Automatic Compensation of Dead Time in Pulse Analysis Equipment D. F. COVELL, M. M. SANDOMIRE,’ and M. S. EICHEN2 U. S. Naval Radiological Defense laboratory, San Francisco 24, Calif.
b In pulse height analyzers of the amplitude-to-time conversion type, pulse height is converted to a proportionate interval of time. As a result, the dead time characteristic of the instrument depends upon the actual distribution being measured, and the correction for dead time is difficult to determine and laborious to apply. A device has been constructed and evaluated which automatically applies a correction for dead time. Gross counting rates to 1,000,000 per minute can be analyzed without additional dead time correction. Use of this method reduces labor and increases precision in evaluating pulse analysis data. The technique is applicable to problems of dead time in equipment generally.
T
o DO high-precision counting on radioactive samples of a wide range of counting rates, one must ordinarily estimate the dead time characteristics of the counting system over the ranges of energies and counting rates of interest, and use this to correct an observed rate. Dead time here is defined as any interval of time during a normal counting period for which the counting system is incapable of recognizing the occurrence of an event. Such paralysis may be introduced into the counting system as a result of performance characteristics inherent in the detector, amplifier, or recording apparatus. Correction factors may be difficult to determine; once determined, they may be laborious to apply, and, in any event, detract from the precision of the data obtained because of the uncertainty of the correction factor. The present discussion is concerned with a problem and a proposed solution based on usage requirements for an Argonne type 256-channel pulse height analyzer ( 5 ) , in which a correction for dead time during any counting interval depends not only on counting rate but Present address, Kestern Regional Research Laboratory, U. S. Department of Agriculture, Albany, Calif. Present address, General Atomic Division, General Dynamics Corp., San Diego, Calif. 1086
ANALYTICAL CHEMISTRY
also on the distribution of pulse amplitudes. A device has been constructed which is similar in principle to that described ( 3 ) . This device provides an estimate of the actual sampling time during any counting interval, so that a dead time estimate and its subsequent application are unnecessary in obtaining an unbiased estimate of a counting rate. The solution proposed and the analysis made, while specific for the problem inherent in the pulse height analyzer equipment, are applicable t o problems of dead time in equipment generally. DESCRIPTION OF PROBLEM
The following discussion is based on technical detail (4). In the Argonne type pulse-height analyzer, pulse height is converted to a proportionate time interval. An input pulse which is accepted for analysis is “stretched” in such a manner that voltage proportional to the peak amplitude of the input pulse is held constant for as long as required by other circuits. This voltage is applied t o one input of a voltage amplitude comparison circuit. Shortly after the input pulse has charged the stretcher fully, the output voltage from a linear ramp function generator (linear sweep) is caused to increase linearly. This voltage is applied to the second input of the voltage amplitude comparison circuit. The time interval between the start of the linear sweep and the time a t which the two voltages are equal is proportional to the amplitude of the input signal. The same circuits which control the start and end of this interval also control a gate following a 2-Mc. oscillator. Thus a pulse train is generated containing pulses, the number of which is proportional to the amplitude of the input pulse, and the estimate of amplitude becomes a matter of determining the number of pulses in these pulse bundles. The analyzer contains 256 separate storage channels (memory), and once the number of pulses in a pulse bundle has been determined, a storage pulse is entered in the storage channel whose number corresponds with the number of pulses in the pulse bundle. In addition to the time required for the linear sweep to be terminated, a fixed interval of time of approximately
16 psec. is required to complete the storage and clear the computer circuits preparatory to the acceptance of another pulse. Thus, the dead time, T , , associated with the amplitude analysis of a particular pulse is 7% =
16
+ e.2 Msec.
where c, is the number of the memory channel where the storage pulse is placed. At moderate counting rates, an input pulse may occur during the conversion of the preceding pulse. If this pulse is larger than the pulse in the conversion operation, the stretcher output voltage will be increased to a level proportional to the larger pulse and the smaller pulse will be ignored. Thus a distortion of the spectrum will be introduced. To prevent this distortion, a linear gate has been placed ahead of the pulse stretcher. This gate is closed during each conversion. At high input rates, the linear gate might be opened during the period that an input pulse exists, which also would result in a spectral distortion. An additional control circuit referred to as trigger pair one (TP-1) has been included, such that it must trigger for a pulse to be accepted for analysis. TP-1 cannot be triggered by an input pulse unless the effect of any previous pulse has vanished. The total arrangement of gates is such that it is impossible for any distorted pulse to be analyzed, regardless of counting rate, with the exception of those pulses occurring within 0.4 psec. of each other. If a large amount of high energy radiation is present during the examination of low energy radiation, time would be consumed analyzing the “off-scale” or “overflow” pulses. To reduce this unnecessary dead time, an upper level discriminator has been included, such that analysis of these overflow pulses is prevented. Thus, for overflow pulses, 128 = 144 psec. only instead of 16 16 psec. are required to dispose of the pulse. Several factors make individual contributions to the total dead time in any counting period, and any attempt to correct for dead time must take each of these factors into consideration. To estimate correction factors, dead time
+
resulting from the analysis of accepted pulses could be calculated from observed pulse height data, because the time required for the analysis of individual pulses is known. However, the number of pulses rejected because they were distorted by the effects of earlier pulses, and the number of pulses rejected as overflow pulses, will depend upon observation of the exact conditions of rate and pulse height as well as the settings and stability of the recognition points of the several trigger circuits which comprise the basic recognition circuitry of the analyzer. The contribution to the total dead time from these latter effects would be difficult to calculate with precision and such calculations could not be generalized with respect to all situations.
