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Dead-time problems
NUCLEAR
INSTRUMENTS
AND
METHODS
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DEAD-TIME PROBLEMS J O R G W. M(2LLER Bureau International des Poids et Mesures, Pavilion de Breteuil, F-92310 SOvres, France The usual two types of dead times, extended and non-extended, are reviewed and fundamental properties of their effect on the interval distribution and the count rate are discussed briefly. The application of renewal theory to counting processes is sketched and it is shown how the interval distribution, which is distorted by the presence of a dead time, can be used to determine the resulting counting statistics. In particular, the modifications of an original Poisson process, due to a non-extended dead time, are indicated for the case where the origin of the measuring interval has been chosen at random. A simple application then
shows the fallacy of the so-called zero-probability analysis. When renewal processes are superimposed, their convolution property is lost. Therefore, a general formula for the density of multiple intervals is given for the superposition of two component processes. These results have proved useful for studying two recently reported methods of measuring dead times. Finally, formulae are given for the four different ways of arranging two dead times in series. The review is confined to one-channel problems and the emphasis is on exact results.
1. Introduction
of radionuclides. For any detector or electronic circuit there exists a minimum time interval by which two events have to be separated to allow each to be detected. This is called the resolution or the dead time z of the device in question, since the apparatus can be supposed to be "dead % i.e., to ignore further pulses for a period r after the arrival of an accepted event. In general, such "natural" dead times are only approximately constant. Besides, it should be specified whether a dead time is lengthened or not by pulses occurring during its operation. Unfortunately, there is no general solution to these problems and each case must be investigated separately. Since this is often not practical and sometimes impossible, the corresponding corrections remain uncertain. This is why today a different approach is preferable, where a well-defined dead time with known behaviour is inserted artificially into a sequence of pulses, even if it is significantly longer than the natural resolving times. Relatively large but well-known corrections are thus preferred to smaller ones which are uncertain. For traditional and practical reasons, two types of dead times are usually considered. They differ in their response to pulses arriving during a dead time, which (a) for a non-extended dead time: have no effect at all; (b) for an extended (also cumulative or paralysable) dead time: extend the dead time by z, measured from the last arrival time. Accordingly, in the first case dead times are only imposed by the registered output counts, whereas for the second type this happens for all input events. The special interest taken in these two particular types is not so much due to the fact that they approximate rather well the real dead-time behaviour of an apparatus, but because electronic circuits are now readily
The literature on problems related to dead times has increased so much in the last ten or twenty years (at least in volume), that any attempt to review the work done and the progress achieved would require a book rather than a modest state-of-the-art report and is therefore out of the question here. The latest survey in this field 1) that we know of dates from 1948 and is nowadays of only historical interest. No comprehensive and up-to-date bibliography exists. The more recent literature is scattered throughout a great number of journals and in various internal reports, and duplications are quite frequent. Accordingly, our goal will be modest: apart from some general considerations which have been included for pedagogical reasons, we can only mention a small selection of problems dealt with recently, in which the emphasis will be on exact results. Furthermore, our choice will be heavily biased towards work performed at the BIPM, but this should obviously not be taken as a hidden evaluation of merit; it is much more a confession of not being familiar enough with more interesting work done elsewhere, if we know of it at all. 2. Preliminary remarks Since the counting of events is a basic operation, especially in all branches of nuclear physics, where individual interactions between elementary particles can be distinguished by the available counters, the problem of counting losses is of vital importance in any type of precision work. While it is often sufficient to determine ratios, where many corrections cancel, other measurements must be absolute, as is normally the case in the standardization 47
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available which are capable of simulating such performance to a very high degree and because the corresponding corrections can then often be determined exactly. We shall not discuss the important practical problem o f producing such a well-defined dead time since a detailed description will soon be available2). It might also be emphasized that a dead time should always be inserted as early as possible into an electronic circuit in order to reduce pile-up effects. Elaborate attempts to deal with a more general type o f dead time, for instance a "linear c o m b i n a t i o n " of the types mentioned, are available3'4), where an extension o f the dead time takes place with a probability p, successive r a n d o m choices being independent. The two types defined above then turn out to correspond to the special cases p = 0 and p - - 1, respectively. Although they allow, in principle at least, a more elegant and coherent treatment, we shall not discuss such generalizations further.
3. On dead-time effects in general The most important effect due to a dead time is clearly the lowering of the count rate by the loss o f pulses. However, the loss can hardly ever be measured directly since the true count rate is normally unknown. Therefore, other effects caused by a dead time have to be used to determine its presence and to estimate its value. Generally speaking, such effects can be subdivided into two separate classes: one shows up as a change in the time distribution and the other in the statistical behaviour o f the process. They are obviously not independent. A modification of the time distribution, i.e., the intervals between successive pulses, is the more direct consequence o f a dead time and will therefore be discussed first. This effect can be conveniently studied ... ........ ........ F(t)
by the use of time-to-amplitude converters or, for very small count rates, by a multi-channel analyzer used in the time mode. The corresponding change in the statistics, which may be described by a probability law for a given number of events in a certain measuring time, is a more difficult problem that will be tackled (at least under some simplifying conditions) in the following section. Experimentally, information about the counting statistics m a y be obtained directly by repeated measurements or by correlation techniques. In general, the interval density F(t) for a given process must be known before the corresponding count rate R can be evaluated, since
R -1 =
tF(t) dt,
where the passage from the original density f ( t ) to the dead-time modified density F(t) may be complicated. However, there is an interesting exception to this rule. It happens that for the case o f an extended dead time, where a determination of F(t) would be awkward, if not impossibleS), the new rate R can be calculated directly from the original density f(t). In this case we notice that only counts with an original interval exceeding r will survive, which would not be true for a non-extended dead time. If the original process with rate p is described by the interval density f ( t ) , we then obtain at once for the count rate R of the remaining pulses R = p
If0 l 1-
( t ) dt .
where U is the unit-step function, this leads to the well-known result R = pe -°~. (4)
"'.
F(t) = Z "'" '"..,......
O
~,
2~
3~,
J
A j,
(5)
j=l • "., .....
t
the interval distribution original Poisson process, time r leads to a modified is given by the sum 5)
J
"....
i
(2)
F o r the important special case o f an original Poisson process with f ( t ) = U(t)pe -°t, (3)
In some cases, however, also has to be known. For an insertion of an extended dead interval density F(t) which (see fig. 1)
i
(l)
0
where ~t
4T,
Fig. 1. Experimental interval density F(t) for a Poisson process which is deformed by an extended dead time (for/3~2000 s-1 and v ~ 4 0 0 ps).
Aj = U(t-jr)[( - 1)J- a/(j_ 1)]]pJ(t _ j r ) J - l e - jp~' and J is the largest integer below t/v. It can be shown 6) that in the limit F(t)'~[p/(l - p r ) ] e -p' for t >>r.
DEAD-TIME
In the case of a non-extended dead time, no similar " s h o r t c u t " is known and one has to evaluate first the new interval distribution. This explains why count-rate corrections are often easier to determine for an extended than for a non-extended dead time. The general formula which allows us to pass from f ( t ) to F ( t ) for a non-extended dead time r is known to be given by 7) F(t) = U ( t - z )
f(t) +
fk(x)f(t--x) k=l
dx ,
(6)
0
whereJk(t) is the density for a k-fold interval (fl = f ) . The evaluation of F ( t ) may be quite complicated, but again for an original Poisson process the following simple result is obtained F(t) = U(t-z)
pe -p(t-~)
(7)
Hence, the exponential is just shifted by the amount of the dead time r. The resulting count rate is now given by R- 1 = P
f;
te - ' " - ~ ) dt = p - 1 _{_~ ,
which corresponds to the familiar correction formula R = p/(1 +Or)
(S)
for a non-extended dead time, a result which can also be obtained for this special case by simpler reasoningS). For the original rate this gives
49
PROBLEMS 4. Renewal processes
The concept of a renewal process9), created and developed by mathematicians some thirty years ago, has proved to be very useful for dealing with counting problems. The early contributions by Jost 1°) and in particular by Feller ~l) still make very worthwhile reading. Renewal theory can be applied in all those cases where a statistical process, developing in time for instance, is fully described by the distribution function of the interval between two successive arrivals, and this is indeed the case for the majority of the onechannel problems. Let us briefly sketch how this theory can be used for determining the probability distribution for a given process of the renewal type. For more details compare refs. 11 or 12. If the basic interval density is described by f l ( t ) , then multiple intervals (of order k) are given by
L(t) = {f, (t)} *~,
i.e., by the k-fold self-convolution of f~(t), since the contributing individual intervals are independent random variables following the same law. The waiting time for the first event, however, may be different and depends also on the way the time origin is chosen. In the important case that this is done completely at random (for a stationary or equilibrium process), the corresponding density is given by 9,,3) g I (t) = ~/
j(', (X) d x ,
where ~, the mean count rate, is now
The dead time may also be expressed by (9)
F r o m time to time attempts are made to define on the basis of its effect alone an "average dead time" "?, without having recourse explicitly to any specific type. At first sight, a relation such as eq. (9) using the difference between input and output rates might look promising for this purpose. In practice, however, this is not feasible for the simple reason that the true count rate p is unknown, so that the "definition" cannot be applied. Furthermore, such a procedure would be at variance with the definition given above for an extended dead time, for then we would obtain with eq. (4): -
P - P e - p~ -
p2e-Pr
~
( 11 )
t
p = R/(1-Rz).
z = (p-g)/pR.
(10)
z(l+½p~).
It thus appears to be unlikely that a useful definition of a dead time can be found which is independent of an assumption about the specific behaviour.
/t-1 =
tfl (t) dt.
