of 𝑛 in Poisson distribution is called the Fano factor (F). By using the Fano factor, the limit of the energy resolution of actual detectors is expressed as
(1.18)
In semiconductor, gas, and scintillation detectors, F is ~0.1, 0.1–0.4, and 1, respectively. Therefore, semiconductor detectors such as Si, Ge, and CdTe are known to have a superior energy resolution compared to scintillation detectors, and the best energy resolution so far is ~2% at 662 keV [60, 61]. In practical detectors, energy resolution is not only affected by statistics but also by non‐uniformity of the scintillator. Especially in luminescence center doped scintillators, because we generally use bulky larger material to interact with ionizing radiations effectively, non‐uniform distribution of dopant ions cannot be avoided. The non‐uniform distribution causes differences of light output at each point on the scintillator, and in such a case, the photoabsorption peak or some other features caused by ionizing radiation become, for example, a superposition of multiple Gaussian. Eventually, the shape of the peak in the pulse height spectrum becomes broad, and the energy resolution becomes worse. Such a non‐uniformity is also observed in photodetectors, and the energy resolution observed in practical detectors depends on the non‐uniform response of scintillators and photodetectors.
In order to evaluate the energy resolution of scintillators fundamentally, sometimes intrinsic energy resolution [62] is evaluated. Before the twenty‐first century, the energy resolution was limited to 6–7% at 662 keV in common scintillators. After the invention of Ce‐doped LaCl3 [63] and LaBr3 [45], some new scintillators with high energy resolution (2–4% at 662 keV) appeared. If we say ΔE/E = 0.01 (1%), then n is 5590. If we use typical Si‐based photodetectors, quantum efficiency at visible wavelength is ~80%, and 5590/0.8 ~7000 photons are required to achieve ~1% energy resolution. However, actual scintillation detectors show a large discrepancy with this simple calculation, and in order to explain such a discrepancy, the intrinsic energy resolution was introduced [64]. Although there are several expressions about the intrinsic energy resolution, logically, the energy resolution can be divided into several terms, for example
(1.19)
where δsc, δcir, and δst represent the intrinsic energy resolution, resolution due to circuit noise, and resolution expected by Poisson statistics, respectively. In pulse height spectrum, we observe ΔE/E directly, and we can estimate δst by the number of scintillation photons and the quantum efficiency of the photodetector. The circuit noise δcir can be directly estimated by the injection of a test pulse into the electrical circuit. Thus, we can calculate δsc by the subtraction of δst and δcir from ΔE/E. Under most experimental conditions, estimation must consider a contribution not only by the scintillator but also by the photodetector, and the simplest measurement can be possible by Si‐PD because it has no internal gain. When we measured and calculate the intrinsic energy resolution, that at 662 keV of Tl‐doped NaI was 2.5% [64], Tl‐doped CsI was 5% [65], Ce‐doped LaBr3 was 2.2% [66], Ce‐doped LuAG was 4.5% [66], and Ce‐doped LSO was 7.7% [66]. At present, the remaining problem is whether the intrinsic energy resolution we are observing is a fundamental limit or not, and whether the intrinsic energy resolution is the physical property of each material or not (detector property). The detailed explanations on energy resolution and intrinsic energy resolution are described in Chapter 12.
In an ideal scintillation detector, energy response (signal output intensity vs. incident radiation energy) should have a simple proportional relationship; however, the energy responses of actual detectors are not simple. The main reason is a non‐proportional energy response of scintillators. Figure 1.6 (top) shows a non‐proportional response plot, which shows a relationship between relative light output (pulse height) per unit energy and irradiated γ‐ray energy. Although Eu‐doped SrI2, Ce‐doped CLYC, and Pr‐doped LuAG show a relatively flat response against γ‐ray energy, BaF2, Ce‐doped GAGG, and Tl‐doped NaI exhibit a large fluctuation against γ‐ray energy. When we use the latter three types of scintillators as actual detectors, we must apply the gain correction function to measure the energy accurately. The main origin of the non‐proportional response is considered as related to the K‐edge of the main element of scintillator. This was first pointed out in Ce‐doped GSO [67]. Now, we understand that the flat response against γ‐ray energy is required to achieve high energy resolution. In practical scintillation detectors, we generally use a function between γ‐ray energy and pulse height, as shown in the bottom of Figure 1.6. In a simple case, we fit the relationship by a linear function with the least squares method, and the bottom panel in the bottom of Figure 1.6 shows the residual from linear fitting at each energy. After we prepare such a fitting function, we can convert pulse height channel to energy of ionizing radiation focused on each measurement. Although these two figures have the same physical meaning, people in basic science prefer a non‐proportionality plot, and those using the actual detector prefer a linearity plot.
Figure 1.6 (Top) Relationship between the scintillation decay time (ns) and emission wavelength (nm) and (bottom) relationship betweenγ‐ray energy and photoabsorption peak channel.
1.3.4 Timing Properties
Timing properties are also important for scintillation detectors, and depend on the rise and decay times of scintillators and photodetectors. Generally, we analyze the decay time profile by the sum of multi exponential functions, which can be understood as a typical rate equation. Here, we assume radiative and non‐radiative rate constants of kr and knr, respectively. N(t) is the number of excited states, and it shows a relaxation with a rate of –ΔNr during a very short time of Δt. This can be formulated as
The number of the decrease of excited states –ΔNnr is expressed as
Totally, the decrease of the excited states –ΔN is a sum of Equations (1.20) and (1.21), and it can be expressed as
(1.22)
Thus, we can obtain
(1.23)