Solving for T gives T =
n' $
-
~
t
)
(9)
Substitution of this value of T in Equation 5 gives T: = n' 2 F Figure 1 . Typical counting system under control of d e a d time-compensating timer
- nlO r #
(10)
This system then provides that the count rate in Equation 4 can be estimated as
Figu clock wiih
SYSTEM FOR DEAD TIME COMPENSATION
A device has been constructed which provides a train of pulses to a time recorder through a normally open gate. Voltage wave forms which correspond in duration to the individual dead time intervals taking place a-ithin the analyzer are extracted and applied to this gate in such a manner that possible timing pulses are blocked. Thus, from the number of timing pulses which are recorded, one can estimate the actual sampling time of the analyzer. Figure 1 shows the basic block diagram of this system.
The term (T - n&) in the denominator of Equation 3 is total counting time less total dead time, and may be defined as expected live time or analysis time of the system, so that (4)
RELATIONS INHERENT IN THE SCHEME
The estimation of counting rates as well as the expected precision of the results obtained with the counting system described above is based on the following derivations. Estimation of Counting Rates. Let R' counts per srcond be the true rate a t which randomly occurring pulses are presented t o the counting system. Let Ri counts per second be the expected (or average) value of the rate recognized by the counting system, with an average dead time of T seconds per count. Then the expected value of the fractional dead time of the system is Ri T seconds per second. As a result, a total of R ' R ~ Tevents per second will be missed. Thus the incoming rate consists of the number recorded plus the number missed, or R' = RA f R'RAT
(1)
where Ti
- nhr
R: = FR& -
(5)
(6)
rt)
Let ni = RAT, and then Equation 6 can be rewritten as n:
(3)
T
To provide for compensation of 7, as described above, a pulse train of frequency F pulses per second, with a pulse width of T~ seconds per pulse is applied to a recording device through a normally open gate. Whenever an event is recognized by the counting system, a closure command of duration T is presented to the gate which is effectively closed for ( T - T ~ seconds. ) The average fra,ctional closure time of the gate is R , ( T - T ~ ) . As a result, Ri timing pulses per second will be cancelled, where
or
In time T , the expected number of events to be recorded will be RiT = no', and Equation 2 may be written as
=
=
F
?L~(T
-
~
t
)
(7)
Thus, of a total of FT timing pulses available in time T, nb is the expected number of timing pulses blocked from entering the time recorder. The number of timing pulses entered in the time recorder is then n' = F T
- n: = F ( T - nA(r - T t ) ]
in which no and nt are the observed values of nd and nt, respectively. In this expression it is noted that T and 7 do not appear. r l is a constant of the system, the value of which is specified by design. Expected Precision of Rate Determinations. An estimate of the variance of R is obtained from Equation 3 by the rules for propagation of error, as follows:
Using Equation 3
(13)
Because the incoming counts, no,are subjected to discontinuous sampling in short interrals of time, their variance is suppressed below the value given by the simple Poisson process. From Equation 1 RT
no = ___
1f R r
(14)
and from the m-ork of Evans and Feller (1, 2) the variance of is estimated as
(8) VOL. 32, NO. 9, AUGUST 1960
1087
in which R is the estimated incoming count rate and T is the average dead time interval associated with each of the no events. From Equations 14 and 15
4 s2(no)= R2T2 Substituting this value into Equation 13,
znd the standard error of the rate can be written
For constant dead time, T, this expression reduces to the Poisson error of a rate, because in this case the fractional error of a number of counts, n, and of the corresponding rate equals I/&* If the timing accessory in this system is used as a predetermined timer, the counting period is started prior to the occurrence of the first recorded timing pulse. The maximum analysis time between the start of the counting period and the recording of the first timing pulse is I j F and the average time is 1/2F. Thus, the value of T , as determined from Equation 10 should actually be
In the situation where the timing accessory is used as an elapsed time indicator, the counting period is started prior and stopped subsequent to the recording of a timing pulse. As a result, the correct value of analysis time would be, on the average,
The values of T, given by Equations 19 and 20, as compared to the values of T. given by Equation 10, represent biases. In using this system of timing, it would be most desirable to be able to disregard these biases. Since
?‘. > T. then
k
