Luminescence Efficiency of GAGG:Ce Inorganic Scintillators for X-Ray Imaging Applications

Abstract

Single-crystal scintillators are used to convert ionizing radiation into optical photons in various medical imaging applications. A promising material is cerium (Ce)-doped gadolinium aluminum gallium garnet (GAGG:Ce) inorganic scintillator. Three GAGG:Ce 10 × 10 × 10 mm³ crystals of different light yield (LY) were exposed in X-ray tube voltage range of 50–140 kVp. Their absolute luminescence efficiency (AE) was experimentally calculated. A theoretical model was employed to simulate the propagation of photons traversing through the crystal mass. The model was utilized to estimate the detector quantum gain (DQG) and the percentage of transmission of the optical photons per elementary thickness $\mathrm{k}$. Their suitability with various optical photodetectors was evaluated by means of the spectral matching factor (SMF). GAGG:Ce presented AE values reaching 60.72 E.U. (where 1 E.U. = 1 μWm⁻²/(mRs⁻¹)) at 140 kVp. The parameter $\mathrm{k}$ ranged from 0.99973 to 0.99980. GAGG:Ce emission spectrum is highly compatible with charged-coupled devices (CCD), complementary metal-oxide semiconductors (CMOS) and silicon photomultipliers (SiPM). These findings may further consolidate the use of GAGG:Ce and could contribute to the future optimization of this inorganic scintillator when applied in X-ray imaging modalities, or as a radiation detector.

1. Introduction

Inorganic scintillator materials are highly important in ionizing radiation imaging modalities, especially in the detector array of high-end applications such as single photon emission computed tomography (SPECT) and positron emission tomography (PET) [1,2,3], as well as in X-ray computed tomography (CT). These luminescent materials convert high-energy ionizing radiation into optical photons [4,5,6,7,8,9]. The amount of optical photon production per ionizing photon energy incident to the inorganic crystal affects the sensitivity of the detector and, as a consequence, the radiation burden, that is, the radiation dose. In recent years, many studies have focused on the optimization of the performance of inorganic scintillators [10,11,12,13,14,15,16,17,18,19]. Their suitability in X-ray imaging systems depends on various parameters; thus, different materials have been designed with characteristics tailored to the intended application. One such example is gadolinium aluminum gallium garnet doped with cerium, Gd₃Al₂Ga₃O₁₂:Ce (GAGG:Ce), which exhibits high light yield (LY) (between 40,000 and 60,000 photons/MeV) [20,21,22,23,24,25] and a decay time of ~90 ns [26]. However, as reported in [27], GAGG:Ce shows high afterglow. It was first developed in 2011 as a 2-inch diameter GAGG:Ce single crystal of 1% Ce concentration and LY of 46,000 photons/MeV [28] using the conventional Czochralski method [20]. GAGG:Ce has a high density of 6.63 g/cm³ [21,29,30,31], ensuring high X-ray attenuation, and its effective atomic number (Zeff) is equal to 54.4 [29,30,32,33]. Other advantages are its energy resolution of 4.9% at 662 keV [20,21,33], its chemical stability due to Gd³⁺ [34,35] and the absence of hygroscopicity [21,24,29,33]. Its peak emission wavelength at approximately 544 nm is within the range of photomultiplier tubes (PMT) and silicon photomultipliers (SiPM) [12], ascribed to the 5d → 4f transitions of Ce³⁺ [20,22]. It is characterized by the absence of intrinsic radioactivity, in contrast to lutetium-based inorganic scintillator materials [34]. It finds applications like medical imaging, homeland security [36], industry, geophysical exploration, space exploration [37,38,39] and high-energy physics [40].

Some of the limitations of GAGG:Ce are its persistent afterglow [27], its cost [41] and the inconsistencies during manufacturing that may result in defects and impurities in the crystal lattice. Also, its decay time [26] is inferior to some of the other Ce-based scintillators used in time-of-flight (ToF) PET. The presence of gadolinium (Gd) in the scintillator results in high thermal neutron absorption [42], which may cause background noise in spectroscopy applications. In order to maximize LY output and efficiency, the optimal dopant concentration must be chosen. Moreover, GAGG:Ce efficiency decreases with temperature, as reported in [22,43]. Scintillator non-proportionality must also be taken into consideration [44] as it affects the crystal’s energy resolution.

