Chapter No. 17

Radiation Detection and Measurements, Glenn T. Knoll,

Third edition (2000), John Willey .

Pulse Processing

IX-System Involving Pulse Timing B. Measurement of Timning Properties1.MULTICHANNEL TIME SPECTROSCOPYdiscussion of pulse timing systems by considering some basic concepts oftime interval measurement. The time-to-amplitude converter (TAC) is a that produces an output pulse with an amplitude proportional to the time interval between input "start" and "stop" pulses. The differential amplitude distribution of the output pulses as recorded by a multichannel analyzer is thusa measure of the distribution of time intervals between start and stop pulses and is often called a time spectrum.The time spectrum bears a close relation to the pulse height spectrum (or differential pulse height distribution). The abscissa, rather than pulse height. Is the time interval length T. The ordinate is dN/dT, the differential number of intervalswhose length lies within dT about the value T. The area under the time spectrum between any two limits is the number of intervals of length between the same limits. The discrete approximation to the ordinate, N/T, is the number of intervals of length within a finiteincrement T, normalized to the length of that increment. During the discussion that follows. it is also convenient to divide the number of recorded intervals by the measurement time to obtain the corresponding rate. The ordinate then becomes dr/dT for the time spectrum, to be interpreted as the differential rate of occurrence of intervals whose length lies

A simple system for recording a time spectrum is shown in Fig. 17.42. For the sake of illustration we assume that a single source of pulses such as a radiation detector provides a split output that is simultaneously sent down two signal branches. A time pick-off unit ineach branch provides timing logic pulses that are supplied to the start and stop inputs of a TAC. A fixed time delay is also present in the stop branch. The multichannel analyzer (MCA) records the number of time intervals that lie within many contiguous increments or channels of width AT, ranging from zero to some maximum interval length set by therange of the TAC. The maximum time range of the TAC is assumed to be small compared with the average spacing between signal pulses, so that the probability of more than one signal pulse per TAC interval is small. Alternatively, the output of a pulse generator maybe substituted for the random pulse source and adjusted to have a period between pulses which is larger than the TAC maximum range.Under these conditions, the time spectrum recorded by the MCA is exceedingly simple. In the absence of time jitter or walk, each timing pulse is produced at precisely ~same time in each branch, and therefore the start and stop pulses are always separated by the fixed delay value. The TAC always produces a constant-amplitude output that is therefore stored in a single channel of the MCA. If the delay is made smaller, the interval between start and stop pulses also decreases, producing a TAC pulse of smaller amplitude which will be stored in a lower-numbered channel.

A more realistic time spectrum shown in Fig. 17.43 takes into account the fact that all}fjtime pick-off method will always involve some degree of timing jitter and walk. If th~ timing uncertainties in each branch are independent of each other, the time spectrum wilt! display a distribution as shown in the sketch. Because the time pick-off in each branch iiilikely to behave similarly with regard to time walk caused by amplitUde or shape variations, most of the distribution spread measured in this conceptual experiment would be caused by those random sources of uncertainty or jitter that are independently generated in each branch.The full width at half maximum of the time distribution is often used as a measure of the overall timing uncertainty in the measurement system and is called the time resolution TAnother widely quoted specification is the full width at -to maximum, which more fairly accounts for tails sometimes observed at either extreme of the distribution. The recording of the time interval distribution in the manner just described is often called multichannel time spectroscopy.

