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Structure and photoluminescence properties of KSr4(BO3)3:Eu3+ red-emitting phosphor

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Abstract

We have synthesized a Eu3+-activated KSr4(BO3)3 red phosphor by solid state reactions. Rietveld refinement on X-ray diffraction data indicates that Eu3+ ions are inclined to occupy Sr(2) (8c) site in the structure of KSr4(BO3)3. The composition-optimized KSr4(BO3)3:Eu3+ exhibits a dominant emission peak at 612 nm (5D0-7F2) with CIE coordinates of (0.64, 0.35) under the excitation at 394 nm. By codoping M+ ions (M = Li, Na, and K) in KSr4(BO3)3:Eu3+ to compensate the charge unbalance, the intensities of emission spectra at 612 nm can be increased greatly, but the CIE coordinates will not be changed. The red-emitting KSr4(BO3)3:Eu3+ phosphor may be potential candidate in the fabrication of white light-emitting diodes (LEDs).

©2011 Optical Society of America

1. Introduction

In the past few years, energy-efficient solid-state lighting and high-power white light emitting diodes (LEDs), which are regarded as the next-generation light source, have been attracted increasing attention for use in display lighting sources and illuminating systems [16]. In the case of the phosphor converted white-light LEDs, the phosphor materials play an important role. For example, the most common and simple LED-based white light source is combined of a blue-emitting InGaN chip and a Ce3+-doped yttrium aluminium yellow phosphor (YAG:Ce3+) [7], which is very stable and exhibits high luminescence efficiency. However, in some respects YAG:Ce3+-based dichromatic systems often suffer from reduced thermal stability and exhibit a poor color-rendering index (CRI) caused by the color deficiency in the red- and blue-green of the phosphor. To overcome this weakness, one of the alternative ways is coupling a blue LED with long-wavelength yellow-emitting phosphors, which are with high down conversion efficiency. Xie’s group has made a great progress on these luminescent materials [811]. Another method is using an UV LED (λem = 350-410 nm) with RGB (red, green, and blue) phosphors or coupling blue LED with RG phosphors for a high CRI and a high power output. Therefore, the efficient red phosphors are required. Plenty of works has been devoted to find out a good red phosphor [1218]. Though rare-earth-doped sulfides and oxysulfides have been used as the red phosphors [1921], they cannot be used widely because these compounds are sensitive to moisture and not easy to be synthesized. Recently, nitrides and oxynitrides have been demonstrated to be good potential as red phosphors because of their good thermal stability [10,22,23]. However, the production cost of the phosphors is very high because the synthesis of the phosphors requires very high firing temperatures and high nitrogen pressures.

From the viewpoint of stability, environmental friendliness, convenient synthesis conditions and low cost, it is urgent to search for new red phosphors with high chemical stability. Borates, chemical stable and easily productive at temperatures below 800°C in air [2427], have attracted much attention on exploring new phosphors because of being suitable host for rare earth ions [2831]. Recently, luminescence properties of Ba2Mg(BO3)2:Eu2+,Mn2+ and ZnB2O4:Bi3+,Eu3+ are investigated and conclude that they are highly efficient, red-emitting phosphors, which prove that borates are good candidates to find out potential new red phosphors [14,28].

KSr4(BO3)3, synthesized by our group, crystallizes with space group Ama2 [32]. It contains [BO3]3- isolated triangles and 4 crystallographically independent cation sites. Jiang et al. have indicated that this compound is considered most suitable hosts for Ce3+ ion luminescence [33]. To our knowledge, the luminescence properties of Eu3+ doped KSr4(BO3)3 have not been studied yet, though the red luminescence of Eu3+ ion has been extensively used in the lighting and displays. Therefore, motivated by the above investigations and the attempt to develop new red phosphors for the potential applications of white-light LED, we have investigated the luminescence and color chromaticity properties as well as the structure of KSr4(BO3)3:Eu3+ phosphor in the present work. Our investigation has demonstrated that doping Eu3+ occupy Sr(2) (8c) site in the strucrture of KSr4(BO3)3 and KSr4(BO3)3:Eu3+ can emit red light with high color saturation. Besides, in order to eliminate the charge unbalance because of Eu3+ doped into KSr4(BO3)3, The effect of charge compensation is also studied in this paper.

