to proceed with the calculation, we assume a polaron model where a polaron accompanies α/2 phonons, then α can be expressed as
(1.9)
where K∞ and K0 are static and high‐frequency dielectric constants, respectively. Under this condition, the optical phonon generation rate is proportional to
(1.10)
Thus, K can be expressed as
(1.11)
By using these equations, we can estimate the scintillation emission efficiency semi‐empirically. Following this first approach, the model was modified for actual use in daily experiments. In 1994 [57], the scintillation light output L per unit energy was expressed as
(1.12)
where ne‐h is the number of electron–hole pairs under γ‐ray with energy of Eγ irradiation, nmax is the number of electron hole pairs which would be generated if there were no losses to optical phonons, S stands for transfer efficiency from the host to luminescence centers, Q is luminescence quantum efficiency at localized luminescence centers, and η is total scintillation efficiency.
Here, we will proceed with further consideration that this assumption is Ei = 1.5Eg. In the Robbins approach [56], the average energy used to produce one electron–hole pair under assumption of no optical phonon loss is ξmin = 2.3Eg, and in this case, nmax = 106 Eγ/2.3Eg where we use the energy unit of eV. Because ne‐h = 106 Eγ/2.3ξ and nmax = 106 Eγ/ξmin, we can deduce that βs = ξmin/ξ. In Equation (1.5), if we assume K = 0, we have ξmin = 1.5Eg(1 + 2Lf), and if we use the relation of ξmin = 2.3Eg, Lf = 0.27. If we assume Lf does not depend on K, we can use Lf = 0.27 in Equation (1.5), and the average energy consumed per electron–hole pair can be expressed as ξ = Eg(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
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 βs = 2.5 is proposed, based on the data of various scintillators [59], and most research uses this formula which is described as
In this formula, we can determine L, Eg, and Q experimentally, and we can deduce S by using these values. L can be evaluated by the pulse height spectrum, Eg 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 Eg. 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