(11 ')
In general, the interval density of event number k(~>l) is thus gk(t) = Or(t) * f k - , ( t ) ,
(12)
where in particular/o(t) = 5(t). The corresponding cumulative distributions defined as usual by Fk(t) =
f,
fk(X) d x
and
Gk(t) =
0
;,
are
gk(X) d x ,
o
(13)
with F o ( t ) = Go(t ) = U(t). If kt is the total number of events occurring in a time interval t, and t k gives the arrival time of event k, then the inequality k > k t implies necessarily t k > t . In terms of the corresponding probabilities we have I. G E N E R A L
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therefore
This shows that for any equilibrium expectation of k is simply
f7
gk(X) dx = 1 --Gk(t).
Prob(kt < k) = Prob(tk > t) =
This now allows us to make the step from the time distributions &(t) to the probabilities Wk(t ) for a given n u m b e r k of counts within t, since
Wk(t) = Prob(kt < k + 1) - P r o b ( k , < k)
(14)
= G k ( t ) -- G g + , ( t ) .
The frequent occurrence of the convolution operation suggests the use o f integral transforms, where this is k n o w n to correspond to a simple product. Taking the usual definition o f the Laplace transform off(t),
f(s) - ~ ° { f ( t ) } -
f(t)e -~' dt,
15)
we can now write for eq. (12)
Ok(s) = 01 ( s ) L - , (s)
06)
E(k, t) = t~t.
.#k(S) = O,(S)
S
(23)
These general relations may now be applied to specific counter models (dead times). In particular, the original process will be assumed to be Poissonian with constant count rate p, which corresponds to an exponential interval density [eq. (3)]. Let us now insert a non-extended dead time r into such a sequence o f pulses. F o r the simple case considered here this results in a mere shift of the exponential density, as has been seen earlier [eq. (7)], and in a mean count rate
(83
= p/(l +p~).
The corresponding Laplace transform is easily found to be f~ (s) = pe-S~/(p + s). (24) For multiple densities (k ~> 1), the original function is a shifted g a m m a density
and for the cumulative distribution
Gk(S) = !
process the
fk(t) = U(Tk)[pTk-~/(k--l)!]e-rk,
[f,(s)]t-,
(17)
S
Similar simplifications occur for the m o m e n t s (of order r), defined by
E(U,t) = ~ UWk(t).
(25)
where T k - p ( t - kr). F o r a r a n d o m start, the interval density o f the first event is given by eq. (21); thus for the present conditions
(18)
k=O
the original of which is
F o r the first m o m e n t one can show that
E(k,t) = ~ Gk(t),
P
(19)
l+p~
k=l
1
(27) [l-~preXp[-p(,-z)],
ff,(
-
S 1
S
for0
91 (t) =
and the corresponding transform is =
,
)l k
0
s [ I - - f (s)] (20)
F o r a given d e n s i t y f ( t ) and a definite policy for choosing the time origin, this relation can be used to determine the average n u m b e r of events• Similar but more complicated expressions can be found for the higher moments• Especially, it follows f r o m eq. (11) that
9, (s) = (~/0 [1 - ) ( 0 ] ,
l b r t ~ > z.
_
(20
F o r the arrival time o f event n u m b e r k, this leads to a density
#k(s) = (~/s) [1 - L (s)] L - , (s) = ~ [ & - , (s)- &(~)], (28) and hence to the original
gk(t) = [p/(I + p * ) ] [Fk-, (t)--Vt(t)].
(29)
Since it is easy to show that
thus eq. (20) becomes E(k,s)
= ~/s 2 ,
(22)
Fk(t) = U(Tk) II - I~I Pg(J)],
(30)
DEAD-TIME
where Pk(J) =- (TJ~/!) e-T~ is a "shifted" Poisson probability, gk(t) is also fully determined. After some lengthy rearrangements one thus arrives at the final expressions for Wk(t) which generalize the well-known Poisson probabilities to the case of a non-extended dead time. If K is defined as the largest integer below t/r, and 2 = (1 + p r ) -1 ,
(31)
the general formula for the stationary case can be brought into the following form: (I)for0~
Wk(t ) = 2,
:
( k - 1 - j ) Pk-, (J) -(j =0 k--1
- 2 ~
(k-j)Pk(j) +
j=o
PROBLEMS
51
difference 1 - V can be used either for an estimate of the dead time z or of the original count rate p. 5. A p p l i c a t i o n s
Les us first illustrate the usefulness of eqs. (32) derived above by a simple example. This might also be helpful in settling the old debate about the so-called zero-probability analysis15), which periodically is reintroduced16). This method is based on the assertion that, in order to determine the true count rate p of a series of pulses, it is sufficient to measure the probability for no registered counts, since for a non-extended dead time, losses could be caused only by surviving pulses. In other words, the Poisson probability Po = exp ( - p t ) should not be affected by the presence of any dead times, thus permitting an unbiased estimate of p. As a check, we apply eq. (32) to the special case k = 0. As before, x is short for pr. For0
+ j =~'0 ( k + l - j ) P k + ~ ( j ) } ,
1 -p~/(l
Wo(O =
(2) for k = K :
(32)
whereas for t ~> r,
+x),
(35)
i.e., K~> l, we obtain
Wo(t) = (1 + x ) -1 e - p ( ' - ° . Wk(t ) = ,~
(K--I
--j) PK-I(J)
(j=o K-l
-2
Therefore, evaluating the count rate by means of P0, one obtains
/
~
(K--j)PK(j)--pt
+K+l,
j=o
(3) f o r k = K + l
Wk(t ) = 2
p = - (l/t)lnPo :
=
- ½(Po-l)
o-1)
2 + .--3
(31)
instead of
Exact expressions for the moments as well as formulae corresponding to other choices of the origin (e.g., coincident with a pulse) can be found in the literature 14). A simple formula for the variance a2(k, t) of the number of pulses in t is obtained in the limiting case t >> r, namely =
-- (l/t)[(P
--~ [ ( 1 - P o ) / t ] [I + ½ ( l - - P 0 ) ] ,
( K - - j ) PK(J) + 0 t -- K . [j=O
aZ(k,t)
(36)
--
2 3
x + - x 1 2) 1 , [ p t + x 2 2 ( 1 + - 32 6
(33)
where x = pr is dimensionless. Since according to eq. (23) the expectation E(k, t) is exactly equal to 2pt, the relative variance V is given by
V =- a 2 ( k , t ) / E ( k , t ) ~- 1 - 2 x + [3 + (pt) -~] X 2 , (34) and is thus always smaller than for a pure Poisson process where the relative variance is unity. The
p = [(1-Wo)/t] (l+x),
for t < z .
(38)
Similarly, for t > r, the real probability for no event is W0 = [eX/(l + x ) ] Po,
(39)
or
14%/Po ~ 1 +½x 2 .
(39)
The count rate is then approximately given by p --~ - - ( 1 / t ) [ l n
Wo--½x2],
(40)
which differs again by a dead-time correction from eq. (37) obtained on the basis of the zero-probability analysis. This correction is obviously due to the fact that pulses can also be lost by a dead time which is present at the very beginning of a measuring interval. For a continuous process, it would only disappear if each interval t were started immediately after the end of a dead time. 1. G E N E R A L
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However, this would suppose an exact knowledge of the length and type of the dead time involved, then permitting the application of the more reliable traditional method that makes better use of the information available. Another application would be a general method for testing an experimental counting distribution for the presence of possible dead-time distortionX7). Here we take advantage of the fact that the deviations from the exact Poisson probabilities P(k), calculated on the basis of the experimental count rate, are expected to be positive in a central region only, but negative for lower or higher values of k. Since for pr ~ 1 the limits of the different domains do not depend on the value of the dead time nor on its type, they can be determined in advance, and a sign test on the empirical differences allows one to decide whether or not a dead time was present, if this method is applied to the famous data of Rutherford and GeigerlS), which are frequently quoted as an example for a Poisson process, one arrives at the somewhat embarrassing conclusion that they have to be rejected at about the 99 % confidence level because their Poissonian character was distorted by the finite resolving time of the observer's eye. 6 . Superpesition
1
k
x }-" {jf~(t) 3 _jZ~Gk+ 1-s(t) + 3 - j g k + 1 _,(t) jAF,(t)}, S=I
(41) where
jAF,(t) =- jF,_, (t) - jF~(t) and
jAG,(t) =- jG,_ t(t) - iG,(t). We recall that, e.g.,
2
~ ~i x p l "-~/~2 j = i
jFk(t) =
J k ( t ) = {ff'(t)} *k and
t
t
jj~,(X) dx, 0
as defined earlier in section 4, and that /lj are the count rates. A simple result is obtained for the densityf(t) of the time interval between succeeding pulses if only two processes are superimposed, namely f(t)
-
/ p'--1 l j ' ( t ) 2 h ( t ) - 1 I"/1 + / / 2 t~z
"~- 2 l ~ ( t ) 2 ~ ( t ) -[-
+/*2 2f(t ) 117(01 , / Pl
of processes
Although renewal processes play a predominant r61e in counting statistics, there are also many cases of practical interest which fall outside this category. Most of them can be thought of as being formed by some kind of superposition. Let us mention here only two such cases which have received some special attention in recent times, namely the random superposition of independent pulse trains and the delayed-state or " p a r e n t - d a u g h t e r " problem. First of all we should mention that in general the superposition of renewal processes does not result in a new renewal process, the only exception being the case of Poisson processes. This feature is the main reason for the additional complexity and means that the density of multiple intervals is no longer given by selfconvolutions of the density for a single interval, as earlier was the case, but must be evaluated separately each time13). In particular, we obtain for the k-fold interval resulting from two superimposed independent renewal processes, where each is characterized by its single interval density if(t) and 2f(t), respectively, the formula
fk(t) -
MOLLER
(42)
where
jg(t) = I~j
;
jh (t) = ktj
;
oo
j f (x) dx
and oc
jg (x) dx.
