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Modulo-2 counting — An alternative to the coincidence method
202
Nuclear Instruments and Methods in Physics Research 224 (l 984~ 202- 206 North-Holland. Amsterdam
MODULO-2 COUNTING - AN ALTERNATIVE TO THE COINCIDENCE METHOD L. S I A D A T , A.P. J U P P and G . W . M c B E T H Department of Physical Sciences, Brighton Polytechnic, Brighton BN2 4GJ, England
Received 27 May 1983 and in revised form 24 October 1983
The theory and practice of activity determination by means of a modulo-2 counting technique is discussed. The method, although statistically less precise than the coincidence method, does not suffer from the time jitter and random coincidence effects which can cause problems with the coincidence method.
I. Introduction
2. Principle of the modulo-2 technique
The coincidence counting method for the determination of activity [1], although simple in concept, has significant disadvantages, not the least of which is that the method demands a determination of the simultaneity or otherwise of events. Such a determination is always fraught with difficulties, owing to uncertainties regarding system delays, dead-time and time resolution effects. To overcome the problems, alternative techniques have been developed by Baerg [2] and Miiller [3]. The method of the former authors is a development of a system described by De Carlos and Granados [4] and makes use of anticoincidence techniques, in which each channel has an imposed extending dead-time. This method can be extended to several counting channels
The modulo-2 method for activity determination is shown in fig. 1. In order to compare the technique with the conventional coincidence counting technique, let us suppose that the transition is a simple /3-y prompt transition with no branching and that the background can be neglected. In this case, measurement of the mean /3, ~, and coincidence channel rates N#, Nv and N~ respectively give in terms of the source activity N0
[51. The method developed by Mialler called selective sampling contains some similarities to that of Baerg, in that an extending dead-time and anticoincidence techniques are employed. Selective sampling, although simple in concept and capable of high precision, requires a "speed converter" [6] if data acquisition rates comparable to that attainable in the coincidence technique are required. Although the selective sampling technique is suited for the determination of nuclides with an isomeric state, the analysis and interpretation of the data can be more complex than that for the case of prompt events. In the following, we describe a modulo-2 [7] correlation technique for measuring activity, which is simple to realise in practice. The statistical precision of the modulo-2 method is inferior to that of the coincidence technique, it is less susceptible to errors arising from the effects of time jitter, finite resolving times and delay mismatching than is the coincidence method. The modulo-2 technique is also suited to the assay of nuclides with delayed (isomeric) states. 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
AT#= ca (1 - , v ) N o + ' f l c v N o , Nv = ,v(1 - ,a)Uo + , a , vN0, Nc
(1)
ca%N0,
where in the above, the correlated and uncorrelated terms have been separated and c# and % are the respective counting efficiencies of the/3 and 3' channels for the appropriate radiation. Eqs. (1) enable N0, ca and % to be obtained (via measurements of Na, Nv and N~) from N O= N a N v / N c ,
(2)
Ca = N j N v " c ~ = Uc/ U~.
The problems of techniques based on have been discussed by Mhller [3]. In the modulo-2 technique, rates determined as before and the outputs branches are mixed (see fig. 1), to give R ~=
eqs. (1) and (2) N0 and Nv are of the /3 and ~, a rate
N~ + N~,
where the mean values of N0 and Nv are given by .Na = ca (1 - p c v ) N o + p ~ t ~ c v N o ,
203
L. Siadat et aL / Modulo-2 counting
N~
NI~
J
RI3v : NI~.N¥ ADD
t
MODULO-2
> PE(Rs)
N¥
T
NY
PERIOD AI
Fig. 1. Arrangement for activity determination using modulo-2.
and
ce = ~ e l N o ,
Nv = 'v (1 - p , # ) U0 + p , # , v N0,
(3)
respectively. In eqs. (3), p is the probability that the correlated radiations are emitted in the same time interval of length a T so that p = 1 for prompt events, and p=l
~ = ~J&.
