The prediction of lattice constants in orthorhombic perovskites

Journal of Alloys and Compounds 488 (2009) 374–379

Contents lists available at ScienceDirect

Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom

The prediction of lattice constants in orthorhombic perovskites R. Ubic a,∗ , G. Subodh b a b

Department of Materials Science & Engineering, Boise State University, 1910 University Drive, Boise, ID 83725, USA Physikalisches Institut, Univerität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany

a r t i c l e

i n f o

Article history: Received 21 July 2009 Received in revised form 27 August 2009 Accepted 28 August 2009 Available online 2 September 2009 Keywords: Ceramics Ferroelectrics Crystal structure

a b s t r a c t The correlation between ionic radii and the lattice constants of orthorhombic perovskites is examined and new empirical formulae for the relationship are derived. As for cubic/pseudocubic perovskites, the lattice constants are largely a function of A–X and B–X bond lengths rather than ionic charge. The average absolute relative error in the predicted a, b, and c lattice constants so derived is expected to be about 0.616%, 1.089%, and 0.714%, respectively; therefore, these formulae are in better agreement with experimental data than those derived by earlier researchers. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Perovskite is a term used to describe an ABX3 arrangement of cations and an anion (typically oxygen) that is isomorphous with CaTiO3 . In the ideal cubic form, the A-site is coordinated to 12 anions to form cuboctahedral coordination polyhedra. The B-site is coordinated to six anions, forming octahedra. The anions are coordinated to two B-site cations and four A-site cations about 41% further away. The anion octahedra are corner-shared, which is a key feature of all perovskites. An example of the ideal structural model is shown in Fig. 1, and others abound in the literature [1,2]. Perovskites abound both in nature and in the laboratory, and their wide compositional range renders a variety of useful properties such that perovskites are encountered in applications as disparate as electroceramics, superconductors, refractories, catalysts, magnetoresistors, and proton conductors. The design of such advanced materials requires an understanding of the relationship between chemical composition and crystal structure. Lattice constants can be measured by experimental means such as X-ray, electron, or neutron diffraction; however, these techniques are typically complicated, difficult, time-consuming, and expensive. As a consequence, predictive models based on empirical relations of easily obtainable structural parameters have gradually gained in popularity and remain an important area of solid-state chemistry research.

∗ Corresponding author. Tel.: +1 208 426 2309; fax: +1 208 426 2470. E-mail addresses: [email protected] (R. Ubic), [email protected] (G. Subodh). 0925-8388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2009.08.139

Due to the wide range of applications in which perovskites find uses, much effort has been expended in order to understand the stabilization and distortions of the perovskite structure. Goldschmidt [3] introduced the concept of the perovskite tolerance factor, t: rA + rX t= √ 2(rB + rX )

(1)

where rA and rB are the ionic radii of the A-site and B-site cations and rX is the ionic radius of the anion. In general, the perovskite structure is stable when the value of t is close to unity, which is true for SrTiO3 , for which t = 1.00. If t is very far from unity, a rhombohedral sesquioxide forms instead, as in MgTiO3 (t = 0.81) which adopts the ilmenite structure [4]. Reaney et al. [5] observed that A1+ B5+ (O2− )3 perovskites with t < ∼0.985 contained axes about which oxygen octahedra were tilted in an anti-phase arrangement, causing cell doubling in the three pseudocubic directions. Similarly, perovskites for which t < ∼0.965 undergo a further tilt transition whereby octahedra are tilted in-phase about one or more axes as well. Perovskites for which t > ∼0.985 were not observed to contain a tilt superlattice; however, many researchers have reported aluminates [6], cuprates [7], nickelates [8], and ferrites [9] with t ∼ 1 and ¯ tilted structures. For example, LaAlO3 has been reported in the R3c

system [6] with octahedra tilted 5.0◦ about the [1 1 1]c , yet it has a t = 1.0166. According to Woodward [10], the rhombohedral a− a− a− tilt system is stabilized by highly charged A-site cations and small tilt angles. At higher tilt angles, the orthorhombic a+ b− b− tilt system should be preferred, as in the case of La(Zn1/2 Ti1/2 )O3 , which has an in-phase tilt angle of 8.3◦ [11]. Historically, relationships between composition and structure have been based on the simple theory of Coulombic interactions, considering close-packing, space-filling, or symmetry arguments.

R. Ubic, G. Subodh / Journal of Alloys and Compounds 488 (2009) 374–379

375

Fig. 2. Calculated pseudocubic lattice constant, ao , as a function of experimental lattice constant, aexptl . Data from Refs. [19,20] are compared. The solid line represents a perfect fit (ao = aexptl ).

Fig. 1. Cubic perovskites (eight unit cells are shown). Atoms between octahedra are A-site cations; B-site cations are at the center of the octahedra; and atoms at the vertices of octahedra are anions.

