KAHIBARO
Table of Contents
What efficiency means in radiation detection
Detector efficiency tells us how well a radiation detector actually records the radiation that reaches it. In real experiments, not every particle or photon emitted by a source produces a count in the detector. Some miss the detector completely, some pass through without interacting, and some interact but still fail to produce a usable signal.
So efficiency is the fraction of radiation events that are successfully detected under specified conditions. It is one of the most important ideas in radiation measurements because the number of counts recorded by a detector is usually smaller than the number of radiation quanta actually emitted.
Detector efficiency is always a ratio between what is detected and what was available to be detected.
$$
\text{Efficiency} = \frac{\text{number detected}}{\text{number incident or emitted}}
$$
The exact meaning depends on which type of efficiency is being discussed.
Why efficiency is less than 100%
A detector can fail to record radiation for several reasons. A photon may travel in a direction away from the detector. A particle may enter the detector but not deposit enough energy to trigger the electronics. A gamma ray may pass through the detector material without interacting. Sometimes the detector produces a signal, but the signal is too small or too distorted to be counted.
Because of these effects, efficiency depends not only on the detector itself, but also on the radiation type, radiation energy, detector size, detector material, source position, and the electronic threshold used in the measurement.
Absolute and intrinsic efficiency
Two common ideas are absolute efficiency and intrinsic efficiency. They are related, but they are not the same.
Absolute efficiency compares the number of counts recorded with the total number of radiation quanta emitted by the source.
$$
\varepsilon_{\text{abs}} = \frac{N_{\text{detected}}}{N_{\text{emitted}}}
$$
Intrinsic efficiency compares the number of counts recorded with the number of radiation quanta that actually reach the detector.
$$
\varepsilon_{\text{int}} = \frac{N_{\text{detected}}}{N_{\text{incident}}}
$$
Absolute efficiency is always affected by geometry, because only some of the emitted radiation travels toward the detector. Intrinsic efficiency removes that geometric effect and focuses on the detector's response once radiation arrives at it.
Do not confuse absolute efficiency with intrinsic efficiency.
$$
\varepsilon_{\text{abs}} = \frac{N_{\text{detected}}}{N_{\text{emitted}}}, \qquad
\varepsilon_{\text{int}} = \frac{N_{\text{detected}}}{N_{\text{incident}}}
$$
Absolute efficiency includes geometric losses. Intrinsic efficiency does not.
Geometric efficiency
If a source emits radiation in many directions, only part of it reaches the detector. This fraction is called geometric efficiency. It depends on the detector area and its distance from the source.
For a point source emitting uniformly in all directions, the fraction intercepted by the detector is approximately
$$
\varepsilon_{\text{geom}} = \frac{\Omega}{4\pi}
$$
where $\Omega$ is the solid angle subtended by the detector as seen from the source.
This leads to a useful relation:
$$
\varepsilon_{\text{abs}} = \varepsilon_{\text{geom}} \, \varepsilon_{\text{int}}
$$
This equation shows that the overall counting success depends on both the geometry and the detector material response.
A simple picture
The detector covers only a limited range of directions from the source. Radiation emitted outside that range is lost geometrically.
Total efficiency and peak efficiency
In many detectors, especially gamma ray detectors, not every detected event is equally useful. A detector may respond to a gamma ray, but the gamma ray may deposit only part of its energy. This still gives a count, but not in the full-energy peak.
For this reason, one often distinguishes between total efficiency and peak efficiency.
Total efficiency counts every event that produces a recorded signal.
$$
\varepsilon_{\text{total}} = \frac{N_{\text{all recorded events}}}{N_{\text{emitted or incident}}}
$$
Peak efficiency, often called full-energy peak efficiency for gamma detectors, counts only those events in which the full radiation energy is deposited and appears in the photopeak.
$$
\varepsilon_{\text{peak}} = \frac{N_{\text{full-energy peak events}}}{N_{\text{emitted or incident}}}
$$
Peak efficiency is smaller than total efficiency because it is a stricter requirement.
For spectroscopy, the most useful efficiency is often the full-energy peak efficiency, not just the total counting efficiency.
