- Email: [email protected]
Modification of the Cox-Isham formula for 4πβ-γ-coincidence counting by the “Gandy”-effect and its automatic correction by electronic circuits
Nuclear Instruments and Methods 175 (1980) 509-514 © North-Holland Publishing Company
MODIFICATION OF THE C O X - I S H A M FORMULA FOR 4n/3-"y-COINCIDENCE COUNTING BY THE "GANDY"-EFFECT AND ITS AUTOMATIC CORRECTION BY ELECTRONIC CIRCUITS E. FUNCK Physikalisch- Technische Bundesanstalt, Braunsch weig, FR G
Received 25 March 1980
The correction formula of Cox and Isham for accidental coincidences and dead time losses in a 4zr/3-,,/-coincidence system is modified for the relative delay between ~- and -r-channels ("Gandy effect"). This extended formula is used to study the problems connected with an automatic measurement and compensation of the relative delay. Basic delay matching circuits for this purpose are proposed.
have used a circuit which permits the adjustment of the delay to a good approximation for symmetrical time distributions by equalizing the counting frequency of 13- and "y-pulses arriving first at the coincidence mixer. The method described in this paper can be understood as a realization of the numerical calculation by means of an electronic circuit offering the advantage that no restrictions to the shape of the time distribution have to be made. The principle of this method was mentioned by Williams in a discussion [7], and a circuit derived from it has been realized in the AECL [8]. The circuits proposed here are further developed to realize self-adjustment of the delay, making any correction unnecessary.
1. Introduction Unequal delay of pulses in/3- and "y-channels of a 4n/3-7-coincidence system requires a correction in the accidental coincidence rate [1]. In normal application this correction, usually called "Gandy effect" [2], is small, but it cannot be neglected, especially when high count rates are used. It has therefore been found helpful to allow for this effect in the correction formula published by Cox and Isham [3] in order to extend it to the case of non-zero relative delay between /3- and 7-channels. This is in fact the typical situation, as the relative delay cannot be adjusted to zero prior to the measurements. As the relative time of arrival of/3- and "y-pulses at the coincidence mixer is spread over a range of 0 . 2 - 1 . 0 /as due to "time jitter" [4], it is necessary to pick up the time distribution in a multi-channel analyzer and to calculate from it the mean relative delay by numerical evaluation. This procedure has to be frequently repeated, as the coincidence distribution varies with nuclide, f'filing gas and high voltage of the proportional counter, thickness of the absorber on the source, and energy level of the discriminator in both channels. These circumstances ask for continuous and careful adjustments, and make the relative delay a more critical parameter than the dead time or the coincidence resolving time [5]. An automatic adjustment of the delay, or at least an automatic measurement, would therefore considerably simplify the measuring procedure. Munzenmayer and Baerg [6] as well as Gostely and Noverraz [12]
2. Extension of the Cox-lsham formula by relative delay between/3- and "y-channels For the treatment of the correction due to the relative delay between/3- and 7-channels it is assumed that the relative delay is constant, and not subject to "time-jitter" as real /3-7-coincidences are (time distribution). The exclusion of this problem is necessary to simplify the treatment and is justified, since preliminary results of investigations still going on show that the effect from "time-jitter" is much smaller than that from the "Gandy effect" [9]. Let the delay imposed on the 7-channel exceed the one on the /3-channel by 5"r" This, then, will not change the dead time correction, as can be proved by discussing the effect on the equilibrium eqs. (11) and 509