A factor that influences GAGG:Ce LY, energy resolution and decay time is the concentration of Ce³⁺. It is reported in [45] that an inorganic crystal with 1% Ce concentration presents higher LY values than others with 2% and 3% due to concentration quenching, resulting in decreased Ce³⁺ 4f–5d emissions. Also, higher Ce³⁺ concentrations present faster decay times but lower energy resolution [45]. Moreover, LY is affected by the aluminum (Al)/gallium (Ga) ratio of the inorganic scintillator [45]. Studies indicate that Al/Ga ratios of 2/3 and 2.3/2.7 result in LYs as high as 55,000 photons/MeV [46,47]. It is also reported that higher Gd concentrations improve the luminescence of ceramic GAGG:Ce scintillators [48]. The inorganic crystals in this research had different LY levels of 45,000 photons/MeV, 49,000 photons/MeV and 54,000 photons/MeV, as provided by the vendor. During the manufacturing process of the GAGG:Ce crystals with the Czochralski method [20], different segregation coefficients for Ga and cerium (Ce) may cause compositional shifts along the crystal. It has been reported that this variation degrades the crystal’s LY as the solidification fraction increases [45,49]. This non-uniformity is a major challenge for the development of novel scintillating garnet materials. After GAGG:Ce production, non-uniform oxygen (O) vacancies may occur [50]. Furthermore, intrinsic defects act as carrier traps reducing the LY. Co-doping with silver cations (Ag⁺) is proposed in [50] as a means to mitigate this. Co-doping with magnesium cations (Mg²⁺) is used to improve the decay time and reduce the afterglow of GAGG:Ce. However, it partially oxidizes active Ce³⁺ to inactive Ce⁴⁻ ions, causing the LY to decrease [50]. Alongside crystal defects and impurities, the LY is also affected by surface roughness. Chemical polishing for 60 min is stated to improve GAGG:Ce light output [51].

The novelty of this study lies in the investigation of the optical properties and efficiency of GAGG:Ce inorganic crystals of different photon yields, applicable to medical imaging radiation detectors. For this reason, two 10 × 10 × 10 mm³ GAGG:Ce single crystals corresponding to two different values of light yield, namely 49,000 photons/MeV and 54,000 photons/MeV, were experimentally examined. Their absolute luminescence efficiency (AE) (emitted light flux over incident exposure rate) and spectral matching factor (SMF) (spectral matching to optical sensors) values were compared with the corresponding previously published results of a GAGG:Ce inorganic crystal [12] with light yield of 45,000 photons/MeV. Furthermore, an empirical theoretical model is used to describe the light output per incident radiation exposure. The model was compared with the experimentally determined AE of the inorganic scintillating crystals and the fraction of the optical photons propagating per elementary layer was determined. These results can be used towards explaining GAGG:Ce crystalline scintillator performance and contribute to the optimization of GAGG:Ce-based detectors. The corresponding experimental X-ray tube voltage ranged from 50 to 140 kVp. Their emission spectra were experimentally determined under UV excitation. The SMF of the aforementioned inorganic scintillators with several photodetectors was calculated in order to study their compatibility. The radiographic efficiency of the inorganic scintillators was evaluated through AE experimental measurements and the theoretically determined detector quantum gain (DQG) (emitted optical photon flux over X-ray photon flux, i.e., gain in number of photons).

The results presented in this work may facilitate the optimization of this inorganic scintillator material in X-ray applications, such as industrial radiography and medical imaging. By considering the fact that GAGG:Ce is non-hygroscopic [21,29,33,41], unlike other high efficiency materials like LaBr₃:Ce and LaCl₃:Ce, its utilization does not need enclosed apparatuses that minimize humidity. This suggests that the detection schemes utilizing GAGG:Ce may be lighter and can be employed in various modalities. The results demonstrate the optimum energy with the highest optical photon output as well as the appropriate photoreceptors. In addition, the theoretical investigation can be used to explain the ionizing radiation absorption process as well as the optical photon propagation characteristics. Hence, they can offer useful insights and a framework for the design of GAGG:Ce-based radiation detectors within the studied X-ray tube voltage range.

2. Results

2.1. Experimental Results

The intrinsic conversion efficiency ${\mathrm{n}}_{\mathrm{c}}$ values, expressing the produced optical photon power per absorbed ionizing radiation power, as calculated by Equation (7) for Crystal 1, Crystal 2 and Crystal 3, were ${{\mathrm{n}}_{\mathrm{c}}}_1=0.1030$, ${{\mathrm{n}}_{\mathrm{c}}}_2=0.1118$ and ${{\mathrm{n}}_{\mathrm{c}}}_3=0.1232, \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}$. The normalized measured emission spectra of the GAGG:Ce inorganic crystals are shown in Figure 1. Their emission peak was 544 ± 2 nm and their spectral width was assessed to be 157 nm, ranging from 496 nm to 653 nm.

Figure 2 and Figure 3 display the SMF values of the 54,000 photons/MeV GAGG:Ce crystal with several optical detectors, namely photocathodes, flat panel systems, SiPM, charged-coupled devices (CCD) and complementary metal-oxide semiconductors (CMOS). SMF evaluates the overlap of the emitted optical photon spectrum with the optical response of the photoreceptor and is used as an index of the scintillator and photoreceptor combinations. The graphs are indicative of the crystal’s ability to be combined with the aforementioned photon detectors. SMF evaluates the spectral response of GAGG:Ce and how well it matches the sensitivity of the photon detector. Their spectra data in Figure 1 ensure that the examined inorganic crystals presented similar spectral compatibility with different photodetectors regardless of their LY. A moderate response of over 0.70 can be observed in Figure 2 for the inorganic scintillator when integrated with CCDs and CMOS. The maximum SMF value was 0.99 for a hybrid CMOS with anti-reflective coating. As illustrated in Figure 2, the SMF values of CCDs and CMOS fluctuate from 0.70 to 0.99 for a crystal of LY = 54,000 photons/MeV. The maximum SMF was calculated for the gallium arsenide (GaAs) photocathode. The crystal presented low SMF values, reaching 0.13 for the flat panel PMT H8500D-03 and for the flat panel PMT H10966A, which were the lowest out of all the light detectors of the test group. Also, GAGG:Ce did not provide an adequate response for all the SiPMs examined. The corresponding SMF values ranged from 0.41 to 0.79, as reported in Figure 3. The spectral matching of an inorganic scintillator with the optical sensor’s sensitivity highly depends on the electronic structure of the material and the dopant used, which, in our case, is Ce³⁺. The fast and highly luminescent 5d → 4f transition of Ce³⁺ has a direct impact on the wavelength of the maximum emission of the crystal and, as a consequence, on the SMF and the ability of GAGG:Ce to be integrated with the photodetectors. High SMF values are achieved when the intrinsic wavelength of the material matches the photodetector’s sensitivity.