As a non-trivial example let us take the case where a constant frequency is superimposed on a Poisson process. Here we have (1) for a source with count rate Pl = P:
i f ( t ) = U ( t ) p e -t'', and therefore ,9(t) = ,h(t)=
,f(t);
(2) for an oscillator with frequency v = 1 / T = P2: 2f(t) = 6(t-T),
i.e.,
2g(t) = U(t) U ( T - t ) / T 2h(t) = U(t) U ( T - - t ) [ ( T - O / T 23 .
DEAD-TIME
Inserting these functions yields f ( t ) = [ p/(1 + p T ) ] { U ( t ) U ( T - t ) [ 2 + p ( T - t ) ]
+ - e-°rf(t-T)
.
(43)
P This interval distribution is of special practical interest and will be used in the next section on new methods for measuring dead times. Another quite different and equally important case where renewal theory cannot be applied is in the analysis of the parent-daughter problem, which arises, in principle, in all two-step decays where the second disintegration is delayed because the lifetime of the intermediate level is not negligible. If we assume that the two radiations cannot be clearly distinguished experimentally, as is often the case, then the observable sequence of pulses consists of a superposition of two types of events which are coupled to each other by the statistical nature of the decay of the excited level. We suppose that this can be described by an exponential of the type exp ( - At), where A - 1 is the mean lifetime. Again, the first task is to determine the interval density for the superposition. For a source rate p and detection efficiencies el.z for the two radiations, this can be shown to be 19" 20)
f(t) = (~+/3) -1 [(~+fl e-at) 2 + ABe -at ] x
xexp{--=t----fl(1--e-At)} ' A
(44)
where 0~-----p(/31"-k~32--/;1/~2) and f l = p e~e2. An experimental background count rate b can simply be added to ~. As for the corresponding dead-time correction, an exact formula is too difficult to obtain, at least for the non-extended case, since this would first require a general expression for the multiple interval densities. Approximations, however, can be obtained 2°) which may prove sufficient for practical applications. Once more the problem is much simpler for an extended dead time, for then all we need is an integration o f f ( t ) as given by eq. (44) from 0 to r, which can be performed numerically to any desired precision.
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PROBLEMS
dead time. Various methods for measuring dead times have been known for a long time, in particular the "decaying source method" or the "double source method". For recent discussions see, e.g., refs. 21-24. However, both methods are quite cumbersome and time-consuming, their precision is rather poor (perhaps 5-10%), and the type of dead time (normally non-extended) has to be known in advance. Some years ago Baerg 25) suggested an interesting variation on the paired source method by substituting artificial periodic pulses (with frequency v) for one of the sources, thereby avoiding the problem of ensuring reproducible geometry for each source when counted singly and in combined form. He has shown that the value of the dead time is then given by z = N,,-'
Accurate corrections demand both reliable formulae and precise knowledge of the parameters involved, i.e., in particular of the type and numerical value of the
N,,)/v]'},
(45)
where Np = observed source count rate and Nov = observed combined rate (source and oscillator). As a matter of fact, this simple formula does not hold rigorously, since some simplifying assumptions have been made in deriving it. It seems, however, that this is not serious provided that we choose v < 1/(3r). A more detailed analysis will have to be based on the exact interval distribution, eq. (43), derived in the previous section. This work is under way, but not yet completed. Here again things are considerably simpler for an extended dead time, for in this case the probability P for losing a pulse can be readily determined by 26) P =
f
~
f(t)
dt = I - (1-
[pv/(p+v)]z}
e-'~ .
(46)
o
This formula as well as the deviation of the interval distribution from a pure exponential have been verified by careful measurements27). In another recent method used for measuring dead times accurately28), the departure from the classical two-source method is even more radical since both sources are replaced by independent oscillators with frequencies v 1 = lIT 1 and v2 I / T 2 . A simple application of eq. (42) for two superimposed renewal processes leads to =
2
7. New measuring methods
{1 - [(N,,,.-
J(/)
--
-
-
U(t) U(T-t)
T1+'12
+ IT, - TzI(T, + T2)-~,5
(t-T),
(47)
with T = 1Iv =- min (Tl, T2). 1. G E N E R A L
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The interval density is thus c o n s t a n t between 0 a n d T w h e r e it ends in a delta function. The extreme simplicity o f this d i s t r i b u t i o n permits an exact evaluation o f the losses p r o d u c e d by a d e a d time as well as the calculation o f the resulting interval density, b o t h o f which are very well confirmed by precise measurements. If vs, 0 is the m e a n frequency o f the s u p e r i m p o s e d oscillations after having passed a d e a d time r, it can be easily shown that (see fig. 2) (!) for a n o n - e x t e n d e d dead time:
VSUp
{
V l + V2--2"rVl V2,
for 0 < r <
T/2,
v,
for T / 2 < r
< T;
(48)
(2) for an extended d e a d time:
JVlq-V2--2"CVl!.'2, Vsup
for
0,
0
(49)
for r > T .
In the region v < 1/(227) this m e t h o d thus yields a value for the d e a d time which is i n d e p e n d e n t o f the type. If n l , n2 and nsu p a r e the n u m b e r o f pulses c o u n t e d in a c o m m o n m e a s u r i n g time t, the dead time is given simply by z -
t nl 2
-~-n2--tlsup
(50)
nln2
However, if v exceeds the value 1/(227), the observed c o u n t rate vsup provides us with a simple means o f d e t e r m i n i n g or checking the type o f dead time. It is evident that the pulse shapes should simulate closely the detector pulses a n d that q u a r t z crystals m u s t be used for the two oscillators; the frequency is changed by choosing different scaling factors. P r o v i d e d that the d e a d - t i m e circuits are o f sufficient quality, it is easy then to o b t a i n 27 with an accuracy better than 10 -3 in a few minutes. This d o u b l e - o s c i l l a t o r m e t h o d , now
V
sup
11+ 12 ~
0
~"r . . . . . . lendod
0.5
I
Fig. 2. Experimental count rate v,,, v in the double-oscillator method for measuring a dead time ,r, as a function of yr.
used also at other laboratories29), has proved to be very practical and reliable. 8. Dead times in series Strictly speaking, in most cases d e a d times occur in series o f two or more. Provided that the earlier dead times in such an a r r a n g e m e n t are much smaller than the last one, their influence can often be neglected or a r o u g h correction m a y be sufficient. The first time 271 m a y be i m p o s e d naturally (e.g., by the p r o p o r t i o n a l counter), whereas the second (27z) is inserted artificially and can be chosen at will. In o r d e r to keep the corrections within reasonable limts, one would like to have a r a t i o 272/271 which is as small as possible. In any case, this requires a reliable estimate o f the a d d i t i o n a l effect p r o d u c e d by 27~ on the o u t p u t count rate. I f we confine ourselves to two dead times in series, four different cases must be distinguished according to the two types o f d e a d times involved. Since the calculations are often rather lengthly, no a t t e m p t has been m a d e to r e p r o d u c e them here and we shall simply state the final results. F o r all cases we assume that the two dead times 27~ and 27z (in that order) are inserted into an original sequence o f pulses forming a Poisson process with count rate p. The o u t p u t count rate will be designated by R, whereas R o is the o u t p u t rate for 27~ = 0. The d e a d - t i m e ratio c~ = 271/272 is always between 0 and 1, and K is defined as the largest integer below 1/~. To simplify notation, we p u t P272 = x . Thus Ro = p / ( l + x ) or pe x, d e p e n d i n g on whether 272 is o f the non-extended or the extended type. If we introduce a t r a n s m i s s i o n factor T by T - R/Ro,
(51)
which thus shows the influence due to 271 alone on the output, it p r o d u c e s the interesting general result that T~< 1, if r 2 is non-extending, b u t T ~> 1, if 272 is extending. F u r t h e r m o r e , the transmission T, as a function o f the d e a d - t i m e r a t i o c~ = 271/r2, reaches an extreme at a = 1, if the d e a d times are o f different types, but at ~ = 2/3, if they are o f the same type, where the last statement h o l d s only for x,~ 1. The result T > 1 m a y be s o m e w h a t surprising since it means that with the insertion o f 27! the losses are smaller than they would be for 272 alone. This effect is difficult to u n d e r s t a n d without a closer l o o k at the c o r r e s p o n d i n g interval distributions. Les us now e n u m e r a t e m o r e specifically the results o b t a i n e d for the various d e a d - t i m e c o m b i n a t i o n s .
DEAD-TIME
(a) Both rl and 712 are non-extending. The t r a n s m i s s i o n f a c t o r T can be written a s 30)
T
=
1-1
(1 +x)
t~ k=O
,
PROBLEMS
55
the b e h a v i o u r becomes m o r e c o m p l i c a t e d for x g 4 where the curves begin to show a d d i t i o n a l m a x i m a . (52)
(b) Both r l a n d T2 are extending. F o r this case it has been shown that 31)
where
K
T = e -:x Z {[--(1--jcOx]klk!} e(i-k~°x
tk' = (e-'~/k!) {[1 + (1 +~) x-sk]s~ +e'"sk+,)k+'"+
(53)
k=O
N u m e r i c a l values can be f o u n d in fig. 4.
+ ( k + l) (1 + ~ x ) [Q(k, Sk+i)-Q(k,s~)],
(c)
with
Ti
non-extending, "c2 extending.
This is the simplest case to calculate and gives
s k = max {(l - k~) x, 0}
R = [p/(1 +c~x)] e-,,(1-~),
and n
thus
~j
Q(n,2) =
(54)
e
.
T = e'X/(1 +c~x) -~ 1 + ½(ctx) 2 - ½(ctx) 3 .
F o r the sake o f convenience, some numerical values o f T are given in fig. 3. It should be noted, however, that
(55)
I f allowance is m a d e for printing errors, eq. (54) has a l r e a d y been o b t a i n e d by Jostl°).