5 i0
E. Funck/Modification of the Cox-lsham formula
(12) of [3], (further referred to as I). They are not changed by the relative delay and hence the corresponding probability functions q(t) containing the dead time correction are the same, leaving the dead time correction for the coincidence rate untouched. This situation is different for the rate of accidental coincidence. By elementary argumentation it can be shown that the time interval producing an accidental coincidence is shortened by 67 when it is formed by an unpaired /3-pulse followed by a pulse from the 7-channel. The interval is lengthened by the same amount when 13 and 7 are interchanged [1,4]. This argument is still valid in the case where the first pulse arrives while the other channel is blocked by a dead time period that ends before the second pulse enters the coincidence mixer [fourth and fifth term of eq. (38) of I]. Thus the introduction of a delay has the effect that the resolving time r r is replaced by 7R -- 67 or rR + 67, depending on whether a/3- or a 7-pulse goes first in a pair forming an accidental coincidence. When eq. (38) of I for the accidental coincidence rate is modified in such a way and the probability functions q(t) for equal dead times in both channels are taken from eq. (15) of I an expression for the disintegration rate No will be obtained that is similar to eq. (4) of the paper of Smith [10] (further referred to as II) containing the correction for a delay in the 7-channel: N o = N'cAe7 ( N o e NOr
fore approximated on the basis of the formalism described in section 3.5 of II. The relative delay &r is introduced into the expression for accidental coincidences by splitting the formula in parts containing accidental coincidences from leading/3- and leading 7-pulses. The rate of accidental coincidences will then read: N'c - P ~ N c =p#~(No - Nc)(1 - e -NT(r R-67) +p#v(N7 - Nc)(1 - e-N~(rR+87) + ~(rR -- 5.r)2N~JV7q~(r~) + ½(rR + 57)2NcIN3,qT(TT) - ~ (rR - 6-r) a [/V~N~q~(r#) I
+ W~flVvq~(r~3)]
-
I
+ N # T q 7 (r.r)] • Zeroth-order approximations are used for the probability functions qo(rO), qT(r7) of C o x - I s h a m and its derivatives qt~rg), q~(%) according to II to obtain the final form of the approximated correction for unequal dead times: Y - N ' c ) / ( N c - p~pTX2) No = N'flV~r( X1 + r 7 P#P'r
=
1 2 1 - NO(T R + 6~) - NT(T R - 5~,) + ~N~(TR + 5~,)2 1
- N, r eN7 r
(2)
where: X1
e(NT-No)(rR-87)
(TR + 6 )3
2
+ ~N~(rR -- 5,r)2 _ ~N~Nv(rR + 57) 2
e(No-NT)(rR+87)
- ~N~V7(TR - 6-t)26(r - 1) + N'c' (eNffr P#P7
eNTr)),/N'e(Nft eN~ r
e%
+ ~ (Wt~hP~ - N}NT[1 - 6(r - 1)] - N ~ } ( r k
(1) where: p# = 1 - N~r,p7 = 1 - N ~ r and
+ ~ [N~N76 (r - 1) + N ~ V ~ 6 ( r - 2) - W~l(rR - 5v) 3 X2 = 2NoWTr R + 2rR6v(NoN ~ - W~jV7) 1 3 3 + (r~57 + ~Sv)(W~W. r - N~N.~)
and Y taken from If, eq.
(8):
Arc = U 'c _ 2N't~V~TR written in a slightly different notation from II. NO, N 7 and Arc are the true count rates in the/3-, 3'- and coincidence channels; N~, N-~, N~ are the observed ones corrected for background, r is the dead time for both channels; rR is the coincidence resolving time. The general case of unequal dead times in both channels (r 7 = rr0, r > 1) does not lead to a solution in closed algebraic form as before. The result is there-
+ 67)3
Y = I ---}r'l {N~+ [ 2 - 5 ( r -
I)1N.~}
+ ~ - {N~ + [6 - 25(r - 1)l N~V 7 + [6 - 5(r - 2) - 56(r - 1)1N.~} r.~ {N~ + [14 - 35(r - 1)] NI~V7 24 '
E. Funck / Modification o f the Cox-Isham formula
+ [36 - 26(r - 2) - 256(r - 1)] N#V~ + [12 - 6(r - 3) - 66(r - 2) - l 1 6 ( r - 1)] N.~) t
po = 1 - N ' ~ r # ,
6(x)
=(l- Ixl
511
according to eqs. (1) or (2) can be neglected, at least as long as the width of the time distribution is not greater than 1/Is.
