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rel="nofollow" href="#ulink_bc907f1f-58bc-5017-9294-a5ac0cda9280">Equation (1.5), and the average energy consumed per electron–hole pair can be expressed as ξ = E_g(2.3 + 1.5 K). Under this condition, the parameter β is approximated to

      (1.13)

      The number of electron–hole pairs is

      (1.14)

      Thus we can obtain

(1.15)

Equation (1.15) is a convenient form because it contains a parameter of optical phonon loss, and the evaluation of the model under various temperatures was made in 2010 [58].

      Generally, experimental evaluations of effects of optical phonon (thermal) loss is difficult. For experimental research a very convenient model assuming β_s = 2.5 is proposed, based on the data of various scintillators [59], and most research uses this formula which is described as

(1.16)

In this formula, we can determine L, E_g, and Q experimentally, and we can deduce S by using these values. L can be evaluated by the pulse height spectrum, E_g by optical absorption or reflection spectrum, and Q by PL quantum yield measurement. Although we cannot predict the scintillation light yield by these formulae, we can understand why halide and sulfide materials show higher light yields than oxide materials by E_g. In most cases, Q of commercial scintillators is close to 100%. As described in Chapter 11, in the case of the integration‐type detector, a factor representing the absorption of ionizing radiation is multiplied to Equation (1.16).

Typical ways to determine L is to compare with the pulse height of a single photoelectron (~single photon) of photodetector with high quantum efficiency, and we can know the number of photoelectrons from the scintillator. In this manner, we can directly measure L after dividing the number of photoelectrons by the quantum efficiency of the photodetector at the emission wavelength of the scintillator. The other common way is to compare the pulse height of ⁵⁵Fe 5.9 keV X‐ray measured directly by Si‐based photodetectors. The photoabsorption peak due to 5.9 keV X‐ray corresponds to ~1640 electron–hole pairs, and we can evaluate the number of photoelectrons generated by scintillation photons if we use the same experimental conditions. Another way is to compare relative pulse height with scintillators with known scintillation light yield. In these evaluation techniques, finally, we must divide the observed number of photoelectrons by the quantum efficiency of photodetectors at emission wavelength, and it contains a certain amount of error because the quantum efficiency of photodetectors has a wavelength dependence. Typically, the error for the estimation is 5–10% for experts of these kind of experiments. In some research, the scintillation light yield is calculated by the area intensity of the radioluminescence spectrum, and this is generally incorrect because radioluminescence intensity is not a quantitative but a qualitative value. The main reason is that we cannot correct the absorption probability of ionizing radiations in scintillators. For example, we have two samples with the same light yield, but one is light and the other is heavy. If we irradiated X‐rays to these samples, the latter would show higher radioluminescence intensity. The other reason is the effect of TSL at room temperature, and we also cannot correct any effects from TSL. If we are to measure scintillation light yield quantitatively, pulse height measurements must be conducted. At present, we cannot measure the pulse height of scintillators with very slow decay (> 0.1 ms), and further technical development is required to measure pulse height.

      The following topics are limited to photon counting‐type detectors, since integration‐type detectors cannot measure the energy of ionizing radiation, except in some special cases. The scintillation light yield is one of the most important properties of scintillation detectors because it directly relates to the energy resolution. Generally, the energy resolution obeys Poisson statistics. If we represent the quantum efficiency of the photodetector as q, the number of electron–hole pairs after photodetector output n is a product of q and the number of scintillation photons. The energy resolution under the absorbed energy of E is expressed as

      (1.17)

      Therefore, we can obtain a better energy resolution in bright scintillators. In actual detectors, the energy resolution is different from the resolution derived from simple Poisson statistics, and the gap between actual dispersion of 𝑛 and dispersion 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 δ_sc, δ_cir, and δ_st 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 δ_st by the number of scintillation photons and the quantum efficiency of the photodetector. The circuit noise δ_cir can be directly estimated by the injection of a test pulse into the electrical circuit. Thus, we can calculate δ_sc by the subtraction of δ_st

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