2. PROMPT AND CHANCE COINCIDENCE SPECTRAWe now imagine the system to be reconfigured as shown in Fig. 17.44. Two independentdetectors are irradiated by a common radioisotope source that is assumed to emit at leasttwo detectable quanta in true coincidence; that is, both radiations arise from the samenuclear event within the source. We further assume that for all true coincidences, thenuclear decay scheme is such that there is no appreciable time delay between the emissionof both radiations.The time spectrum taken under these conditions will have the general appearanceshown in Fig. 17.45. Some fraction of all the true coincidence events will give rise to radiationsthat are detected simultaneously in both detectors. These true coincidence events willappear in the same region of the time spectrum as did the split pulse output in the previousexample, producing a prompt coincidence peak. If there were no delay differencebetween the two branches, this peak would be centered about time zero, and therefore onlyabout half of its shape would be measured. Introducing the fixed delay into the stop chanChapter17 Systems Involving Pulse Timing 667• Radioi$Otopesource" StartTACStopFigure 17.44 A simplified system to record multichannel time spectra from aradioisotope source emitting coincident radiation.nel moves the entire time spectrum to the right by an amount equal to the delay and allowsboth sides of the prompt coincidence peak to be recorded. The area under the peak, aftersubtraction of the continuum discussed later in this section, gives the total number ofdetected coincidences.The width of the prompt distribution is likely to be somewhat greater than in the previousexample because separate detectors are now involved. Amplitude walk, for example,will no longer be identical in both branches and therefore will also contribute to the widthof the distribution together with the time jitter. If detectors, timing electronics, and triggeringconditions are nearly identical in both branches, then all sources of time jitter andwalk should be symmetric. Under these conditions the prompt coincidence peak shouldalso be symmetric with a width that indicates the total contribution of all sources of timeuncertainty. If one branch differs substantially from the other, then asymmetric promptcoincidence peaks will often result. For example, the effect of amplitude walk in leadingedge triggering is to produce a small number of timing pulses that occur substantially laterthan the majority. If a system subject to amplitude walk is used in the stop branch in conjunctionwith a system with little time uncertainty in the start branch, the measured promptcoincidence distribution will have an asymmetric tail in the direction of longer intervals.In addition to the coincident events, each detector will typically produce a much largernumber of pulses that correspond to the detection of one quanta for which there may notbe a corresponding coincident emission, or for which the coincident radiation escapesdetection in the opposite detector. For these events there can be no true coincidence.Because of their random distribution in time, however, some sequences will occur bychance in which a stop pulse will be generated within the TAC time range following anCountsper 1+ _____ Fixed _____ ~channel delayFWHMTime resolution Chance intervalcontinuumChannel numberor timeFigure 17.45 The multichannel time spectrum for a radioisotope source emittingsome radiation in prompt coincidence. The cross-hatched area gives the total numberof recorded coincident events. The time resolution of the system is conventionallydefined as the FWHM of the prompt coincidence peak.668 Chapter 17 Linear and Logic Pulse Functionsunrelated start pulse. These events are called chance intervals, and their intensityon the rates at which pulses are generated in either branch leading to therates, often called singles ,ates, are not high compared with the reciprocal of theof the TAC, the chance interval distribution will be uniform over the entire timeshown in Fig. 17.45.The amplitude of the chance distribution can be derived as follows: Let '1 andresent the rates of arrival of uncorrelated start and stop pulses, respectively, at theinputs. For typical applications, '1 and '2 will be much larger than the true coincidenceand thus are essentially equal to the singles rate in either branch. (If '1 is not smallpared with the inverse of the TAC time range, account must be taken of those startthat are lost because the TAC is busy when the start pulse arrives. We ignore suchhere.) After each start, the probability that an interval of length Twill elapse without apulse is simply e-Trz. The differential probability of a stop pulse arriving within thedifferential time dT is just '2 dT. Because both independent events must occur, thedifferential probability of generating an interval within dT about T is simply '2e-TTZThe differential rate of these intervals is then simply the product of the rate of arrivalstart pulses multiplied by this probability, or 'l'2e-TT2 dT. Now as long as '2 is notcompared with the reciprocal of the TAC time range, '2 Twill be small and the "AtJVLI."llLcan be approximated by unity. The differential distribution d,/dT is then constantequal to '1'2' Ifthe output of the TAC is recorded by an MCA with a time width of.1.Tchannel, then the chance interval rate per channel is Simply 'l'2.1.T.In a multichannel time spectroscopy measurement, one would normally like to m~'~'the true coincidence peak compared with the chance interval background. For equal are '.peaks with the narrowest width will be most prominent, so that improvements in the f .,accuracy of either