8.4.6.3. True Coincidences

Table of Contents

What a true coincidence means

In coincidence measurements, two or more detector signals are compared in time. A true coincidence happens when those signals come from the same physical event. The detectors respond separately, but the recorded pulses are related because they were created by one common interaction or decay.

For example, a nucleus may emit two gamma rays in cascade. If two detectors each register one of those gamma rays within the chosen timing window, and both signals really came from that same nucleus, the event is a true coincidence. The key idea is physical connection, not just closeness in time.

Physical origin

True coincidences appear when one process naturally produces multiple detectable products. Common cases include gamma cascades, pair annihilation photons, and correlated particles from a nuclear reaction. In all of these, the emissions are linked by the underlying event.

Suppose an excited nucleus de-excites in two steps,
$$
A^* \to A' + \gamma_1,
$$
followed by
$$
A' \to A + \gamma_2.
$$
If detector 1 sees $\gamma_1$ and detector 2 sees $\gamma_2$, and both came from the same decay sequence, this is a true coincidence.

Timing condition

A true coincidence must satisfy the electronic timing requirement. If the arrival times at the two channels are $t_1$ and $t_2$, the system usually accepts the event when
$$
|t_1 - t_2| \le \Delta t,
$$
where $\Delta t$ is the coincidence resolving time or accepted timing window.

But timing alone is not enough. Signals that fall inside the window may still be unrelated. Those belong to random coincidences, which are treated separately. A true coincidence is both temporally accepted and physically correlated.

A true coincidence is not defined only by two pulses arriving close together. It must also come from the same physical event.

How true coincidences appear experimentally

In an experiment, true coincidences increase the number of accepted paired events above the random background. If the geometry is favorable, the source is appropriate, and the detectors have good timing, the coincidence spectrum shows a clear excess due to these correlated events.

The measured coincidence count rate often contains more than one contribution:
$$
R_{\text{measured}} = R_{\text{true}} + R_{\text{random}}.
$$
The goal of coincidence analysis is often to determine $R_{\text{true}}$.

Example with two gamma rays

Consider a radioactive source that emits two gamma rays in cascade. A detector placed on one side and another detector placed on another side can register them nearly simultaneously. If one gamma ray goes to detector 1 and the other to detector 2, the electronics may form a coincidence signal.

If the cascade is real and both photons come from the same nucleus, the event is a true coincidence. If detector 1 records a gamma ray from one decay and detector 2 records an unrelated gamma ray from another decay, the event may still fall in the timing window, but it is not true.

Relation to detector efficiency

Not every correlated physical event becomes a recorded true coincidence. Each emitted particle or photon must be detected successfully. If the efficiencies of the two detectors are $\varepsilon_1$ and $\varepsilon_2$, then the observed true coincidence rate depends on those efficiencies, along with source activity, branching ratios, and geometry.

In a simplified picture,
$$
R_{\text{true}} \propto A \, B \, \varepsilon_1 \varepsilon_2,
$$
where $A$ is the activity and $B$ represents the probability that the source emits the correlated pair of radiations relevant to the measurement.

This shows that true coincidences are reduced by missed detections, even when the physical event occurs.

Energy and timing selection

True coincidences are often identified more cleanly by using both time and energy information. If each detector is set to accept only certain pulse heights corresponding to expected gamma-ray energies, unrelated events are reduced.

For example, if detector 1 is gated around energy $E_1$ and detector 2 around energy $E_2$, then accepted coincidences are much more likely to represent the desired correlated transition.

Good true coincidence measurements usually rely on both timing selection and energy selection. Timing alone does not guarantee that the event is physically correlated.

Practical importance

True coincidences are valuable because they reveal relationships between emissions. They are used to study decay schemes, identify cascade transitions, suppress background, and improve measurement specificity. In many nuclear experiments, the coincidence requirement allows one to keep events of interest while rejecting a large number of unrelated single counts.

This is especially useful when the source produces many different radiations, but only certain pairs belong to the process under study.

Simple comparison

Feature	True coincidence	Unrelated accepted event
Same physical origin	Yes	No
Arrive within timing window	Yes	Often yes
Useful for correlation studies	Yes	No
Represents real event linkage	Yes	No

Visual picture

Rendering scientific sketch…

True coincidence from a single decay event

Key point

A true coincidence is the detection of two or more signals that are accepted in time and that originate from the same physical event. This makes coincidence methods powerful tools for identifying correlated radiation and reducing background.

For true coincidences, the essential rule is:
$$
\text{common physical event} + \text{accepted timing relationship}.
$$
Without the common origin, the coincidence is not true.

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