2. Experimental

Polycrystalline samples of KSr4(BO3)3:Eu3+ and KSr4(BO3)3:Eu3+,M+ (M = Li, Na, K) were prepared by a solid-state reaction at high temperature starting from analytical purity K2CO3, SrCO3, H3BO3, Li2CO3, Na2CO3, and high purity Eu2O3(99.99%). The raw materials were weighed out, mixed and ground together in an agate mortar, and then sintered under ambient atmosphere at 600 °C for 24 h. After this step, the sintered powder mixtures were ground and sintered at 800 °C for 24 h twice.

The X-ray diffraction (XRD) data for phase identification and structural refinements were collected at ambient temperature by a PANalytical powder X-ray diffractometer X’Pert Pro with Cu Kα radiation (40 kV, 40 mA). The data were collected over a 2θ range from 10þ to 135þ at intervals of 0.017þ with a counting time of 1 s per step. The photoluminescence (PL) and photoluminescence excitation (PLE) spectra were measured by a spectrofluorometer (Edinburgh Instruments, FLS920) equipped with a Xe light source and double excitation monochromators. The powder samples were compacted and excited under 45° incidence and emitted fluorescence was detected by a photomultiplier (R928P) perpendicular to the excitation beam. To eliminate the second-order emission of the source radiation, a cutoff filter was used in the measurement. A μF900 lamp (100 W) was used as a light source for the emission lifetimes measurement. The slits were set at 2 nm, and a photomultiplier (R928P) was used as detector.

3. Results and discussion

3.1 Structure of KSr4 (BO3)3:Eu3+ phosphors

The XRD patterns of KSr4(BO3)3:Eu3+ phosphors with different doping Eu3+ contents are found to be in good agreement with that reported of KSr4(BO3)3 [32] regardless of the contents of dopants as certificated in the following Rietveld refinement (Fig. 1 is the Rietveld refinement plot with the XRD pattern of KSr4(BO3)3:0.03Eu3+ phosphor). This indicates that no impurity phase is present in the as-prepared powders and the host crystal structure is not changed by doping Eu3+ ions.The host crystal KSr4(BO3)3 crystallizes in the noncentrosymmetric space group Ama2. There are four crystallographic positions of cations in the unit cell: eight-fold coordinated K+(4b) sites, eight-fold coordinated Sr(1) (4b) sites, eight-fold coordinated Sr(2) (8c) sites, and nine-fold coordinated Sr(3) (4a) sites. Based on the effective ionic radii (r) of cations with different coordination number (CN) reported by Shannon [34], it is proposed that Eu3+ ions are expected to occupy the Sr sites preferably, because the ionic radii of Eu3+ (r = 1.25 Å when CN = 8 and r = 1.30 Å when CN = 9) is very close to that of Sr2+ (r = 1.26 Å when CN = 8 and r = 1.31 Å when CN = 9). Though the ionic radii of K+ (r = 1.51 Å when CN = 8 and r = 1.55 Å when CN = 9) is also close to that of Eu3+, however, compared with the radii of Sr2+, the radii of K+ is much larger than that of Eu3+. The compound structure will not be stable in crystallography when Eu3+ occupy K+ site because the difference of effective ionic radii between these two ions is too large. Therefore, Eu3+ will not prefer to occupy K+ sites in this compound. In order to prove the Eu3+ ions occupy Sr2+ sites and also further to make clear which Sr sites are occupied by Eu3+, refinement of the XRD patterns of KSr4(BO3)3:Eu3+ are performed by Rietveld method [35,36] within the Fullprof Program [37]. The K+ sites are found could not be occupied because of the final agreement factors is very high and the occupancy of Eu3+ on this site is far away from the nominal doping content. As for Sr sites, it is found that all the doping Eu3+ ions are preferred to occupy Sr(2) (8c) sites during the refinement. Setting the structure of KSr4(BO3)3 as the initial structure, we try to refine the XRD patterns supposing that Eu3+ occupy the three different Sr sites, as shown in Table 1 (selected XRD pattern of KSr4(BO3)3:0.03Eu3+ as example), in which A–G represents Eu3+ occupy Sr(1) site, Sr(2) site, Sr(3) site, Sr(1) and Sr(2) site, Sr(1) and Sr(3) sites, Sr(2) and Sr(3) sites, and Sr(1), Sr(2) and Sr(3) sites, respectively. Obviously, Eu3+ cannot occupy Sr(1), Sr(2), and Sr(3) sites at the same time because of the negative occupancy. As for the other cases, the lowest final agreement factors and the reasonable final compositions which are in good agreement with the nominal compositions of the starting materials is obtained in case of B. Therefore, doped Eu3+ ions prefer to occupy Sr(2) (8c) site. Figure 1 is the final Rietveld refinement plot of KSr4(BO3)3:0.03Eu3+, and the inset is the structure projection of KSr4(BO3)3. The selected refinement results are listed in Tables 2 and 3 . It is found that the distance between Sr(2) (8c) sites is much shorter than that of the other two sites, which might be the reason for the prefer occupancy. Therefore, we conclude that doped Eu3+ ions prefer to occupy Sr(2) (8c) sites.