branch, which diminish the overall timing uncertainty, are always benefi,;]cial. Other experimental factors can also help improve the ratio of the prompt peak to th~chance continuum. For example, use of energy selection criteria in each branch can limit 'Iiand'2 by discarding any events that cannot correspond to true coincidences. The chance continuumwill therefore be reduced without affecting the area under the true coincidence dis-"tribution. The true rate also scales linearly with source activity, whereas the chance rate is:proportional to the product of '1 and '2 or the square of the source activity. Consequently,using as Iowa source activity as pennitted by reasonable counting statistics will also enhance,the prominence of the true coincidence peak. Varying the rates by changing the countinggeometry, however, affects both the true and chance rates by an equal factor.3. MEASUREMENTS USING A COINCIDENCE UNITAn equivalent single-channel method is available to carry out the type of time spectroscopyjust described, but at the price of a somewhat increased total measurement time.The single-channel method, in effect, consists of setting a time window to accept only thosesequences in which the interval between start and stop pulses lies within a narrow band.The situation is somewhat analogous to pulse height analysis, where both single-channeland multichannel methods are used. As discussed earlier in this chapter, the unit used toperform narrow band pulse height selection is called a single-channel analyzer.For time spectroscopy, the coincidence unit performs the equivalent function andselects from all intervals only those for which the time difference between inputs is lessthan a circuit parameter, known as the ,esolving time. A coincidence unit, in its simplestform, has two identical logic inputs. Whereas the TAC must receive a start pulse and stoppulse within the time range in that specific order, a coincidence unit will produce a logicoutput if pulses arrive at either input within the resolving time of a second pulse at theopposite input. The order of the arrival is not significant.The system shown in Fig. 17.46 illustrates the application of a coincidence unit to timespectroscopy. The fixed delay is assumed to be the same as in the previous example, and a• RadioisotopesourceChapter 17 Systems Involving Pulse Tuning 669oncidenunitFigure 17.46 A simplified system to record coincidence-delay curves from a radioisotopesource emitting coincident radiation.variable delay has been inserted into the opposite branch. We initially assume that the coincidenceunit is set with a resolving time equal to one-half the time width of one channel inthe multichannel time spectroscopy example. We now record the rate from the coincidenceunit as a function of the variable delay value. If the delay is set to a value that correspondsto a time interval at the midpoint of one of the channels in the multichannel spectrum, thenthe rate measured by the coincidence unit will be exactly the same as the rate recorded bythat specific channel. Under these conditions, the coincidence unit produces an output forpredelay intervals that range from (tv - tf - 1') to (tv - tf + 1') where tf and tv are the fixedand variable delay times, and 7 is the coincidence resolving time. If a series of measurementsis now made in which the coincident rate is measured as tv is varied in increments of27, then a curve that is exactly equivalent to the mUltichannel time spectrum of Fig. 17.45will be generated by plotting this rate versus the delay setting. This type of plot is oftencalled a coincidence-delay curve and can be used to reproduce the multichannel time spectrumas just described, provided the resolving time l' is small compared with the overalltime resolution of the system.In common coincidence measurements, however, the object is not to map out fully thetime interval spectrum but rather simply to record the number of true coincidence events.The coincidence resolving time is therefore chosen to be larger than the system time resolution,so that the acceptance time window can fully encompass all true coincidences.Figure 17.47 graphically illustrates the interrelation between the measured coincidencerate and the differential distribution dr/dTversus Tfor the case of a relatively large resolvingtime l' . This differential distribution is the same as the multichannel spectrum ofFig. 17.45, normalized by the measurement time so that the distribution now is in units ofrate. At a given delay setting tv the measured coincidence rate corresponds to the areaunder the differential spectrum between the time window limits of tv - l' and tv + l' .Ideally, coincidence measurements are carried out with delay tv adjusted to the pointindicated as (j) on the figure. Here, the acceptance time window is centered around theprompt coincidence peak in the spectrum. The measured coincidence rate corresponds tothe cross-hatched area and consists of two additive terms. The true coincidence rate correspondsto the net area under the prompt coincidence peak, whereas the chance coincidencerate corresponds to the area of the flat chance continuum on which the peak is superimposed.If the delay is changed to a point well away from the prompt peak (such as point @ onFig. 17.47), only chance coincidences are measured. The rate can be deduced from theresults of the previous section, where it was shown that the amplitude of the differentialdistribution dr/dT for chance intervals is the product of the two singles rates r1r2.Therefore, the area under the differential spectrum in the chance continuum region is this670 Chapter 17 Linear and Logic Pulse FunctionsdrIffMeasuredcoincidencerater,h ~-__ ..