KAHIBARO
Table of Contents
Why dead time matters
When radiation is detected, the detector and its electronics need a short recovery time after each event. During this recovery time, the system cannot correctly record another event, or may record it incorrectly. This interval is called dead time.
Dead time is important because if radiation arrives too quickly, some events are missed. The measured count rate then becomes smaller than the true count rate. This effect is especially important in high-rate measurements.
Dead time is the time after a detected event during which the detector system is unable, fully or partly, to record another event correctly.
If the true event rate is low, dead time has little effect. If the true event rate becomes comparable to the inverse of the dead time, the losses become significant.
Physical origin of dead time
Dead time can come from different parts of the detection system. In a gas detector, ions and electrons need time to move and for the electric field to recover. In scintillation systems, the light pulse and electronic pulse processing take time. In semiconductor systems, charge collection and pulse shaping also require time.
The electronics can also introduce dead time. After one pulse is accepted, amplifiers, discriminators, counters, or digital acquisition systems may need a short interval before they are ready for the next pulse.
So, dead time is not always just a property of the detector material. It is often a property of the whole measurement chain.
What dead time does to counting
Suppose the true rate of incoming events is $R_{\text{true}}$, and the measured rate is $R_{\text{meas}}$. Because some events arrive during dead periods and are lost, we usually have
$$
R_{\text{meas}} < R_{\text{true}}
$$
As the true rate increases, the difference becomes larger.
A useful way to think about this is to imagine a detector that becomes briefly blind after each recorded event. If many events arrive close together in time, some fall into blind intervals and are not counted.
Two ideal models of dead time
Two simple models are commonly used, nonparalyzable dead time and paralyzable dead time. These are idealized descriptions, but they are very useful.
Nonparalyzable model
In the nonparalyzable model, after one event is recorded, the system is dead for a fixed time $\tau$. Any event arriving during that interval is ignored. These lost events do not extend the dead time.
If the true rate is $R_{\text{true}}$, then the measured rate is
$$
R_{\text{meas}} = \frac{R_{\text{true}}}{1 + R_{\text{true}}\tau}
$$
This can be rearranged to estimate the true rate from the measured rate,
$$
R_{\text{true}} = \frac{R_{\text{meas}}}{1 - R_{\text{meas}}\tau}
$$
This formula is valid only when $R_{\text{meas}}\tau < 1$.
For a nonparalyzable system,
$$
R_{\text{meas}} = \frac{R_{\text{true}}}{1 + R_{\text{true}}\tau}
$$
and
$$
R_{\text{true}} = \frac{R_{\text{meas}}}{1 - R_{\text{meas}}\tau}
$$
In this model, the measured rate cannot exceed $1/\tau$.
Paralyzable model
In the paralyzable model, each event creates a dead interval of duration $\tau$. If another event arrives during that dead interval, it is also not recorded, but it starts a new dead interval. In this way, high event rates can keep the system dead for long periods.
For this model,
$$
R_{\text{meas}} = R_{\text{true}} e^{-R_{\text{true}}\tau}
$$
This model behaves very differently at high rates. If $R_{\text{true}}$ becomes very large, the measured rate can actually decrease because the system is almost always dead.
For a paralyzable system,
$$
R_{\text{meas}} = R_{\text{true}} e^{-R_{\text{true}}\tau}
$$
At very high true rates, the measured rate may drop because repeated events keep extending the dead time.
Comparing the two models
The two models agree approximately at low rates, but differ strongly at high rates.
| Model | Event during dead time | Does dead time get extended? | High-rate behavior |
|---|---|---|---|
| Nonparalyzable | Lost | No | Measured rate approaches $1/\tau$ |
| Paralyzable | Lost | Yes | Measured rate can fall toward zero |
Visual picture
A time-line picture helps show the difference.
In the top line, events inside the shaded region are simply ignored. In the bottom line, an event inside the shaded region starts a new shaded region.
Dead time fraction and live time
The system is not dead all the time. The fraction of time during which it can accept events is called the live time. The fraction during which it is unavailable is the dead-time fraction.
If the total measurement time is $T$ and the total dead time accumulated is $T_d$, then
$$
\text{live time fraction} = 1 - \frac{T_d}{T}
$$
and
$$
\text{dead time fraction} = \frac{T_d}{T}
$$
In practical instruments, data acquisition systems may report live time and real time separately. Real time is the total elapsed clock time. Live time is the time during which the system was able to record events.
Real time is the total elapsed time.
Live time is the effective counting time when the system is ready to accept events.
These are not the same when dead time is present.
Example of dead-time loss
Suppose a detector has dead time $\tau = 100 \, \mu\text{s} = 1.0 \times 10^{-4}\,\text{s}$, and the true rate is
$$
R_{\text{true}} = 2000\,\text{s}^{-1}
$$
For a nonparalyzable model,
$$
R_{\text{meas}} = \frac{2000}{1 + 2000(1.0\times10^{-4})}
= \frac{2000}{1.2}
\approx 1667\,\text{s}^{-1}
$$
So about $333\,\text{s}^{-1}$ are lost.
For the same numbers in the paralyzable model,
$$
R_{\text{meas}} = 2000 e^{-2000(1.0\times10^{-4})}
= 2000 e^{-0.2}
\approx 1637\,\text{s}^{-1}
$$
The two values are similar here because the rate is not extremely high.
How dead time is measured
Dead time can be estimated in several practical ways. One common method compares measured count rates under known conditions. Another classic approach is the two-source method, where the count rate is measured for source 1 alone, source 2 alone, and then both together. If there were no dead time, the combined rate would equal the sum of the separate rates. The difference gives information about count losses.
In modern systems, dead time may also be determined from pulser signals or from the electronics specifications.
When dead time becomes serious
Dead time becomes serious when the average time between events is not much larger than $\tau$. Since the average time between events is roughly $1/R$, the condition for noticeable dead-time effects is
$$
R\tau \not\ll 1
$$
If $R\tau \ll 1$, corrections are usually small. If $R\tau$ approaches $1$, corrections become large and uncertain.
A useful rule is to check the dimensionless quantity $R\tau$.
If $R\tau \ll 1$, dead-time losses are small.
If $R\tau$ is not small, dead-time correction is essential.
Practical consequences
Dead time affects more than the total count rate. It can distort measured spectra, reduce apparent activity, and create errors in coincidence work and timing measurements. At very high rates, pulse pile-up and dead time often appear together, making measurements harder to interpret.
Because of this, experimental setups are often designed to keep count rates moderate, use faster electronics, or apply reliable dead-time corrections.
Summary
Dead time is the recovery interval after a detected event during which the system cannot properly record another one. It causes the measured count rate to be lower than the true rate. The two main ideal models are nonparalyzable and paralyzable dead time, with formulas
$$
R_{\text{meas}} = \frac{R_{\text{true}}}{1 + R_{\text{true}}\tau}
$$
for the nonparalyzable case, and
$$
R_{\text{meas}} = R_{\text{true}} e^{-R_{\text{true}}\tau}
$$
for the paralyzable case. Understanding dead time is essential whenever radiation measurements are made at moderate or high counting rates.