Incorporation of Trivalent Cations in Synthetic Garnets A3B5O12 (A

Italy, and Scientific and Technological Department, UniVersity of Verona, and INSTM UdR Verona,. Ca' Vignal 1, Strada le Grazie 15, I-37134 Verona, It...
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J. Phys. Chem. B 2006, 110, 6561-6568

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Incorporation of Trivalent Cations in Synthetic Garnets A3B5O12 (A ) Y, Lu-La, B ) Al, Fe, Ga) Filippo Maglia,*,† Vincenzo Buscaglia,‡ Silvia Gennari,† Paolo Ghigna,† Monica Dapiaggi,§ Adolfo Speghini,| and Marco Bettinelli| IENI/CNR, INSTM, and Department Physical Chemistry, UniVersity of PaVia, Viale Taramelli, 16, I-27100 PaVia, Italy, Institute for Energetics and Interphases, National Research Council, Via De Marini 6, I-16149 GenoVa, Italy, Department Earth Sciences, UniVersity of Milano, Via Botticelli 23, I-20133 Milano, Italy, and Scientific and Technological Department, UniVersity of Verona, and INSTM UdR Verona, Ca’ Vignal 1, Strada le Grazie 15, I-37134 Verona, Italy ReceiVed: October 7, 2005; In Final Form: January 20, 2006

Static-lattice atomistic calculations have been used to study the solution energy for the incorporation of 13 foreign cations at 3 different lattice positions of 12 synthetic garnets. Trends have been obtained as a function of the ionic radius of the dopant cation, and the predictions about site preference have been compared with both literature and experimental data. The preferred substitution site is mainly determined by the ionic size and has been correctly predicted in all cases. Moreover, the energy difference between the preferred substitution site and the next favored site is relatively small in several cases, and hence the foreign ions can be inserted at two different positions by using the correct stoichiometry. A remarkably different behavior has been encountered for Al garnets, due to the smaller size of the unit cell. In particular, some cations, such as Fe3+ and Ga3+, can be inserted at the dodecahedral position usually occupied by the rare-earth ion. Despite the limitations of the static-lattice approach, the results of the present simulations help in the understanding of the defect chemistry of garnets, which is strongly responsible for the physicochemical properties (such as luminescence and ferrimagnetism) that make garnets interesting for technological applications. Such results lead to the possibility of tuning the optical and luminescence properties of garnets by the formation of different types of solid solutions.

1. Introduction Among complex oxides, synthetic garnets (A3B2C3O12) have attracted a lot of attention as new materials due to their remarkable electrical and magnetic properties, leading to several interesting technological applications.1 According to the scientific community, synthetic garnets are generally divided into three classes: aluminum garnets, iron garnets, and gallium garnets. Among all of them, the most investigated are undoubtedly the yttrium aluminum garnet (Y3Al5O12, YAG) and the yttrium iron garnet (Y3Fe5O12, YIG). The former is, as a matter of fact, a well-known laser host material, and the latter is a ferrimagnetic crystal important in microwave applications.2 Similar to YAG, other garnets (Y3Ga5O12, Gd3Ga5O12, Lu3Al5O12, etc.) are well-known hosts for rare-earth and transitionmetal ions in lasers and phosphors.3,4 Such physical properties have been demonstrated to be strongly related to the formation of lattice defects, and for this reason, a great deal of work has been recently oriented toward the study of energetic changes and the stability of crystal structures as a function of intrinsic defects, the introduction of doping elements, and nonstoichiometry. In detail, site occupancy in the garnet lattice, the ionic radius difference between the dopant and the host cation, as well as the difference in their * Author to whom correspondence should be addressed. Phone: +39 0382 987208. Fax: +39 0382 987575. E-mail: [email protected]. † INSTM and University of Pavia. ‡ National Research Council. § University of Milano. || University of Verona and INSTM UdR Verona.