These results indicate that GAGG:Ce can be used in several imaging applications in which these sensors are an integral part of. Some of them, such as small-animal PET systems [52] and SPECT scanners [53], already have commercial uses. Other implementations of GAGG:Ce are in early stages of development and under research for use in systems such as PET [54,55,56], SPECT [1,54] and CT [17,57].

The experimental AE of the three inorganic crystals in E.U. units along with the experimental error is presented in Figure 4. The AE values of Crystal 1 were obtained from [12] and are displayed here along with the AE of Crystal 2 and Crystal 3 for comparison purposes. The AE was found increasing with the tube voltage. Considering the 54,000 photons/MeV crystal, a maximum value of 60.72 E.U. is presented for 140 kVp. Similarly, for the 49,000 photons/MeV crystal, the maximum AE was found to be 56.65 E.U. at 140 kVp. As the number of the emitted photons is greater for high-LY inorganic crystals, an increase in the absolute efficiency is to be expected. When compared with the efficiency of other inorganic crystalline scintillator materials, GAGG:Ce is highly advantageous due to its LY and high density (6.63 g/cm³). It attains high incident energy deposition and photon emission, thus achieving high efficiency. For the X-ray tube voltages considered in this work, the minimum values of AE for Crystals 2 and 3 were found at 50 kVp, as illustrated in Figure 4, and are 25.81 E.U. and 28.07 E.U., respectively. For comparison purposes with other inorganic crystalline materials, we report that the experimental AE values at 130 kVp for Crystals 1–3 in E.U. were 25.18, 54.60 and 58.83, respectively. These values are higher than the maximum AE of most Ce-doped inorganic scintillators also at 130 kVp, namely 12.35 E.U. for lutetium-yttrium oxyorthosilicate (LYSO:Ce), 7.90 E.U. for gadolinium oxyorthosilicate (GSO:Ce), 25.22 E.U. for lutetium gadolinium oxyorthosilicate (LGSO:Ce), 17.69 E.U. for lutetium oxyorthosilicate (LSO:Ce), 30.22 E.U. for lutetium aluminum garnet (LuAG:Ce) and 38.72 E.U. for lanthanum chloride (LaCl₃:Ce), as well as other inorganic scintillators such as 2.28 E.U. for barium fluoride (BaF₂) and 0.81 E.U. for cerium fluoride (CeF₃) already reported in literature [10,11,18,58] under the same radiographic conditions.

In addition, the maximum experimental AE values of Crystal 2 (56.65 E.U. at 140 kVp) and Crystal 3 (60.72 E.U. at 140 kVp) are equivalent to those of lanthanum bromide (LaBr₃:Ce) (60.78 E.U. at 140 kVp) [59]. Furthermore, the calculated $\mathrm{k}$ values reported here are close to unity for $\mathsf{\Delta }\mathrm{t}=1 \mathsf{\mu }\mathrm{m}$. LY and crystal efficiency depend, among others, on the properties of the crystal. Dense scintillating materials have increased the probability of absorbing incident radiation. Especially in X-rays where the photoelectric effect is more dominant, a dense crystal’s high interaction probability is crucial. In

Table 1. Scintillating properties of some Ce-based inorganic scintillators.

Crystal	Light Yield (Photons/MeV)	Density (g/cm3)	% Ce Concentration
LYSO:Ce	33,200 [60]	7.1 [60,61]	~0.2 [62,63]
GSO:Ce	8000 [11]	6.7 [11,61]	~0.4–1.5 [64,65]
LGSO:Ce	29,000 [66]	6.9 [66]	~0.2–0.6 [66,67]
LSO:Ce	26,000 [11]	7.4 [11]	~0.02–0.12 [68]
LuAG:Ce	26,000 [69]	6.7 [69]	~0.55–1.0 [69,70]
LaCl3:Ce	49,000 [10]	3.86 [10]	~10.0 [71,72]
LaBr3:Ce	63,000 [10]	5.2 [10]	~0.5 [73]