1-T 0"05 j T--1
0.05 I
X =0.7(
i
0.01
~-. . . . '
/
/
__
0.02
: /'
0.01
~)/I/1// I/,/~i ~\"~ll~,\\l //
///
/
/
,.~"---F-"~
tl///il/
0.005
\
t/////7//
\
J
y
///////
~
I ~
,\\\~
o.oos
\ \\1\\l
"DO"2t
0.002
_2o _ \,
0.001
0.001
~k/-/ 5 z-- .---i .... ---:-i. ,~, ! //iHllI/ / / " 4 _ i',,i\ I /' / ~ 7 0.0005 -IIIIJlIIM/ /i/i/Ill/ / I i." ~ :,
IN/HI~~,i/
.."
/
~, 0.0002
o.ooo, o
/"
0.5
1
Fig. 3. T r a n s m i s s i o n f a c t o r T for x = p T 2 f r o m 0.05 to 0.7, in the case o f t w o n o n - e x t e n d e d d e a d t i m e s in series, w i t h :~ = v t / T 2
0.0001
0
/
ll/Ih?/
/
l 0.5
1\2
;
\. 1
Fig. 4. T r a n s m i s s i o n f a c t o r T for x f r o m 0.05 to 0.5, in the case o f t w o e x t e n d e d d e a d t i m e s in series. I. G E N E R A L
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J . W . MULLER
(d) rl extending, T2 non-extending.
References
This combination is fairly difficult to treat but leads to the surprisingly simple result R = p / [ ( 1 - ~ ) x+e~X],
(56)
hence T = (1 + x ) / [ ( 1 - c t )
x + e "x]
1 - ½(~x) 2 + l(o~x)3 [(3/0 0 - 1].
(57)
9. Final remarks
We are well aware that this short survey of deadtime problems is very incomplete. Time does not permit the inclusion of the effects due to a finite lifetime of the source (which would change the interval densities, the counting statistics and the dead-time corrections), variable dead times as they normally occur, for instance, in a multi-channel analyzer, and the dead-time problems connected with pulsed sources. However, there is a much more serious omission, and I suspect that many of you would have been especially interested in hearing something about the dead-time corrections for two-channel problems. More specifically, we would like to be able to correct the experimental count rate of the coincidences. This is a well-known and quite old problem ; it may be sufficient to recall the important contributions by Gandy 32) and Bryant33), for instance. The reason for this omission is simple: to my mind, no decisive progress in these problems has been achieved during the last decade. In spite of repeated attempts, no convincing way has been found to determine, e.g., the interval distribution of the coincidences, and it is difficult to see how some related important questions, such as the evaluation of the random coincidences, could be approached without knowing this function. In the meantime, one will go on using approximate solutions which should be sufficient for most practical purposes. Besides, Bryant's formula 33) for the deadtime coincidence correction in the case of equal dead times in both channels is probably rigorous, a conclusion which is supported by the results of extensive recent Monte Carlo simulations. Furthermore, there is always the experimental possibility of using a livetimer for the coincidence channel 34) or some similar device 29) to circumvent the problem. Nevertheless, we would still prefer to have a general formula at hand and we hope to be able to tell you more about that at some later time--perhaps at the Second Summer School.
1) M. Blackman and J. L. Michiels, Proc. Phys. Soc. 60 (1948) 549. 2) p. Br6once and J. W. Mtiller, An accurate self-measuring dead-time generator, to be published. z) G. E. Albert and L. Nelson, Ann. Math. Stat. 24 (1953) 9. 4) A. T. Bharucha-Reid, Elements of the theory of Markov processes and their application (McGraw-Hill, New York, 1960) ch. 6.3. 5) j. W. Mailer, Interval densities for extended dead times, Report BIPM-112 (1971). 6) id., On the limiting behaviour of the interval density for an extended dead time, Report BIPM-72/7 (1972). 7) id., On the interval distribution for recurrent events with a non-extended dead time, Report BIPM-105 (1967). 8) R. D. Evans, The atomic nucleus (McGraw-Hill, New York, 1955) ch. 28. 9) D. R. Cox, Renewal theory (Methuen, London, 1962). 10) R. Jost, Helv. Phys. Acta 20 (1947) 173. 11) W. Feller, On probability' problems in the theory of counters, Courant Anniversary Volume (Interscience, New York, 1948) p. 105. as) j. W. Miiller, Counting statistics of a Poisson process with dead time, Part I, Report B1PM-111 (1970). la) id., Interval distributions for superimposed renewal processes, Report BIPM-107 (1969). 14) id., Summary of formulae for the dead-time-distorted Poisson distribution, Report BIPM-110 (1970). 15) G. I. Coats, IEEE Trans. Nucl. Sci. NS-13 (1966) 301. 16) N. Pacilio, Nucl. Instr. and Meth. 73 (1969) 167. 17) j. W. Miiller, A general test for detecting dead-time distortions in a Poisson process, Report BIPM-72/10 (1972). 18) E. Rutherford and H. Geiger, Phil. Mag. 20 (1910) 698. 19) j. W. Miiller, Intervalles entre impulsions corr616es, Report BIPM-72/3 (1972). z0) V. E. Lewis (NPL), personal communication (1972). 21) W. Schulze, Z. Elektrochem., Ber. Bunsenges. phys. Chem. 64 (1960) 1089. 22) G. A. Brinkman, Radiochim. Acta 2 (1963) 41. 2a) H. W. Kirby, Determination of coincidence corrections in a proportional counter, I. Double source methods, U.S.A.E.C. Report MLM-1197 (1964). 24) H. W. Kirby and J. Z. Braun, Determination of coincidence corrections in a proportional counter, It. The known half-life method, U.S.A.E.C. Report MLM-1202 (1964). 25) A. P. Baerg, Metrologia 1 (1965) 131, and personal communication (1972). 26) j. W. Miiller, Remarques sur une m6thode propos6e par Baerg pour la mesure de temps morts, Report BIPM-68/8 (1968). 27) id., D6termination d'une d6riv6e pour une r6partition empirique, Report BIPM-69/13 (1969). 28) Id., Une m6thode simple pour mesures pr6cises de temps morts, Report BIPM-69/3 (1969); Une nouvelle m6thode pour la mesure des temps morts, Report BIPM-69/I 1 (1969). 29) J. E. De Carlos and C. E. Granados, Informe preliminar del sistema SAMAR, Report JEN 247 (1972). ao) J. W. Mfiller, On the influence of two consecutive dead times, Report BIPM-106 (1968). For numerical values see: Proc6s-Verbaux S6ances Comit6 Inter. Poids Mesures 36 (1968) 71. aa) id., On the effect of two extended dead times in series, Report BIPM-72/9 (1972).
DEAD-TIME a2) A. Gandy, Intern. J. Appl. Radiation Isotopes 13 (1962) 501. aa) j. Bryant, Intern. J. Appl. Radiation Isotopes 14 (1963) 143. 34) A. Gandy, Intern. J. Appl. Radiation Isotopes 14 0963) 385.
Discussion Houtermans: I would like to stress two points f r o m the experimentalist's point o f view: Firstly, the quickest method by far for the accurate determination of dead times (and coincidence resolution times, by the way) is the mixing o f two sequences of oscillator pulses. Secondly, whatever correction formula and parameters (dead-time values) one intends to use, it is advisable to check the procedure for the highest count rates to be used by the two-source method. The sum o f the corrected count rates of sources 1 and 2 separately should be equal to the corrected count rate of both counted simultaneously. Taylor: I agree with Dr H o u t e r m a n s that it is necessary to measure the dead time over the full range of counting rates to make sure it is not rate dependent. I f it is rate dependent, none o f the dead-time measurement methods discussed will give a meaningful result. F o r example, with the two-source method, if the two sources count at approximately equal rates, nl ~ n2, and the dead time is 0 at rate nl (or n2) and O+d at rate nl+n2, then the method yields an apparent dead time o f 0 + 2A [J.G.V. Taylor, Report AECL-1831 (1963) p. 31]. MMler: Let me first try to make clear what has probably been blurred or completely forgotten in the talk, namely, that the two-oscillator method does not measure any dead-time contribution stemming from the detector or the amplifier. At high count rates, however, their influence may become large enough that it has to be taken into account. A possible solution may then be the classical two-source method, although I would not be entirely happy with such an approach. The reason is that for doing this one has to assume a specific type for the dead-time as well as a pure exponential interval density for the original
57
PROBLEMS
process, what may be doubtful. Perhaps, an acceptable solution could be based on the empirical distribution o f the measured n u m b e r of counts, but unfortunately the corresponding analysis for an extended dead time has not yet been done. On the other hand, the substitution method, as elaborated, e.g., by T h o m a s , is suitable only as an overall check of rate-dependent effects as a whole. Thus it normally does not allow us to find the real physical reason in the case of observed discrepancies since there are too many possible causes. Lowenthal: It would be desirable to have a statement on the magnitude of the error which results from the fact, that the formulae represent approximations only. It is such error estimates which are of interest to experimenters. The results of the 1963 BIPM intercomparison with solid sources o f different activity suggest that dead-time errors are relatively insignificant. MMler: You are certainly right in stressing that it is not the corrections themselves, but rather their uncertainties which an experimentalist should be interested in when judging the quality of a measurement or a method. But it seems to me that it is just here that the conventional approaches fall short since for a realistic error estimation we would have to know the effective interval densities. In the two-oscillator method, for instance, this problem does not arise since here the interval density is produced artificially and depends only on the stability and independence of the quartz oscillators, which is easy to achieve. Finally I might add that eqs. (48)-(50) for determining a dead time do not use any approximations, again in contrast to, say the two-source method. As for the intercomparisons, it is not too surprising that errors due to dead times have never shown up clearly there, since they had not been organized for this purpose. On the other hand, this obviously does not prove that they are always negligible. They might be just hidden by other larger uncertainties. Nevertheless, I would probably agree with your opinion that nowadays there often exist more serious sources of errors than dead times.
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(I973) 47-57;
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NORTH-HOLLAND
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CO.