Such reasoning led to radius-ratio rules. Roth [12], who studied A3+ B3+ (O2− )3 compounds, used this approach to map structural stability fields for A-M2 O3 , B-M2 O3 , bixbyite, ilmenite, corundum, orthorhombic perovskite, rhombohedral perovskite, and three different hexagonal forms; however, the regions bordering different structural types were not well defined. Newnham [13] produced a similar map for A2+ B4+ (O2− )3 compounds, while Giaquinta and Loye [14] used a combination of ionic radii and bond ionicities to improve the accuracy of the structural plot for A3+ B3+ (O2− )3 . The concept of the global instability index (GII) was introduced by Salinas-Sanchez et al. [15] to calculate the stability of the perovskite structure and the occurrence of octahedral tilting based on differences between experimental and calculated bond valence sums. The GII is defined as:

 ⎤1/2 ⎡ N  2 (di ) ⎥ ⎢ ⎢ i=1 ⎥ ⎢ ⎥ GII = ⎢ ⎥ N ⎣ ⎦

(2)

where the discrepancy factor, di , is the difference between the formal valence and calculated bond valence for the ith ion and N is the number of ions in the unit cell. The GII is typically <0.1 v.u. (valence units) for untilted perovskites and <0.2 v.u. for tilted perovskites. Crystal structures for which GII > 0.2 are generally unstable. Woodward [16] later developed the POTATO program, which used geometrical considerations to predict the structure of tilted perovskites but required the tilt system, tilt angles, and octahedral bond distances as inputs. Lufaso and Woodward [17] later developed an algorithm based on GII-minimization to predict the structures and lattice parameters of perovskites from composition alone and introduced the concept of a bond valence based tolerance factor. Their procedure was automated in the Structure Prediction Diagnostic Software (SPuDS). A recent study by Jiang et al. [18] established an empirical equation relating the ionic radii (A- and B-site cations and the X-site anion) to the pseudocubic perovskite lattice constant with an aver-

age absolute error of about 0.63%. This geometrical approach first assumes that ac = 2(rB + rX ) and then uses the tolerance factor to account for the variation in ac resulting from the size of the A-site species. In their work, a sixfold coordination was assumed for all ions involved. While such an error results in an underestimation of the A-site species radius and tolerance factor, it nevertheless resulted in an impressively accurate predictive formula. Ubic [19] later identified this error and used a linear regression in two variables to derive a more accurate and slightly simpler formula, also assuming sixfold coordination: ac = 0.06741 + 0.49052(rA + rX ) + 1.29212(rB + rX )

(3)

A subsequent report by Verma et al. [20] attempted to relate the lattice constant of pseudocubic perovskites to a combination of ionic charge and radii; however, even after making the same corrections for inaccurate experimental lattice constants made in Ref. [19], the results were still less accurate than those calculated from Eq. (3) in 83% of cases, with an average absolute relative error of 2.52% as opposed to 0.60% for the 130 compounds analyzed. A comparison is demonstrated in Fig. 2. Perovskites adopt a tilted structure partly to accommodate Asite species which are too small for the cuboctahedral sites which they would otherwise occupy. Octahedral tilting has the effect of reducing the coordination of the A-site cation while retaining the octahedral coordination of the B-site cation. One of the most common tilt configurations is a+ b− b− , which corresponds to in-phase tilting about one pseudocubic axis and anti-phase tilting of equal magnitude about the other two axes. This tilt system accounts for a majority of known perovskites [17]. Such a tilt system gives rise to an orthorhombic supercell in Pbnm [21], although other structural details like cation ordering may further reduce the symmetry to P21 /n [11,22]. Although larger orthorhombic supercells are also possible (e.g., in Imma, Cmcm, Pbcm, or Cmmm), supercells in Pbnm account for >90% of the observed orthorhombic perovskites [17]. This orthorhombic supercell is related to the cubic aristotype perovskite as shown in Fig. 3. The√lattice constants are related to the cubic ac thus: ao ∼ 2ac , bo ∼co ∼ 2ac . This class of materials has useful dielectric, ferroelectric, and optical properties of their own; however, the prediction of their lattice constants is also of importance for their use as substrates or buffer layers for compound semiconductor epitaxy [23,24]. Consequently, structural prediction based on easily obtainable parameters for compounds with novel compositions remains an important problem in the area of solid-state chemistry.

376

R. Ubic, G. Subodh / Journal of Alloys and Compounds 488 (2009) 374–379

Fig. 3. The relationship between the simple cubic perovskite unit cell (eight are ¯ and the orthorhombic Pbnm supercell (shaded). shown) in Pm3m

Kumar and Verma [25] recently reported equations describing the relationship between composition and lattice constants based on a combination of ionic charge and radii. Unfortunately, their list of orthorhombic perovskites contains only oxides, so it is unclear if their approach would work for the more generic case of ABX3 ; and the errors reported in lattice constants were up to 10% in some cases. While such an approach may be worthwhile phenomenologically, simpler geometrical models have proven more effective as predictive tools. Based on the results of Ref. [19] and the geometrical relationship of the orthorhombic supercell to the aristotype cubic perovskite illustrated in Fig. 3, a predictive tool more accurate than any reported previously has now been developed for the lattice constants of orthorhombic (Pbnm) perovskites. 2. Theory, results, and discussion Ideally, A-site cations and anions are packed along 1 1 0 pseudocubic directions in perovskite. The pseudocubic lattice constant, a , can therefore be derived from the equation: √ a = 2(rA + rX ) (4) In addition, B-site cations and anions are packed along 1 0 0 pseudocubic directions. The lattice constant, a , can therefore alter-

Fig. 4. Calculated lattice constant, ao , as a function of measured lattice constant, aexptl . The solid line represents a perfect fit (ao = aexptl ). The fit parameter for the data from this work is R2 = 0.93403, and the average absolute relative error is 0.617%.