(5)
Eqns. (5) are the modulo-2 equivalents of the coincidence counting eqs. (2). In order to understand how R s is obtained from the count rate R#r by means of a modulo-2 filter, eqs. (3) can be written in the form R#~ = R # + R~ + 2R c,
1 - e - x a r for events proceeding through an inhaT termediate state of decay constant k.
where R# = , ~ ( 1 - p % ) N o ,
In the system of fig. 1, the logic signals in the/3 and y branches are kept as short as possible ( - 3 ns) and the minimum delay necessary to prevent overlap of prompt fl-~, events is inserted in the y channel. Under these circumstances, rate R#v is given by eqs. (3) above. Mi~ller [8] has pointed out that at high count rates, the use of delays may cause problems owing to "random coincidences" being generated due to the time interval AT not being exactly coincident for both channels and that this problem can be overcome if the parity of the counts in each channel is determined. The parity of the total counts is then the joint parity of the beta and gamma channels. To realise this is practice, additional modulo-2 correlators are required. The output of the modulo-2 filter gives, through a measurement of the probability of an even number of events occurring in the /3 + y channel in a modulo-2 period AT, the uncorrelated count rate, R s, [9,10] which is from eq. (3) R~ = , # ( 1 - p , v ) N o
+,~,(1 - p , o ) N O.
Thus No, c# and %, can be obtained via N O= 2 p N o N v / ( ~ l s + N v - R s ) ,
(4)
and (6)
Rc = 2 p c # % N o.
Count rates Ra, R v and Re above are statistically independent [11] and Poisson in nature and the probability of an even number of pulses being presented to the modulo-2 filter in a time interval AT is the joint probability that the numbers of pulses arising from R# and R v are either both odd or both even in the interval a T. The probability, PE, of an even number of pulses arising from Ray in time a T can be determined from modulo-2 theory [12] to be PE =
½(1 + e-2~.'~r). ½(1 + e-2~ at) + ½(1 - e-2
Y). ½(1 - e-2
= ½(1 + e -2k s a t ) .
,
(7)
Measurement of the number, hE, of occurrences of an even number of pulses arising from Ray in n time intervals each of length AT enables k s to be calculated via eq. (7) with PE = n E/n. The modulo-2 theory is symmetrical under the inter-
L. Stadat et aL / Modulo-2 counting
204
change of the/3 and 3' channels and as such is similar to coincidence counting theory and differs from the asymmetrical selective sampling technique [3].
1000 I "
.,...
........ ...... .
3. Precision of the modulo-2 method ,
•~
The coincidence method for activity determination requires for its successful implementation, that the necessary corrections for time jitter effects and random coincidence be applied. In the case of the modulo-2 method, timing uncertainties are of lesser importance and the concept of random coincidences does not apply, since the modulo-2 filter determines the uncorrelated component of the radiation field. All coincidence systems are susceptible to random coincidences and the relative probability of a coincidence being random can be deduced to be [1]
100 x \
, \ x\ x \
50
,..
"-.. -,
........-....
',, ".,
N ,•" xx \ \ -,,, x
~ ! i x x, \
.,....
,
\ •
-.., .,
x\ \
%)+ 1'
where this probability is always finite for % and c# ~ 1 and increases as the product of activity and resolving time (i.e. Nor ) increases• Both the modulo-2 and coincidence techniques are subject to statistical uncertainty, owing to the statistical uncertainties inherent in the determination of rates Nv, N~, R~ and No. However, it is only in the modulo-2 method that statistical imprecision is the major contributor to the uncertainty in the activity determination• The statistical uncertainty of the modulo-2 method depends on the choice of RsAT and is a minimum when R A T--- 0.4 [9]. The fractional standard deviation of the activity determination is proportional to ( NOn A T) - ] / 2 where from the previous definition of n as being the number of time periods, nAT is seen to be the total measuring time. Fig. 2 shows the fractional standard deviation of the modulo-2 activity determination for NonAT= 1 as a function of the efficiencies cp and %. The derivation of the formula for this statistical uncertainty in the modulo-2 activity determination is given for prompt events ( p = 1) in the appendix. In the curves shown in fig. 2, it has been assumed that RsAT has been adjusted for optimum precision. Fig. 2 clearly demonstrates how the statistical uncertainty of the modulo-2 method increases rapidly with decreasing detection efficiency• Measured relative to the statistical uncertainty for t,0 = % = 1, the statistical uncertainty for fl-7 counting (c~ - 0.9, % - 0.1) is 6.2 and for 7 - 7 counting is 27. The statistical uncertainty of the coincidence method exhibits a similar behaviour and fig. 3 shows the relative uncertainties of the modulo-2 and coincidence methods as a function of the detection efficiencies. For fl-~,
-.