t
p ~ = 1 - N'~T v ,
for ] x [ ~ l
3. Principles of an automatic measurement of the mean delay
for Ixl > 1 . For r. r > r#, r. r = rTt~, the indices/3 and 7 have to be interchanged for all values. Equations (1) and (2) thus describe the effects of a relative delay between /3- and 7-channels on the correction formula for dead time losses and accidental coincidences. To illustrate the importance of the effect the value o f the correction is listed in table 1 for both the modified C o x - I s h a m formula for equal dead times o f eq. (1) (column 1), as well as the original Gandy formula (column 2). F r o m the table below it can be seen that the difference between the correction due to the modified C o x - I s h a m formula and the Gandy formula is not very great. It surpasses the limit o f 0.1% only when the delay has reached 350 ns - a value that will certainly be avoided in normal application. Another interesting feature is the linearity o f the correction. If this condition is not fulfilled the correction will not follow a linear function and hence the calculation of the mean delay from a coincidence time distribution must be corrected for non-linearity. This, obviously, would hinder an automatic calculation or measurement by electronic devices. However, the deviation from linearity proves to be small. It does not exceed 2 × 10 -4 up to a delay o f 450 ns. Without serious disadvantage the non-linearity
The mean delay o f a coincidence time distribution (for a typical example, see fig. 1) is calculated by: =
fw(t) t dr,
(3)
where W ( t ) is the normalized probability for the relative delay t between /3- and 7-channels. When the/3channel is delayed in such a way that every/3-pulse o f a coincident /3-7-pair arrives later than the 7-pulse, the time distribution can be picked up by a time-topulse-height converter or a time-to.digital converter and stored in a multichannel analyzer. The foregoing
5,5 xl 0a 5,0
I
4,5
.~ 4,0 t~ I.z 3,5 i ~- 3,0
~_ 2,5'
Table 1 The value o f the correction for the relative delay according to •eq. (1) and Gandy. Activity of source: 100 000 Bq: p/~/efficiency: 0.9/0.2; dead time: ~'t~ = r6 = 3.0 ~s; coinc, time: r R = 0.5/,ts
Delay (ns)
Eq. (1)
Gandy
0 50 100 150 200 250 300 350 400 450
0.0000 0,0027 0.0055 0.0082 0.0110 0.0138 0.0167 0.0195 0.0224 0.0253
0.0000 0.0026 0.0052 0.0078 0.0105 0.0132 0.0258 0.0185 0.0212 0.0240
u u
2,0
1,5 m Z 1,0
0,5
0 0 ~
'o.o
0.2
J
,,,,,,
....
o.4
i ....
k h,
0.6
~.8
|
~,,
i
i.o
ARRIVAL TIME OF COINCIDENT ~,-PULSES
Fig. 1. Coincidence time distribution of 139Ce in a 4nfl-~/system with pressure proportional counter representing the probability of the arrival time of/~-pulses relative to the coincident y-pulse. The splitting into two peaks is due to the heights of K- and L-capture pulses that trigger the discriminator at different times.
512
E. Funck / Modification o f the Cox-Isham formula
integral should then be transformed into:
=~
iKif
Ki.
(4)
i=1
There are two summations, one over the channel contents K i in the denominator and the other over the channel contents multiplied by channel address i and channel width f ( i f = delay time) in the nominator. Instead of calculating the mean delay the summations of eq. (4) can be done by electronics. The digital information of the channel address i is then stored in an addition unit. Every delay t registered for a fl-3,-pair adds its channel number i to the one before, thus building up the sum of the nominator. At the same time the event is counted by a scaler representing the sum of the denominator. The electronic circuit thus consists mainly of an addition unit and a scaler. After a certain time long enough to register a sufficiently large number of pulse pairs in order to keep the statistical error within reasonable limits, .the summing-up is stopped. By dividing the number shown in the addition unit by the number of the scaler, the mean delay is obtained in units of the channel width f. The addition unit may be replaced by a scaler, when the information of the channel address is available as a pulse train containing as many pulses as are indicated by the number of the address. This is the case for a Wilkinson-type analog-to-digital converter.
DUB ECDU TAC/ADC DAC CUN CUD PU BUD
This ADC has an oscillator whose pulses are counted for a time interval that is determined by the pulse height. An alternative is a time-to-digital converter in which the pulse pair determines the gating period of a scaler counting the pulses of a high frequency crystal oscillator [ 11 ]. The system for an automatic measurement consists of two scalers: one for the channel address and the other for the "BUSY" signal (fig. 2). The division, finally, can be circumvented when the summing-up in the address-scaler is stopped after the second scaler has counted a number of pulse pairs that equals an integer power of ten. The value read in the address scaler is equal to the mean delay, apart from a factor for the channel width, when an appropriate number of divide-by-ten circuits is put into the line of the address scaler.
4. Design of an electronic circuit for automatic delay
matching By straightforward design the circuit for the determination of the automatic delay may be extended to realize automatic delay matching. The digital information of the mean delay has only to be transferred to a delay unit that is set by that value. This externally controlled delay unit may be composed of digital or analog electronics and be combined to the delay measuring circuit to form an open loop or a feedback control. More reliable function can be expected from digital electronics and this then will not necessitate feedback control. The circuit proposed here is made solely from digital electronic circuits. The block diagram is shown in fig. 3. The outline of the circuitry is the same as shown in fig. 2 except for the DAC which can be omitted. Details of
Delay Unit Beta Channel Externally Controlled Delay Unit Time Amplitude Cony. and Analog Digital Cony. Digital Analog Converter Scaler for Address Pulse Trains Scaler for Number of Registered Pulse Pairs Preset for CUD Buffer for Delay Value
Fig. 2. Open loop controlled delay matching.
