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NUCLEAR INSTRUMENTS AND METHODS 165 (1979) 325-332; © NORTH-HOLLAND PUBLISHING CO.

**DEAD TIME CORRECTIONS IN COINCIDENCE
MEASUREMENTS BY TIME-TO-PULSE-HEIGHT CONVERTERS OR STANDARD COINCIDENCE SYSTEMS**

Istituto di Fisica dell’Università di Catania,

Gruppo Nazionale di Struttura della Materia del CNR, Catania,

Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Italy

* *

* *

Received 7 March 1979

A
definite probabilistic model for a time-to-pulse-height converter is studied
taking into account dead time effects. Formulae are given for the time spectra
and the coincidence rate. Two models for standard coincidence systems are also
studied considering dead time either of the prolonging or non-prolonging type.
All the theoretical models have been checked by Monte Carlo simulation.

*1. Introduction*

In many physical experiments either
time‑to‑pulse‑height converters (TPHC) or standard coincidence
systems (SCS) are employed in order to extract useful information related to
the intensity of the coincidences or to the time distributions of correlated
physical events.

In order to improve on the
statistical accuracy it is often necessary to increase the intensity of the
source of the physical events beyond the limits where effects connected with
the finite time resolution of the apparatus can no longer be neglected^{1
,2}).

The main effects connected with high
intensities are dead time losses and pile-up distortions.

In such cases it is necessary to
correct for such disturbing effects by the help of an exact theory developed
for a realistic probabilistic model of the experiment.

In this paper we develop such
theories for some models of TPHC and SCS.

*2. Theory of the TPHC*

2.1. general
considerations

To be definite we shall consider a
probabilistic model for the TPHC, having in mind the combination of the TPHC
Ortec Model 467 together with the Gate and Delay Generator Ortec model 416A
(GDG).

We consider trains of electrical
pulses of negligible duration and with exponential distributions in two
separate channels.

We assume that the pulses on the
first channel act directly on the start input of the TPHC, while the pulses on
the second channel act on the stop input of the TPHC after being delayed by the
GDG.

Start-to-stop conversion is
accomplished only after a valid start has been identified and after a stop
pulse has arrived within the selected time range *R.*

The start input is disabled during
the busy interval to prohibit pile-up; this introduces a dead time in the
start channel equal to *ξ* + τ* _{ξ}* ;
here

_{}

Also the GDG input is disabled for a
constant time interval τ after eaeh passed stop pulse.

Because of the dead times introduced
by the TPHC and the GDG the actual distributions of the interarrival times of
the pulses in the two channels will no longer be exponential, however the new
distributions can be easily deduced.

We shall be mainly interested in the
time spectrum, i.e. in the distributions of the time intervals between a valid
start and the first stop pulse.

2.2. accidental
coincidences

We first consider only the time spectrum of the accidental coincidences, i.e. assume that the distributions of the interarrival times of the pulses in the two channels are independent.

The two previously named cases (a)
and (b) will be treated separately.

Referring to fig. 1, let us first consider case
(a) and let *X*, *Y _{k}*,

- the time length of the
previous conversion process,

- the interarrival time between
two effective consecutive pulses in the stop channel,

- the waiting time for the first
start pulse following the instant in which the start input circuit of the
TPHC is again free to accept pulses after the preceding conversion,

- the time interval between the
instant in which the GDG is again free to accept other incoming stop pulses,
after the last stop pulse arrived before start, and the instant in which the
next stop pulse has arrived to complete the conversion process.

We adhere to the methods and the
terminology of the renewal theory as expounded in ref. 3.

We look for the probability *H _{ξ}*

This event occurs if a start pulse
arrives with *W = w*,

*U _{m} = *

* *

and finally the next pulse on the stop channel
arrives with *Z = z *and

*(*τ_{1}* + w)* - *(u + *τ*)*** **£ *z *£* (*τ_{1}* + w) *- *(u *+* *τ*) + x .*

All this for some possible
combinations *w*,* u*,* **z*,*
m* .