---..,.= 2r'.'2True coincidencepeakTime interval T"Variable delay'.Figure 17.47 Relation between the differential time spectrum (upper plot) and thecoincidence-delay curve recorded with the system of Fig. 17.46 (lower plot). Therecorded coincidence rate at any specific delay value tv is equal to the area undel thedifferential spectrum between the limits tv - T and tv + T. In the lower plot, 'ch representsthe chance coincidence rate, and 'I is the true coincidence rate.-----~~~------~---- --------------------amplitude multiplied by the width of the time window 2'1'. The general result for anytwofold coincidence unit is therefore that the chance coincidence rate from uncorrelatedinputs at rates '1 and '2 is given by(17.28)This chance contribution must be subtracted from the rate measured at point <.D to derivethe net true coincidence rate.The remainder of the coincidence-delay curve shown at the bottom of Fig. 17.47 can betraced out by measuring the total coincidence counting rate as tv is varied. This is a veryuseful calibration procedure during the initial setup of a standard coincidence measurement.Starting at <.D, assume that the delay is changed very gradually. For a while nothingchanges because the amount of area under the chance distribution that is lost by movingone limit of the acceptance window is made up by that area gained at the opposite limit.However, when the delay is changed sufficiently so that some portion of the true coincidencepeak begins to be lost, the measured coincidence rate starts to drop off. When theChapter 17 Systems Involving Pulse Tuning 671delay has been changed sufficiently, the entire peak is excluded and only the chance rateremains. The transition region at either side extends over a range of delay equal to the fullwidth (at its base) of the .true coincidence peak. The midpoint of a transition region correspondsto the delay setting at which one edge of the time window exactly bisects theprompt coincidence peak. The time difference between these two midpoints (as illustratedon the figure) is therefore a measure of the time window width and is equal to 21', twice thecoincidence unit resolving time.Coincidence-delay curves generated by assuming different values for l' are shown inFig. 17.48. The minimum value of l' for which all true coincidences can be recorded.is-halfthe total width (at its base) of the true coincidence peak. The curve labeled C in the figurecorresponds to this case. With the resolving time set to this minimum, only one specificvalue of the delay will lead to counting all true coincidences. Should any time drift occur ineither branch, true coincidence events would begin to be lost. Therefore, in most coincidencemeasurements one would like to have the resolving time l' somewhat larger than thisminimum to allow some leeway for such drifts or other timing changes. On the other hand,the chance coincidence rate increases linearly with 1', and the resolving time should be keptas small as possible to maximize the true-to-chance coincidence ratio. The usual compromisebetween these conflicting considerations is one in which l' is chosen to be severaltimes the system time resolution as in curve B. The coincidence-delay curve then has a flattop or plateau, which represents the range in delay settings that can be tolerated withoutlosing true coincidences. In the initial calibration of the coincidence system, delays are thenadjusted to choose an operating point near the middle of this plateau. Because of slight differencesin the inherent delay of pulse-processing components, this point may correspondto some apparent delay difference between the two branches.In practice, the fixed delay ttin Fig. 17.46 is usually omitted and the coincidence-delaycurve is obtained by using only the variable delay tv' The curve is then approximately centeredabout zero, and only the right half is recorded with tv in the lower signal branch. Theremainder of the curve is then obtained by shifting tv to the upper branch, where its valuerepresents "negative delay."Coincidence measurements are not necessarily confined to two signal branches butcan, in general, involve inspecting any number of signals for true coincidence. For such multipleinputs, all signals must arrive within a total time interval corresponding to the resolvingtime of the unit for an output logic pulse to be produced. The measured rate will againTotalcoincidencecounting:-ate'_ _____ c'-------DVariable delay tvFigure 17.48 Coincidence-delay curves for different values of the coincidence resolving timeT. CurveA corresponds to the largest value of T, curve D to the smallest. Values of T larger thanhalf the width (at its base) of the prompt coincidence peak in the differential time spectrum ofFig. 17.47 lead to the flat-topped plateau shape of curves A and B. In curve C, T is just equal tothis value and the full true coincidence rate is obtained only at one specific delay value. Incurve D, T is too small to obtain all the true coincidences at any delay value.672 Chapter 17 Linear and Logic Pulse Functionsbe a. mixture .of true and c.hance c.oincidences, but correction for the latter is more d.""as dIscussed In the follOWIng sectIOn. "4. CHANCE COINCIDENCE CORRECTIONSIn standard coincidence measurements it is essential to correct the recordedrate for the contribution due to chance events to derive the rate due to truealone. For twofold coincidence systems, Eq. (17.28) allows calculation of the chancecidence contribution if each singles rate is measured and the resolving time of the .dence unit is known. Alternatively, the chance rates can be measured directly byporarily inserting a large delay