" - . -..
-..
~x'-, "=
j
10
xx~ \ \ .. \\
N\-~\,
............. '-o;o5 . . . . . . . . . . . . . . . . . 011
5
...........................
. . . . . . . .
0,01
.....
-.
~'-.. "', "-. \ Q \ "-. \ - . ~ - . . . "-,
0,3
0,8
. . . .
"-.
"~....'.. " .
0,5
. . . .
2Nor (1 - ,/~)(1 - %) -
.
x\\ \ \
........... 2No'r(1 - , ~ ) ( I
...
"", -.
"-.~\ "..
,,0
"-.
"%g.i"
0,05
0,1
0,5
1,0
Ep Fig. 2. Relative fractional standard deviation of the modulo-2 activity determination as a function of detection efficiencies. To obtain absolute fractional standard deviations, multiply the ordinate scale by (NonAT)-1/2 where the symbols have the meanings as defined in the text.
counting, the modulo-2 relative uncertainty is 4.4 times that of the coincidence method and for ~'-7 counting, the corresponding factor is 3. The above clearly demonstrates that the modulo-2
_k
. . . . . . . . . . . . . . . . . . . . . . . . . .
cv =°,°1
I
~--~.~ ..............
1
0,01
I
I
I
|
i llll
0,05
I
0,1
E~
I
0,5
1,0
Fig. 3. The ratio of the standard deviations of the activity determinations of the modulo-2 technique to that of the coincidence technique as a function of detection efficiencies.
L. Siadat et al. / Modulo-2 counting
205
Table 1 Summary of measurements on 22Na for a total measurement time of 1000 s. K/v, [S- l ]
Nv2 [S- 1]
Rs [S- 1]
R A T a)
NO [Bq]
ONo [Bq]
( Yl (calc.)
E¥2 (calc.)
194 194 144 145 144
34 34 122 122 121
226 224 258 260 258
0.72 0.36 0.83 0.21 0.41
5.3 × 103 3.1 × 103 4.3 x 103 4.8 x 103 4.1 x 103
3.6 × 103 0.6 × 103 0.9 x 103 0.4 x 103 0.4x 103
0.029 0.050 0.029 0.026 0.030
0.006 0.011 0.029 0.025 0.029
1) Note that the smallest uncertainty is not necessarily obtained for the condition RsAT --- 0.4, since the variance of R s is proportional to n -1.
technique is inferior as regards statistical precision to the coincidence technique; however, it does not suffer from the previously mentioned defects of the coincidence method. The rate of data acquisition is comparable with that of the coincidence method and is superior to that of the selective sampling technique.
ciently precise to warrant consideration as an alternative to coincidence counting, or (when high data acquisition rates are required) selective sampling. The authors wish to thank Dr. J.W. Mhller, Bureau Internationale Des Poids et Mesures, for his helpful comments on our manuscript.