 figure: Fig. 1

Fig. 1 Final Rietveld refinement plots of the KSr4(BO3)3:0.03Eu3+. Small circles (o) correspond to experimental values, and the continuous lines, the calculated pattern; vertical bars (|) indicate the position of Bragg peaks. The bottom trace depicts the difference between the experimental and the calculated intensity values. Inset is the structure projection of KSr4(BO3)3.

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Tables Icon

Table 1. Occupancy and Lattice Parameters of Eu3+ Occupying Different Sites for KSr4(BO3)3:0.03Eu3+ Refined by Rietveld Method, in which A–G Represents Eu3+ Occupy Sr(1) Site, Sr(2) Site, Sr(3) Site, Sr(1) and Sr(2) Site, Sr(1) and Sr(3) Sites, Sr(2) and Sr(3) Sites, and Sr(1), Sr(2) and Sr(3) Sites, Respectively

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Table 2. Fractional Atomic Coordinates and Equivalent Isotropic Displacement Parameters (Å2) for KSr4(BO3)3:0.03Eu3+

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Table 3. Interatomic Distances (Å) and Angles (deg) for KSr4(BO3)3:0.03Eu3+

3.2 Photoluminescence properties of KSr4(BO3)3:Eu3+

The PLE and PL spectra of KSr4(BO3)3:0.03Eu3+ are shown in Fig. 2 . The excitation spectrum detected at 612 nm consists of a broad band which is attributed to the O2--Eu3+ charge transfer band (CTB) and several linear peaks centering at 318 nm, 361 nm, 366 nm, 377 nm, 382 nm, 394 nm, 416 nm, and 465 nm, corresponding to the typical f-f transitions, i.e., the transitions from the ground state 7F0 to the exciting state of 4f6 configuration. The strongest peak in the excitation spectrum locates at 394 nm which comes from the 7F0-5L6 transition.

 figure: Fig. 2

Fig. 2 PL/PLE spectra of as–synthesized KSr4(BO3)3:0.03Eu3+ (red, λem = 612 nm; black, λex = 394 nm). The inset is the effect of Eu3+ concentration on the PL intensity.

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The emission spectrum is composed of a series sharp peaks from 560 nm to 720 nm, which are attributed to the transitions of 5D07FJ (J = 0–5). It is accepted that the selection rule of electric dipole (ED) transition is △J6, when J or J = 0, △J = 2, 4, 6, and the selection rule of magnetic dipole (MD) transition is △J = 0, ± 1. Thus, the highly intense line at 5D07F1 is because of magnetic dipole transition. The main emission bands centered at 612 nm and 701 nm are attributed to the electric dipole transition of 5D07F2 and 5D07F4 respectively, which are ascribed to the lifting of the spin and parity prohibitions of Eu3+ f – f transitions in KSr4(BO3)3 host lattice in the asymmetric C2v point group [28,38]. It is consisted with the structural study that Eu3+ occupies the noncentrosymmetric position of Sr2 (8c). Because of the crystal field effect, 7FJ energy level will appear Stark levels. Therefore, the energy level transition 5D07F1 is split into 589 nm and 593 nm emission peaks and 5D07F2 is split into 612 nm, 617 nm, 620 nm and 623 nm; Moreover, a transition of 5D07F0 (579 nm) is observed in the PL spectra. It does not belong to the selection rule of ED transition. This transition is coming from the relaxation of selection rule somewhat because the crystal field plays a role in mixing states.