Fig. 3. Delay matching by digital circuit.
E. Funck / Modification o f the Cox-lsham formula
the TDC-ECDU-combination are shown in fig. 3. The time base of the circuit is the oscillator, the pulses of which are fed to the scalers TCU and DCU. The scaler TCU is part of the time-to-digital converter. It is started by an incoming 7-pulse and is stopped by a /3-pulse that has been accepted by the unit SAC. This unit releases a signal "Stop Accepted" to DR for transmission of the address into the addition unit ADD when the/3-pulse arrives, before more time has elapsed than given by the range chosen (RANGE PS). The "Stop Accepted" signal is counted in the scaler CUD for the number of pulse pairs registered. As soon as a preset value of PVD has been reached, the summing up is stopped, marking the end of a counting cycle. The sum of ADD is then stored in the buffer BUD and offered to PUD which sets the delay of the T-pulses. The T-pulse that has started the counting in the scaler DCU is released to the coincidence mixer when the number given by PVD has been reached. That means, the stop signal ending the counting is identical with the delayed T-pulse. A critical parameter for the TDC is the frequency of the oscillator or the time resolution. It has to be adapted to the time distribution of the l~-7-coincidences. Fig. 1 gives an example of such a distribution. The first peak has a fwhm of 32 ns. The time resolution of the TDC should be equal to this value or better, which means that the frequency of the oscillator should not be lower than 30 MHz to allow for adequate compensation of the mean delay. The accuracy of the delay unit ECDU is then limited by this value. To put it more precisely, it is erroneous by I d (d = 1/f, f = frequency of the oscillator) since a 7-pulse thathas to be delayed falls between two oscillator pulses. This deviation, however, does not impair the delay matching. The mean delay may have been determined by the TDC as a value of (n + e)d (n = integer, 0 ~
513
5. Comparison with other methods The above-mentioned circuit of Munzenmayer and Baerg [6] as well as the one of Gostely [12] are the only systems which were found to have been published. Both are based on the same principle and offer substantially the possibility of indicating a delay mismatch between /3- and 7-channels. The adjustment is made by hand. Neither method produces precise results for any time distribution. The delay matching is assumed to have been reached when, on the average, 1~- and 3'-pulses open the coincidence gate with equal frequency. In most cases this leads to good approximations. There are considerable deviations, however, when the time distribution is asymmetrical.
6. Conclusion It has been shown that the relative delay between /3- and 7-channels introduced into the formulas of Cox-Isham [3] for dead time losses and accidental coincidences changes the activity value by a factor which, in normal application, can be described by a linear function. The mean delay can therefore be determined by the weighted mean of the time distribution without correcting for non-linearity due to eqs. (1) or (2). The automatic measurement of the mean delay, applying the principle explained here, will therefore be free from systematic deviations. The only drawback of this method, compared with numerical calculation is the neglect of the influence of the background from accidental coincidences. If the ~- and the 7-efficiencies differ considerably and are relatively small, the distribution of accidental coincidences cannot be neglected. They have to be subtracted from the true coincidences before the mean delay is calculated. This, obviously, cannot be done by elementary electronic circuits, and care has to be taken not to overestimate the quality of an automatic measurement. The author is indebted to Dr. D. Smith from NPL for his helpful comments on the manuscript, and to Dr. H.M. Wei13 PTB, for supporting this work.
References [1] A. Gandy, Int. J. App. Rad. Isot. 11 (1961) 75. [2] J.W. Miiller, BIPM Working Party Note 207 (1977).
514
E. Funck /Modification of the Cox-lsham formula
[3] D.R. Cox and V. Isham, Proc. Roy. Soc. Lond. A35 (1977) 149. [4] A. Williams and P.J. Campion, Int. J. App. Rad. Isot. 16 (1965) 555. [5] D. Smith, A. Williams and M.J. Woods, Rapport BIPM77/7, (1977). [6] K. Munzenmayer and A.P. Baerg, Nucl. Instr. and Meth. 70 (1969) 157. [7] A. Williams, IAEA Symp. on Standardization of Radionuclides (Vienna, 1967) p. 147.
[8] [9] [10] [11]
F. Gibson, private communication, AECL (1978). E. Funck, PTB-Jahresbericht 1978 (1978) 191. D. Smith, Nucl. Instr. and Meth. 152 (1978) 505. I. Bialkowski, L. Harms-Ringdahl and I. Sztarkier, Nucl. Instr. and Meth. 123 (1975) 605. [12] J.-J. Gostely and O. Noverraz, Nucl. Instr. and Meth. 131 (1975) 69.