* *

If *g*_{1}*(αw)*,* h _{m}(βu)*,

(1) _{}

It remains to write down the
expression for the probability densities:

- for *W *and

Fig.
1. Schematical diagram of the pulse sequence in the two channels of the TPHC.

(exponential distribution for the incoming
pulses on both channels), we can write

*g*_{1}*(αw) *º *α e*^{- }* ^{α w}* ,

*g*_{1}*(βz) *º *β e*^{- }* ^{β z}* ,

with *α* and *β* the
incoming intensities in the start and stop channel respectively and g_{1 }*(x)* º 0 for *x* < 0;

- for *U _{m} *we must write the convolution
integral of

Remembering that the *Y _{k} *represent the interarrival times
of the effective pulses on the stop input after leaving the GDG, which remains
blocked during an interval τ after each effective pulse has been passed
on, we can write

*Y _{k} = V_{k} + *τ

* *

After performing the convolution
integrals we have

*h _{m}(β*

*h _{m}(β*

Here

_{}

is the derivative of the modifìed incomplete
gamma function^{4}’^{5}), namely

_{}.

We note that the summation in eq.
(1) is extended to *m *= 0 to take into account the possibility of having a stop
pulse after start without any previous pulse in the stop channel between the
preceding conversion and the new start pulse, by assuming an atom of unit
weight at the origin, i.e. *h*_{0}*(βu) *º* δ(u) *^{3}).

The infinite sum and the integrals
in eq. (1) can be easily performed by the properties of the incomplete gamma
functions^{4}’^{5}), we do not give here the lengthy formal
manipuiations.

The final result is

(2) _{}

where

_{},

_{} ,

_{}

__ __

_{}

τ* _{m}* º τ

__ __

*M* º** **integral value of the ratio τ

*G _{m+}*

*G _{m+}*

In case (b) *(ξ ***= ***R) *we may consider a delayed renewal process

* *

*U _{m} = U*

* *

under the hypothesis τ *<**R *(this can practically always be
achieved),

_{} .

In order to find out the
distribution independently of the length of the preceding conversion we note
that *H _{ξ} (x)
*can be regarded as the transition probability of a simple Markov
process as defined in ref. 6 and because it depends on

(3) *H(x) *@ *H(R)*×*H _{ξ}*

This also because it can be
demonstrated by the properties of the incomplete gamma functions, that *H _{ξ}*

*H(R) *can be
easily determined by putting *x* **= ***R *in formula (3) and fìnally we may
write

(4) _{}

which expresses the time distribution of the
accidental coincidences in our model of TPHC in terms of the conditional
probability distributions *H _{ξ }(x) *previously
derived in the two cases (a) and (b).

2.3. correlated
events

Of course before studying the effect
of correlated events in the two channels we must postulate a specific
conditional probability distribution of having a pulse in the stop channel at
time *t*′ given a pulse on the start channel at time *t*.

We assume that this distribution
admits a density which we indicate by ρ* _{t }(t*′

We first consider the limit case

(5) ρ* _{t }(t*′

with δ*(x)* the Dirac function.

However we shall assume also uncorrelated
pulses arriving with an intensity α - γ in the first channel
and β - γ in the second channel,
γ being the intensity of the correlated events.

So four different possibilities must be
considered:

1) The start pulse has no correlated event in
the stop channel, and will give an accidental coincidence with any pulse in the
stop channel (either of the correlated stock or else) arriving within the time
range.

2) The start pulse has a correlated event in
the stop channel delayed by *r *[according to eq. (5)], however
the true coincidence will be lost because of the arrival of any other pulse
before the arrival of the correlated one.

3) The start pulse has a correlated event in
the stop channel but the true coincidence will be lost by the dead time of the
GDG, so an accidental coincidence will result by the arrival of any other
pulse within the time range.

4) The start pulse has a correlated event on
the stop channel; events under (2) or (3) do not occur, so we have a true
coincidence.