electronic structure have been shown to be the most important factors that lead to the formation of luminescent centers.5-13 For such a reason, the study of the incorporation of trivalent dopants into different garnets is the main interest for a finetuning of the optical properties of these materials and can also give indications for the synthesis of new active materials and even help in the prediction of new applications. In this context, static-lattice atomistic simulation techniques, based on interatomic potentials, have been applied to the study of many simple and complex structures over the last 20 years with remarkable results and are nowadays recognized as a powerful tool to study the energetics of solids, lattice defects, defect processes in crystals and ionic transport in oxides, as well as the structure of solid surfaces and the interaction of surfaces with the surrounding environment.14-22 In every case, the general procedure is based on the adjustment of the position of the constituent atoms in the structure, in a search for the minimum energy configuration. Interactions are described using interatomic potentials, and full relaxation of the lattice is allowed. Thus, the lattice energy is minimized with respect to all relevant structural variables, i.e., cell constants and atomic coordinates in crystal structure simulation. Recently, the same static-lattice method has been used by the authors for the study of disorder and deviation from stoichiometry (intrinsic disorder) in synthetic garnets as a function of the ionic radius of the cations A3+ (Y3+, Lu3+La3+) and B3+ (Al3+, Ga3+, Fe3+).23 Since previous atomistic and quasi-chemical modeling studies of the defect chemistry in garnets have been focused only on

10.1021/jp055713o CCC: $33.50 © 2006 American Chemical Society Published on Web 03/16/2006

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Y3Fe5O12 and on Y3Al5O1224-30 and scant information is available on rare-earth garnets, in the present work the authors systematically consider the incorporation of 13 foreign cations (M3+ ) Fe3+, Al3+, Ga3+, Y3+, La3+, Nd3+, Sm3+, Eu3+, Gd3+, Tb3+, Er3+, Yb3+, and Lu3+) into 12 garnets (Lu3A5O12, Gd3A5O12, La3A5O12, and Y3A5O12, with A ) Fe3+, Al3+, and Ga3+). In such a way, a wide range of ionic radii of both the doping cations and the garnet constituents is explored. It should be here recalled that the garnet structure is described by the space group Ia3hd; oxide anions occupy the 96(h) general position while the cations are all in special positions with no positional degrees of freedom. In particular, the rare earth (RE ) A) cations occupy the 24(c) dodecahedral site, while the B cations occupy the 16(a) octahedral and the 24(d) tetrahedral sites. Therefore there are in principle three different sites on which the doping cation can reside. In the present study, our aim is to find the most energetically favorable sites for each of the doping cations in the abovementioned garnets. The predicted trends are compared with the available literature data and, in some selected cases, with the results of a Rietveld analysis of the X-ray powder diffraction pattern of purposely prepared samples.

TABLE 1: Interatomic Potential Parameters23 (A) Short Range interaction O -O Al3+(a)-O2Al3+(d)-O2Fe3+(a)-O2Fe3+(d)-O2Ga3+(a)-O2Ga3+(d)-O2Y3+-O2Lu3+-O2Yb3+-O2Er3+-O2Tb3+-O2Gd3+-O2Eu3+-O2Sm3+-O2Nd3+-O2La3+-O22-

Eij )

( )

2 rij Cij 1 ZiZje + Aij exp - 6 4πn0 rij Fij rij

(1)

where 0 is the dielectric constant of vacuum, rij is the distance between a given pair of ions, Zi and Zj the formal valence, and e the electron charge, while Aij, Fij, and Cij are the “adjustable” parameters of the potential. The first term in eq 1 represents the long-range electrostatic interaction, the second term the short-range repulsion between neighboring charge clouds, and the third term the van der Waals attraction. The electronic polarizability of ions is accounted for by the shell model of Dick and Overhauser.32 According to this approach, each ion is described in terms of a massive core of charge Xi|e| connected by a harmonic spring with force constant ki to a massless shell of charge Yi|e| on which all short-range pair potentials act. Then, the formal charge of the ion corresponds to Zi ) Xi + Yi. The method for obtaining the parameters Aij, Fij, Cij, Yi, and ki is to fit the parameters empirically to reproduce the observed crystal structure and lattice properties (elastic constants, dielectric constants, lattice energy).33 Despite its simplicity and limitations, the empirical pair potential has been successfully applied to a number of problems concerned with crystalline solids.14-22 More specific examples include the partitioning of trace impurities in garnet-melt silicate systems34 and cation ordering in natural garnets.35 The detailed analysis of cation ordering in pyrope and grossular garnets35 has been confirmed by more sophisticated quantum mechanical calculations based on the density-functional theory36 and