Factors that affect detector efficiency
Efficiency changes with the physical setup. Some important influences are shown below.
| Factor | Effect on efficiency |
|---|---|
| Detector size | Larger detectors usually intercept more radiation and allow more interactions |
| Detector material | Dense, high atomic number materials are often better for gamma detection |
| Radiation energy | Some energies interact more easily than others |
| Distance from source | Greater distance usually reduces geometric efficiency |
| Source position | Off-center placement can reduce efficiency |
| Detector thickness | Thicker detectors often increase probability of absorption |
| Electronic threshold | If set too high, small signals may be ignored |
For charged particles, efficiency can be very high if the particle enters the sensitive region and loses enough energy there. For gamma rays, efficiency is often much lower because gamma rays can pass through matter without interacting.
Energy dependence of efficiency
Efficiency is usually not a single constant for a detector. It often depends strongly on radiation energy.
For gamma rays, low and medium energies may be detected efficiently if the detector material has a high interaction probability. At higher energies, photons may pass through more easily, reducing efficiency. The exact shape of the efficiency curve depends on the detector type and size.
This means that if a radioactive source emits several gamma ray energies, the detector may record them with different efficiencies. Measured peak heights must therefore be corrected using the proper efficiency at each energy.
Measuring efficiency experimentally
Efficiency is often determined using a calibrated source whose activity and emission probability are known. If the source emits radiation at a known rate, and the detector count rate is measured, the efficiency can be found from the ratio.
If a source has activity $A$ and emits the radiation of interest with branching fraction $P$, then the emission rate is
$$
R_{\text{emitted}} = A P
$$
If the measured count rate in the detector is $R_{\text{det}}$, then the absolute efficiency is
$$
\varepsilon_{\text{abs}} = \frac{R_{\text{det}}}{A P}
$$
If background is present, it must be subtracted first.
$$
R_{\text{net}} = R_{\text{measured}} - R_{\text{background}}
$$
Then use $R_{\text{net}}$ in the efficiency formula.
Always use net count rate, not raw measured count rate, when calculating efficiency from experimental data.
$$
R_{\text{net}} = R_{\text{measured}} - R_{\text{background}}
$$
Example calculation
Suppose a source has activity $A = 5.0 \times 10^4 \,\text{s}^{-1}$ and emits one gamma ray of interest in 40 percent of its decays, so $P = 0.40$. The detector records a net count rate of $2.0 \times 10^3 \,\text{s}^{-1}$.
Then
$$
R_{\text{emitted}} = AP = (5.0 \times 10^4)(0.40) = 2.0 \times 10^4 \,\text{s}^{-1}
$$
and the absolute efficiency is
$$
\varepsilon_{\text{abs}} = \frac{2.0 \times 10^3}{2.0 \times 10^4} = 0.10
$$
So the detector has an absolute efficiency of 10 percent for that radiation in that setup.
Efficiency as a practical correction factor
Efficiency is essential when converting measured counts into physical quantities such as source activity or emission rate. If a detector records only a fraction of the emitted radiation, the true emission rate is larger than the observed count rate.
If the efficiency is known, then
$$
N_{\text{true}} = \frac{N_{\text{detected}}}{\varepsilon}
$$
or in terms of rates,
$$
R_{\text{true}} = \frac{R_{\text{detected}}}{\varepsilon}
$$
This is why efficiency calibration is a central part of quantitative radiation measurement.
Summary table
| Efficiency type | Definition | Includes geometry |
|---|---|---|
| Absolute efficiency | $N_{\text{detected}}/N_{\text{emitted}}$ | Yes |
| Intrinsic efficiency | $N_{\text{detected}}/N_{\text{incident}}$ | No |
| Total efficiency | Counts all recorded events | Depends on definition used |
| Peak efficiency | Counts only full-energy events | Depends on definition used |
Key idea
Detector efficiency describes how effectively a detector turns incoming radiation into useful recorded counts. It is not just a property of the detector alone, but of the whole measurement arrangement.
Efficiency depends on radiation type, radiation energy, detector material, detector geometry, and measurement conditions.
A quoted efficiency value is meaningful only for the specific setup in which it was measured.