Fig. 5. Calculated lattice constant, bo , as a function of measured lattice constant, bexptl . The solid line represents a perfect fit (bo = bexptl ). The linear fit parameter for the data from this work is R2 = 0.77138, and the average absolute relative error is 1.089%.

natively be defined thus: a = 2(rB + rX )

(5) a /a

The ratio of is the tolerance factor, t. It is a very well/ a and so t = / 1. At low demonstrated fact that, in general, a = values of t, a tends to be underestimated and a overestimated; whereas at high values of t, the reverse is true. This discrepancy implies stretching/compressing of the A–X and B–X bonds, which has been demonstrated in bond valence (BV) calculations [19]. It is only around t ∼ 1.0 that errors in both a and a approach zero and the pseudocubic lattice constant, ac ≈ a ≈ a . Ubic’s formula [19], which is a linear regression of a and a , achieves an average absolute relative error in predicted values of ac of only 0.60%, better than values obtained using the equation of Jiang et al. [18] in 64% of cases. It can easily be modified to predict the three lattice parameters of orthorhombic (Pbnm) perovskites. To obtain a good first approximation of the three orthorhombic lattice constants, it is only necessary√to consider the geometry in Fig. 3 and assume ao ∼ 2ac , bo ∼ co ∼ 2ac . One can then define the average absolute errors |a¯ o |, |b¯ o |, |¯co | as well as a generalized error parameter which combines errors in ao , bo , and co . As error parameters which involve the products of ao , bo , and/or co , such as errors in cell volume, can be underestimated when one of the errors is very small while one or both of the others can be arbitrarily large, such parameters are not appropriate measures of error in the orthorhombic system. A more reliable diagnostic is the sum of absolute errors |ao | + |bo | + |co | = |error|. Based on Table 1 from Ref. [25], which contains the structural details of 152 orthorhombic oxide perovskites, |a¯ o |, |b¯ o |, |¯co |, and |error| are equal to 0.915%, 4.185%, 3.181%, and 8.281, respectively; however, simply modifying the results from Ref. [19] as above one obtains better overall results: 1.880%, 2.576%, 1.087%, and 5.543. In compiling these data, it becomes immediately obvious that the large error for LaYO3 (|error| = 449) is the result of an incorrect ionic radius for Y3+ VI (r = 0.9 Å not 0.645 Å) [26] used in Ref. [25]. Simi(r = 0.62 Å not larly, correcting the errors in the ionic radius of Ru4+ VI 0.68 Å) [26] as well as the c parameter of DyCrO3 (c = 5.520 Å not 5.2 Å) [27] improves the fit overall. Finally, there are several inconsistencies in the table compiled by Kumar and Verma [25]. While the lattice constants of most compounds are listed assuming a Pbnm model, several are listed in Pnam (CaVO3 , SrRuO3 , SrCeO3 , LaCrO3 , LaGaO3 , LaTiO3 , and SmAlO3 ). When the unit cells are re-oriented such that they can all be described in Pbnm, the fit improves such that |a¯ o |, |b¯ o |, |¯co |, and |error| are equal to 1.832%, 2.506%,

R. Ubic, G. Subodh / Journal of Alloys and Compounds 488 (2009) 374–379

377

Table 1 Experimental and calculated values of lattice constants for orthorhombic perovskites. Compound

rAVI

rBVI

aexptl (Å)

bexptl (Å)

cexptl (Å)

ao (Å)

|ao | (%)

bo (Å)

|bo | (%)

co (Å)

|co | (%)

|Error|

NaUO3 NaTaO3 NaNbO3 NaPaO3 CaMnO3 CaCrO3 CaVO3 CaTiO3 CaRuO3 CaMoO3 CaNbO3 CaSnO3 CaHfO3 CaZrO3 CaUO3 BaPbO3 SrRuO3 SrUO3 SrIrO3 SrHfO3 SrZrO3 SrPbO3 SrCeO3 LaCrO3 LaGaO3 LaFeO3 LaVO3 LaMnO3 LaRhO3 LaTiO3 LaScO3 LaInO3 LaYO3 CeCrO3 CeFeO3 CeVO3 CeMnO3 CeTiO3 PrCrO3 PrGaO3 PrFeO3 PrVO3 PrMnO3 PrRhO3 PrTiO3 PrCoO3 PrScO3 PrAlO3 NdCrO3 NdGaO3 NdFeO3 NdVO3 NdMnO3 NdRhO3 NdTiO3 NdCoO3 NdScO3 NdInO3 PmCrO3 PmScO3 PmInO3 SmAlO3 SmCrO3 SmGaO3 SmFeO3 SmVO3 SmMnO3 SmRhO3 SmTiO3 SmCoO3 SmScO3 SmInO3 EuAlO3 EuCrO3

1.02 1.02 1.02 1.02 1 1 1 1 1 1 1 1 1 1 1 1.35 1.18 1.18 1.18 1.18 1.18 1.18 1.18 1.032 1.032 1.032 1.032 1.032 1.032 1.032 1.032 1.032 1.032 1.01 1.01 1.01 1.01 1.01 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.983 0.983 0.983 0.983 0.983 0.983 0.983 0.983 0.983 0.983 0.97 0.97 0.97 0.958 0.958 0.958 0.958 0.958 0.958 0.958 0.958 0.958 0.958 0.958 0.947 0.947