2- a

F (Å)

0.4

22764.3 1084.8536 1031.6334 965.2979 965.1945 1116.9655 1102.2420 1116.0742 1063.1279 1040.2604 990.3213 893.5328 858.4231 804.9372 854.8898 781.3372 711.6994

0.14900 0.32346 0.32346 0.34240 0.34240 0.32977 0.32977 0.35451 0.35187 0.35424 0.35978 0.37124 0.37568 0.38183 0.37910 0.38931 0.40580

0.645 0.62 1.015 0.97 0.98 1.00 1.045 1.06 1.07 1.09 1.12 1.18

(B) Shell Model

2. Computational and Experimental Details A wider description of the computational details concerning the methodology and the algorithm employed in the present calculations and in the derivation of the interatomic parameters can be found in a previous paper by some of the authors23 and in the references therein. Briefly, from the general theory developed by Catlow and Mackrodt,31 the position of the ions is adjusted until the minimum energy configuration is found. At each step, the energy of the system is obtained by summing over the various interactions using specified interatomic potentials. A pairwise potential (Buckingham) is employed here, with the simple analytical form

A (eV)

cation radius (Å)

a

species

Y (eV)

k (eV Å-2)

O2Al3+ Fe3+ Ga3+

-2.7803 2.9825 5.6122 2.9804

37.9081 181.6557 188.1816 77.9508

For the O2--O2- interaction, C is 27.8 eV Å.6

experimental NMR measurements. The direct use of ab initio calculations in the present case would have required prohibitively long computational times, because of the large number of dopants and structures investigated as well as by the large number of atoms (160) in the garnet unit cell. It should be remarked that the potential parameters used here have been simultaneously fitted to the structure and the lattice properties of several garnets using the relaxed fitting strategy described by Gale.37 The multistructural fitting approach results in potentials reflecting the ion-ion interactions over a wide range of interatomic distances. This strategy generally produces more reliable potentials, in comparison to the fit of a single garnet structure, in particular when defect calculations are involved and a substantial lattice relaxation is expected to produce atomic separations around lattice defects that are very different from those in the perfect crystal. The potential parameters are reported in Table 1; short-range repulsive interactions have been considered only for oxygen-cation and oxygen-oxygen pairs, due to the shorter interatomic separation in comparison to cation-cation distances. For Al-O, Fe-O, and Ga-O short-range interactions, the parameter A has been allowed to take a different value depending on the site considered (octahedral or tetrahedral). Small differences between the A parameters have been found for Al-O and Ga-O, while for Fe-O the A parameter is almost identical. The reliability and quality of the adopted potential has been verified by calculations of the lattice parameters, lattice properties, and mean thermal expansion coefficient in the temperature range 25800°C.23 The calculated and experimental values of the lattice parameters of the garnets under investigation are reported in Table 2. The experimental data were taken from the compilation reported in the Landolt-Bo¨rnstein handbook38 and are in agreement with the ICSD crystallographic database.39 Energy minimization on perfect crystals was performed at constant pressure using periodic boundary conditions and making full use of crystal symmetry. Defect calculations were carried out by means of the two-region strategy originally introduced by Catlow and Mackrodt,31 which has become one

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TABLE 2: Lattice Parameters of the 12 Garnets under Investigationa cell edge (Å)

cell edge (Å)

cell edge (Å) a

b

mounted on aluminum holders to minimize any eventual preferential orientation. Standard NBS silicon (NBS 640c) was used to evaluate the goniometer zero, with the aim of obtaining absolute values of the cell parameter.