in either branch of the system so that the true . .peak occurs well away from the acceptance time window of the coincidence unit.ter approach is usually preferable because it can be applied more easily towhich the singles counting rate may not be constant over the period of the me,aslIremSeveral electronic schemes have been suggested79-8! for simultaneously measuringtotal coincidence and chance coincidence rates throughout the course of the me:asllrelmby periodically excluding the true coincidences in a similar manner.It is possible that there may be some confusion in attempting to reconcile the rjust obtained for the two·fold chance coincidence rate with the results for pile-up robtained earlier in this chapter. Figure 17.49 illustrates the comparison. The input ratand '2 are assumed to be uncorrelated Poisson random processes. The top diagram shthe result for the (first order) chance coincidence rate from Eq. (17.28) as 27'1'2' wheis the coincidence resolving time. In the lower diagram, we imagine combining the stwo input lines with the same singles pulse rates into a common line. We then ask whatpile-up rate will be, assuming that the pile-up and coincidence resolving times are the s.The first-order result for the rate of piled-up counts from Eq. (17.23) is 7,2, where 7 is tIt:pile-up resolving time. For this example, the predicted pile-up rate is 7('1 + 'V2, which ijJclearly different from the chance coincidence rate in the top diagram.There is a fundamental difference between the cases of chance coincidence and pile~up that resolves this apparent contradiction. In the case of the coincidence unit, pulses from\both branches are required to occur within the resolving time. For pile-up, it is possible foripulses from either branch to generate a pile-up event with a pulse from the same brancf4jas well as to generate a pile-up event between the two branches. By expanding the expres"'lsion for the pile-up rate, we see that it is consistent with the sum of three separate rat~the pile-up of pulses from line 1, the pile-up of pulses from line 2, and the pile-up of puls~Jfrom opposite lines. This cross term (27'1 '2) does indeed follow the prediction of the;jchance coincidence rate, and therefore both representations are consistent.'I ~ coinc. t---, = 2'tr ,, 't ch! 22Figure 17.49 At the top, a coincidence unit with resolving time 7 results in a chancecoincidence rate 0(27 '1'2' In the bottom diagram, two input lines with equivalent singlesrates are combined in a system with pile-up resolving time of 7. Expansion of theresulting pile-up rate shows three components: pile-up of pulses in line 1, pile-up ofpulses in line 2, and the pile-up involving pulses from opposite lines.Chapter 17 Systems Involving Pulse Timing 673In multiple coincidence systems, it is usually not possible to make an analytic correctionfor chance contributions. Instead, supplemental measurements must be carried out inwhich the separate twofold coincidences between various inputs are individually determinedand applied in a more complex analysis, such as that given in Refs. 82 and 83. Forthe example of a triple coincidence unit, the chance rate from totally uncorrelated inputscan be calculated from the resolving time and individual singles rates as 3'T2'l'2'3' If thiswere the only source of chance events, the correction could be made analytically as in thecase of twofold coincidences. However, the added complexity arises because of partiallycorrelated events in which a true coincidence in two branches is accompanied ~ chanceevent in the third branch within the system resolving time. Because the prolJa'6llity of theseevents depends on the specifics of each experiment, they must, in general, be experimentallyevaluated for each case.5. DETERMINATION OF COINCIDENCE RESOLVING TIMESeveral methods are available to the experimenter for the detennination of the resolvingtime of a coincidence circuit. One is to provide totally uncorrelated inputs and to measuresimultaneously both the singles rates and the resulting chance coincidence rate. The resolvingtime can then be Calculated from Eq. (17.28). In setting up such a measurement, caremust be taken to exclUde any possibility of true coincidences between the two branches.The source must be incapable of generating coincident radiations that can interact in bothdetectors, or alternatively, separated sources with adequate shielding should be used toprevent true coincidences. Consideration should also be given to the possibility that truecoincidences may arise from the scattering of radiation from one detector to the other.The resolving time can also be measured by recording the coincidence-delay curve. Asshown in Fig. 17.47, the FWHM of the portion of the curve that corresponds only to truecoincidences is equal to twice the resolving time. For this method' to be practical, a sourcewith a sufficiently high probability of true coincidence emission must be used to ensurethat the true coincidence rate stands out well above the chance background.6. DELAYED COINCIDENCE AND OTHER INTERVAL MEASUREMENTSIn the previous example, the time spectrum was illustrated for a source that emittedprompt coincidence radiation. One way of defining prompt coincidence is to include anyevents that are separated by a delay time that is small compared with the instrumental timeresolution. There are many occasions in which radiations are emitted in the same nucleardecay but are separated in time because of an intermediate nuclear state of finite lifetime.The time distribution should then show an exponential tail to the right of the prompt peakshown previously in Fig. 17.45. By measuring the time constant of this tail, the