0.76 0.64 0.64 0.78 0.53 0.55 0.58 0.605 0.62 0.65 0.68 0.69 0.71 0.72 0.89 0.775 0.62 0.89 0.625 0.71 0.72 0.775 0.87 0.615 0.62 0.645 0.64 0.645 0.665 0.67 0.745 0.8 0.9 0.615 0.645 0.64 0.645 0.67 0.615 0.62 0.645 0.64 0.645 0.665 0.67 0.545 0.745 0.535 0.615 0.62 0.645 0.64 0.645 0.665 0.67 0.545 0.745 0.8 0.615 0.745 0.8 0.535 0.615 0.62 0.645 0.64 0.645 0.665 0.67 0.545 0.745 0.8 0.535 0.615

5.775 5.494 5.51 5.82 5.27 5.287 5.326 5.381 5.36 5.45 5.5338 5.519 5.568 5.587 5.78 6.024 5.53 6.01 5.58 5.785 5.792 5.86 5.986 5.479 5.473 5.556 5.54 5.529 5.524 5.546 5.678 5.723 5.877 5.475 5.519 5.486 5.532 5.513 5.479 5.458 5.495 5.487 5.545 5.414 5.499 5.331 5.615 5.322 5.425 5.431 5.441 5.451 5.38 5.378 5.487 5.336 5.574 5.627 5.4 5.56 5.7 5.285 5.367 5.369 5.394 5.393 5.359 5.321 5.468 5.289 5.53 5.589 5.271 5.34

5.905 5.513 5.57 5.92 5.275 5.316 5.352 5.443 5.53 5.58 5.6541 5.668 5.732 5.758 5.97 6.065 5.57 6.17 5.6 5.786 5.814 5.958 6.125 5.515 5.526 5.565 5.54 5.662 5.629 5.753 5.787 5.914 6.087 5.475 5.536 5.486 5.557 5.757 5.484 5.49 5.578 5.562 5.787 5.747 5.724 5.373 5.776 5.347 5.478 5.499 5.573 5.579 5.854 5.755 5.707 5.336 5.771 5.891 5.49 5.79 5.9 5.29 5.508 5.52 5.592 5.588 5.843 5.761 5.665 5.354 5.76 5.886 5.292 5.515

8.25 7.751 7.77 8.36 7.464 7.486 7.547 7.645 7.67 7.8 7.9131 7.885 7.984 8.008 8.29 8.506 7.847 8.6 7.89 8.182 8.196 8.331 8.531 7.753 7.767 7.862 7.83 7.715 7.9 7.832 8.098 8.207 8.493 7.74 7.819 7.74 7.812 7.801 7.718 7.733 7.81 7.751 7.575 7.803 7.798 7.587 8.027 7.481 7.694 7.71 7.753 7.734 7.557 7.775 7.765 7.547 7.998 8.121 7.69 7.94 8.2 7.473 7.643 7.65 7.711 7.672 7.482 7.708 7.737 7.541 7.95 8.082 7.458 7.622

5.630 5.473 5.473 5.656 5.298 5.324 5.364 5.396 5.416 5.455 5.495 5.508 5.534 5.547 5.770 6.149 5.689 6.043 5.695 5.807 5.820 5.892 6.016 5.458 5.465 5.497 5.491 5.497 5.524 5.530 5.628 5.700 5.831 5.425 5.464 5.457 5.464 5.497 5.394 5.401 5.434 5.427 5.434 5.460 5.466 5.303 5.565 5.290 5.384 5.390 5.423 5.417 5.423 5.449 5.456 5.292 5.554 5.626 5.364 5.534 5.607 5.241 5.346 5.353 5.385 5.379 5.385 5.411 5.418 5.254 5.516 5.588 5.224 5.329

2.513 0.389 0.678 2.817 0.535 0.708 0.708 0.288 1.047 0.100 0.706 0.202 0.610 0.713 0.175 2.082 2.872 0.542 2.068 0.376 0.481 0.544 0.507 0.382 0.153 1.055 0.888 0.572 0.008 0.286 0.874 0.394 0.775 0.918 0.996 0.520 1.229 0.294 1.543 1.045 1.115 1.090 2.006 0.849 0.591 0.530 0.895 0.608 0.759 0.748 0.328 0.631 0.802 1.327 0.567 0.822 0.356 0.014 0.664 0.459 1.640 0.830 0.392 0.307 0.162 0.265 0.490 1.700 0.914 0.657 0.248 0.012 0.883 0.201

5.854 5.602 5.602 5.896 5.369 5.411 5.474 5.526 5.558 5.621 5.684 5.705 5.747 5.768 6.125 5.912 5.573 6.140 5.583 5.762 5.783 5.898 6.098 5.550 5.561 5.613 5.603 5.613 5.655 5.666 5.823 5.939 6.149 5.548 5.611 5.601 5.611 5.664 5.547 5.557 5.610 5.599 5.610 5.652 5.662 5.399 5.820 5.378 5.546 5.556 5.609 5.598 5.609 5.651 5.662 5.399 5.819 5.935 5.545 5.818 5.934 5.376 5.544 5.554 5.607 5.596 5.607 5.649 5.659 5.397 5.817 5.933 5.375 5.543