Before proceeding to the derivation
of the relevant formulae it must be pointed out that in this case of
correlated events the conditional probability may not in general have the
simple Markov property (as defined in ref. 6), because the probability
distribution of the waiting time for the first stop pulse after start depends
on the probability distribution of the incoming pulses in the stop channel which
in turn, because of the delayed correlation, depends on the previous occurrence
of the correlated pulse in the start channel and, because of the dead time of
the anti-pile-up blocking circuit of the start input, on the preceding history
of the process.

However, if the intensity ratio of
the correlated events is small as compared to the total intensity in the stop
channel (γ_{ }/_{ }β << 1)
the process will be approximately simple Markov, because we do not have to
worry, in this case, about the detailed distribution of the correlated events
in the stop channel.

In such an hypothesis we can write for the
conditional probability distribution of the waiting time of an effective stop
pulse after a valid start, given a preceding conversion length *ξ*.

(6) _{}

with

_{} the contribution of process (1).

_{} the contribution of process (2),

_{} the contribution of process (3),

_{} the contribution of process (4),

and

(7) _{}

(8) _{}

_{}is
obviously identical to the integral appearing in formula (1) and can be
expressed by the right hand side of formula (2).

The right hand side of formula (8)
may be integrated by the same methods and gives

(9) _{}

[obviously _{} if *r *>* *τ* *or *x* < *r*],

* *

here the function _{} is given by the right
hand side of formula (2) with, however, *M *º* *integral
value of the ratio *(*τ_{1 }*+ r) / *τ* *and

(10) _{},

** **

with

_{}

and the other quantities as previously defined.

Obviously also in this case the a
priori probability can be obtained by formula (4).

In the more general case of an
arbitrary density ρ_{t}_{′ }*(t)*,
to find out the general formulae we can derive the distribution function for
the limit densities eq. (5), _{},** **and use this as the Green function of a randomized process

(11) _{}

We have here made the usual assumption of
translational invariance:

ρ* _{t }(t*′

* *

Also in this case the a priori
probability distribution will be given by formula (4) in terms of the a
posteriori probabilities given by eq. (11) for the two cases (a) and (b) after
the usual sobstitution _{}.

2.4. monte
carlo simulation

In order to get some insight in the
more general case we have simulated our TPHC model by a Monte Carlo method.

This allowed us also to check all
previously derived formulae.

The model consists in generating, by
standard method, the interarrival times in the two channels with initial
exponential distribution and checking the eventual losses for dead time.

In figs. 2 and 3 the comparison of
the Monte Carlo results with the exact formulae previously derived is shown.
The theoretical curves given by formulae (2) and (4) are normalized to the
relative Monte Carlo results by their respective maximum values.

The Monte Carlo curves were computed
by their absolute values, but in the figures they are normalized to the linear
approximation, in order to give an idea of the dead time losses (i.e. the
ordinate *H _{ }/_{ }H*

The agreement is very good within
the statistical error introduced by the Monte Carlo computations: this
statistical error, even if not signifted in the

Fig. 2. Time spectra of the TPHC** **according to the exact formulae
derived for the accidental coincidences (continuous lines) compared with the
results of the Monte Carlo simulation (points). The theoretical curves are
normalized at the maximum value to the respective Monte Carlo calculations. The
Monte Carlo curves are normalized to the linear approximation (see text) in
order to show the dead time losses. Fixed parameters:
α = β = 1 μs^{-}^{1} , τ_{1} = 5 μs,
τ_{2} = 4 μs, *R* = 4 μs.
Varying parameter: τ = 0, 1, 2, 3.3 μs. See text for the
meaning of the parameters.

Fig. 3. Same as fig. 2. Fixed parameters: β = 1 μs^{-}^{1} , * *τ_{1} = 5 μs,
τ_{2} = 4 μs,
τ = 3.3 μs, *R* = 4 μs.
Varying parameter: α = 0.01, 0.1, 0.5* *μs^{-}^{1} . The ordinate value for
α = 0.01 should be multiplied by a factor 1.5.

figure, can be estimated by the scattering of
the points around the theoretical curves.