Lu3Al5O12

Y3Al5O12

Gd3Al5O12

La3Al5O12

11.916 (11.912)

12.019 (12.008)

12.101 (12.111)

12.390

Lu3Ga5O12

Y3Ga5O12

Gd3Ga5O12

La3Ga5O12

12.192 (12.188)

12.293 (12.280)

12.380 (12.377)

12.671

Lu3Fe5O12

Y3Fe5O12

Gd3Fe5O12

La3Fe5O12

The incorporation of trivalent cations into the garnet structure can be described, according to the Kroger-Vink notation, by the quasi-chemical reaction

12.278 (12.277)

12.378 (12.376)

12.468 (12.470)

12.761 (12.767)b

M2O3 + garnet f 2MCx + C2O3

Available experimental values Hypothetical.

38

are also reported (in brackets).

of the most widely used approaches to the calculation of defect formation energies in crystals.14-19 The energy-minimized perfect lattice is used as the starting structure. The first region (I) includes a spherical portion of the crystal surrounding the defect, while the second region (II) includes the rest of the crystal. Interactions in region I are explicitly treated at the atomistic level using the empirical potential (eq 1), and full relaxation of the lattice around the foreign ion is allowed. The response of region II, where the perturbations induced by the defect are small, is evaluated using a more approximate approach.31 The radius of region I was set to 10-12 Å and contained about 400-770 ions. The use of larger regions has produced only negligible variations of the defect energy; i.e., the energy has converged with respect to region size. Since the effective charge of M3+ substituting at a regular cation site of the garnet lattice is zero, there are not energy contributions from region II coming from the polarization of the lattice. In contrast, this contribution is always significant in the case of charged defects because of the long-range electrostatic perturbation due to the defect. All calculations have been performed using the GULP code implemented by Gale.40 To perform a comparison with simulations, some selected doped synthetic garnets (Lu2.7Fe0.3Al5O12 and Lu2.7Ga0.3Al5O12) have been prepared by using a solution combustion (propellant) synthesis procedure that allows a relatively simple preparation of homogeneous garnet single phases.41,42 This process involves the exothermic reaction between the metal nitrates (oxidizer) and an organic fuel, such as urea, glycine, or carbohydrazide. An aqueous solution containing appropriate quantities of carbohydrazide (NH2NH)2CO (Aldrich, 98%) and metal nitrates (Aldrich, >99.9%) was prepared. A carbohydrazide-to-metal nitrate molar ratio of 2.5 was employed. The precursor solution was heated with a Bunsen flame, and after the evaporation of the solvent, the autocombustion process took place with the evolution of a brown fume. After a few seconds elapsed, a very porous voluminous mass of the powder was formed. After combustion, the fluffy powders were fired for 1 h at 500 °C to decompose the residual carbohydrazide and nitrate ions. The nano-powders obtained in this way were then fired at 800 °C in air for 12 h and then at 1300 °C for further 12 h. At the end of this thermal treatment the synthetic garnet materials are very well crystallized, as demonstrated by XRPD. The data were collected using a Panalytical Xpert-Pro MPD diffractometer, with an incident slit of 0.5°, an antiscatter slit of 0.5°, a detector slit of 5 mm, a nickel filter on the diffracted side, and with an Xcelerator opening of 2.154° 2θ. The data range was 10-120° 2θ, with a step scan of about 0.008° 2θ and a counting time of 50 s/step. The samples were back-

3. Results and Discussion

(2)

where MCx indicates a foreign M3+ ion that replaces one of the constituent C3+ cations in either a dodecahedral (c), an octahedral (a), or a tetrahedral (d) site. The solution energy (per M2O3 formula unit), ES(M2O3), was calculated as the lattice energy of C2O3 plus twice the substitution energy of M3+ at the given cation site minus the lattice energy of M2O3. Therefore, the solution energy represents the heat evolved (or adsorbed) during the dissolution of 1 mol of the binary oxide M2O3 into the garnet lattice. Evolution of 1 mol of C2O3 is required to conserve the number of sites in the garnet crystal. The substitution energy corresponds to the energy required to remove a C3+ cation from a regular lattice site to infinity and to bring the foreign M3+ ion from infinity into the vacant lattice position. This quantity is directly provided by GULP as a result of a two-region defect calculation. The lattice energy of the binary oxides was obtained by energy minimization using the potential parameters of Table 1 and the crystal structure stable at room temperature. The computed solution energies correspond to the limit of infinite dilution, i.e., non-interacting defects, and can be related to the equilibrium concentration of foreign ions on the given site. Assuming that the Gibbs free energy variation of reaction 2 is mainly determined by the energy term ES(M2O3) (i.e., neglecting contributions from vibrational and configurational entropy), it is possible to obtain the equilibrium fraction of lattice sites occupied by the foreign ion using the mass-action equation