MODIFICATION OF THE C O X - I S H A M FORMULA FOR 4n/3-"y-COINCIDENCE COUNTING BY THE "GANDY"-EFFECT AND ITS AUTOMATIC CORRECTION BY ELECTRONIC CIRCUITS E. FUNCK Physikalisch- Technische Bundesanstalt, Braunsch weig, FR G
Received 25 March 1980
The correction formula of Cox and Isham for accidental coincidences and dead time losses in a 4zr/3-,,/-coincidence system is modified for the relative delay between ~- and -r-channels ("Gandy effect"). This extended formula is used to study the problems connected with an automatic measurement and compensation of the relative delay. Basic delay matching circuits for this purpose are proposed.
have used a circuit which permits the adjustment of the delay to a good approximation for symmetrical time distributions by equalizing the counting frequency of 13- and "y-pulses arriving first at the coincidence mixer. The method described in this paper can be understood as a realization of the numerical calculation by means of an electronic circuit offering the advantage that no restrictions to the shape of the time distribution have to be made. The principle of this method was mentioned by Williams in a discussion [7], and a circuit derived from it has been realized in the AECL [8]. The circuits proposed here are further developed to realize self-adjustment of the delay, making any correction unnecessary.
1. Introduction Unequal delay of pulses in/3- and "y-channels of a 4n/3-7-coincidence system requires a correction in the accidental coincidence rate [1]. In normal application this correction, usually called "Gandy effect" [2], is small, but it cannot be neglected, especially when high count rates are used. It has therefore been found helpful to allow for this effect in the correction formula published by Cox and Isham [3] in order to extend it to the case of non-zero relative delay between /3- and 7-channels. This is in fact the typical situation, as the relative delay cannot be adjusted to zero prior to the measurements. As the relative time of arrival of/3- and "y-pulses at the coincidence mixer is spread over a range of 0 . 2 - 1 . 0 /as due to "time jitter" [4], it is necessary to pick up the time distribution in a multi-channel analyzer and to calculate from it the mean relative delay by numerical evaluation. This procedure has to be frequently repeated, as the coincidence distribution varies with nuclide, f'filing gas and high voltage of the proportional counter, thickness of the absorber on the source, and energy level of the discriminator in both channels. These circumstances ask for continuous and careful adjustments, and make the relative delay a more critical parameter than the dead time or the coincidence resolving time [5]. An automatic adjustment of the delay, or at least an automatic measurement, would therefore considerably simplify the measuring procedure. Munzenmayer and Baerg [6] as well as Gostely and Noverraz [12]
2. Extension of the Cox-lsham formula by relative delay between/3- and "y-channels For the treatment of the correction due to the relative delay between/3- and 7-channels it is assumed that the relative delay is constant, and not subject to "time-jitter" as real /3-7-coincidences are (time distribution). The exclusion of this problem is necessary to simplify the treatment and is justified, since preliminary results of investigations still going on show that the effect from "time-jitter" is much smaller than that from the "Gandy effect" [9]. Let the delay imposed on the 7-channel exceed the one on the /3-channel by 5"r" This, then, will not change the dead time correction, as can be proved by discussing the effect on the equilibrium eqs. (11) and 509
5 i0