Normally, with different amounts of Eu3+ doped into the host lattice, the local surroundings around a substituted site will be changed, which eventually makes it possible to tune the luminescence properties. By investigating the emission intensity of KSr4(BO3)3 doped with different Eu3+ contents excited at 394 nm, we get the dependence of the corresponding PL intensity as a function of Eu3+ concentration, which is shown in the inset of Fig. 2. Obviously, the optimal dopant concentration of Eu3+ is 3 mol.%, which means the PL intensity of emission peaks decrease while the doped content is above 3 mol.% because of the quenching concentration of Eu3+.

Based on the Dexter and Schulman theory [39], concentration quenching is due to energy transfer from one activator to another until an energy sink in the lattice is reached. Critical distance (Rc) is the critical separation between donor (activator ion) and acceptor (quenching ion), where the nonradiative rate equals that of the internal single ion relaxation. Thus the critical distance (Rc) is an important parameter which can be estimated from the concentration quenching data. Blasse proposes that the average shortest distance between the nearest activator ions for the critical concentration is equal to the critical distance (Rc) [40]. The Rc can be expressed by

Rc2[3V4πxcZ]13
where xc denotes the critical concentration of dopants; Z is the number of host cations in the unit cell, and V is the volume of the unit cell. For KSr4(BO3)3 host, Z is 16, V is 909.03 Å3 obtained from the XRD refinement. xc is 0.03 in our case. Thus, the critical distance (Rc) of KSr4(BO3)3:Eu3+ is calculated by the Eq. (1) to be about 15.3 Å.

Dexter theory indicates that the concentration quenching of inorganic materials is due to the electric multipolar interaction or magnetic dipolar interaction among the activator ions [39]. In this study, the concentration quenching of the KSr4(BO3)3:Eu3+ phosphor is probably from the non-radiative transition among the Eu3+ ions. It is well know that the crystal field in non-centrosymmetric C2v sites mixes some 5d character into the 4f wavefunction, which will make electric multipolar interactions as the probable transfer mechanism. In addition, from the data of fluorescence lifetime (discussed later), there is no any evidence of lengthening lifetime with increasing concentration. Therefore, the process of energy transfer of Eu3+ ions in KSr4(BO3)3 would be due to electric multipolar interaction [39,41,42].

Because the lifetimes are nearly the same for the all samples and the curves are overlapped, the selected decay properties by time-resolved measurement of the KSr4(BO3)3:0.03Eu3+ phosphor are shown in Fig. 3 as an example (excited at 394nm, monitored at 612 nm). The curve fits a first-order exponential decay model described by equation [43]:

I=I0exp(t/τ)
where I0 and I are the luminescence intensities at time 0 and t, respectively, and τ is fluorescence lifetime. The decay curves of KSr4(BO3)3:Eu3+ can be fitted by Eq. (2) very well, which implies that there is only one type of Eu3+ emission site in the host. This is in agreement with the structural study. The decay time value of 1.9 ms is derived by fitting the curve with Eq. (2), which implies the decay time of KSr4(BO3)3:Eu3+ is in the order of millisecond.

 figure: Fig. 3

Fig. 3 Selected decay curve of KSr4(BO3)3:xEu3+ (x = 0.03) (excited at 394 nm, monitored at 612 nm).

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Charge compensation is an important factor to the emission efficiency of the luminescent materials. In this study, alkali metal ions, Li+, Na+ and K+ (1 mol.%), are codoped into the KSr4(BO3)3:0.01Eu3+ to compensate the charge unbalancing brought by Eu3+. Examining the phosphors doped with Eu3+ and M+ (M = Li, Na, or K) by X-ray diffraction, no other peaks appear besides KSr4(BO3)3 (as shown in Fig. 4(a) ), which means the doped alkali ions will not change the host structure. Figure 4(b) is the dependence of volume of KSr4(BO3)3:0.01Eu3+ to the type of 1 mol.%-doped alkali metal ions, in which the volumes of samples are obtained from XRD data. A slightly difference is observed for the volumes in accordance with the change of the ion radii of Li+ (0.96 Å), Na+ (1.18 Å), and K+ (1.38 Å). When K+ ions doped into host compounds, might not all of them substitute the Sr2+ sites. A part of them will occupy some vocations of K+ formed during crystallization, and the bonding environment is changed because K+ ions will combine with O2- tightly. That is why the volume of KSr4(BO3)3:0.01Eu3+ doped by K+ ions shrinks compared with that of host compound. Figure 4(c) is the PL spectra and 4(d) is the intensities of peaks at 612 nm for the emission spectra of KSr4(BO3)3:0.01Eu3+, 0.01M+(M = Li, Na and K). Obviously the codoping Li+, Na+, or K+ significantly enhances the luminescence of in KSr4(BO3)3:Eu3+. It yields 3.9, 2.3, and 4.5 times of the emission intensity of the sample for the KSr4(BO3)3:Eu3+ doping 1 mol.% Li+, Na+ and K+, respectively.