0.869 1.606 0.566 0.410 1.777 1.783 2.276 1.531 0.504 0.733 0.528 0.652 0.261 0.174 2.600 2.515 0.049 0.485 0.299 0.417 0.536 1.000 0.440 0.635 0.624 0.863 1.129 0.865 0.463 1.520 0.625 0.418 1.016 1.337 1.359 2.091 0.976 1.620 1.140 1.221 0.566 0.666 3.066 1.660 1.081 0.492 0.757 0.588 1.240 1.045 0.646 0.349 4.185 1.807 0.797 1.178 0.834 0.742 1.000 0.484 0.570 1.622 0.652 0.623 0.267 0.151 4.040 1.945 0.098 0.799 0.991 0.792 1.566 0.507

8.064 7.767 7.767 8.113 7.463 7.512 7.586 7.648 7.685 7.759 7.833 7.858 7.908 7.932 8.352 8.637 7.978 8.645 7.990 8.200 8.225 8.361 8.595 7.725 7.737 7.799 7.787 7.799 7.848 7.861 8.046 8.182 8.429 7.689 7.763 7.751 7.763 7.825 7.657 7.669 7.731 7.718 7.731 7.780 7.792 7.484 7.978 7.459 7.645 7.658 7.719 7.707 7.719 7.769 7.781 7.472 7.966 8.102 7.624 7.945 8.081 7.407 7.605 7.617 7.679 7.666 7.679 7.728 7.740 7.432 7.926 8.062 7.389 7.587

2.260 0.208 0.037 2.955 0.016 0.350 0.521 0.041 0.198 0.522 1.007 0.341 0.958 0.946 0.751 1.542 1.667 0.522 1.270 0.222 0.353 0.357 0.756 0.363 0.384 0.802 0.554 1.088 0.653 0.367 0.642 0.306 0.754 0.658 0.714 0.140 0.625 0.307 0.796 0.829 1.016 0.421 2.055 0.294 0.071 1.363 0.614 0.295 0.634 0.680 0.435 0.350 2.148 0.081 0.207 0.991 0.396 0.231 0.858 0.066 1.450 0.885 0.503 0.433 0.419 0.074 2.628 0.260 0.044 1.451 0.305 0.252 0.925 0.464

5.642 2.203 1.282 6.181 2.328 2.841 3.506 1.860 1.749 1.355 2.240 1.195 1.829 1.833 3.525 6.139 4.588 1.549 3.637 1.016 1.369 1.902 1.703 1.380 1.161 2.720 2.571 2.525 1.124 2.173 2.140 1.118 2.545 2.913 3.068 2.751 2.829 2.221 3.480 3.094 2.696 2.178 7.128 2.802 1.744 2.385 2.265 1.491 2.634 2.473 1.409 1.330 7.135 3.214 1.571 2.991 1.585 0.986 2.521 1.009 3.660 3.336 1.547 1.363 0.848 0.490 7.159 3.906 1.056 2.907 1.544 1.056 3.374 1.172

378

R. Ubic, G. Subodh / Journal of Alloys and Compounds 488 (2009) 374–379

Table 1 (Continued ) Compound

rAVI

rBVI

aexptl (Å)

bexptl (Å)

cexptl (Å)

ao (Å)

|ao | (%)

bo (Å)

|bo | (%)

co (Å)

|co | (%)

|Error|

EuGaO3 EuFeO3 EuMnO3 EuRhO3 EuScO3 EuInO3 GdAlO3 GdCrO3 GdGaO3 GdFeO3 GdVO3 GdMnO3 GdRhO3 GdTiO3 GdCoO3 GdScO3 GdInO3 TbAlO3 TbCrO3 TbGaO3 TbFeO3 TbMnO3 TbRhO3 TbTiO3 DyAlO3 DyCrO3 DyGaO3 DyFeO3 DyVO3 DyMnO3 DyRhO3 DyTiO3 DyScO3 DyInO3 HoAlO3 HoCrO3 HoGaO3 HoFeO3 HoMnO3 HoRhO3 HoTiO3 HoScO3 ErAlO3 ErCrO3 ErGaO3 ErFeO3 ErVO3 ErMnO3 ErRhO3 ErTiO3 TmAlO3 TmCrO3 TmGaO3 TmFeO3 TmMnO3 TmRhO3 TmTiO3 YbAlO3 YbCrO3 YbGaO3 YbFeO3 YbMnO3 YbTiO3 LuCrO3 LuGaO3 LuFeO3 LuMnO3 LuRhO3 LuTiO3 YAlO3 YCrO3 YGaO3 YFeO3 YVO3

0.947 0.947 0.947 0.947 0.947 0.947 0.938 0.938 0.938 0.938 0.938 0.938 0.938 0.938 0.938 0.938 0.938 0.923 0.923 0.923 0.923 0.923 0.923 0.923 0.912 0.912 0.912 0.912 0.912 0.912 0.912 0.912 0.912 0.912 0.901 0.901 0.901 0.901 0.901 0.901 0.901 0.901 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.89 0.88 0.88 0.88 0.88 0.88 0.88 0.88 0.868 0.868 0.868 0.868 0.868 0.868 0.861 0.861 0.861 0.861 0.861 0.861 0.9 0.9 0.9 0.9 0.9