4. Determination of the activity of a 22Na source
The problems inherent in the determination of source strengths by 3`-3` coincidence counting [1] apply also to the modulo-2 technique. In particular, if the gammas are labelled 3`~ and "1'2, the sensitivity of detector 2 for 3`~ and the corresponding sensitivity of detector 1 for 3`2 have to be allowed for in correcting the apparent count rates Nv~ and Nv2. The activity of a point 22Na source encapsulated in lead foil (to ensure positron annihilation at the source) was determined using the modulo-2 technique, with a variety of counting geometries. Corrections for background and efficiency effects were made using pulse height spectra, which were recorded during the modulo-2 measurements. The most important correction was to allow for a count rate R12 in channel 3'1, caused by 3`2 recording in the 3`a channel. If the fraction of such events in the 3`~ channel is f, then to allow for this, the expression for NO in eq. (5) should be multiplied by f and count rate R12 =lTVr~ included in the fight hand side of eq. (4). The results obtained in table 1 with 3`~ = 0.51 MeV and 3`2 = 1.275 MeV were mutually consistent and were in agreement with the recorded activity of the 22Na source. Table 1 shows that the modulo-2 method can be used to determine source strengths by 3`-3` counting, providing a relatively large statistical uncertainty (approximately 10 times greater than the coincidence statistical uncertainty under the same geometrical conditions) is acceptable. At high activities and with high efficiencies (e.g. /3-3` counting), the method should be suffi-
Appendix: Statistical precision of the modulo-2 and coincidence methods
A.1. Modulo-2 From eqs. (5) (assuming p = 1 for simplicity) 5/04 Var[ NO] = ~
+
1
NN/
1
Var[Rs]
N~ N~ ~ +
( -Rs/2Var[N,l N~]
~)cov{N,,Nfl}
2--~[1--~#)cov{N.r,Rs)
+N~,N~ 1
Rs
cov{Nfl, R~}.
(A.1)
Using eqs. (4) and (5) and making use of the relationship between the covariance of two quantities and the variance of the correlated component [11], it is easy to show that cov{ N~, Na } = ev, IjNo/nAT, coy{ N.r, R~ ) = 'v (1 - ,~)No~nAT , cov{ N#, R s } = ¢p (1 - ¢ v ) No/n A r.
L. Skadat et al. / Modulo-2 cmmtmg
1206
The variance of the rate R S is given by [9,10]
From eqs. (1),
Var[R~] = (e ' R ~ j r - 1 ) / 4 n A T 2.
cov{ N~, N~ } = ~o~rNo/nAT,
and
Substituting for the variances and covariances in eq. (A.3) gives
Var[N~] = N s / n A T , where 6 = fl or ¥. Substitution of the above results into eq. (A.1) yields after simplification Var[N0] = F [ 06 _ 4 ~~, _ 4 c Nf2
1 cv
with ~ =/3, V.
Var[Nol_4F{l -%(1-~)-~0(1 -c~)} Ug
1 - % - Ep+ 2 % ~
1 c0 +8cyc 0
=
NcnA r
(A.4)
Eq. (A.4) is in agreement with the result of a previous analysis of the statistical precision of the coincidence system [13].
+nF(e 4R'zT- 1)}, where
f = (4naTNoco, v) -~.
(A.2)
A.2. Coincidence counting The statistical precision of the coincidence method can be computed in a similar manner to that used in the determination of the statistical imprecision of the modulo-2 method, Var[N0 ] _
N~
Var[Nv]+
N2
Nv2 Var[N0]
NgNg NrZN~ Var[Nc ] + ,#--;?c'
2NrN~ cov{N~,Nc) -
2NvZNo cov{ N0, N~ }
NgU) +
2NvNo cov{ Nv, N0 }
(1.3)
References
[1] R.A. Allen, in: Alpha, beta and gamma ray spectroscopy, ed., K. Siegbahn (North-Holland, Amsterdam, 1965) pp. 425-465. [2] A.P. Baerg, Metrologia 12 (1976) 77. 13] J.W. Mialler, Nucl. Instr. and Meth. 189 (1981) 449. [4] J.E. De Carlos and C.E. Granados, Nucl. Instr. and Meth. 112 (1973) 209. [5] A.P. Baerg, Nucl. Instr. and Meth. 190 (1981) 345. [6] P. Br6once, Rapport BIPM-76/14 (1976) (in French) 9pp. [7] J.W. Mi~ller, Rapport BIPM-72/4 (1972). [8] J.W. Mailer, personal communication (1983). [9] J.W. Mt~ller, Rapport BIPM-75/7 (1975). [10] K.R.D. Mylon, G.W. McBeth and L. Siadat, Nucl. Instr. and Meth. 205 (1983) 197. [11] A handbook of radioactivity measurements procedures, NCRP Report No. 58, (1978) p. 75. [12] L. Siadat, G.W, McBeth and K.R.D. Mylon, Nucl. Instr. and Meth. 205 (1983) 203. [13] P.J. Campion and J.G.V. Taylor, Int. J. Appl. Rad. Isotopes 10 (1961) 131.