 figure: Fig. 4

Fig. 4 (a) Selected X-ray diffraction pattern of KSr4(BO3)3:0.01Eu3+,0.01Li+; (b) volume of KSr4(BO3)3:0.01Eu3+ doped by 0.01M+ (M = Li, Na, K), none in the figure refers to the volume of KSr4(BO3)3:0.01Eu3+. (c) the emission spectra of KSr4(BO3)3:0.01Eu3+, 0.01M+ (M = Li, Na, K) excited at 394 nm. (d) relation between the intensities of peaks at 612nm of the emission spectra in (c) and the compensation charge of alkali ions M+ (M = Li, Na, K).

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When a trivalent cation Eu3+ is incorporated into a KSr4(BO3)3 lattice and occupies the divalent Sr2+ site, charge balance is destroyed. In order to maintain the charge balance, the host has to capture O2 in the air. That is, not all Eu3+ ions go into the lattice and occupy Sr sites. In this study, 1 mol.% Eu3+ ions are doped into the host, the sites of 3 Sr2+ ions will be occupied by 2Eu3+ ions. Thus, only about 0.67 mol.% Eu3+ ions occupy the sites of Sr2+ ions, and about 0.16 mol.% Eu3+ ions exists in the sample as Eu2O3 (which is below the detection limit of XRD, so it cannot be shown in the XRD data), and the sample is possible denoted as KSr3.96(BO3)3:0.0067 Eu3+/0.0016 Eu2O3. Then the existence of Eu2O3 phase will restrain the KSr4(BO3)3 grains growth during the sintering process, leading to the decrease of luminescence intensity. However, in the charge compensated phosphors KSr4(BO3)3:0.01Eu3+, 0.01M+(M = Li, Na and K), the charge unbalance will be compensated directly by this possible mechanisms: 2Sr2+→Eu3+ + K+. Therefore, suitable amounts of Eu3+ (and M+) are able to occupy Sr sites. For example, in KSr4(BO3)3:0.01Eu3+, two Sr2+ ions are replaced by one Eu3+ and one alkali ion, resulting in the decrease of Eu2O3 content. In consequence, the phosphors with efficient charge compensation exhibit higher emission [44,45]. Because the radius of Li+ ion is smaller than that of Na+ and K+, the impurity phase is formed hardly in the KSr4(BO3)3:0.01Eu3+,0.01Li+ phosphor and therefore the KSr4(BO3)3:0.01Eu3+, 0.01Li+ phosphor shows comparatively good photoluminescence. As for the KSr4(BO3)3:0.01Eu3+,0.01K+, in addition to occupy Sr2+ site, some doped K+ will occupy some vacancy caused by K+ loss during the sintering process, which will promote the growth of KSr4(BO3)3. As a result, the phosphor charge compensated by K+ show an intensive emission.The Commission International de I’Eclairage (CIE) chromaticity coordinates for KSr4(BO3)3:Eu3+ excited at 394 nm are measured and shown in Fig. 5 . The CIE chromaticity coordinates of Eu3+ activated KSr4(BO3)3 are (0.64, 0.35), which is very close to that of NSTC standard red color (0.66, 0.33). This means that the red phosphor found in this study is high purity and high color saturation. Furthermore, charge compensation with Li+, Na+, or K+ does not change the chromaticity coordinates of KSr4(BO3)3:Eu3+, as shown in Table 4 . This discloses that charge compensation will just enhance the luminescence intensity. The narrow emission band, high color saturation, and short decay time make KSr4(BO3)3:Eu3+ a very attractive red phosphor in white LEDs.

 figure: Fig. 5

Fig. 5 (color online) CIE chromaticity diagram for KSr4(BO3)3:Eu3+ excited at 394 nm.