DEAD-TIME PROBLEMS J O R G W. M(2LLER Bureau International des Poids et Mesures, Pavilion de Breteuil, F-92310 SOvres, France The usual two types of dead times, extended and non-extended, are reviewed and fundamental properties of their effect on the interval distribution and the count rate are discussed briefly. The application of renewal theory to counting processes is sketched and it is shown how the interval distribution, which is distorted by the presence of a dead time, can be used to determine the resulting counting statistics. In particular, the modifications of an original Poisson process, due to a non-extended dead time, are indicated for the case where the origin of the measuring interval has been chosen at random. A simple application then
shows the fallacy of the so-called zero-probability analysis. When renewal processes are superimposed, their convolution property is lost. Therefore, a general formula for the density of multiple intervals is given for the superposition of two component processes. These results have proved useful for studying two recently reported methods of measuring dead times. Finally, formulae are given for the four different ways of arranging two dead times in series. The review is confined to one-channel problems and the emphasis is on exact results.
1. Introduction
of radionuclides. For any detector or electronic circuit there exists a minimum time interval by which two events have to be separated to allow each to be detected. This is called the resolution or the dead time z of the device in question, since the apparatus can be supposed to be "dead % i.e., to ignore further pulses for a period r after the arrival of an accepted event. In general, such "natural" dead times are only approximately constant. Besides, it should be specified whether a dead time is lengthened or not by pulses occurring during its operation. Unfortunately, there is no general solution to these problems and each case must be investigated separately. Since this is often not practical and sometimes impossible, the corresponding corrections remain uncertain. This is why today a different approach is preferable, where a well-defined dead time with known behaviour is inserted artificially into a sequence of pulses, even if it is significantly longer than the natural resolving times. Relatively large but well-known corrections are thus preferred to smaller ones which are uncertain. For traditional and practical reasons, two types of dead times are usually considered. They differ in their response to pulses arriving during a dead time, which (a) for a non-extended dead time: have no effect at all; (b) for an extended (also cumulative or paralysable) dead time: extend the dead time by z, measured from the last arrival time. Accordingly, in the first case dead times are only imposed by the registered output counts, whereas for the second type this happens for all input events. The special interest taken in these two particular types is not so much due to the fact that they approximate rather well the real dead-time behaviour of an apparatus, but because electronic circuits are now readily
The literature on problems related to dead times has increased so much in the last ten or twenty years (at least in volume), that any attempt to review the work done and the progress achieved would require a book rather than a modest state-of-the-art report and is therefore out of the question here. The latest survey in this field 1) that we know of dates from 1948 and is nowadays of only historical interest. No comprehensive and up-to-date bibliography exists. The more recent literature is scattered throughout a great number of journals and in various internal reports, and duplications are quite frequent. Accordingly, our goal will be modest: apart from some general considerations which have been included for pedagogical reasons, we can only mention a small selection of problems dealt with recently, in which the emphasis will be on exact results. Furthermore, our choice will be heavily biased towards work performed at the BIPM, but this should obviously not be taken as a hidden evaluation of merit; it is much more a confession of not being familiar enough with more interesting work done elsewhere, if we know of it at all. 2. Preliminary remarks Since the counting of events is a basic operation, especially in all branches of nuclear physics, where individual interactions between elementary particles can be distinguished by the available counters, the problem of counting losses is of vital importance in any type of precision work. While it is often sufficient to determine ratios, where many corrections cancel, other measurements must be absolute, as is normally the case in the standardization 47
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available which are capable of simulating such performance to a very high degree and because the corresponding corrections can then often be determined exactly. We shall not discuss the important practical problem o f producing such a well-defined dead time since a detailed description will soon be available2). It might also be emphasized that a dead time should always be inserted as early as possible into an electronic circuit in order to reduce pile-up effects. Elaborate attempts to deal with a more general type o f dead time, for instance a "linear c o m b i n a t i o n " of the types mentioned, are available3'4), where an extension o f the dead time takes place with a probability p, successive r a n d o m choices being independent. The two types defined above then turn out to correspond to the special cases p = 0 and p - - 1, respectively. Although they allow, in principle at least, a more elegant and coherent treatment, we shall not discuss such generalizations further.
3. On dead-time effects in general The most important effect due to a dead time is clearly the lowering of the count rate by the loss o f pulses. However, the loss can hardly ever be measured directly since the true count rate is normally unknown. Therefore, other effects caused by a dead time have to be used to determine its presence and to estimate its value. Generally speaking, such effects can be subdivided into two separate classes: one shows up as a change in the time distribution and the other in the statistical behaviour o f the process. They are obviously not independent. A modification of the time distribution, i.e., the intervals between successive pulses, is the more direct consequence o f a dead time and will therefore be discussed first. This effect can be conveniently studied ... ........ ........ F(t)
by the use of time-to-amplitude converters or, for very small count rates, by a multi-channel analyzer used in the time mode. The corresponding change in the statistics, which may be described by a probability law for a given number of events in a certain measuring time, is a more difficult problem that will be tackled (at least under some simplifying conditions) in the following section. Experimentally, information about the counting statistics m a y be obtained directly by repeated measurements or by correlation techniques. In general, the interval density F(t) for a given process must be known before the corresponding count rate R can be evaluated, since
R -1 =
tF(t) dt,
where the passage from the original density f ( t ) to the dead-time modified density F(t) may be complicated. However, there is an interesting exception to this rule. It happens that for the case o f an extended dead time, where a determination of F(t) would be awkward, if not impossibleS), the new rate R can be calculated directly from the original density f(t). In this case we notice that only counts with an original interval exceeding r will survive, which would not be true for a non-extended dead time. If the original process with rate p is described by the interval density f ( t ) , we then obtain at once for the count rate R of the remaining pulses R = p
If0 l 1-
( t ) dt .
where U is the unit-step function, this leads to the well-known result R = pe -°~. (4)
"'.
F(t) = Z "'" '"..,......
O
~,
2~
3~,
J
A j,
(5)
j=l • "., .....
t
the interval distribution original Poisson process, time r leads to a modified is given by the sum 5)
J
"....
i
(2)
F o r the important special case o f an original Poisson process with f ( t ) = U(t)pe -°t, (3)
In some cases, however, also has to be known. For an insertion of an extended dead interval density F(t) which (see fig. 1)
i
(l)
0
where ~t
4T,
Fig. 1. Experimental interval density F(t) for a Poisson process which is deformed by an extended dead time (for/3~2000 s-1 and v ~ 4 0 0 ps).
Aj = U(t-jr)[( - 1)J- a/(j_ 1)]]pJ(t _ j r ) J - l e - jp~' and J is the largest integer below t/v. It can be shown 6) that in the limit F(t)'~[p/(l - p r ) ] e -p' for t >>r.
DEAD-TIME
In the case of a non-extended dead time, no similar " s h o r t c u t " is known and one has to evaluate first the new interval distribution. This explains why count-rate corrections are often easier to determine for an extended than for a non-extended dead time. The general formula which allows us to pass from f ( t ) to F ( t ) for a non-extended dead time r is known to be given by 7) F(t) = U ( t - z )
f(t) +
fk(x)f(t--x) k=l
dx ,
(6)
0
whereJk(t) is the density for a k-fold interval (fl = f ) . The evaluation of F ( t ) may be quite complicated, but again for an original Poisson process the following simple result is obtained F(t) = U(t-z)
pe -p(t-~)
(7)
Hence, the exponential is just shifted by the amount of the dead time r. The resulting count rate is now given by R- 1 = P
f;
te - ' " - ~ ) dt = p - 1 _{_~ ,
which corresponds to the familiar correction formula R = p/(1 +Or)
(S)
for a non-extended dead time, a result which can also be obtained for this special case by simpler reasoningS). For the original rate this gives
49
PROBLEMS 4. Renewal processes
The concept of a renewal process9), created and developed by mathematicians some thirty years ago, has proved to be very useful for dealing with counting problems. The early contributions by Jost 1°) and in particular by Feller ~l) still make very worthwhile reading. Renewal theory can be applied in all those cases where a statistical process, developing in time for instance, is fully described by the distribution function of the interval between two successive arrivals, and this is indeed the case for the majority of the onechannel problems. Let us briefly sketch how this theory can be used for determining the probability distribution for a given process of the renewal type. For more details compare refs. 11 or 12. If the basic interval density is described by f l ( t ) , then multiple intervals (of order k) are given by
L(t) = {f, (t)} *~,
i.e., by the k-fold self-convolution of f~(t), since the contributing individual intervals are independent random variables following the same law. The waiting time for the first event, however, may be different and depends also on the way the time origin is chosen. In the important case that this is done completely at random (for a stationary or equilibrium process), the corresponding density is given by 9,,3) g I (t) = ~/
j(', (X) d x ,
where ~, the mean count rate, is now
The dead time may also be expressed by (9)
F r o m time to time attempts are made to define on the basis of its effect alone an "average dead time" "?, without having recourse explicitly to any specific type. At first sight, a relation such as eq. (9) using the difference between input and output rates might look promising for this purpose. In practice, however, this is not feasible for the simple reason that the true count rate p is unknown, so that the "definition" cannot be applied. Furthermore, such a procedure would be at variance with the definition given above for an extended dead time, for then we would obtain with eq. (4): -
P - P e - p~ -
p2e-Pr
~
( 11 )
t
p = R/(1-Rz).
z = (p-g)/pR.
(10)
z(l+½p~).
It thus appears to be unlikely that a useful definition of a dead time can be found which is independent of an assumption about the specific behaviour.
/t-1 =
tfl (t) dt.