Nuclear Instruments and Methods in Physics Research 224 (l 984~ 202- 206 North-Holland. Amsterdam
MODULO-2 COUNTING - AN ALTERNATIVE TO THE COINCIDENCE METHOD L. S I A D A T , A.P. J U P P and G . W . M c B E T H Department of Physical Sciences, Brighton Polytechnic, Brighton BN2 4GJ, England
Received 27 May 1983 and in revised form 24 October 1983
The theory and practice of activity determination by means of a modulo-2 counting technique is discussed. The method, although statistically less precise than the coincidence method, does not suffer from the time jitter and random coincidence effects which can cause problems with the coincidence method.
I. Introduction
2. Principle of the modulo-2 technique
The coincidence counting method for the determination of activity [1], although simple in concept, has significant disadvantages, not the least of which is that the method demands a determination of the simultaneity or otherwise of events. Such a determination is always fraught with difficulties, owing to uncertainties regarding system delays, dead-time and time resolution effects. To overcome the problems, alternative techniques have been developed by Baerg [2] and Miiller [3]. The method of the former authors is a development of a system described by De Carlos and Granados [4] and makes use of anticoincidence techniques, in which each channel has an imposed extending dead-time. This method can be extended to several counting channels
The modulo-2 method for activity determination is shown in fig. 1. In order to compare the technique with the conventional coincidence counting technique, let us suppose that the transition is a simple /3-y prompt transition with no branching and that the background can be neglected. In this case, measurement of the mean /3, ~, and coincidence channel rates N#, Nv and N~ respectively give in terms of the source activity N0
[51. The method developed by Mialler called selective sampling contains some similarities to that of Baerg, in that an extending dead-time and anticoincidence techniques are employed. Selective sampling, although simple in concept and capable of high precision, requires a "speed converter" [6] if data acquisition rates comparable to that attainable in the coincidence technique are required. Although the selective sampling technique is suited for the determination of nuclides with an isomeric state, the analysis and interpretation of the data can be more complex than that for the case of prompt events. In the following, we describe a modulo-2 [7] correlation technique for measuring activity, which is simple to realise in practice. The statistical precision of the modulo-2 method is inferior to that of the coincidence technique, it is less susceptible to errors arising from the effects of time jitter, finite resolving times and delay mismatching than is the coincidence method. The modulo-2 technique is also suited to the assay of nuclides with delayed (isomeric) states. 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
AT#= ca (1 - , v ) N o + ' f l c v N o , Nv = ,v(1 - ,a)Uo + , a , vN0, Nc
(1)
ca%N0,
where in the above, the correlated and uncorrelated terms have been separated and c# and % are the respective counting efficiencies of the/3 and 3' channels for the appropriate radiation. Eqs. (1) enable N0, ca and % to be obtained (via measurements of Na, Nv and N~) from N O= N a N v / N c ,
(2)
Ca = N j N v " c ~ = Uc/ U~.
The problems of techniques based on have been discussed by Mhller [3]. In the modulo-2 technique, rates determined as before and the outputs branches are mixed (see fig. 1), to give R ~=
eqs. (1) and (2) N0 and Nv are of the /3 and ~, a rate
N~ + N~,
where the mean values of N0 and Nv are given by .Na = ca (1 - p c v ) N o + p ~ t ~ c v N o ,
203
L. Siadat et aL / Modulo-2 counting
N~
NI~
J
RI3v : NI~.N¥ ADD
t
MODULO-2
> PE(Rs)
N¥
T
NY
PERIOD AI
Fig. 1. Arrangement for activity determination using modulo-2.
and
ce = ~ e l N o ,
Nv = 'v (1 - p , # ) U0 + p , # , v N0,
(3)
respectively. In eqs. (3), p is the probability that the correlated radiations are emitted in the same time interval of length a T so that p = 1 for prompt events, and p=l
~ = ~J&.
