- net D
W h e n either detector a p p r o a c h e s 100 per cent efficiency, the angular correlation factor, f ( O ) , a p p r o a c h e s unity (4), while the genuine chance coincidences given by the last t e r m in the n u m e r ator of expressions (5) or (7a) a p p r o a c h zero. E q u a t i o n (7a) has been derived without considering b a c k g r o u n d effects. W h e n these effects are i n c o r p o r a t e d we can rewrite the expression as
require m u c h more operating time t h a n would normally be required to see that the e q u i p m e n t is in p r o p e r operating order. It m a y be instructive to see that the total coincidence rate expected from equation (5) for a typical 7-7 coincidence m e a s u r e m e n t w h e r e the overall
(n1 -- nl Ba) (n~, -- nfO)f(O) 1 + rn(n, + n 2 -- 2ne) + ( 2 m i n e ) ( n I -- ne)(n 2 -- no) nO 1 --
* A very suitable source for this application would be Cs asT.
neq" ~
(7b)
150
Letter to the editors
efficiency of each 7-counter is 0.03, ~-9 = 10 #sec, *n = l # s e c , and n o = 5 × 104 d.p.s, is about 0.3 per cent smaller than the coincidence rate that would be obtained using conventional low-efficiency expressions. This discrepancy is somewhat greater than the statistical accuracy that is usually obtained in a routine disintegration-rate determination.
National Bureau of Standards Washington 25, D.C.
R. W. HAYWARD
References
1. CAMPION P. J. Int. J. appl. Rad. Isotopes 4, 232 (1959). 2. DUNWORTHJ. V. Rev. sci. Instrum. 11, 167 (1940). 3. PUTMAN J. L. Beta and Gamma Ray Spectroscopy (SmGBAHN K. Ed.) Chap. 26, Interscience, N.Y.
(1955). 4. HAYWARDR. W., HoPPES D. D. and MANN W. B. J. Res. Nat. bur. Stand. 54, 47 (1955). 5. PRICE W. J. Nuclear Radiation Detection, M c G r a w Hill, N.Y. (1958).