0.62 0.645 0.645 0.665 0.745 0.8 0.535 0.615 0.62 0.645 0.64 0.645 0.665 0.67 0.545 0.745 0.8 0.535 0.615 0.62 0.645 0.645 0.665 0.67 0.535 0.615 0.62 0.645 0.64 0.645 0.665 0.67 0.745 0.8 0.535 0.615 0.62 0.645 0.645 0.665 0.67 0.745 0.535 0.615 0.62 0.645 0.64 0.645 0.665 0.67 0.535 0.615 0.62 0.645 0.645 0.665 0.67 0.535 0.615 0.62 0.645 0.645 0.67 0.615 0.62 0.645 0.645 0.665 0.67 0.535 0.615 0.62 0.645 0.64

5.351 5.371 5.338 5.298 5.51 5.567 5.247 5.312 5.322 5.346 5.343 5.313 5.277 5.407 5.228 5.487 5.548 5.22 5.291 5.307 5.326 5.297 5.254 5.388 5.21 5.263 5.282 5.302 5.302 5.275 5.245 5.361 5.43 5.519 5.18 5.243 5.251 5.278 5.255 5.23 5.339 5.42 5.16 5.223 5.239 5.263 5.262 5.24 5.216 5.318 5.15 5.209 5.224 5.251 5.23 5.203 5.306 5.12 5.195 5.208 5.233 5.22 5.293 5.176 5.188 5.213 5.205 5.186 5.274 5.179 5.241 5.257 5.283 5.284

5.528 5.611 5.842 5.761 5.76 5.835 5.304 5.525 5.537 5.616 5.614 5.853 5.761 5.667 5.404 5.756 5.842 5.28 5.518 5.531 5.602 5.831 5.749 5.648 5.31 5.52 5.534 5.598 5.602 5.828 5.731 5.659 5.71 5.751 5.33 5.519 5.531 5.591 5.831 5.726 5.665 5.71 5.32 5.516 5.527 5.582 5.604 5.82 5.712 5.657 5.33 5.508 5.515 5.576 5.81 5.697 5.647 5.33 5.51 5.51 5.557 5.8 5.633 5.497 5.505 5.547 5.79 5.67 5.633 5.329 5.521 5.536 5.592 5.605

7.628 7.686 7.453 7.68 7.94 8.078 7.447 7.606 7.606 7.668 7.637 7.432 7.658 7.692 7.436 7.925 8.071 7.41 7.576 7.578 7.635 7.403 7.623 7.676 7.38 7.552 7.556 7.623 7.601 7.375 7.6 7.647 7.89 8.041 7.36 7.538 7.536 7.602 7.354 7.582 7.626 7.87 7.33 7.519 7.522 7.591 7.578 7.335 7.561 7.613 7.29 7.5 7.505 7.584 7.32 7.543 7.607 7.31 7.49 7.49 7.57 7.3 7.598 7.475 7.484 7.565 7.31 7.512 7.58 7.37 7.532 7.533 7.603 7.587

5.336 5.369 5.369 5.395 5.500 5.572 5.211 5.316 5.322 5.355 5.348 5.355 5.381 5.388 5.224 5.486 5.558 5.188 5.293 5.299 5.332 5.332 5.358 5.365 5.171 5.276 5.283 5.316 5.309 5.316 5.342 5.348 5.447 5.519 5.155 5.260 5.266 5.299 5.299 5.325 5.332 5.430 5.138 5.243 5.249 5.282 5.276 5.282 5.308 5.315 5.123 5.228 5.234 5.267 5.267 5.293 5.300 5.105 5.210 5.216 5.249 5.249 5.282 5.199 5.206 5.238 5.238 5.265 5.271 5.153 5.258 5.265 5.297 5.291

0.283 0.045 0.573 1.827 0.188 0.084 0.689 0.069 0.004 0.168 0.101 0.790 1.974 0.357 0.078 0.018 0.181 0.611 0.037 0.142 0.117 0.665 1.988 0.427 0.740 0.252 0.016 0.256 0.133 0.769 1.845 0.236 0.306 0.006 0.487 0.317 0.289 0.396 0.836 1.819 0.137 0.183 0.424 0.382 0.200 0.366 0.260 0.806 1.772 0.056 0.525 0.361 0.198 0.307 0.709 1.736 0.116 0.297 0.281 0.157 0.304 0.554 0.214 0.445 0.338 0.486 0.640 1.514 0.056 0.497 0.326 0.145 0.273 0.130

5.553 5.606 5.606 5.648 5.816 5.932 5.374 5.542 5.553 5.605 5.595 5.605 5.647 5.658 5.395 5.815 5.931 5.373 5.541 5.552 5.604 5.604 5.646 5.657 5.372 5.540 5.551 5.603 5.593 5.603 5.645 5.656 5.813 5.929 5.371 5.539 5.550 5.602 5.602 5.644 5.655 5.812 5.370 5.538 5.549 5.601 5.591 5.601 5.643 5.654 5.369 5.537 5.548 5.600 5.600 5.643 5.653 5.368 5.536 5.547 5.599 5.599 5.652 5.536 5.546 5.599 5.599 5.641 5.651 5.371 5.539 5.550 5.602 5.592

0.461 0.089 4.039 1.961 0.975 1.657 1.322 0.312 0.284 0.191 0.343 4.232 1.974 0.162 0.164 1.032 1.523 1.759 0.417 0.371 0.036 3.892 1.791 0.152 1.167 0.364 0.300 0.092 0.167 3.859 1.498 0.059 1.808 3.092 0.771 0.366 0.338 0.201 3.924 1.428 0.181 1.792 0.943 0.404 0.394 0.346 0.236 3.758 1.202 0.056 0.738 0.535 0.597 0.439 3.606 0.957 0.106 0.719 0.480 0.671 0.765 3.457 0.338 0.707 0.752 0.936 3.300 0.513 0.327 0.788 0.328 0.246 0.181 0.239