In figs. 4 and 5 the approximate formulae for processes, not simple-Markov, as
given by formulae (4), (7) and (8), are compared with the Monte Carlo
computations.

We note that for accidental
coincidences with low intensities in the start channel the interarrival times
between consecutive and effective start pulses will be long enough and if
ατ << 1 the distribution of the interarrival times on
the stop channel may be considered as stationary, in this case we know that
because of the renewal theorem applied to persistent renewal processes^{3}),
the distribution of the residual waiting time can be expressed in terms of the
distribution of the interarrival times on the stop channel by^{3})

_{} ,

* *

Fig. 4. Same as fig. 2 for the approximated formulae derived in case of
partially correlated events. Fixed parameters: τ_{1} = 5 μs,
τ_{2} = 4 μs,
τ = 3.3 μs, *r* = 3 μs, *R* = 4 μs,
γ/α = 0.1. Varying parameter:
α = β = 0.l, 0.5,** **1, 2 μs

and, because in our case

_{}

we obtain

_{}

If we also have β*R* << 1
we get the often quoted linear approximation

*H(x) = *β*x *, for any *x* < *R*.

* *

The validity of these approximations
can be checked in fig. 2.

As a final point we note that the
renewal theorem can be used to obtain the coincidence rate for the limit
distribution eq. (5).* *In fact we may consider

Fig. 5.* *Same as fig. 4.*
*Fixed parameters: τ_{1} = 5 μs, τ_{2} = 4 μs,
τ = 3.3 μs, *r* = 3 μs, *R* = 4 μs.
Varying parameters: α = β = l μs^{-}^{1} and γ = 0.1,
l μs^{-}^{1} ;^{ }α = β = 0
μs^{-}^{1} and γ = l μs^{-}^{1}

the process originated by the random variable

_{}

and apply the renewal theorem to the number of
renewal epochs^{3}). So we may affirm that the number of effective
start pulses in a long counting interval *t
*is approximately normally distributed^{3}) with expectation *t _{ }*/

_{} ,

and _{} ; _{}

TABLE
1

Coincidence rate per *μs*, integrated over the entire time
range of the converter and ratio of the number of the events under the peak and
the integral spectrum. Fixed parameters (see text): τ_{1} = 5 μs,
τ_{2} = 4 μs, *r* = 3 μs,
*R* = 4 μs.

In practical calculations for *μ*** **we have used the approximate formula

(12) _{}

Here _{}.

* *

In the stationary approximation the measured
true coincidence rate obtained after subtraction of the accidental coincidence
rate is given by

_{}.

This last formula allows to correct
in a simple way for dead time losses whenever the linear approximation is
justified.

In table 1 the absolute value given
by the approximated formula (12) is compared with the Monte Carlo computation.
A comparison of the peak-to-integral ratio between the model and the Monte
Carlo simulation is also shown in table 1, as deduced by the spectra. This
ratio is independent of the approximations of formula (12) and so gives an idea
of the error introduced by the non-Markovian character of the process.

*3. Standard coincidence system*

3.1. dead
time of the non-prolonging type

We here consider trains of
electrical pulses of negligible duration and with exponential distribution in
two separate channels as before. However, the pulses act on two time pick-up
units which generate pulses of duration τ_{1} in channel one and
τ_{2 }in channel two.

The two time pick-up units remain
blocked for a time equal to the duration of the output pulses. This introduces
in the two channels dead times of the non-prolonging type^{7}).

The effective pulses coming out of
the two time pick-up units will be processed by the coincidence circuit which
will give an output pulse whenever an effective pulse arrives in channel one
(two) during a time interval equal to the duration of the pulse in channel two
(one) starting from the arrival of a pulse in channel two (one), plus the
symmetrical case.