(11 ')
In general, the interval density of event number k(~>l) is thus gk(t) = Or(t) * f k - , ( t ) ,
(12)
where in particular/o(t) = 5(t). The corresponding cumulative distributions defined as usual by Fk(t) =
f,
fk(X) d x
and
Gk(t) =
0
;,
are
gk(X) d x ,
o
(13)
with F o ( t ) = Go(t ) = U(t). If kt is the total number of events occurring in a time interval t, and t k gives the arrival time of event k, then the inequality k > k t implies necessarily t k > t . In terms of the corresponding probabilities we have I. G E N E R A L
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MULLER
therefore
This shows that for any equilibrium expectation of k is simply
f7
gk(X) dx = 1 --Gk(t).
Prob(kt < k) = Prob(tk > t) =
This now allows us to make the step from the time distributions &(t) to the probabilities Wk(t ) for a given n u m b e r k of counts within t, since
Wk(t) = Prob(kt < k + 1) - P r o b ( k , < k)
(14)
= G k ( t ) -- G g + , ( t ) .
The frequent occurrence of the convolution operation suggests the use o f integral transforms, where this is k n o w n to correspond to a simple product. Taking the usual definition o f the Laplace transform off(t),
f(s) - ~ ° { f ( t ) } -
f(t)e -~' dt,
15)
we can now write for eq. (12)
Ok(s) = 01 ( s ) L - , (s)
06)
E(k, t) = t~t.
.#k(S) = O,(S)
S
(23)
These general relations may now be applied to specific counter models (dead times). In particular, the original process will be assumed to be Poissonian with constant count rate p, which corresponds to an exponential interval density [eq. (3)]. Let us now insert a non-extended dead time r into such a sequence o f pulses. F o r the simple case considered here this results in a mere shift of the exponential density, as has been seen earlier [eq. (7)], and in a mean count rate
(83
= p/(l +p~).
The corresponding Laplace transform is easily found to be f~ (s) = pe-S~/(p + s). (24) For multiple densities (k ~> 1), the original function is a shifted g a m m a density
and for the cumulative distribution
Gk(S) = !
process the
fk(t) = U(Tk)[pTk-~/(k--l)!]e-rk,
[f,(s)]t-,
(17)
S
Similar simplifications occur for the m o m e n t s (of order r), defined by
E(U,t) = ~ UWk(t).
(25)
where T k - p ( t - kr). F o r a r a n d o m start, the interval density o f the first event is given by eq. (21); thus for the present conditions
(18)
k=O
the original of which is
F o r the first m o m e n t one can show that
E(k,t) = ~ Gk(t),
P
(19)
l+p~
k=l
1
(27) [l-~preXp[-p(,-z)],
ff,(
-
S 1
S
for0
91 (t) =
and the corresponding transform is =
,
)l k
0
s [ I - - f (s)] (20)
F o r a given d e n s i t y f ( t ) and a definite policy for choosing the time origin, this relation can be used to determine the average n u m b e r of events• Similar but more complicated expressions can be found for the higher moments• Especially, it follows f r o m eq. (11) that
9, (s) = (~/0 [1 - ) ( 0 ] ,
l b r t ~ > z.
_
(20
F o r the arrival time o f event n u m b e r k, this leads to a density
#k(s) = (~/s) [1 - L (s)] L - , (s) = ~ [ & - , (s)- &(~)], (28) and hence to the original
gk(t) = [p/(I + p * ) ] [Fk-, (t)--Vt(t)].
(29)
Since it is easy to show that
thus eq. (20) becomes E(k,s)
= ~/s 2 ,
(22)
Fk(t) = U(Tk) II - I~I Pg(J)],
(30)
DEAD-TIME
where Pk(J) =- (TJ~/!) e-T~ is a "shifted" Poisson probability, gk(t) is also fully determined. After some lengthy rearrangements one thus arrives at the final expressions for Wk(t) which generalize the well-known Poisson probabilities to the case of a non-extended dead time. If K is defined as the largest integer below t/r, and 2 = (1 + p r ) -1 ,
(31)
the general formula for the stationary case can be brought into the following form: (I)for0~
Wk(t ) = 2,
:
( k - 1 - j ) Pk-, (J) -(j =0 k--1
- 2 ~
(k-j)Pk(j) +
j=o
PROBLEMS
51
difference 1 - V can be used either for an estimate of the dead time z or of the original count rate p. 5. A p p l i c a t i o n s
Les us first illustrate the usefulness of eqs. (32) derived above by a simple example. This might also be helpful in settling the old debate about the so-called zero-probability analysis15), which periodically is reintroduced16). This method is based on the assertion that, in order to determine the true count rate p of a series of pulses, it is sufficient to measure the probability for no registered counts, since for a non-extended dead time, losses could be caused only by surviving pulses. In other words, the Poisson probability Po = exp ( - p t ) should not be affected by the presence of any dead times, thus permitting an unbiased estimate of p. As a check, we apply eq. (32) to the special case k = 0. As before, x is short for pr. For0
+ j =~'0 ( k + l - j ) P k + ~ ( j ) } ,
1 -p~/(l
Wo(O =
(2) for k = K :
(32)
whereas for t ~> r,
+x),
(35)
i.e., K~> l, we obtain
Wo(t) = (1 + x ) -1 e - p ( ' - ° . Wk(t ) = ,~
(K--I
--j) PK-I(J)
(j=o K-l
-2
Therefore, evaluating the count rate by means of P0, one obtains
/
~
(K--j)PK(j)--pt
+K+l,
j=o
(3) f o r k = K + l
Wk(t ) = 2
p = - (l/t)lnPo :
=
- ½(Po-l)
o-1)
2 + .--3
(31)
instead of
Exact expressions for the moments as well as formulae corresponding to other choices of the origin (e.g., coincident with a pulse) can be found in the literature 14). A simple formula for the variance a2(k, t) of the number of pulses in t is obtained in the limiting case t >> r, namely =
-- (l/t)[(P
--~ [ ( 1 - P o ) / t ] [I + ½ ( l - - P 0 ) ] ,
( K - - j ) PK(J) + 0 t -- K . [j=O
aZ(k,t)
(36)
--
2 3
x + - x 1 2) 1 , [ p t + x 2 2 ( 1 + - 32 6
(33)
where x = pr is dimensionless. Since according to eq. (23) the expectation E(k, t) is exactly equal to 2pt, the relative variance V is given by
V =- a 2 ( k , t ) / E ( k , t ) ~- 1 - 2 x + [3 + (pt) -~] X 2 , (34) and is thus always smaller than for a pure Poisson process where the relative variance is unity. The
p = [(1-Wo)/t] (l+x),
for t < z .
(38)
Similarly, for t > r, the real probability for no event is W0 = [eX/(l + x ) ] Po,
(39)
or
14%/Po ~ 1 +½x 2 .
(39)
The count rate is then approximately given by p --~ - - ( 1 / t ) [ l n
Wo--½x2],
(40)
which differs again by a dead-time correction from eq. (37) obtained on the basis of the zero-probability analysis. This correction is obviously due to the fact that pulses can also be lost by a dead time which is present at the very beginning of a measuring interval. For a continuous process, it would only disappear if each interval t were started immediately after the end of a dead time. 1. G E N E R A L
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PAPERS
52
J.W.
However, this would suppose an exact knowledge of the length and type of the dead time involved, then permitting the application of the more reliable traditional method that makes better use of the information available. Another application would be a general method for testing an experimental counting distribution for the presence of possible dead-time distortionX7). Here we take advantage of the fact that the deviations from the exact Poisson probabilities P(k), calculated on the basis of the experimental count rate, are expected to be positive in a central region only, but negative for lower or higher values of k. Since for pr ~ 1 the limits of the different domains do not depend on the value of the dead time nor on its type, they can be determined in advance, and a sign test on the empirical differences allows one to decide whether or not a dead time was present, if this method is applied to the famous data of Rutherford and GeigerlS), which are frequently quoted as an example for a Poisson process, one arrives at the somewhat embarrassing conclusion that they have to be rejected at about the 99 % confidence level because their Poissonian character was distorted by the finite resolving time of the observer's eye. 6 . Superpesition
1
k
x }-" {jf~(t) 3 _jZ~Gk+ 1-s(t) + 3 - j g k + 1 _,(t) jAF,(t)}, S=I
(41) where
jAF,(t) =- jF,_, (t) - jF~(t) and
jAG,(t) =- jG,_ t(t) - iG,(t). We recall that, e.g.,
2
~ ~i x p l "-~/~2 j = i
jFk(t) =
J k ( t ) = {ff'(t)} *k and
t
t
jj~,(X) dx, 0
as defined earlier in section 4, and that /lj are the count rates. A simple result is obtained for the densityf(t) of the time interval between succeeding pulses if only two processes are superimposed, namely f(t)
-
/ p'--1 l j ' ( t ) 2 h ( t ) - 1 I"/1 + / / 2 t~z
"~- 2 l ~ ( t ) 2 ~ ( t ) -[-
+/*2 2f(t ) 117(01 , / Pl
of processes
Although renewal processes play a predominant r61e in counting statistics, there are also many cases of practical interest which fall outside this category. Most of them can be thought of as being formed by some kind of superposition. Let us mention here only two such cases which have received some special attention in recent times, namely the random superposition of independent pulse trains and the delayed-state or " p a r e n t - d a u g h t e r " problem. First of all we should mention that in general the superposition of renewal processes does not result in a new renewal process, the only exception being the case of Poisson processes. This feature is the main reason for the additional complexity and means that the density of multiple intervals is no longer given by selfconvolutions of the density for a single interval, as earlier was the case, but must be evaluated separately each time13). In particular, we obtain for the k-fold interval resulting from two superimposed independent renewal processes, where each is characterized by its single interval density if(t) and 2f(t), respectively, the formula
fk(t) -
MOLLER
(42)
where
jg(t) = I~j
;
jh (t) = ktj
;
oo
j f (x) dx
and oc
jg (x) dx.