, the density, light yield and some published possible % Ce concentrations are reported for comparison. Heavy inorganic silicates containing lutetium (Lu) have densities of over 6.7 g/cm³. Because of their high effective atomic number, they present good stopping power and are used in PET applications where high spatial resolution is required. The lanthanide halides in Table 1 are less dense compared to other scintillators. LaCl₃:Ce and LaBr₃:Ce are hydroscopic but present higher LY and spatial resolution [10]. Therefore, they find use in applications such as SPECT and nuclear imaging. Scintillator non-proportionality [44] is evident in dense silicates. Incident photons interact within the crystal lattice, creating secondary electrons. The excited electrons cause lattice vibrations. Their energy is not transferred to the Ce³⁺ ions, thus hindering light production. This non-radiative process is what describes the quenching phenomenon in dense scintillators such as LYSO:Ce and LSO:Ce, resulting in lower LY values compared to the lanthanide halides. It is reported that as Ce³⁺ concentration increases, LaBr₃ rise time decreases [15]. In low Ce³⁺ concentrations the incident photons travel longer paths before reaching and activator ion having increased chances of getting trapped at crystal impurities and defects. However, when Ce³⁺ concentration overcomes a certain threshold, LY decreases due to Ce³⁺ concentration quenching phenomena (Ce³⁺ ions interacting with each other).

2.2. Theoretical Model

The theoretical AE in Equation (9) was fitted into the experimental AE data and an optimum $\mathrm{k}$ (fraction of optical photons transmitted through an elementary layer) value was calculated for each inorganic crystal. Figure 5 depicts the range of parameter $\mathrm{k}$ for all the inorganic crystals examined so that the theoretical model corresponds to the exact experimental AE value. Results indicate that the value of parameter $\mathrm{k}$ remains relatively stable across the tube voltage range of 50 kVp to 140 kVp as it exhibits values of over 0.99. This can be interpreted as the 99% of the incident photons from the ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ layer is transmitted to the $\mathrm{n}+1$ layer. Specifically, the values ranged from 0.999726 to 0.999806. The $\mathrm{k}$ values utilized were $\mathrm{k}=0.999730749$ for Crystal 1, $\mathrm{k}=0.999800745$ for Crystal 2 and $\mathrm{k}=0.999799461$ for Crystal 3, indicating an optical photon transmission over 99.9% by each layer of thickness $\mathsf{\Delta }\mathrm{t}=1 \mathsf{\mu }\mathrm{m}$. For the 45,000 photons/MeV crystal, the variation between the experimental AE results and those determined by the theoretical model was between 0.04% to 4.51%, for the 49,000 photons/MeV crystal 0.22% to 5.44% and for the brightest crystal of 54,000 photons/MeV was 0.48% to 4.61%. As shown in Figure 4, the differences found were within the experimental error.

The predicted normalized values to the overall maximum DQG value for the examined 10 mm inorganic crystals are displayed in Figure 6. DQG exhibited higher values as the X-ray tube voltage increased. The highest value, used for normalization, was calculated for the crystal with LY 54,000 photons/MeV at 140 kVp. As shown by Equation (10), DQG is affected by energy absorption phenomena within the crystal mass. Higher X-ray tube voltages are expected to deposit their energy closer to the crystal exit, resulting in higher light output and less losses during their propagation towards the scintillator output.

3. Discussion

The wavelength of maximum emission of the inorganic crystals was measured at ~544 nm. This makes a good match for most SiPMs [74], which are proven to be highly advantageous for potential uses with GAGG:Ce in gamma detectors [26,34,60] and PET scanners [3,55]. GAGG:Ce presented a suboptimal spectral match with most of the PMTs compared to other Ce-doped inorganic crystals such as LYSO:Ce, LSO:Ce and LGSO:Ce that are highly compatible with most PMTs. This is the result of their lower sensitivity [2,11] in contrast to other silicon-based detectors. The sensitivity of most PMTs used in medical applications drops considerably after 500 nm, as depicted from the SMF results in Figure 3. The best overall performance, in terms of spectral matching, was found for the CCDs and CMOS, as shown in Figure 2. SMF depends on how well the emission peak matches the optical detector’s spectral sensitivity and is not affected by the amount of optical flux emitted by the scintillator. GAGG:Ce has a near optimal overlap with CMOS sensors due to their silicon properties and their wavelength band that ranges from 400 to 700 nm [75], while GAGG:Ce was measured at 496–653 nm. The same compatibility can be deduced for the CCD sensors [76] as they present an excellent spectral match when coupled with GAGG:Ce as a consequence of their maximum sensitivity lying close to the measured emission peak of the inorganic crystal (~544 nm).

GAGG:Ce remains in a transitional stage and has yet to be implemented commercially in hybrid devices, namely SPECT/CT and PET/CT. Due to its afterglow [27,37] and decay time (~90 ns) [26], it has not replaced the relatively faster LSO crystal (~40 ns) [11] in the aforementioned systems. Considering the decay time, LYSO and LSO still have the edge, especially in ToF PET applications. Another limiting variable is its cost. The 15 × 15 × 15 mm GAGG:Ce current market value (USD 710) [41] is much higher compared to that of LYSO (USD 260) of the same dimensions [77]. However, GAGG:Ce is utilized pre-commercially in small-animal imaging, where the energy resolution is more important than ToF. It has been reported that GAGG:Ce is employed in a small-animal PET system with dimensions of 0.3 × 0.3 × 20 mm³ [52] and small-animal SPECT scanners with dimensions of 0.84 × 6 × 20 mm³ [53]. A corresponding hybrid PET/CT and SPECT/CT system may utilize the theoretical results of this work. Additionally, GAGG inorganic crystals of 0.9 × 0.9 × 6 mm³ have been applied to inorganic scintillator arrays of Compton camera detector rings [78]. Also, GAGG-based materials for CT detectors are studied in [17], where a cerium-doped gadolinium yttrium gallium aluminum garnet (GYGAG) ceramic scintillator of 2 mm was evaluated as a substitute for Gd₂O₂S in a CT scanner. GAGG:Ce achieved high efficiency in terms of spectral matching with the sensors evaluated in this study. Along with the existing literature, these results contribute towards the development for future commercial applications of this scintillating inorganic crystal in hybrid modalities such as SPECT/CT and PET/CT.