E. Funck/Modification of the Cox-lsham formula
(12) of [3], (further referred to as I). They are not changed by the relative delay and hence the corresponding probability functions q(t) containing the dead time correction are the same, leaving the dead time correction for the coincidence rate untouched. This situation is different for the rate of accidental coincidence. By elementary argumentation it can be shown that the time interval producing an accidental coincidence is shortened by 67 when it is formed by an unpaired /3-pulse followed by a pulse from the 7-channel. The interval is lengthened by the same amount when 13 and 7 are interchanged [1,4]. This argument is still valid in the case where the first pulse arrives while the other channel is blocked by a dead time period that ends before the second pulse enters the coincidence mixer [fourth and fifth term of eq. (38) of I]. Thus the introduction of a delay has the effect that the resolving time r r is replaced by 7R -- 67 or rR + 67, depending on whether a/3- or a 7-pulse goes first in a pair forming an accidental coincidence. When eq. (38) of I for the accidental coincidence rate is modified in such a way and the probability functions q(t) for equal dead times in both channels are taken from eq. (15) of I an expression for the disintegration rate No will be obtained that is similar to eq. (4) of the paper of Smith [10] (further referred to as II) containing the correction for a delay in the 7-channel: N o = N'cAe7 ( N o e NOr
fore approximated on the basis of the formalism described in section 3.5 of II. The relative delay &r is introduced into the expression for accidental coincidences by splitting the formula in parts containing accidental coincidences from leading/3- and leading 7-pulses. The rate of accidental coincidences will then read: N'c - P ~ N c =p#~(No - Nc)(1 - e -NT(r R-67) +p#v(N7 - Nc)(1 - e-N~(rR+87) + ~(rR -- 5.r)2N~JV7q~(r~) + ½(rR + 57)2NcIN3,qT(TT) - ~ (rR - 6-r) a [/V~N~q~(r#) I
+ W~flVvq~(r~3)]
-
I
+ N # T q 7 (r.r)] • Zeroth-order approximations are used for the probability functions qo(rO), qT(r7) of C o x - I s h a m and its derivatives qt~rg), q~(%) according to II to obtain the final form of the approximated correction for unequal dead times: Y - N ' c ) / ( N c - p~pTX2) No = N'flV~r( X1 + r 7 P#P'r
=
1 2 1 - NO(T R + 6~) - NT(T R - 5~,) + ~N~(TR + 5~,)2 1
- N, r eN7 r
(2)
where: X1
e(NT-No)(rR-87)
(TR + 6 )3
2
+ ~N~(rR -- 5,r)2 _ ~N~Nv(rR + 57) 2
e(No-NT)(rR+87)
- ~N~V7(TR - 6-t)26(r - 1) + N'c' (eNffr P#P7
eNTr)),/N'e(Nft eN~ r
e%
+ ~ (Wt~hP~ - N}NT[1 - 6(r - 1)] - N ~ } ( r k
(1) where: p# = 1 - N~r,p7 = 1 - N ~ r and
+ ~ [N~N76 (r - 1) + N ~ V ~ 6 ( r - 2) - W~l(rR - 5v) 3 X2 = 2NoWTr R + 2rR6v(NoN ~ - W~jV7) 1 3 3 + (r~57 + ~Sv)(W~W. r - N~N.~)
and Y taken from If, eq.
(8):
Arc = U 'c _ 2N't~V~TR written in a slightly different notation from II. NO, N 7 and Arc are the true count rates in the/3-, 3'- and coincidence channels; N~, N-~, N~ are the observed ones corrected for background, r is the dead time for both channels; rR is the coincidence resolving time. The general case of unequal dead times in both channels (r 7 = rr0, r > 1) does not lead to a solution in closed algebraic form as before. The result is there-
+ 67)3
Y = I ---}r'l {N~+ [ 2 - 5 ( r -
I)1N.~}
+ ~ - {N~ + [6 - 25(r - 1)l N~V 7 + [6 - 5(r - 2) - 56(r - 1)1N.~} r.~ {N~ + [14 - 35(r - 1)] NI~V7 24 '
E. Funck / Modification o f the Cox-Isham formula
+ [36 - 26(r - 2) - 256(r - 1)] N#V~ + [12 - 6(r - 3) - 66(r - 2) - l 1 6 ( r - 1)] N.~) t
po = 1 - N ' ~ r # ,
6(x)
=(l- Ixl
511
according to eqs. (1) or (2) can be neglected, at least as long as the width of the time distribution is not greater than 1/Is.