As a non-trivial example let us take the case where a constant frequency is superimposed on a Poisson process. Here we have (1) for a source with count rate Pl = P:
i f ( t ) = U ( t ) p e -t'', and therefore ,9(t) = ,h(t)=
,f(t);
(2) for an oscillator with frequency v = 1 / T = P2: 2f(t) = 6(t-T),
i.e.,
2g(t) = U(t) U ( T - t ) / T 2h(t) = U(t) U ( T - - t ) [ ( T - O / T 23 .
DEAD-TIME
Inserting these functions yields f ( t ) = [ p/(1 + p T ) ] { U ( t ) U ( T - t ) [ 2 + p ( T - t ) ]
+ - e-°rf(t-T)
.
(43)
P This interval distribution is of special practical interest and will be used in the next section on new methods for measuring dead times. Another quite different and equally important case where renewal theory cannot be applied is in the analysis of the parent-daughter problem, which arises, in principle, in all two-step decays where the second disintegration is delayed because the lifetime of the intermediate level is not negligible. If we assume that the two radiations cannot be clearly distinguished experimentally, as is often the case, then the observable sequence of pulses consists of a superposition of two types of events which are coupled to each other by the statistical nature of the decay of the excited level. We suppose that this can be described by an exponential of the type exp ( - At), where A - 1 is the mean lifetime. Again, the first task is to determine the interval density for the superposition. For a source rate p and detection efficiencies el.z for the two radiations, this can be shown to be 19" 20)
f(t) = (~+/3) -1 [(~+fl e-at) 2 + ABe -at ] x
xexp{--=t----fl(1--e-At)} ' A
(44)
where 0~-----p(/31"-k~32--/;1/~2) and f l = p e~e2. An experimental background count rate b can simply be added to ~. As for the corresponding dead-time correction, an exact formula is too difficult to obtain, at least for the non-extended case, since this would first require a general expression for the multiple interval densities. Approximations, however, can be obtained 2°) which may prove sufficient for practical applications. Once more the problem is much simpler for an extended dead time, for then all we need is an integration o f f ( t ) as given by eq. (44) from 0 to r, which can be performed numerically to any desired precision.
53
PROBLEMS
dead time. Various methods for measuring dead times have been known for a long time, in particular the "decaying source method" or the "double source method". For recent discussions see, e.g., refs. 21-24. However, both methods are quite cumbersome and time-consuming, their precision is rather poor (perhaps 5-10%), and the type of dead time (normally non-extended) has to be known in advance. Some years ago Baerg 25) suggested an interesting variation on the paired source method by substituting artificial periodic pulses (with frequency v) for one of the sources, thereby avoiding the problem of ensuring reproducible geometry for each source when counted singly and in combined form. He has shown that the value of the dead time is then given by z = N,,-'
Accurate corrections demand both reliable formulae and precise knowledge of the parameters involved, i.e., in particular of the type and numerical value of the
N,,)/v]'},
(45)
where Np = observed source count rate and Nov = observed combined rate (source and oscillator). As a matter of fact, this simple formula does not hold rigorously, since some simplifying assumptions have been made in deriving it. It seems, however, that this is not serious provided that we choose v < 1/(3r). A more detailed analysis will have to be based on the exact interval distribution, eq. (43), derived in the previous section. This work is under way, but not yet completed. Here again things are considerably simpler for an extended dead time, for in this case the probability P for losing a pulse can be readily determined by 26) P =
f
~
f(t)
dt = I - (1-
[pv/(p+v)]z}
e-'~ .
(46)
o
This formula as well as the deviation of the interval distribution from a pure exponential have been verified by careful measurements27). In another recent method used for measuring dead times accurately28), the departure from the classical two-source method is even more radical since both sources are replaced by independent oscillators with frequencies v 1 = lIT 1 and v2 I / T 2 . A simple application of eq. (42) for two superimposed renewal processes leads to =
2
7. New measuring methods
{1 - [(N,,,.-
J(/)
--
-
-
U(t) U(T-t)
T1+'12
+ IT, - TzI(T, + T2)-~,5
(t-T),
(47)
with T = 1Iv =- min (Tl, T2). 1. G E N E R A L
REVIEW
PAPERS
54
J.W.
MULLER
The interval density is thus c o n s t a n t between 0 a n d T w h e r e it ends in a delta function. The extreme simplicity o f this d i s t r i b u t i o n permits an exact evaluation o f the losses p r o d u c e d by a d e a d time as well as the calculation o f the resulting interval density, b o t h o f which are very well confirmed by precise measurements. If vs, 0 is the m e a n frequency o f the s u p e r i m p o s e d oscillations after having passed a d e a d time r, it can be easily shown that (see fig. 2) (!) for a n o n - e x t e n d e d dead time:
VSUp
{
V l + V2--2"rVl V2,
for 0 < r <
T/2,
v,
for T / 2 < r
< T;
(48)
(2) for an extended d e a d time:
JVlq-V2--2"CVl!.'2, Vsup
for
0,
0
(49)
for r > T .
In the region v < 1/(227) this m e t h o d thus yields a value for the d e a d time which is i n d e p e n d e n t o f the type. If n l , n2 and nsu p a r e the n u m b e r o f pulses c o u n t e d in a c o m m o n m e a s u r i n g time t, the dead time is given simply by z -
t nl 2
-~-n2--tlsup
(50)
nln2
However, if v exceeds the value 1/(227), the observed c o u n t rate vsup provides us with a simple means o f d e t e r m i n i n g or checking the type o f dead time. It is evident that the pulse shapes should simulate closely the detector pulses a n d that q u a r t z crystals m u s t be used for the two oscillators; the frequency is changed by choosing different scaling factors. P r o v i d e d that the d e a d - t i m e circuits are o f sufficient quality, it is easy then to o b t a i n 27 with an accuracy better than 10 -3 in a few minutes. This d o u b l e - o s c i l l a t o r m e t h o d , now
V
sup
11+ 12 ~
0
~"r . . . . . . lendod
0.5
I
Fig. 2. Experimental count rate v,,, v in the double-oscillator method for measuring a dead time ,r, as a function of yr.
used also at other laboratories29), has proved to be very practical and reliable. 8. Dead times in series Strictly speaking, in most cases d e a d times occur in series o f two or more. Provided that the earlier dead times in such an a r r a n g e m e n t are much smaller than the last one, their influence can often be neglected or a r o u g h correction m a y be sufficient. The first time 271 m a y be i m p o s e d naturally (e.g., by the p r o p o r t i o n a l counter), whereas the second (27z) is inserted artificially and can be chosen at will. In o r d e r to keep the corrections within reasonable limts, one would like to have a r a t i o 272/271 which is as small as possible. In any case, this requires a reliable estimate o f the a d d i t i o n a l effect p r o d u c e d by 27~ on the o u t p u t count rate. I f we confine ourselves to two dead times in series, four different cases must be distinguished according to the two types o f d e a d times involved. Since the calculations are often rather lengthly, no a t t e m p t has been m a d e to r e p r o d u c e them here and we shall simply state the final results. F o r all cases we assume that the two dead times 27~ and 27z (in that order) are inserted into an original sequence o f pulses forming a Poisson process with count rate p. The o u t p u t count rate will be designated by R, whereas R o is the o u t p u t rate for 27~ = 0. The d e a d - t i m e ratio c~ = 271/272 is always between 0 and 1, and K is defined as the largest integer below 1/~. To simplify notation, we p u t P272 = x . Thus Ro = p / ( l + x ) or pe x, d e p e n d i n g on whether 272 is o f the non-extended or the extended type. If we introduce a t r a n s m i s s i o n factor T by T - R/Ro,
(51)
which thus shows the influence due to 271 alone on the output, it p r o d u c e s the interesting general result that T~< 1, if r 2 is non-extending, b u t T ~> 1, if 272 is extending. F u r t h e r m o r e , the transmission T, as a function o f the d e a d - t i m e r a t i o c~ = 271/r2, reaches an extreme at a = 1, if the d e a d times are o f different types, but at ~ = 2/3, if they are o f the same type, where the last statement h o l d s only for x,~ 1. The result T > 1 m a y be s o m e w h a t surprising since it means that with the insertion o f 27! the losses are smaller than they would be for 272 alone. This effect is difficult to u n d e r s t a n d without a closer l o o k at the c o r r e s p o n d i n g interval distributions. Les us now e n u m e r a t e m o r e specifically the results o b t a i n e d for the various d e a d - t i m e c o m b i n a t i o n s .
DEAD-TIME
(a) Both rl and 712 are non-extending. The t r a n s m i s s i o n f a c t o r T can be written a s 30)
T
=
1-1
(1 +x)
t~ k=O
,
PROBLEMS
55
the b e h a v i o u r becomes m o r e c o m p l i c a t e d for x g 4 where the curves begin to show a d d i t i o n a l m a x i m a . (52)
(b) Both r l a n d T2 are extending. F o r this case it has been shown that 31)
where
K
T = e -:x Z {[--(1--jcOx]klk!} e(i-k~°x
tk' = (e-'~/k!) {[1 + (1 +~) x-sk]s~ +e'"sk+,)k+'"+
(53)
k=O
N u m e r i c a l values can be f o u n d in fig. 4.
+ ( k + l) (1 + ~ x ) [Q(k, Sk+i)-Q(k,s~)],
(c)
with
Ti
non-extending, "c2 extending.
This is the simplest case to calculate and gives
s k = max {(l - k~) x, 0}
R = [p/(1 +c~x)] e-,,(1-~),
and n
thus
~j
Q(n,2) =
(54)
e
.
T = e'X/(1 +c~x) -~ 1 + ½(ctx) 2 - ½(ctx) 3 .
F o r the sake o f convenience, some numerical values o f T are given in fig. 3. It should be noted, however, that
(55)
I f allowance is m a d e for printing errors, eq. (54) has a l r e a d y been o b t a i n e d by Jostl°).