(5)
Eqns. (5) are the modulo-2 equivalents of the coincidence counting eqs. (2). In order to understand how R s is obtained from the count rate R#r by means of a modulo-2 filter, eqs. (3) can be written in the form R#~ = R # + R~ + 2R c,
1 - e - x a r for events proceeding through an inhaT termediate state of decay constant k.
where R# = , ~ ( 1 - p % ) N o ,
In the system of fig. 1, the logic signals in the/3 and y branches are kept as short as possible ( - 3 ns) and the minimum delay necessary to prevent overlap of prompt fl-~, events is inserted in the y channel. Under these circumstances, rate R#v is given by eqs. (3) above. Mi~ller [8] has pointed out that at high count rates, the use of delays may cause problems owing to "random coincidences" being generated due to the time interval AT not being exactly coincident for both channels and that this problem can be overcome if the parity of the counts in each channel is determined. The parity of the total counts is then the joint parity of the beta and gamma channels. To realise this is practice, additional modulo-2 correlators are required. The output of the modulo-2 filter gives, through a measurement of the probability of an even number of events occurring in the /3 + y channel in a modulo-2 period AT, the uncorrelated count rate, R s, [9,10] which is from eq. (3) R~ = , # ( 1 - p , v ) N o
+,~,(1 - p , o ) N O.
Thus No, c# and %, can be obtained via N O= 2 p N o N v / ( ~ l s + N v - R s ) ,
(4)
and (6)
Rc = 2 p c # % N o.
Count rates Ra, R v and Re above are statistically independent [11] and Poisson in nature and the probability of an even number of pulses being presented to the modulo-2 filter in a time interval AT is the joint probability that the numbers of pulses arising from R# and R v are either both odd or both even in the interval a T. The probability, PE, of an even number of pulses arising from Ray in time a T can be determined from modulo-2 theory [12] to be PE =
½(1 + e-2~.'~r). ½(1 + e-2~ at) + ½(1 - e-2
Y). ½(1 - e-2
= ½(1 + e -2k s a t ) .
,
(7)
Measurement of the number, hE, of occurrences of an even number of pulses arising from Ray in n time intervals each of length AT enables k s to be calculated via eq. (7) with PE = n E/n. The modulo-2 theory is symmetrical under the inter-
L. Stadat et aL / Modulo-2 counting
204
change of the/3 and 3' channels and as such is similar to coincidence counting theory and differs from the asymmetrical selective sampling technique [3].
1000 I "
.,...
........ ...... .
3. Precision of the modulo-2 method ,
•~
The coincidence method for activity determination requires for its successful implementation, that the necessary corrections for time jitter effects and random coincidence be applied. In the case of the modulo-2 method, timing uncertainties are of lesser importance and the concept of random coincidences does not apply, since the modulo-2 filter determines the uncorrelated component of the radiation field. All coincidence systems are susceptible to random coincidences and the relative probability of a coincidence being random can be deduced to be [1]
100 x \
, \ x\ x \
50
,..
"-.. -,
........-....
',, ".,
N ,•" xx \ \ -,,, x
~ ! i x x, \
.,....
,
\ •
-.., .,
x\ \
%)+ 1'
where this probability is always finite for % and c# ~ 1 and increases as the product of activity and resolving time (i.e. Nor ) increases• Both the modulo-2 and coincidence techniques are subject to statistical uncertainty, owing to the statistical uncertainties inherent in the determination of rates Nv, N~, R~ and No. However, it is only in the modulo-2 method that statistical imprecision is the major contributor to the uncertainty in the activity determination• The statistical uncertainty of the modulo-2 method depends on the choice of RsAT and is a minimum when R A T--- 0.4 [9]. The fractional standard deviation of the activity determination is proportional to ( NOn A T) - ] / 2 where from the previous definition of n as being the number of time periods, nAT is seen to be the total measuring time. Fig. 2 shows the fractional standard deviation of the modulo-2 activity determination for NonAT= 1 as a function of the efficiencies cp and %. The derivation of the formula for this statistical uncertainty in the modulo-2 activity determination is given for prompt events ( p = 1) in the appendix. In the curves shown in fig. 2, it has been assumed that RsAT has been adjusted for optimum precision. Fig. 2 clearly demonstrates how the statistical uncertainty of the modulo-2 method increases rapidly with decreasing detection efficiency• Measured relative to the statistical uncertainty for t,0 = % = 1, the statistical uncertainty for fl-7 counting (c~ - 0.9, % - 0.1) is 6.2 and for 7 - 7 counting is 27. The statistical uncertainty of the coincidence method exhibits a similar behaviour and fig. 3 shows the relative uncertainties of the modulo-2 and coincidence methods as a function of the detection efficiencies. For fl-~,
-.