In the absence of correlated pulses
in the two channels, by the often quoted renewal theorem^{3}’^{6})
the distribution of the waiting time of a pulse in the second channel after the
arrival of a pulse in the first channel can be considered as stationary, for
sufficiently long counting times, so that the accidental coincidence rate will
be given by the product of the expectation for the number of pulses in one
channel during the counting time *t*,*
*by the expectation for the number of pulses in the other channel during a
period τ_{1 }+ τ_{2}, in the stationary
approximation, i.e.

* *

(13) _{} ,

α and β being the intensities of the
pulses in the two channels before the time pick-up units.

In case we have correlated pulses in
the two channels with intensity γ, we note that such pulses will always be
counted without dead time losses, unless the preceding pulse blocking the time
pickup unit with shorter dead time is also of the correlated stock.

This is because any correlated pulse
which arrives within the dead time originated by a preceding pulse of the
non-correlated stock, to the effect of the coincidence output will be replaced
by the killing pulse, so that, if τ_{1 }£ τ_{2}, the total (i.e. true
plus random) rate will be

(14) _{}

If, as usual, the random
coincidences are measured only partly by inserting a convenient delay in one
channel, the intensity γ can be deduced by formulae (13) and (14) by
measuring independently, besides *n*_{A}** **and *n*_{T},** **the intensities
α and β and the ratio τ_{1 }/ τ_{2}.

If γ << α
and γ << β we obviously have for the true coincidence
rate

_{} .

** **

3.2. dead
time of the prolonging type

The case in which the time pick-up
units prolong the output high voltage level, when another pulse arrives during an
interval of time equal to the processing time for a single pulse, has already
been considered by other authors^{7}’^{8}).

However, the formulae they give are
not correct, in fact one formula gives wrong results for higher incoming
intensities^{8}), the other for longer resolving times^{7}).

In order to get the correct formula
for the random coincidence rate we observe that if in channel one (two) arrives
a pulse able to pass through the time pick-up unit [i.e. no previous pulse has
arrived in a time interval τ_{1 }*(*τ_{2}*)*
before it], then any pulse whatever arriving in channel two (one) in a time
interval τ_{2 }*(*τ_{1}*)* before it,
will produce an accidental coincidence. So the accidental coincidence rate
will be given by the product of the number of pulses arriving in channel one
(two), during the counting time *t*** **by the probability of having no other
pulse in the same channel within a interval τ_{1 }*(*τ_{2}*)*
before the instant of arrival of the pulse which originates the coincidence, by
the probability of having a pulse in channel two (one) within an interval
τ_{2 }*(*τ_{1}*)* before it, plus the
symmetrical case, i.e.

(15) _{}

* *

In case of correlated pulses in the
two channels with intensity γ by the same considerations made in sect. 3.1
we have for the total coincidence rate (τ_{1 }< τ_{2})

(16) _{}

We note that formulae
(13) – (16) can be generalized to the case of an *n*-fold
coincidence system by the methods considered in refs. 7, 8.

Formulae (13) – (16) have also been checked by Monte-Carlo simulation with an accuracy better than 1%.

We thank Prof. V. I. Goldanskii for
useful discussions.

** **

1) J. Radeloff, N. Buttler, W.
Kesternich and E. Bodenstedt, Nucl. Instr. and Meth. **47** (1967) 109.

2) J. Glatz, Nucl. Instr. and Meth. **79
**(1970) 277.

3) W.
Feller, *Introduction to
probability theory and its applications *(J. Wiley, New York, 1971).

4) M. Abramovitz and I. A. Stegun, *Handbook of mathematical functions
*(Dover Publ, New York, 1965).

5) A. Erdelyi, *Higher transcendental functions *(McGraw-Hill, New-York, 1953).

6) J. L. Doob, *Stochastic processes *(J.
Wiley, New York, 1953).

7)** **V. I.** **Goldanskii, A. V.** **Kutsenko and M. I. Podgoretskii, *Counting statistics of nuclear particles *(Hindustan Publ. Co., Delhi, 1962).

8) L. I. Schiff, Phys. Rev.** 50 **(1936) 88.