AE is steadily increasing with kVp, as illustrated in Figure 4. The two inorganic crystals with LY of 49,000 photons/MeV and 54,000 photons/MeV, which were experimentally examined in this work, present higher AE compared to a previous literature-reported crystal [12]. This is expected given the fact that the X-ray energy absorption properties of the 10 × 10 × 10 mm³ inorganic crystals are similar, and the change in efficiency is directly associated with the differences in LY and the corresponding intrinsic conversion efficiency. In high-LY crystals, an efficient energy conversion is achieved, and non-radiative losses are reduced. More electron–hole pairs are created from the incident photons interacting with the crystal and reach Ce³⁺ as they migrate from the ionization track, thus resulting in higher efficiency. The increase in X-ray tube voltage raises the energy of the incident X-ray photons. This reduces the probability of X-ray interaction and absorption, but this is balanced by the fact that the optical photons are generated near the exit of the inorganic scintillator. In addition, for higher X-ray energies, the AE values, providing some insight into the X-ray interaction properties of GAGG, display uniformity [12]. This leads to smaller differences in the optical photon output for 140 kVp in the GAGG:Ce inorganic crystals under investigation in this work. The Ce³⁺ concentration also affects the scintillating properties of GAGG:Ce [79]. Increasing the dopant concentration introduces more luminescent centers to the crystal. However, LY is reported to decrease after a certain threshold of 1% Ce³⁺ concentration [45]. This is due to thermal quenching of the emission centers, the Ce concentration quenching, the introduction of more lattice imperfections and the subsequent reabsorption effects. Moreover, an increased Ce³⁺ concentration results in energy transfer between the dopant ions, which leads to photon losses and a differentiation of the emission wavelength. Therefore, the inorganic scintillator’s LY decreases. In addition, the presence of Ce alone results in phosphorescence phenomena. The phosphorescence may be reduced with the introduction of Mg atoms. The presence of Mg²⁺ ions [27] presents improved performance in terms of afterglow, decay time and rise time due to the faster relaxation of the electron traps. It is stated that Mg Co-doped inorganic crystals present 20 times less phosphorescence than Ce-doped GAGG inorganic crystals [27], which is an important factor for their potential use in ToF PET applications. Other factors that also affect detector performance are the crystal thickness [80] and the presence of defects [81]. Another important factor of the excellent efficiency of GAGG:Ce is its density (6.63 g/cm³). High-density inorganic scintillator materials like bismuth germanate (BGO) (7.1 g/cm³), cadmium tungsten (CdWO₄) (7.9 g/cm³), gadolinium oxysulfide (Gd₂O₂S:Tb) (7.3 g/cm³) and LSO (7.35 g/cm³) [11,82] have improved radiation absorption efficiency as the incident photon is more probable to interact within the inorganic scintillator. The experimental results indicated that the GAGG:Ce inorganic crystal with a LY of 54,000 photons/MeV has higher AE under the same radiographic conditions compared to other Ce-doped inorganic scintillators [11] like recently studied LaCl₃ [59]. Its AE values are also comparable to those of LaBr₃ [59]. The “as low as reasonably achievable” (ALARA) principle dictates that in medical imaging the radiation exposure should be minimized as low as possible at levels that also ensure the desired image quality. The radiation dose is to be optimized below the regulatory limits. For this reason, high efficiency is essential for an inorganic scintillator in medical applications because it contributes to a high-quality diagnostic image to be produced with a subsequent smaller dose of radiation to the patient.

The calculated $\mathrm{k}$ values reported here are close to unity for $\mathsf{\Delta }\mathrm{t}=1 \mathsf{\mu }\mathrm{m}$, under the assumption that the inorganic crystal exhibited 100% optical transmittance. Therefore, our theoretical model can only be interpreted for GAGG:Ce crystals with the same uniformity as the ones examined in this study and under the assumptions used to calculate ${\mathrm{n}}_{\mathrm{c}}$ and $\mathrm{k}$. The ${\mathrm{n}}_{\mathrm{c}}$ and $\mathrm{k}$ values calculated for 80% and 75% transmittances reported in the literature for GAGG:Ce inorganic crystals [83,84,85] are reported in

Table 2. The calculated n_c and k values for 100%, 80% and 75% transmittance for GAGG:Ce inorganic scintillator.

	100% Transmittance		80% Transmittance		75% Transmittance
LY in Photons/MeV	nc	k	nc	k	nc	k
45,000	0.103020	0.999730749	0.128776	0.999707561	0.137360	0.999700853
49,000	0.111765	0.999800745	0.139707	0.999777229	0.149021	0.999770496
54,000	0.123170	0.999799461	0.153963	0.999776300	0.164227	0.999769600

. The 100% transmittance corresponding values are shown in the first two columns for comparison.