7.599 7.661 7.661 7.710 7.908 8.044 7.374 7.572 7.584 7.646 7.634 7.646 7.696 7.708 7.399 7.893 8.029 7.350 7.548 7.560 7.622 7.622 7.671 7.684 7.332 7.530 7.542 7.604 7.592 7.604 7.653 7.666 7.851 7.987 7.314 7.512 7.524 7.586 7.586 7.635 7.648 7.833 7.296 7.494 7.506 7.568 7.556 7.568 7.618 7.630 7.280 7.478 7.490 7.552 7.552 7.601 7.614 7.261 7.458 7.471 7.532 7.532 7.594 7.447 7.459 7.521 7.521 7.570 7.583 7.313 7.510 7.523 7.584 7.572

0.380 0.328 2.788 0.393 0.405 0.424 0.975 0.447 0.284 0.285 0.042 2.881 0.490 0.207 0.497 0.401 0.519 0.810 0.374 0.238 0.174 2.955 0.632 0.098 0.649 0.295 0.184 0.251 0.125 3.103 0.701 0.244 0.495 0.674 0.622 0.347 0.156 0.211 3.154 0.704 0.285 0.470 0.459 0.333 0.208 0.302 0.294 3.178 0.747 0.221 0.136 0.297 0.199 0.424 3.167 0.772 0.087 0.676 0.424 0.260 0.498 3.183 0.051 0.377 0.332 0.582 2.886 0.777 0.036 0.779 0.289 0.138 0.245 0.198

1.124 0.462 7.400 4.181 1.568 2.166 2.987 0.828 0.573 0.644 0.486 7.904 4.438 0.726 0.738 1.452 2.223 3.180 0.828 0.750 0.327 7.513 4.410 0.677 2.556 0.911 0.499 0.599 0.425 7.731 4.044 0.539 2.609 3.772 1.879 1.029 0.783 0.808 7.914 3.951 0.603 2.445 1.826 1.118 0.803 1.013 0.790 7.742 3.722 0.334 1.399 1.192 0.994 1.170 7.483 3.464 0.309 1.693 1.186 1.087 1.566 7.194 0.603 1.529 1.422 2.004 6.826 2.804 0.419 2.064 0.943 0.529 0.699 0.566

R. Ubic, G. Subodh / Journal of Alloys and Compounds 488 (2009) 374–379

379

Table 1 (Continued ) Compound

rAVI

rBVI

aexptl (Å)

bexptl (Å)

cexptl (Å)

ao (Å)

|ao | (%)

bo (Å)

|bo | (%)

co (Å)

|co | (%)

|Error|

YMnO3 YTiO3 YScO3 YInO3

0.9 0.9 0.9 0.9

0.645 0.67 0.745 0.8

5.26 5.34 5.431 5.5

5.83 5.665 5.712 5.787

7.36 7.624 7.894 8.053

5.297 5.330 5.428 5.500

0.711 0.184 0.048 0.009

5.602 5.655 5.812 5.928

3.909 0.182 1.755 2.434

7.584 7.646 7.831 7.967

3.048 0.290 0.793 1.064

7.668 0.657 2.596 3.507

0.714

2.420

Averages: |Error| = |ao | + |bo | + |co |.

rOVI

0.617

1.089

= 1.4 Å [25].

Fig. 6. Calculated lattice constant, co , as a function of measured lattice constant, cexptl . The solid line represents a perfect fit (co = cexptl ). The linear fit parameter for the data from this work is R2 = 0.90936, and the average absolute relative error is 0.714%.

1.009%, and 5.348, respectively. One of the compounds with a large error remaining is CaNbO3 (|error| = 5.530); however, when the unit cell reported by Hervieu et al. [28] (PDF 47-1668) is substituted for the one used in Ref. [25], the value of |error| decreases to just 2.752. A great improvement to the fits can be made now by calculating a linear regression in two variables, (rA + rO ) and (rB + rO ), for each parameter ao , bo , and co instead of simply assuming ao ∼ 2ac , √ bo ∼ co ∼ 2ac . A still better fit is obtained if, like Jiang et al. [18] and Ubic [19], a sixfold coordination for all ions is assumed. The fact that neither A-site cations nor oxygen anions are coordinated in this way is irrelevant. As a predictive tool it turns out to be more accurate to use the sixfold radii, probably because these values are known with greater certainty than the 12-fold radii. The result is a set of three equations (one each for ao , bo , and co ) with |a¯ o |, |b¯ o |, |¯co |, and |error| equal to 0.631%, 1.104%, 0.714%, and 2.449, respectively. A further slight improvement can be achieved by again re-orienting unit cells such that the rule ao < bo < co is always obeyed, even when it means altering the setting of the space group accordingly. The final fits yield results with |a¯ o |, |b¯ o |, |¯co |, and |error| equal to 0.617%, 1.089%, 0.714%, and 2.420, respectively. The equations are listed below: ao = 1.514921(rA + rO ) + 1.310215(rB + rO ) − 0.86631

(6)

bo = 0.082623(rA + rO ) + 2.101275(rB + rO ) + 1.114993

(7)

co = 1.625883(rA + rO ) + 2.470594(rB + rO ) − 1.20754

(8)