1-T 0"05 j T--1
0.05 I
X =0.7(
i
0.01
~-. . . . '
/
/
__
0.02
: /'
0.01
~)/I/1// I/,/~i ~\"~ll~,\\l //
///
/
/
,.~"---F-"~
tl///il/
0.005
\
t/////7//
\
J
y
///////
~
I ~
,\\\~
o.oos
\ \\1\\l
"DO"2t
0.002
_2o _ \,
0.001
0.001
~k/-/ 5 z-- .---i .... ---:-i. ,~, ! //iHllI/ / / " 4 _ i',,i\ I /' / ~ 7 0.0005 -IIIIJlIIM/ /i/i/Ill/ / I i." ~ :,
IN/HI~~,i/
.."
/
~, 0.0002
o.ooo, o
/"
0.5
1
Fig. 3. T r a n s m i s s i o n f a c t o r T for x = p T 2 f r o m 0.05 to 0.7, in the case o f t w o n o n - e x t e n d e d d e a d t i m e s in series, w i t h :~ = v t / T 2
0.0001
0
/
ll/Ih?/
/
l 0.5
1\2
;
\. 1
Fig. 4. T r a n s m i s s i o n f a c t o r T for x f r o m 0.05 to 0.5, in the case o f t w o e x t e n d e d d e a d t i m e s in series. I. G E N E R A L
REVIEW
PAPERS
56
J . W . MULLER
(d) rl extending, T2 non-extending.
References
This combination is fairly difficult to treat but leads to the surprisingly simple result R = p / [ ( 1 - ~ ) x+e~X],
(56)
hence T = (1 + x ) / [ ( 1 - c t )
x + e "x]
1 - ½(~x) 2 + l(o~x)3 [(3/0 0 - 1].
(57)
9. Final remarks
We are well aware that this short survey of deadtime problems is very incomplete. Time does not permit the inclusion of the effects due to a finite lifetime of the source (which would change the interval densities, the counting statistics and the dead-time corrections), variable dead times as they normally occur, for instance, in a multi-channel analyzer, and the dead-time problems connected with pulsed sources. However, there is a much more serious omission, and I suspect that many of you would have been especially interested in hearing something about the dead-time corrections for two-channel problems. More specifically, we would like to be able to correct the experimental count rate of the coincidences. This is a well-known and quite old problem ; it may be sufficient to recall the important contributions by Gandy 32) and Bryant33), for instance. The reason for this omission is simple: to my mind, no decisive progress in these problems has been achieved during the last decade. In spite of repeated attempts, no convincing way has been found to determine, e.g., the interval distribution of the coincidences, and it is difficult to see how some related important questions, such as the evaluation of the random coincidences, could be approached without knowing this function. In the meantime, one will go on using approximate solutions which should be sufficient for most practical purposes. Besides, Bryant's formula 33) for the deadtime coincidence correction in the case of equal dead times in both channels is probably rigorous, a conclusion which is supported by the results of extensive recent Monte Carlo simulations. Furthermore, there is always the experimental possibility of using a livetimer for the coincidence channel 34) or some similar device 29) to circumvent the problem. Nevertheless, we would still prefer to have a general formula at hand and we hope to be able to tell you more about that at some later time--perhaps at the Second Summer School.
1) M. Blackman and J. L. Michiels, Proc. Phys. Soc. 60 (1948) 549. 2) p. Br6once and J. W. Mtiller, An accurate self-measuring dead-time generator, to be published. z) G. E. Albert and L. Nelson, Ann. Math. Stat. 24 (1953) 9. 4) A. T. Bharucha-Reid, Elements of the theory of Markov processes and their application (McGraw-Hill, New York, 1960) ch. 6.3. 5) j. W. Mailer, Interval densities for extended dead times, Report BIPM-112 (1971). 6) id., On the limiting behaviour of the interval density for an extended dead time, Report BIPM-72/7 (1972). 7) id., On the interval distribution for recurrent events with a non-extended dead time, Report BIPM-105 (1967). 8) R. D. Evans, The atomic nucleus (McGraw-Hill, New York, 1955) ch. 28. 9) D. R. Cox, Renewal theory (Methuen, London, 1962). 10) R. Jost, Helv. Phys. Acta 20 (1947) 173. 11) W. Feller, On probability' problems in the theory of counters, Courant Anniversary Volume (Interscience, New York, 1948) p. 105. as) j. W. Miiller, Counting statistics of a Poisson process with dead time, Part I, Report B1PM-111 (1970). la) id., Interval distributions for superimposed renewal processes, Report BIPM-107 (1969). 14) id., Summary of formulae for the dead-time-distorted Poisson distribution, Report BIPM-110 (1970). 15) G. I. Coats, IEEE Trans. Nucl. Sci. NS-13 (1966) 301. 16) N. Pacilio, Nucl. Instr. and Meth. 73 (1969) 167. 17) j. W. Miiller, A general test for detecting dead-time distortions in a Poisson process, Report BIPM-72/10 (1972). 18) E. Rutherford and H. Geiger, Phil. Mag. 20 (1910) 698. 19) j. W. Miiller, Intervalles entre impulsions corr616es, Report BIPM-72/3 (1972). z0) V. E. Lewis (NPL), personal communication (1972). 21) W. Schulze, Z. Elektrochem., Ber. Bunsenges. phys. Chem. 64 (1960) 1089. 22) G. A. Brinkman, Radiochim. Acta 2 (1963) 41. 2a) H. W. Kirby, Determination of coincidence corrections in a proportional counter, I. Double source methods, U.S.A.E.C. Report MLM-1197 (1964). 24) H. W. Kirby and J. Z. Braun, Determination of coincidence corrections in a proportional counter, It. The known half-life method, U.S.A.E.C. Report MLM-1202 (1964). 25) A. P. Baerg, Metrologia 1 (1965) 131, and personal communication (1972). 26) j. W. Miiller, Remarques sur une m6thode propos6e par Baerg pour la mesure de temps morts, Report BIPM-68/8 (1968). 27) id., D6termination d'une d6riv6e pour une r6partition empirique, Report BIPM-69/13 (1969). 28) Id., Une m6thode simple pour mesures pr6cises de temps morts, Report BIPM-69/3 (1969); Une nouvelle m6thode pour la mesure des temps morts, Report BIPM-69/I 1 (1969). 29) J. E. De Carlos and C. E. Granados, Informe preliminar del sistema SAMAR, Report JEN 247 (1972). ao) J. W. Mfiller, On the influence of two consecutive dead times, Report BIPM-106 (1968). For numerical values see: Proc6s-Verbaux S6ances Comit6 Inter. Poids Mesures 36 (1968) 71. aa) id., On the effect of two extended dead times in series, Report BIPM-72/9 (1972).
DEAD-TIME a2) A. Gandy, Intern. J. Appl. Radiation Isotopes 13 (1962) 501. aa) j. Bryant, Intern. J. Appl. Radiation Isotopes 14 (1963) 143. 34) A. Gandy, Intern. J. Appl. Radiation Isotopes 14 0963) 385.
Discussion Houtermans: I would like to stress two points f r o m the experimentalist's point o f view: Firstly, the quickest method by far for the accurate determination of dead times (and coincidence resolution times, by the way) is the mixing o f two sequences of oscillator pulses. Secondly, whatever correction formula and parameters (dead-time values) one intends to use, it is advisable to check the procedure for the highest count rates to be used by the two-source method. The sum o f the corrected count rates of sources 1 and 2 separately should be equal to the corrected count rate of both counted simultaneously. Taylor: I agree with Dr H o u t e r m a n s that it is necessary to measure the dead time over the full range of counting rates to make sure it is not rate dependent. I f it is rate dependent, none o f the dead-time measurement methods discussed will give a meaningful result. F o r example, with the two-source method, if the two sources count at approximately equal rates, nl ~ n2, and the dead time is 0 at rate nl (or n2) and O+d at rate nl+n2, then the method yields an apparent dead time o f 0 + 2A [J.G.V. Taylor, Report AECL-1831 (1963) p. 31]. MMler: Let me first try to make clear what has probably been blurred or completely forgotten in the talk, namely, that the two-oscillator method does not measure any dead-time contribution stemming from the detector or the amplifier. At high count rates, however, their influence may become large enough that it has to be taken into account. A possible solution may then be the classical two-source method, although I would not be entirely happy with such an approach. The reason is that for doing this one has to assume a specific type for the dead-time as well as a pure exponential interval density for the original
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PROBLEMS
process, what may be doubtful. Perhaps, an acceptable solution could be based on the empirical distribution o f the measured n u m b e r of counts, but unfortunately the corresponding analysis for an extended dead time has not yet been done. On the other hand, the substitution method, as elaborated, e.g., by T h o m a s , is suitable only as an overall check of rate-dependent effects as a whole. Thus it normally does not allow us to find the real physical reason in the case of observed discrepancies since there are too many possible causes. Lowenthal: It would be desirable to have a statement on the magnitude of the error which results from the fact, that the formulae represent approximations only. It is such error estimates which are of interest to experimenters. The results of the 1963 BIPM intercomparison with solid sources o f different activity suggest that dead-time errors are relatively insignificant. MMler: You are certainly right in stressing that it is not the corrections themselves, but rather their uncertainties which an experimentalist should be interested in when judging the quality of a measurement or a method. But it seems to me that it is just here that the conventional approaches fall short since for a realistic error estimation we would have to know the effective interval densities. In the two-oscillator method, for instance, this problem does not arise since here the interval density is produced artificially and depends only on the stability and independence of the quartz oscillators, which is easy to achieve. Finally I might add that eqs. (48)-(50) for determining a dead time do not use any approximations, again in contrast to, say the two-source method. As for the intercomparisons, it is not too surprising that errors due to dead times have never shown up clearly there, since they had not been organized for this purpose. On the other hand, this obviously does not prove that they are always negligible. They might be just hidden by other larger uncertainties. Nevertheless, I would probably agree with your opinion that nowadays there often exist more serious sources of errors than dead times.
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