t
p ~ = 1 - N'~T v ,
for ] x [ ~ l
3. Principles of an automatic measurement of the mean delay
for Ixl > 1 . For r. r > r#, r. r = rTt~, the indices/3 and 7 have to be interchanged for all values. Equations (1) and (2) thus describe the effects of a relative delay between /3- and 7-channels on the correction formula for dead time losses and accidental coincidences. To illustrate the importance of the effect the value o f the correction is listed in table 1 for both the modified C o x - I s h a m formula for equal dead times o f eq. (1) (column 1), as well as the original Gandy formula (column 2). F r o m the table below it can be seen that the difference between the correction due to the modified C o x - I s h a m formula and the Gandy formula is not very great. It surpasses the limit o f 0.1% only when the delay has reached 350 ns - a value that will certainly be avoided in normal application. Another interesting feature is the linearity o f the correction. If this condition is not fulfilled the correction will not follow a linear function and hence the calculation of the mean delay from a coincidence time distribution must be corrected for non-linearity. This, obviously, would hinder an automatic calculation or measurement by electronic devices. However, the deviation from linearity proves to be small. It does not exceed 2 × 10 -4 up to a delay o f 450 ns. Without serious disadvantage the non-linearity
The mean delay o f a coincidence time distribution (for a typical example, see fig. 1) is calculated by: =
fw(t) t dr,
(3)
where W ( t ) is the normalized probability for the relative delay t between /3- and 7-channels. When the/3channel is delayed in such a way that every/3-pulse o f a coincident /3-7-pair arrives later than the 7-pulse, the time distribution can be picked up by a time-topulse-height converter or a time-to.digital converter and stored in a multichannel analyzer. The foregoing
5,5 xl 0a 5,0
I
4,5
.~ 4,0 t~ I.z 3,5 i ~- 3,0
~_ 2,5'
Table 1 The value o f the correction for the relative delay according to •eq. (1) and Gandy. Activity of source: 100 000 Bq: p/~/efficiency: 0.9/0.2; dead time: ~'t~ = r6 = 3.0 ~s; coinc, time: r R = 0.5/,ts
Delay (ns)
Eq. (1)
Gandy
0 50 100 150 200 250 300 350 400 450
0.0000 0,0027 0.0055 0.0082 0.0110 0.0138 0.0167 0.0195 0.0224 0.0253
0.0000 0.0026 0.0052 0.0078 0.0105 0.0132 0.0258 0.0185 0.0212 0.0240
u u
2,0
1,5 m Z 1,0
0,5
0 0 ~
'o.o
0.2
J
,,,,,,
....
o.4
i ....
k h,
0.6
~.8
|
~,,
i
i.o
ARRIVAL TIME OF COINCIDENT ~,-PULSES
Fig. 1. Coincidence time distribution of 139Ce in a 4nfl-~/system with pressure proportional counter representing the probability of the arrival time of/~-pulses relative to the coincident y-pulse. The splitting into two peaks is due to the heights of K- and L-capture pulses that trigger the discriminator at different times.
512
E. Funck / Modification o f the Cox-Isham formula
integral should then be transformed into:
=~
iKif
Ki.
(4)
i=1
There are two summations, one over the channel contents K i in the denominator and the other over the channel contents multiplied by channel address i and channel width f ( i f = delay time) in the nominator. Instead of calculating the mean delay the summations of eq. (4) can be done by electronics. The digital information of the channel address i is then stored in an addition unit. Every delay t registered for a fl-3,-pair adds its channel number i to the one before, thus building up the sum of the nominator. At the same time the event is counted by a scaler representing the sum of the denominator. The electronic circuit thus consists mainly of an addition unit and a scaler. After a certain time long enough to register a sufficiently large number of pulse pairs in order to keep the statistical error within reasonable limits, .the summing-up is stopped. By dividing the number shown in the addition unit by the number of the scaler, the mean delay is obtained in units of the channel width f. The addition unit may be replaced by a scaler, when the information of the channel address is available as a pulse train containing as many pulses as are indicated by the number of the address. This is the case for a Wilkinson-type analog-to-digital converter.
DUB ECDU TAC/ADC DAC CUN CUD PU BUD
This ADC has an oscillator whose pulses are counted for a time interval that is determined by the pulse height. An alternative is a time-to-digital converter in which the pulse pair determines the gating period of a scaler counting the pulses of a high frequency crystal oscillator [ 11 ]. The system for an automatic measurement consists of two scalers: one for the channel address and the other for the "BUSY" signal (fig. 2). The division, finally, can be circumvented when the summing-up in the address-scaler is stopped after the second scaler has counted a number of pulse pairs that equals an integer power of ten. The value read in the address scaler is equal to the mean delay, apart from a factor for the channel width, when an appropriate number of divide-by-ten circuits is put into the line of the address scaler.
4. Design of an electronic circuit for automatic delay
matching By straightforward design the circuit for the determination of the automatic delay may be extended to realize automatic delay matching. The digital information of the mean delay has only to be transferred to a delay unit that is set by that value. This externally controlled delay unit may be composed of digital or analog electronics and be combined to the delay measuring circuit to form an open loop or a feedback control. More reliable function can be expected from digital electronics and this then will not necessitate feedback control. The circuit proposed here is made solely from digital electronic circuits. The block diagram is shown in fig. 3. The outline of the circuitry is the same as shown in fig. 2 except for the DAC which can be omitted. Details of
Delay Unit Beta Channel Externally Controlled Delay Unit Time Amplitude Cony. and Analog Digital Cony. Digital Analog Converter Scaler for Address Pulse Trains Scaler for Number of Registered Pulse Pairs Preset for CUD Buffer for Delay Value
Fig. 2. Open loop controlled delay matching.
Fig. 3. Delay matching by digital circuit.