The aforementioned results are important since they express optical photon generation within the crystal and optical photon transmission through the crystal’s mass, which in turn determine the efficiency of scintillators. The maximum DQG values were presented at 140 kVp for the three inorganic crystals. GAGG:Ce was proven superior to LaCl₃:Ce as it presented higher AE values (60.72 E.U. at 140 kVp) under the same radiographic conditions [10] and it is non-hygroscopic contrary to LaCl₃:Ce. However, it is inferior compared to both LaCl₃:Ce and LaBr₃:Ce in terms of decay time. It would be beneficial for further research to experimentally and theoretically assess the optical photon transmission phenomena in inorganic scintillators of various aspect ratios and thicknesses other than the 10 × 10 × 10 mm³ evaluated in this work.

4. Materials and Methods

4.1. Experimental Evaluation

Three 10 × 10 × 10 mm³ GAGG:Ce inorganic crystals with different light yield were purchased by Advatech U.K. Ltd. Woodford Green, UK [24]. Inorganic crystals exhibiting light yields of 45,000, 49,000 and 54,000 photons/MeV were labeled as Crystals 1, 2, and 3, respectively. The efficiency properties of Crystal 1 in terms of SMF and AE have been experimentally reported in previous studies [12]. The SMF was computed according to Equation (1), in order to quantify the spectral compatibility of GAGG:Ce with various optical sensors as [10,12]:

$$\mathrm{S}\mathrm{M}\mathrm{F}={\displaystyle \frac{\int {\mathrm{S}}_{\mathrm{p}}\left(\mathrm{\lambda }\right){\mathrm{S}}_{\mathrm{D}}\left(\mathrm{\lambda }\right)\mathrm{d}\mathrm{\lambda }}{\int {\mathrm{S}}_{\mathrm{p}}\left(\mathrm{\lambda }\right)\mathrm{d}\mathrm{\lambda }}}$$

(1)

where ${\mathrm{S}}_{\mathrm{P}}\left(\mathrm{\lambda }\right)$ indicates the spectrum of the produced light and ${\mathrm{S}}_{\mathrm{D}}\left(\mathrm{\lambda }\right)$ denotes the spectral sensitivity of the respective photodetector. For measurements of the emission spectrum, a grating spectrometer HR2000+ was used (Ocean Optics Inc., Largo, FL, USA).

The setup used to experimentally determine AE values consisted of an X-ray tube, with tube voltages that ranged from 50 kVp to 140 kVp, in increments of 10 kVp. A 21 mm Al filter was introduced as a means to replicate the beam hardening caused by the human body. The exposure time was 1 s. The optical energy flux generated from the inorganic crystal was measured using an Oriel 70451 light integration sphere (Newport Corp., Irvine, CA, USA, Oriel model 70451), an EMI 9798B PMT (EMI, London, UK, EMI model 9798) and a Cary 401 vibrating reed electrometer (Cary instruments/Varian, Palo Alto, CA, USA). The experimental AE or the ratio of the inorganic crystal’s light flux $\dot{{\mathsf{\Psi }}_{\mathrm{\lambda }}}$ per X-ray exposure rate $\dot{\mathrm{X}}$ [12] was determined as follows:

$${\mathrm{A}\mathrm{E}}_{\mathrm{e}\mathrm{x}\mathrm{p}}={\displaystyle \frac{\dot{{\mathsf{\Psi }}_{\mathrm{\lambda }}}}{\dot{\mathsf{{\rm X}}}}}$$

(2)

4.2. Theoretical Model

A generic theoretical empirical model [10,86,87] was employed in order to estimate the DQG of the three GAGG single crystals. The theoretical model assumes that the single crystal can be divided in $\mathrm{N}$ elementary layers of thickness $\mathsf{\Delta }\mathrm{t}=\mathrm{T}/\mathrm{N}$, where $\mathrm{T}$ is the crystal’s thickness. In this case, thickness $\mathsf{\Delta }\mathrm{t}$ is chosen equal to 1 $\mu$m; thus, for $\mathrm{T}=10 \mathrm{m}\mathrm{m}$, $\mathrm{N}$ equals to 10,000. The X-rays are generally assumed to follow exponential attenuation within the crystal mass. A part of them is absorbed in a layer of thickness $\mathsf{\Delta }\mathrm{t}$ at position $\mathrm{n}$, where $\mathrm{n}$ takes values from 1, corresponding to the input elementary layer, up to 10,000, corresponding to the output layer. In the layer at position $\mathrm{n}\mathsf{\Delta }\mathrm{t}$, the incident X-rays are absorbed between $\mathrm{n}\mathsf{\Delta }\mathrm{t}$ and $\left(\mathrm{n}+1\right)\mathsf{\Delta }\mathrm{t}$. The absorbed X-ray energy is transformed into optical photon energy. This process is affected by the intrinsic conversion efficiency of the inorganic scintillator, which is defined as the generated optical photon power over the absorbed X-ray power. The generated optical photons are transmitted bidirectionally through the different layers. It was assumed that from each layer, only a constant fraction, or percentage, of the optical photons, denoted as $\mathrm{k}$, is transmitted to the next layer. Thus, if $\mathrm{M}$ photons are produced at position $\mathrm{n}$, then $\mathrm{M}\mathrm{k}$ will be transmitted to the layer at position $\mathrm{n}+1$, $\mathrm{M}{\mathrm{k}}^2$ at the layer at position $\mathrm{n}=2$ and $\mathrm{M}{\mathrm{k}}^{\mathrm{N}-\mathrm{n}}$ at the exit layer. Assuming that optical photons traverse the crystal mass bidirectionally, and taking into consideration the reflections occurring at the output and input surfaces of the crystal, it is possible to calculate the number of the optical photons generated in the ${\mathrm{n}}^{\mathrm{t}\mathrm{h}}$ layer that exits the inorganic scintillator’s surface as [10]:

$${\mathrm{M}}_{\mathrm{n}}=(1-{\mathrm{R}}_0)\left[{\displaystyle \frac{{\mathrm{k}}^{\mathrm{N}-\mathrm{n}}}{1-{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}}+{\mathrm{R}}_1{\mathrm{k}}^{\mathrm{n}+\mathrm{N}}{\displaystyle \frac{{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}{1-{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}}\right]$$

(3)

where ${\mathrm{R}}_0$ is the output reflectance, indicating the incident photons that are reflected back into the crystal when reaching the exit surface. On the other hand, ${\mathrm{R}}_1$ is the input reflectance indicating the incoming photons that are reflected into the crystal when reaching its entrance surface. In our experimental setup, the three inorganic crystalline scintillators were covered with PTFE (polytetrafluoroethylene) teflon tape on each of their five sides, a technique often used in scintillator detectors [88,89] in order to decrease optical photon losses and improve light collection in the output. The refractive index of GAGG with Teflon and GAGG with air were considered when calculating ${\mathrm{R}}_0$ and ${\mathrm{R}}_1$. Therefore, ${\mathrm{R}}_0$ and ${\mathrm{R}}_1$:

$${\mathrm{R}}_0={\mathrm{R}}_{\mathrm{G}\mathrm{A}\mathrm{G}\mathrm{G},\mathrm{T}\mathrm{e}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{n}}={\displaystyle \frac{{({\mathrm{n}}_1-{\mathrm{n}}_2)}^2}{({{\mathrm{n}}_1+{\mathrm{n}}_2)}^2}}$$

(4)

$${{\mathrm{R}}_1=\mathrm{R}}_{\mathrm{G}\mathrm{A}\mathrm{G}\mathrm{G},\mathrm{a}\mathrm{i}\mathrm{r}}={\displaystyle \frac{{({\mathrm{n}}_2-1)}^2}{({{\mathrm{n}}_2+1)}^2}}$$

(5)

where ${\mathrm{n}}_1=1.9$ is the refractive index of GAGG inorganic crystal at 540 nm [24] and ${\mathrm{n}}_2=1.35$ is the refractive index of teflon tape [90,91]. These account for ${\mathrm{R}}_0=0.029$ and ${\mathrm{R}}_1=0.022$. By taking into account all consecutive layers, the optical photons generated by X-rays of energy E and exiting the inorganic crystal can be computed as follows [10]:

$$\mathrm{M}\left(\mathrm{E}\right)={\sum }_{\mathrm{n}=1}^{\mathrm{N}}\mathrm{f}\left(\mathrm{E}\right){\mathrm{e}}^{-\mathsf{\mu }\mathrm{n}\mathsf{\Delta }\mathrm{t}}\mathsf{\mu }\mathsf{\Delta }\mathrm{t}{\mathrm{n}}_{\mathrm{c}}{\displaystyle \frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{\lambda }}}}\left(1-{\mathrm{R}}_0\right)\left[{\displaystyle \frac{{\mathrm{k}}^{\mathrm{N}-\mathrm{n}}}{1-{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}}+{\mathrm{R}}_1{\mathrm{k}}^{\mathrm{n}+\mathrm{N}}{\displaystyle \frac{{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}{1-{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}}\right],\mathrm{n}=1,\dots ,\mathrm{N}/\mathsf{\Delta }\mathrm{t}$$

(6)

where $\mathrm{f}\left(\mathrm{E}\right)$ is the X-ray spectrum [92], $\mathsf{\mu }$ is the attenuation coefficient of GAGG, and ${\mathrm{E}}_{\mathrm{\lambda }}$ is the energy of optical photons. The factor ${\mathrm{n}}_{\mathrm{c}}$ or the intrinsic conversion efficiency is assumed to be a function of the LY for each inorganic crystal. By taking into consideration that LY is defined as the number of the produced optical photons per deposited ionizing radiation energy [29,93], the multiplication of LY with the optical photon energy can determine ${\mathrm{n}}_{\mathrm{c}}$ as [10]:

$${\mathrm{n}}_{\mathrm{c}}={\displaystyle \frac{\mathrm{L}\mathrm{Y}}{{10}^6}}{\mathrm{E}}_{\mathrm{\lambda }}$$

(7)

where LY is defined in optical photons per MeV and the denominator accounts for the conversion of MeV to eV.