[Mc3+] ) exp -(ES(M2O3)/2RT)

(3)

where R is the gas constant and T the absolute temperature. At the temperature (∼1300 K) at which the formation of solid solutions is observed in the investigated systems, it can be easily shown that the solubility [Mc3+] is 1 eV. The above treatment is very approximate and prevents a quantitative estimate of the solubility; nevertheless it provides a threshold value for ES beyond which the solubility of M2O3 can be considered as negligible for the purpose of this work. It is worth noting that eq 3 only holds for small defect concentrations, i.e for non-interacting defects and a fraction of lattice sites occupied by the other ions close to 1. Furthermore, when the defects strongly interact, the solution energy can change significantly.19 There are neither experimental data nor results from ab initio calculations to use as reference values for the solution energy. Nevertheless, there is a broad literature about the site preference for ions that enter garnets and the formation of solid solutions to compare with the predicted trends. Given the limitations of the static-lattice approach and of the empirical potentials, the relative trends in the solution energies are more meaningful than the absolute values. We calculated the solution energy of 13 oxides M2O3 with M ) Al, Fe, Ga, Y, Lu-La in 12 garnets of the M3Al5O12,

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Figure 1. Solution energy of M2O3 (M ) Al, Ga, Fe, Y, Lu-La) in (A) Lu3Fe5O12, (B) Y3Fe5O12, (C) Gd3Fe5O12, and (D) La3Fe5O12. White squares correspond to the dodecahedral (c) site, black circles to the octahedral (a) site, and white circles to the tetrahedral (d) site. Solid lines are intended as a guide for the eye and were obtained by fitting with a second-order polynomial.

M3Fe5O12, and M3Ga5O12 families (M ) Lu, Y, Gd, and La). The energy corresponding to the incorporation of foreign cations in the garnet structure is expected to be mainly dependent on the ionic size mismatch between the dopant and the host cation. The host garnets therefore were selected to cover the rare-earth cationic radius interval from Lu3+ (radius ) 0.97 Å43) to La3+ (radius ) 1.18 Å43). Although lanthanum garnets are not experimentally observed (they are only metastable because the two-phase LaAO3 + A2O3 mixture has a lower free energy), they were included in the simulation to show the effect of the inclusion of a large cation on the dodecahedral site. The energy corresponding to the incorporation of the dopant cation in all three cationic sites (dodecahedral, octahedral, or tetrahedral) was calculated. In total, 468 energy defect calculations were performed. 3.1. Incorporation of Trivalent Cations in Iron and Gallium Garnets. The solution energies of trivalent cations in M3Fe5O12 and M3Ga5O12 are reported in Figures 1 and 2. The trends of the solution energies of the various dopant oxides are similar in both garnet families. A clear preference of the rareearth ions for the dodecahedral site, with only a few exceptions (Figures 1D and 2D) that will be discussed later on, is observed. However, Fe3+ (radius ) 0.62 Å43) and Ga3+ (radius ) 0.645 Å43) prefer, with virtually no difference, either the octahedral or the tetrahedral sites. Finally Al3+ shows a clear preference for the smaller tetrahedral site. This agrees with the generally accepted idea that, in garnets, site preference depends almost exclusively on the ionic size.44 The solution energy for all of the cations investigated, within their preferred position, is rather small or even slightly negative (most values are in the range from -0.2 to 0.5 eV), suggesting a significant solubility in agreement with the experimental evidence. In particular complete solid solubility of Y3Fe5O12 with Y3Al5O12, Y3Ga5O12, and Gd3Fe5O12 is reported38,44 in agreement with values of ES very close to zero or slightly