" - . -..
-..
~x'-, "=
j
10
xx~ \ \ .. \\
N\-~\,
............. '-o;o5 . . . . . . . . . . . . . . . . . 011
5
...........................
. . . . . . . .
0,01
.....
-.
~'-.. "', "-. \ Q \ "-. \ - . ~ - . . . "-,
0,3
0,8
. . . .
"-.
"~....'.. " .
0,5
. . . .
2Nor (1 - ,/~)(1 - %) -
.
x\\ \ \
........... 2No'r(1 - , ~ ) ( I
...
"", -.
"-.~\ "..
,,0
"-.
"%g.i"
0,05
0,1
0,5
1,0
Ep Fig. 2. Relative fractional standard deviation of the modulo-2 activity determination as a function of detection efficiencies. To obtain absolute fractional standard deviations, multiply the ordinate scale by (NonAT)-1/2 where the symbols have the meanings as defined in the text.
counting, the modulo-2 relative uncertainty is 4.4 times that of the coincidence method and for ~'-7 counting, the corresponding factor is 3. The above clearly demonstrates that the modulo-2
_k
. . . . . . . . . . . . . . . . . . . . . . . . . .
cv =°,°1
I
~--~.~ ..............
1
0,01
I
I
I
|
i llll
0,05
I
0,1
E~
I
0,5
1,0
Fig. 3. The ratio of the standard deviations of the activity determinations of the modulo-2 technique to that of the coincidence technique as a function of detection efficiencies.
L. Siadat et al. / Modulo-2 counting
205
Table 1 Summary of measurements on 22Na for a total measurement time of 1000 s. K/v, [S- l ]
Nv2 [S- 1]
Rs [S- 1]
R A T a)
NO [Bq]
ONo [Bq]
( Yl (calc.)
E¥2 (calc.)
194 194 144 145 144
34 34 122 122 121
226 224 258 260 258
0.72 0.36 0.83 0.21 0.41
5.3 × 103 3.1 × 103 4.3 x 103 4.8 x 103 4.1 x 103
3.6 × 103 0.6 × 103 0.9 x 103 0.4 x 103 0.4x 103
0.029 0.050 0.029 0.026 0.030
0.006 0.011 0.029 0.025 0.029
1) Note that the smallest uncertainty is not necessarily obtained for the condition RsAT --- 0.4, since the variance of R s is proportional to n -1.
technique is inferior as regards statistical precision to the coincidence technique; however, it does not suffer from the previously mentioned defects of the coincidence method. The rate of data acquisition is comparable with that of the coincidence method and is superior to that of the selective sampling technique.
ciently precise to warrant consideration as an alternative to coincidence counting, or (when high data acquisition rates are required) selective sampling. The authors wish to thank Dr. J.W. Mhller, Bureau Internationale Des Poids et Mesures, for his helpful comments on our manuscript.