E. Funck / Modification o f the Cox-lsham formula
the TDC-ECDU-combination are shown in fig. 3. The time base of the circuit is the oscillator, the pulses of which are fed to the scalers TCU and DCU. The scaler TCU is part of the time-to-digital converter. It is started by an incoming 7-pulse and is stopped by a /3-pulse that has been accepted by the unit SAC. This unit releases a signal "Stop Accepted" to DR for transmission of the address into the addition unit ADD when the/3-pulse arrives, before more time has elapsed than given by the range chosen (RANGE PS). The "Stop Accepted" signal is counted in the scaler CUD for the number of pulse pairs registered. As soon as a preset value of PVD has been reached, the summing up is stopped, marking the end of a counting cycle. The sum of ADD is then stored in the buffer BUD and offered to PUD which sets the delay of the T-pulses. The T-pulse that has started the counting in the scaler DCU is released to the coincidence mixer when the number given by PVD has been reached. That means, the stop signal ending the counting is identical with the delayed T-pulse. A critical parameter for the TDC is the frequency of the oscillator or the time resolution. It has to be adapted to the time distribution of the l~-7-coincidences. Fig. 1 gives an example of such a distribution. The first peak has a fwhm of 32 ns. The time resolution of the TDC should be equal to this value or better, which means that the frequency of the oscillator should not be lower than 30 MHz to allow for adequate compensation of the mean delay. The accuracy of the delay unit ECDU is then limited by this value. To put it more precisely, it is erroneous by I d (d = 1/f, f = frequency of the oscillator) since a 7-pulse thathas to be delayed falls between two oscillator pulses. This deviation, however, does not impair the delay matching. The mean delay may have been determined by the TDC as a value of (n + e)d (n = integer, 0 ~
513
5. Comparison with other methods The above-mentioned circuit of Munzenmayer and Baerg [6] as well as the one of Gostely [12] are the only systems which were found to have been published. Both are based on the same principle and offer substantially the possibility of indicating a delay mismatch between /3- and 7-channels. The adjustment is made by hand. Neither method produces precise results for any time distribution. The delay matching is assumed to have been reached when, on the average, 1~- and 3'-pulses open the coincidence gate with equal frequency. In most cases this leads to good approximations. There are considerable deviations, however, when the time distribution is asymmetrical.
6. Conclusion It has been shown that the relative delay between /3- and 7-channels introduced into the formulas of Cox-Isham [3] for dead time losses and accidental coincidences changes the activity value by a factor which, in normal application, can be described by a linear function. The mean delay can therefore be determined by the weighted mean of the time distribution without correcting for non-linearity due to eqs. (1) or (2). The automatic measurement of the mean delay, applying the principle explained here, will therefore be free from systematic deviations. The only drawback of this method, compared with numerical calculation is the neglect of the influence of the background from accidental coincidences. If the ~- and the 7-efficiencies differ considerably and are relatively small, the distribution of accidental coincidences cannot be neglected. They have to be subtracted from the true coincidences before the mean delay is calculated. This, obviously, cannot be done by elementary electronic circuits, and care has to be taken not to overestimate the quality of an automatic measurement. The author is indebted to Dr. D. Smith from NPL for his helpful comments on the manuscript, and to Dr. H.M. Wei13 PTB, for supporting this work.
References [1] A. Gandy, Int. J. App. Rad. Isot. 11 (1961) 75. [2] J.W. Miiller, BIPM Working Party Note 207 (1977).
514
E. Funck /Modification of the Cox-lsham formula
[3] D.R. Cox and V. Isham, Proc. Roy. Soc. Lond. A35 (1977) 149. [4] A. Williams and P.J. Campion, Int. J. App. Rad. Isot. 16 (1965) 555. [5] D. Smith, A. Williams and M.J. Woods, Rapport BIPM77/7, (1977). [6] K. Munzenmayer and A.P. Baerg, Nucl. Instr. and Meth. 70 (1969) 157. [7] A. Williams, IAEA Symp. on Standardization of Radionuclides (Vienna, 1967) p. 147.
[8] [9] [10] [11]
F. Gibson, private communication, AECL (1978). E. Funck, PTB-Jahresbericht 1978 (1978) 191. D. Smith, Nucl. Instr. and Meth. 152 (1978) 505. I. Bialkowski, L. Harms-Ringdahl and I. Sztarkier, Nucl. Instr. and Meth. 123 (1975) 605. [12] J.-J. Gostely and O. Noverraz, Nucl. Instr. and Meth. 131 (1975) 69.