negative in these cases. Several natural and synthetic mixed garnets are known38,44 with general formulas (A3-xBx)C5O12, where A and B represent rare earths. The maximum value of x is generally dependent on the ionic size mismatch between the A and B rare-earth ions and in agreement with the solution energy (Figures 1 and 2) diminishes as the ionic size difference increases.44 Figure 1B allows a direct comparison with the results of Donnerberg and Catlow24 who studied the extrinsic defect formation in the iron aluminum garnet (YIG). These authors reported, in agreement with our results and experimental evidence, that large rare-earth ions show a clear preference for the dodecahedral sites followed by the octahedral site. In general, the stabilization energies (energy difference between the preferred site and next favored site) calculated in the present work are lower than those of ref 24. The main difference concerns the site preference of Al3+; Donnerberg and Catlow found a small (0.35 eV) preference for the octahedral site while our simulation indicates, in agreement with the experimental results, a clear preference (0.98 eV) for the smaller tetrahedral site. The Ga site preference in YIG was also studied experimentally in great detail.44,45 It was reported that Ga enters both (a) and (d) sites with a preference for the tetrahedral position at small Ga contents that decreases as the amount of Ga increases, in agreement with the extremely reduced energy difference (e6 kJ mol-1) predicted by our calculations. By examination of Figure 1 in more detail some trends can be outlined. The site preference shown by the rare-earth cations for the (c) site increases with increasing ionic radius of the dopant; for large cations it is of several electronvolts, ruling out the possibility of a mixed doping mechanism, which is, however, probable when small cations are considered. For a given dopant, however, increasing the ionic radius of the host rare earth (i.e., increasing the size of the garnet unit cell) decreases the site preference for the (c) site. Site preference is,

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Figure 2. Solution energy of M2O3 (M ) Al, Ga, Fe, Y, Lu-La) in (A) Lu3Ga5O12, (B) Y3Ga5O12, (C) Gd3Ga5O12, and (D) La3Ga5O12. White squares correspond to the dodecahedral (c) site, black circles to the octahedral (a) site, and white circles to the tetrahedral (d) site. Solid lines are intended as a guide for the eye and were obtained by fitting with a second-order polynomial.

for instance, much larger in Lu3Fe5O12 than in La3Fe5O12. In this last garnet the combination of the two trends described above produces a cross point between the solution energy curves in the (c) and (a) sites; dopant cations smaller than Tb3+ prefer the octahedral site. Since La3Fe5O12 is a hypothetical garnet, this behavior cannot be verified directly, but there are several experiments that support this result. In ref 24 it is reported that upon expanding the lattice of YIG by the addition of 25% of the larger Bi3+ (i.e., (Y9/4Bi3/4)Fe5O12) the site preference of Lu3+ for the (c) site decreases considerably. The possibility of synthesizing garnets with the general formula Nd3(Ga2-xREx)Ga3O12 was systematically investigated by Suchow et al.46 In ref 46, Nd garnets were considered because of the large radius of Nd3+ that guarantees that it will only enter the dodecahedral sites. Small rare earths such as Lu3+, Yb3+, Tm3+, Er3+, Ho3+, and Dy3+ were investigated. Complete substitution of Lu3+ on the (a) site was achieved, and a single-phase product could be formed for x up to 2. High x values were also obtained for Yb3+, Tm3+, and Er3+ before secondary phases appeared in the product, while smaller x values could be introduced for the two largest rare earths, Ho3+ and Dy3+. For yttrium and gadolinium garnets, even though the preferred substitution site for the rare earths is the dodecaheadral site, the calculated solution energies for incorporation of Lu3+-Gd3+ at the octahedral site (Figure 1) are e1 eV. Therefore, a non-negligible solubility of the small rare-earth ions at the iron or gallium octahedral site is expected, as experimentally found.44,47 Moreover, the observed emission spectra of gadolinium gallium garnets doped with trivalent