In Equation (6), $\mathrm{f}\left(\mathrm{E}\right){\mathrm{e}}^{-\mathsf{\mu }\mathrm{n}\mathsf{\Delta }\mathrm{t}}$ accounts for the number of X-rays of energy E impinging in the layer at position $\mathrm{n}$, $\mathrm{f}\left(\mathrm{E}\right){\mathrm{e}}^{-\mathsf{\mu }\mathrm{n}\mathsf{\Delta }\mathrm{t}}\mathsf{\mu }\mathsf{\Delta }\mathrm{t}$ corresponds to the number of X-rays absorbed in the layer of thickness $\mathsf{\Delta }\mathrm{t}$, while $\mathrm{f}\left(\mathrm{E}\right){\mathrm{e}}^{-\mathsf{\mu }\mathrm{n}\mathsf{\Delta }\mathrm{t}}\mathsf{\mu }\mathsf{\Delta }\mathrm{t}{\mathrm{n}}_{\mathrm{c}}{\displaystyle \frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{\lambda }}}}$ accounts for the number of optical photons produced in the layer at position $\mathrm{n}$. This number is affected by $\mathsf{\mu }$, that is, the X-ray interactions properties as a function of energy.

The theoretical AE is expressed as:

$${\mathrm{A}\mathrm{E}}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}={\sum }_{\mathrm{n}=1}^{\mathrm{N}}\mathsf{\gamma }(\mathrm{E}){\mathrm{e}}^{-\mathsf{\mu }\mathrm{n}\mathsf{\Delta }\mathrm{t}}\mathsf{\mu }\mathsf{\Delta }\mathrm{t}{\mathrm{n}}_{\mathrm{c}}\left(1-{\mathrm{R}}_0\right)\left[{\displaystyle \frac{{\mathrm{k}}^{\mathrm{N}-\mathrm{n}}}{1-{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}}+{\mathrm{R}}_1{\mathrm{k}}^{\mathrm{n}+\mathrm{N}}{\displaystyle \frac{{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}{1-{\mathrm{k}}^{2\mathrm{N}}{\mathrm{R}}_0{\mathrm{R}}_1}}\right]$$

(8)

where the factor $\mathsf{\gamma }\left(\mathrm{E}\right)$ is used to convert the energy fluence (W/m²) into exposure rate (mR/s).

When the energy spectrum is considered, equation ${\mathrm{A}\mathrm{E}}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}$ becomes:

$${\mathrm{A}\mathrm{E}}_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}}={\displaystyle \frac{{\sum }_{\mathrm{E}=\mathrm{E}\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{E}\mathrm{m}\mathrm{a}\mathrm{x}}\mathrm{f}\left(\mathrm{E}\right)\mathrm{A}\mathrm{E}\left(\mathrm{E}\right)}{{\sum }_{\mathrm{E}=\mathrm{E}\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{E}\mathrm{m}\mathrm{a}\mathrm{x}}\mathrm{f}\left(\mathrm{E}\right)}}$$

(9)

The theoretical model was then utilized for the assessment of DQG values of three GAGG inorganic crystalline scintillators, where DQG is defined as the amount of optical photons per incident X-ray photon and is formulated as:

$$\mathrm{D}\mathrm{Q}\mathrm{G}={\displaystyle \frac{{\sum }_{\mathrm{E}=\mathrm{E}\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{E}\mathrm{m}\mathrm{a}\mathrm{x}}\mathrm{M}\left(\mathrm{E}\right)}{{\sum }_{\mathrm{E}=\mathrm{E}\mathrm{m}\mathrm{i}\mathrm{n}}^{\mathrm{E}\mathrm{m}\mathrm{a}\mathrm{x}}\mathrm{f}\left(\mathrm{E}\right)}}$$

(10)

The value of parameter $\mathrm{k}$ was calculated by fitting Equation (9) to the experimental AE values.

5. Conclusions

The AE for X-ray tube voltages 50–140 kVp was experimentally determined. In addition, a theoretical model was employed in this work that describes the propagation of photons passing through the GAGG:Ce scintillating material. The AE estimated by the model was compared with the experimental AE values of the two GAGG:Ce single crystals considered in this work and with already-published GAGG:Ce efficiency data [12]. In this manner, the optical photon fraction propagating per 1 µm layer, that is, parameter $\mathrm{k},$ and DQG were predicted. Moreover, SMF was computed for several commercially available photodetectors, utilized in medical imaging applications with GAGG inorganic crystals of various LY. GAGG:Ce’s superior efficiency, in E.U. units, was confirmed when compared to other Ce-doped inorganic scintillators. Additionally, GAGG presented higher $\mathrm{k}$ values than other single crystals, reaching over 99% optical photon transmission per elementary layer. By employing the theoretical model, the response of the inorganic scintillator was predicted. The 10 mm inorganic crystal with a LY of 54,000 photons/MeV presented the highest DQG values. These predicted DQG values are based in the assumptions considered in the intrinsic conversion efficiency and $\mathrm{k}$ fraction calculations. The GAGG:Ce single crystal, having a light yield of 54,000 photons/MeV, may serve as an efficient inorganic scintillator material in X-ray tube voltages of 50–140 kVp and as a detector in X-ray imaging modalities, especially when coupled with CCD, CMOS and SiPMs.