4. Determination of the activity of a 22Na source
The problems inherent in the determination of source strengths by 3`-3` coincidence counting [1] apply also to the modulo-2 technique. In particular, if the gammas are labelled 3`~ and "1'2, the sensitivity of detector 2 for 3`~ and the corresponding sensitivity of detector 1 for 3`2 have to be allowed for in correcting the apparent count rates Nv~ and Nv2. The activity of a point 22Na source encapsulated in lead foil (to ensure positron annihilation at the source) was determined using the modulo-2 technique, with a variety of counting geometries. Corrections for background and efficiency effects were made using pulse height spectra, which were recorded during the modulo-2 measurements. The most important correction was to allow for a count rate R12 in channel 3'1, caused by 3`2 recording in the 3`a channel. If the fraction of such events in the 3`~ channel is f, then to allow for this, the expression for NO in eq. (5) should be multiplied by f and count rate R12 =lTVr~ included in the fight hand side of eq. (4). The results obtained in table 1 with 3`~ = 0.51 MeV and 3`2 = 1.275 MeV were mutually consistent and were in agreement with the recorded activity of the 22Na source. Table 1 shows that the modulo-2 method can be used to determine source strengths by 3`-3` counting, providing a relatively large statistical uncertainty (approximately 10 times greater than the coincidence statistical uncertainty under the same geometrical conditions) is acceptable. At high activities and with high efficiencies (e.g. /3-3` counting), the method should be suffi-
Appendix: Statistical precision of the modulo-2 and coincidence methods
A.1. Modulo-2 From eqs. (5) (assuming p = 1 for simplicity) 5/04 Var[ NO] = ~
+
1
NN/
1
Var[Rs]
N~ N~ ~ +
( -Rs/2Var[N,l N~]
~)cov{N,,Nfl}
2--~[1--~#)cov{N.r,Rs)
+N~,N~ 1
Rs
cov{Nfl, R~}.
(A.1)
Using eqs. (4) and (5) and making use of the relationship between the covariance of two quantities and the variance of the correlated component [11], it is easy to show that cov{ N~, Na } = ev, IjNo/nAT, coy{ N.r, R~ ) = 'v (1 - ,~)No~nAT , cov{ N#, R s } = ¢p (1 - ¢ v ) No/n A r.
L. Skadat et al. / Modulo-2 cmmtmg
1206
The variance of the rate R S is given by [9,10]
From eqs. (1),
Var[R~] = (e ' R ~ j r - 1 ) / 4 n A T 2.
cov{ N~, N~ } = ~o~rNo/nAT,
and
Substituting for the variances and covariances in eq. (A.3) gives
Var[N~] = N s / n A T , where 6 = fl or ¥. Substitution of the above results into eq. (A.1) yields after simplification Var[N0] = F [ 06 _ 4 ~~, _ 4 c Nf2
1 cv
with ~ =/3, V.
Var[Nol_4F{l -%(1-~)-~0(1 -c~)} Ug
1 - % - Ep+ 2 % ~
1 c0 +8cyc 0
=
NcnA r
(A.4)
Eq. (A.4) is in agreement with the result of a previous analysis of the statistical precision of the coincidence system [13].
+nF(e 4R'zT- 1)}, where
f = (4naTNoco, v) -~.
(A.2)
A.2. Coincidence counting The statistical precision of the coincidence method can be computed in a similar manner to that used in the determination of the statistical imprecision of the modulo-2 method, Var[N0 ] _
N~
Var[Nv]+
N2
Nv2 Var[N0]
NgNg NrZN~ Var[Nc ] + ,#--;?c'
2NrN~ cov{N~,Nc) -
2NvZNo cov{ N0, N~ }
NgU) +
2NvNo cov{ Nv, N0 }
(1.3)
References
[1] R.A. Allen, in: Alpha, beta and gamma ray spectroscopy, ed., K. Siegbahn (North-Holland, Amsterdam, 1965) pp. 425-465. [2] A.P. Baerg, Metrologia 12 (1976) 77. 13] J.W. Mialler, Nucl. Instr. and Meth. 189 (1981) 449. [4] J.E. De Carlos and C.E. Granados, Nucl. Instr. and Meth. 112 (1973) 209. [5] A.P. Baerg, Nucl. Instr. and Meth. 190 (1981) 345. [6] P. Br6once, Rapport BIPM-76/14 (1976) (in French) 9pp. [7] J.W. Mi~ller, Rapport BIPM-72/4 (1972). [8] J.W. Mailer, personal communication (1983). [9] J.W. Mt~ller, Rapport BIPM-75/7 (1975). [10] K.R.D. Mylon, G.W. McBeth and L. Siadat, Nucl. Instr. and Meth. 205 (1983) 197. [11] A handbook of radioactivity measurements procedures, NCRP Report No. 58, (1978) p. 75. [12] L. Siadat, G.W, McBeth and K.R.D. Mylon, Nucl. Instr. and Meth. 205 (1983) 203. [13] P.J. Campion and J.G.V. Taylor, Int. J. Appl. Rad. Isotopes 10 (1961) 131.