lanthanide ions (Nd3+ or Tm3+) indicate the presence of a multisite distribution of the luminescent rare-earth ions.48,49 3.2. Incorporation of Trivalent Cations in Aluminum Garnets. The solution energy of trivalent cations in M3Al5O12 is reported in Figure 3. The introduction of the smaller aluminum ion in the octahedral and tetrahedral positions drives a considerable reduction of the cell volume. This clearly affects solution energy trends. In particular, due to the smaller (a) and (d) positions in the host structure the solution energies of the rareearth ions on aluminum sites increases around 1 eV for all dopants in comparison to gallium and iron garnets. Since the increase of the solution energy of the same cations in the (c) position is less pronounced, the site preference for the dodecahedral site is enhanced, in particular for the small rare earths. One of the consequences is that the crossing point between (c) and (a) site solution energy, observed in La3Fe5O12 and La3Ga5O12, is not observed in La3Al5O12. For the La3Al5O12 garnet, even for Lu3+ the preferred site is (c), even though the energy difference is very small (∼0.1 eV) and some solubility of small RE3+ on the octahedral position can be expected. Another difference between aluminum garnets and the other two series is the larger difference in solubility exhibited by large REs that can be visualized by observing the higher curvature/ slope of their solution energy plots. This is particularly evident in the case of Lu3Al5O12 and Y3Al5O12. While in Ga and Fe garnets the solution energy of RE2O3 on the (c) site is always between -0.3 and +0.5 eV, here solution energies larger than 1 eV are predicted for La3+ and diminish as the RE radius

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Figure 3. Solution energy of M2O3 (M ) Al, Ga, Fe, Y, Lu-La) in (A) Lu3Al5O12, (B) Y3Al5O12, (C) Gd3Al5O12, and (D) La3Al5O12. White squares correspond to the dodecahedral (c) site, black circles to the octahedral (a) site, and white circles to the tetrahedral (d) site. Solid lines are intended as a guide for the eye and were obtained by fitting with a second-order polynomial.

decreases. In La3Al5O12, however, a small negative solution energy is predicted for all REs. Another relevant point to be discussed is the site preference shown by Ga3+ and Fe3+. As already said the site preference of Ga3+ in iron garnets and of Fe3+ in gallium garnets is for both the (a) and (d) sites while the solution energy for the (c) site is several electronvolts higher in particular when garnets with large REs are considered. However, in the aluminum garnets the solution energy of Fe3+ and Ga3+ in the octahedral and tetrahedral sites is no longer equal, but a difference of around 1 eV in favor of the larger octahedral site is found. Again this can be ascribed to the reduced space available at the (d) site in the aluminum garnets with respect to the iron and gallium ones. The absolute values of the solution energies of Fe3+ and Ga3+ in both octahedral and tetrahedral sites increase by reducing the ionic size of the host RE. In La3Al5O12 the solution energy in the (a) site is largely negative for both Ga3+ and Fe3+ and is around zero in the (d) site, while it takes values around zero for the (a) site and around 0.8 eV for the (d) site in Lu3Al5O12, as can be seen by comparing Figures 3A and 3D. Analogously to iron and gallium garnets, the opposite trend is observed for the solution energy of Ga3+ and Fe3+ in the dodecahedral site as expected on the basis of ionic radius mismatch considerations. As the radius of the rare earth diminishes, the energy cost for its replacement by the smaller Ga3+ and Fe3+ ions also diminishes. The above trends produce an interesting situation in the Lu3Al5O12 garnet where the solution energy of Ga3+ and Fe3+ in the (c) site becomes equal