The results are shown in Table 1, which is based on Table 1 from Ref. [25]. The results are slightly better even than those calculated from the SPuDS program [17]. The average errors (|a¯ o |, |b¯ o |, |¯co |, and |error|) as calculated by SPuDS are 0.889%, 1.132%, 0.999%, and 3.020 (excluding the three Pm3+ -containing compounds – PmCrO3 , PmScO3 , and PmInO3 – as SPuDS does not support the Pm3+ cation). While not quite as accurate as Eqs. (5)–(7), SPuDS nevertheless produces far more accurate results than the

equations published by Kumar and Verma [25]; however, in almost ¯ rather than Pbnm as the most all cases it predicts a structure in R3c stable perovskite structure by GII-minimization. These results are illustrated graphically in Figs. 4–6. From Fig. 4 it can be seen that Eq. (6) produces generally more accurate results for ao than that of Kumar and Verma [25], especially at lower values of aexptl , or SPuDS [17], especially at higher values of aexptl . Fig. 5 reveals the comparatively very poor fit of the equation for bo given in Ref. [25]. Both SPuDS [17] and Eq. (7) yield far more accurate results. According to Fig. 6, there is some overestimation in values of co as predicted by either Eq. (8) or SPuDS [17] at lower values of cexptl ; however, the values obtained from Ref. [25] are almost always underestimated. According to Eq. (6), ao is a function of both (rA + rO ) and (rB + rO ) in fairly equal measure; however, Eq. (7) shows that bo is almost independent of (rA + rO ) and is far more sensitive to (rB + rO ). Likewise, Eq. (8) indicates that, while co is a function of both (rA + rO ) and (rB + rO ), it is influenced more strongly by changes in (rB + rO ). 3. Conclusions New empirical formulations describing the relationship between ionic radii and orthorhombic lattice constants for perovskites in space group Pbnm have been derived. These formulae can be used to predict the lattice constants of orthorhombic perovskites with more accuracy than any other method yet reported. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

H.D. Megaw, Proc. Phys. Soc. Lond. 58 (2) (1946) 133. R. Ubic, I.M. Reaney, W.E. Lee, Int. Mater. Rev. 43 (5) (1998) 205. V.M. Goldschmidt, Die Naturwiss. 14 (21) (1926) 477. H.D. Megaw, Crystal Structures: A Working Approach, W.B. Saunders Co. Ltd., London, 1973. I.M. Reaney, E.L. Colla, N. Setter, Jpn. J. Appl. Phys. Part 1 33 (7A) (1994) 3984. C.J. Howard, B.J. Kennedy, B.C. Chakoumakos, J. Phys.: Condens. Matter 12 (2000) 349. G. Demazeau, C. Parent, M. Pouchard, P. Hagenmuller, Mater. Res. Bull. 7 (1972) 913. ˜ J.L. García-Munoz, J. Rodriguez-Carvajal, P. Lacorre, J.B. Torrance, Phys. Rev. B: Condens. Matter 46 (1992) 4414. P.D. Battle, T.C. Gibb, P. Lightfoot, J. Solid State Chem. 84 (1990) 271. P.M. Woodward, Acta Crystrallogr. B53 (1997) 44. R. Ubic, Y. Hu, I. Abrahams, Acta Crystrallogr. B62 (2006) 521. R.S. Roth, J. Res. NBS 58 (1957) 75. R.E. Newnham, J. Mater. Ed. 5 (6) (1983) 947. D.M. Giaquinta, H.-C. zur Loye, Chem. Mater. 6 (1994) 365. ˜ A. Salinas-Sanchez, J.L. García-Munoz, J. Rodriguez-Carvajal, R. Saez-Puche, J.L. Martinez, J. Solid State Chem. 100 (1992) 201. P.M. Woodward, J. Appl. Cryst. 30 (1997) 206. M.W. Lafaso, P.M. Woodward, Acta Crystrallogr. B57 (2001) 725. L.Q. Jiang, J.K. Guo, H.B. Liu, M. Zhou, X. Zhou, P. Wu, C.H. Li, J. Chem. Phys. Solids 67 (2006) 1531. R. Ubic, J. Am. Ceram. Soc. 90 (10) (2007) 3326. A.S. Verma, A. Kuma, R.S. Bhardwaj, Phys. Stat. Sol. (b) 245 (8) (2008) 1520. A.M. Glazer, Acta Crystrallogr. A31 (1975) 756. C.J. Howard, B.J. Kennedy, P.M. Woodward, Acta Crystrallogr. B59 (2003) 463. M.I. Kotelyanski, I.M. Kotelyanski, V.B. Kravchenko, Technol. Phys. Lett. 26 (2) (2000) 163. K. Eisenbeiser, R. Emrickm, R. Droopad, Z. Yu, J. Finder, S. Rockwell, J. Holmes, C. Overgaard, W. Ooms, IEEE Electron Dev. Lett. 23 (6) (2002) 300. A. Kumar, A.S. Verma, J. Alloys Compd. 480 (2009) 650. R.D. Shannon, Acta Crystrallogr. A32 (1976) 751. E.F. Bertaut, J. Mareschal, J. Phys. France 29 (1968) 67. M. Hervieu, F. Studer, B. Raveau, J. Solid State Chem. 22 (3) (1977) 273.