decay constantof the intermediate state can be deduced. These measurements can be carried outeither with the multichannel technique using a TAC or as a series of single-channel measurementsusing a coincidence unit and variable delay. The latter method has historicallybeen known as the delayed coincidence technique, and before the days of multichanneltime spectroscopy was widely used to measure time interval distributions.Multichannel time spectroscopy can be applied quite generally to any situation inwhich a time interval is to be measured. For example, in neutron time-of-flight spectroscopy,the start pulse is supplied from a detector that senses the time at which the neutronis generated, whereas the stop pulse is taken from a detector in which the neutroninteracts after traveling some distance. The time interval between these two events is thena measure of the flight time of the neutron from which its energy can be calculated. Thereare many other examples of physical measurements in which the time interval distributionis important, and methods originally developed for nuclear measurements have beenapplied to a large assortment of determinations in other scientific fields.674 Chapter 17 Linear and Logic Pulse Functions7. MEASUREMENT OF ABSOLUTE SOURCE ACTIVITY USINGCOINCIDENCE TECHNIQUESAs an illustration of one of the common applications of coincidence circuits, wethe problem of measuring the absolute activity of a given radioisotope source.emits two coincident radiations that are distinguishable, methods can be appliedinate the need to know absolute detector efficiencies in order to calculate the source "ity. Because detector efficiencies are often uncertain and hard to determine, use ofmethods can provide a more accurate means of source activity determination thanwise available.Assume that the source activity (disintegrations per second) is given by S. Furt1.• i!assume that each such disintegration gives rise to two coincident radiations with no co:J~lation of the angle of emission of one with respect to the other. We now arrange two det~:tors, such that detector 1 records pulses only from radiation 1, whereas detector 2 recor~~pulses from radiation 2 only. The output of these detectors after appropriate pulse ~4cessing is fed to a coincidence unit with a resolving time 1'. By separately measuring the '., .'singles rates and the coincidence counting rate, the source activity can be determined aS~follows.The singles counting rate in branch 1 ('1)' corrected for background and dead tim~"losses, can be written as the product of the source activity S multiplied by some overall e '",ciency Et. This efficiency includes the solid angle subtended by the detector, the interacti::'probability within the detector, and the fraction of detector pulses accepted by the subse!lquent circuitry. A similar relation can be written for branch 2. Therefore, we have "The true coincidence rate '1 can be predicted by noting that two independent events must:occur: Radiation 1 must be detected in branch 1. whereas at the same time radiation 2 mustbe detected in branch 2. The independent probabilities (they are independent if there is nOangular correlation) of these two events are El and E2. Therefore, the probability that bothoccur is simply their product E1E2• The true coincidence rate is thus the product of this combinedprobability and the source activity S:(17.31)The measured coincidence rate '12 is the sum of the true coincidence rate and thechance coincidence rate. Therefore,'12 = 't + 'eh'12 = E1E2S + 'eh (17.32)Now we can solve the three equations (17.29). (17.30), and (17.32) simultaneously andthereby eliminate any two variables. By eliminating the efficiencies El and E2' we can write(17.33)'12 - 'ehThis expression gives the source activity in terms of directly measured rates and the chancecoincidence rate which can be measured using the methods described earlier. In manyapplications, the two coincident radiations selected are beta (~) and gamma ('Y) rays emittedin the decay of a given isotope, and the method is often called {3--y coincidence. Therequirement of no angular correlation can be relaxed if one of the radiations is detectedover a 4n solid angle. Therefore. a common implementation of the method is to use a 41Cproportional counter as the beta detector in connection with a gamma-ray detectorChapter 17 Systems Involving Pulse Tuning 675subtending a smaller solid angle. Accuracies approaching 1 % can be obtained with thismethod, whereas activity measurements that rely on a prior knowledge of detector efficiencyare nearly always less precise.The analysis above is somewhat oversimplified, because it is often difficult to meet thecondition that each detector responds only to one of the two radiations. In 13-"1 coincidencemeasurements, it is normally quite easy to eliminate the 13 sensitivity of the "1 detector byinterposing an absorber that is thicker than the maximum distance of penetration of thebeta particles involved. However, most beta particle detectors show some response togamma rays. Two gamma rays of distinguishable energies are also sometimes used asjhetwo radiations (-Y-'Y coincidence) by selecting each photopeak separately in two garrrrna:.raydetectors. It is then difficult to avoid including some of the higher-energy gamma rays inthe lower-energy branch because of the contribution of the Compton continuum from thehigher-energy gamma rays. Also, the complications of true and chance summing (see p. 322)that may occur in either detector must be considered. For a more detailed discussion ofthese and other complications in absolute activity measurements using coincidence methods,see Ref. 84.C