or smaller than that in the (d) site and only slightly higher than that in the (a) site. To check the possibility for substitution of Ga3+ and Fe3+ on the dodecahedral site of Lu3Al5O12 we synthesized singlephase garnets with the formulas Lu2.7Fe0.3Al5O12 and Lu2.7Ga0.3Al5O12. The synthesis, performed using the propellant method followed by annealing at 800 and 1300 °C as described above, produced single-phase materials in both cases. Since antisite disorder is well-known to occur in garnets, we investigated the actual cation distribution in both phases. The structure refinement was performed using the software GSAS-EXPGUI;50-51 the number of refinable structural parameters, even though the structure is simple, is quite high, if one considers also the thermal parameters. As the structure is highly symmetric, the number of integrated intensities available (where all of the structural information is hidden) is not so high; for this reason it was chosen to keep the number of refined parameters as low as possible. Bearing this in mind, all of the atoms were forced to have the same isotropic thermal parameter, while oxygen coordinates were left free to vary. The dodecahedral site, where mixing of the cations takes place, is kept fully occupied by bonding the occupancy fraction of Lu and of Fe and Ga. Moreover, before attempting any refinement of the cell parameter, a pattern of standard silicon (NBS640c) was recorded in exactly the same counting and geometrical conditions. In this way, it was possible to independently determine the correction of the goniometer zero for our instrument. In this way, the cell

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TABLE 3: Structural Parameters for Lu2.7Fe0.3Al5O12 and Lu2.7Ga0.3Al5O12 as Obtained by Rietveld Refinement cell parameter (Å) x (oxygen) y (oxygen) z (oxygen) occupancy (Lu) occupancy (Fe, Ga) Uiso (all atoms) Rwp Rp X2 (reduced)

Lu2.7Fe0.3Al5O12

Lu2.7Ga0.3Al5O12

11.93153 (4) 0.0316 (2) 0.0502 (3) 0.6489 (2) 0.904 (5) 0.091 (5) 0.0249 (2) 11.25 8.24 2.847

11.92308 (3) 0.0311 (2) 0.0503 (2) 0.6504 (2) 0.880 (5) 0.120 (5) 0.0242 (2) 9.77 7.16 2.033

parameter refined from the two garnet diffraction patterns was not affected by the strong statistical correlation between the zeropoint correction and the cell parameter itself. The other parameters that were refined are those regarding the peak shape (a pseudo-Voight modified for asymmetry), while the zero-point correction was kept fixed at the value calculated by the cell of standard silicon. The results are reported in Table 3 where the number in brackets represents the standard error on the last digit. The amount of the substitution in the dodecahedral site is 9.1% for the Lu-Fe garnet, while for the Lu-Ga garnet the amount of gallium in the site is 12%; the difference between the refined occupancy and the theoretical stoichiometry of the garnets phases is therefore within 1-2%, confirming the possibility to dope Lu-Al garnets with Fe and Ga on the dodecahedral position as predicted by our simulations. The refinements show consistent goodness of fit, and the thermal parameters refined for each phase are very similar to each other and with an absolute value that is reasonable for such structures at room temperature. The cell parameter of the Lu-Ga garnet is slightly smaller than that of the Lu-Fe garnet, in agreement with ionic radius considerations. The values of the cell parameters of both our garnets turned out to be slightly higher than that reported by Geller44 for Lu3Al5O12. It must however be considered that our lattice parameters are not affected by any instrumental aberration nor by any correlation between cell parameters and goniometer zero, as the former was refined against standard silicon and is therefore absolute. The comparison of these values with those reported in ref 38 is not completely fair, as those were not standardized and therefore not absolute. Moreover, as shown in ref 46, at low dopant content nonideality of the lattice constant variation versus dopant content is often encountered in these materials. 4. Conclusions The incorporation of various trivalent ions (Fe3+, Al3+, Ga3+, La3+, Nd3+, Sm3+, Eu3+, Gd3+, Tb3+, Er3+, Yb3+, and 3+ Lu ) in 12 different garnets (Lu3A5O12, Gd3A5O12, La3A5O12, and Y3A5O12, with A ) Fe3+, Al3+, or Ga3+) has been investigated by means of atomistic modeling using the staticlattice computational approach and pairwise (Buckingham) interatomic potentials. The incorporation of trivalent ions on substitutional sites (tetrahedral, octahedral, and dodecahedral) does not require any charge-compensating defects and is mainly driven by ionic size differences. The predictions about site preference have been compared with data from the literature and experimental results obtained for garnets that were appositely synthesized. The preferred substitution site was correctly predicted in all cases, including the preference of Al3+ for the tetrahedral position. In general, the tetrahedral site, owing to the reduced space available, is not accessible to the rare-earth Y3+,

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