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Tables Icon

Table 4. Chromaticity coordinates of KSr4(BO3)3:Eu3+ and KSr4(BO3)3:0.01Eu3+,0.01M+ (M = Li, Na, and K)

4. Conclusion

A new red-emitting phosphor, KSr4(BO3)3:Eu3+, is synthesized by solid state reactions and the photoluminescence is studied for the first time. The doped Eu3+ ions prefer to occupy Sr(2) (8c) crystallographic sites by Rietveld refinement on XRD patterns. The phosphor can be effectively excited by a near UV light of 394 nm. The emission spectra exhibit the transition from 5D0 level to 7FJ (J = 0-5) with a main emission at 612 nm, which comes from the electro-dipole transition because of the asymmetric point group. The quenching concentration of Eu3+ is about 3%, and the critical distance is about 15.3 Å. The codoped alkali ions (Li+. Na+, K+) increase the intensities of emission spectra at 612 nm effectively but do not change the chromaticity coordinates. The present results indicate that the novel red emitting phosphor is a suitable candidate for the application on white LEDs combine with a near UV chip.

Acknowledgments

This work was financially supported by National Natural Science Foundation of China (50902074, 90922037 and 60906033), Natural Science Foundation of Tianjin (09JCYBJC02500), and the Program for New Century Excellent Talents in University of China. The work was also supported through a Grant-in-Aid from the International Centre for Diffraction Data (ICDD). We thank Mrs. Y. P. Xu of N01 group, Institute of Physics, Chinese Academy of Science; Dr. J. W. Qi and Dr. Z. H. Wang of Applied Physics School, Nankai University for their great help in collecting powder X-ray diffraction data, UV–vis., and PL spectra measurements.

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Figures (5)

Fig. 1
Fig. 1 Final Rietveld refinement plots of the KSr4(BO3)3:0.03Eu3+. Small circles (o) correspond to experimental values, and the continuous lines, the calculated pattern; vertical bars (|) indicate the position of Bragg peaks. The bottom trace depicts the difference between the experimental and the calculated intensity values. Inset is the structure projection of KSr4(BO3)3.
Fig. 2
Fig. 2 PL/PLE spectra of as–synthesized KSr4(BO3)3:0.03Eu3+ (red, λem = 612 nm; black, λex = 394 nm). The inset is the effect of Eu3+ concentration on the PL intensity.
Fig. 3
Fig. 3 Selected decay curve of KSr4(BO3)3:xEu3+ (x = 0.03) (excited at 394 nm, monitored at 612 nm).
Fig. 4
Fig. 4 (a) Selected X-ray diffraction pattern of KSr4(BO3)3:0.01Eu3+,0.01Li+; (b) volume of KSr4(BO3)3:0.01Eu3+ doped by 0.01M+ (M = Li, Na, K), none in the figure refers to the volume of KSr4(BO3)3:0.01Eu3+. (c) the emission spectra of KSr4(BO3)3:0.01Eu3+, 0.01M+ (M = Li, Na, K) excited at 394 nm. (d) relation between the intensities of peaks at 612nm of the emission spectra in (c) and the compensation charge of alkali ions M+ (M = Li, Na, K).
Fig. 5
Fig. 5 (color online) CIE chromaticity diagram for KSr4(BO3)3:Eu3+ excited at 394 nm.

Tables (4)

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Table 1 Occupancy and Lattice Parameters of Eu3+ Occupying Different Sites for KSr4(BO3)3:0.03Eu3+ Refined by Rietveld Method, in which A–G Represents Eu3+ Occupy Sr(1) Site, Sr(2) Site, Sr(3) Site, Sr(1) and Sr(2) Site, Sr(1) and Sr(3) Sites, Sr(2) and Sr(3) Sites, and Sr(1), Sr(2) and Sr(3) Sites, Respectively

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Table 2 Fractional Atomic Coordinates and Equivalent Isotropic Displacement Parameters (Å2) for KSr4(BO3)3:0.03Eu3+

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Table 3 Interatomic Distances (Å) and Angles (deg) for KSr4(BO3)3:0.03Eu3+

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Table 4 Chromaticity coordinates of KSr4(BO3)3:Eu3+ and KSr4(BO3)3:0.01Eu3+,0.01M+ (M = Li, Na, and K)

Equations (2)

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R c 2 [ 3V 4π x c Z ] 1 3
I= I 0 exp(t/τ )
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