Volume-Based Thermoelasticity: Compressibility of Mineral

Tables S1−S5 listing experimental values of ion-pair volumes, Vpr, and compressibilities, β, for respectively clinopyroxenes I and II, for chalcopy...
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Volume-Based Thermoelasticity: Compressibility of Mineral-Structured Materials Leslie Glasser† Nanochemistry Research Institute, Department of Applied Chemistry, Curtin UniVersity of Technology, GPO Box U1987, Perth, WA 6845, Australia ReceiVed: February 18, 2010; ReVised Manuscript ReceiVed: May 21, 2010

Thermodynamic properties, such as entropy among others, have been shown to correlate well with formula volume, thus permitting prediction of these properties on the basis of chemical formula and density alone, with no structural detail required. We here extend these studies to the thermoelastic property of isothermal compressibility, β, of a wide range of materials. We show that compressibility is strongly linearly correlated with formula volume per atom pair, Vpr, within selected groups of materials (silicate clinopyroxenes, chalcopyrites, perovskites, transition metal diborides, spinels;these structurally related groups are selected on the basis of availability of experimental data). The groups are clearly distinguished by their “crumple” coefficients, w;that is, by the slopes of plots of their compressibilities versus their ion-pair volumes. The “crumple” coefficients decrease in the sequence of these groups listed above, and explanations are offered for the observed sequence: from flexible framework structures which crumple readily to rigid close-packed oxides and borides. This is believed to be the first examination of thermoelastic data across a wide range of structure groups, providing some broad insight into the factors influencing the elasticity of materials. The correlations may be used either to estimate or to check the compressibility of a material within a group. Introduction Isothermal compressibility, β, is a fundamental thermoelastic property of materials, related to their chemical constitution, crystallographic structures, and states of aggregation (crystalline or polycrystalline); it has special relevance to the study of the geophysics and geochemistry of the Earth and other planets since it is involved in interpretation of their seismic properties.1-3 However, compressibility (and its reciprocal, bulk modulus) is not trivially measured and values for many materials are unavailable or unreliable. It is, therefore, unsurprising that empirical relations between compressibility and other more readily accessible parameters are widely sought and used. Colleagues and I have, over recent years, developed volumebased thermodynamic (VBT) correlations by which thermodynamic quantities, such as entropies, lattice enthalpies, and formation enthalpies, may be reliably estimated by linear correlation with molar (or formula unit) volume, Vm.4 Similar studies of this kind, as applied to the elastic properties of minerals, are by no means unique, perhaps first introduced by Birch5 (1960) through his work on seismic wave velocities and further developed by Anderson and Nafe6 (1965), who related the logarithm of bulk modulus to the logarithm of mean formula unit volume per ion pair, Vpr. The present work is an extension of correlations already reported for simple ionic solids,7 here further developing the analysis for a first time within groupings of materials selected according to their broad crystallographic characteristics. Isothermal compressibility is represented variously by the symbols β or κ:

β, κ ) -

( )

1 ∂Vm Vm ∂p

)

T

1 1 , B KT

while the bulk modulus, B or KT, which is the reciprocal of the coefficient of isothermal compressibility, represents the rigidity † Phone: + 61 8 9266-3126. Fax: + 61 8 9266-4699. E-mail: [email protected].

or resistance to bulk compression of the material (and is generally favored by geophysicists). It is appropriate to use the volume of the ion pair for reference, rather than the molar volume, since materials of different composition are thereby placed on a comparable volume basis. It would be similarly suitable to normalize the volume to a single ion (or any multiple) basis, but we choose to focus on the ion-pair volume partly because it corresponds to the volume of the many 1:1 ionic compounds (halides, oxides, sulfides, etc.) and because it matches the choice of Anderson and Nafe.6 Bridgman8a (1923), in studying the compressibilities of the elemental metals, suggested a very rough relation of β to V4/3, based on the Born model for ionic solids.8b Anderson and Nafe,6a aware of Bridgman’s suggestion, applied this relation by plotting the logarithm of the bulk modulus, log B, against the logarithm of the corresponding mean formula volume per ion pair, log Vpr. However, the slopes of the log-log plots for alkali halides and for a number of diatomic solids were found to be -1 (instead of -4/3), while those for a wide selection of oxide compounds were between -3 and -4. In spite of these considerable discrepancies between theory and observation, it seems to have become traditional to continue to use log-log plots in analysis of elastic data. We show both that this is unnecessary (since a linear plot of log B versus log Vpr with slope -1 corresponds exactly to a linear plot of β versus Vpr) and that classification according to broad structure types can provide satisfactory linear correlations directly between β and Vpr for a wide range of ionic materials (Anderson and Anderson6b have remarked that the linear relation is “entirely empirical, and it is amazing that it fits so many groups of data”). The groups are distinguished among themselves by exhibiting differing values of their “crumple” coefficients;that is, by the slopes of plots of their compressibilities versus their ion-pair volumes.7 The term “crumple” is here introduced by reference to the generally observed increase in compressibility with volume of materials, with particular relevance to the process of rigid unit rotation under pressure in framework materials (see below). We

10.1021/jp101512f  2010 American Chemical Society Published on Web 06/03/2010

Compressibility of Mineral-Structured Materials

Figure 1. Compressibility, β (in GPa-1), versus ion-pair volume, Vpr (in nm3) for six sets of four groupings of materials: two groups of silicate clinopyroxenes (ABSi2O6, squares and triangles;see text), two groups of chalcopyrites (ABX2, pluses and stars;see text), perovskites (ABO3, crosses), and transition metal diborides (MB2, diamonds) (data in the Supporting Information). Fitted lines: clinopyroxenes I: β ) 1.794Vpr - 0.0263 (R2 ) 0.831); clinopyroxenes II: β ) 0.768Vpr 0.0081 (R2 ) 0.886); chalcopyrites I: β ) 0.314Vpr (R2 ) 0.894). I: β ) 0.336Vpr (R2 ) 0.834); perovskites: β ) 0.198Vpr - 0.001;see Figure 2 for more detail); transition metal diborides: β ) 0.085Vpr + 0.002 (R2 ) 0.697). Values quoted to 3 significant figures are to facilitate (limited) extrapolation; compressibilities are obtained to 2 significant figures.

also here introduce the symbol w (as a visual reminder in its shape) in representing the “crumple” coefficients. Patterns of Correlation among Ionic Materials. Figure 1 of our initial publication,7 which considers compressibility as a function of ion-pair volume, shows that the compressibilities of the alkali halides and of a large group of diatomic solids (including the 4-atom chalcopyrites, ABX2, whose volumes are normalized to binary by multiplication by 2/n, where n ) 4 for these materials;thus treating the properties of the materials as the mean of AX and BX) are each strongly linearly correlated, as separate groups, to formula volume per ion pair. This corresponds with a slope of -1 for the log-log plot of bulk modulus versus volume per ion pair for alkali halides and for many diatomic solids.6 Thermoelasticity versus volume correlations have been represented in a considerable variety of related, but different, forms by various authors, as follows;Bridgman,8 also referred to by McCarthy, Downs, and Thompson:9 B versus Vcell; Anderson and Nafe:6 log B versus log Vpr; Zimmer, Winzen, and Syassen:10 log B versus log Vm; Jayaraman et al.11 and Neumann:12 log B versus log Vcell; Fischer et al.13 and Webb and Dingwell:14 log B versus log Vm; Ross and Angel:15 B versus Vm-1; Penda´s et al.:16 B versus cation volumes; Ross:17 B versus bond length, d; Ledbetter and Kim:8b ln B versus ln Va (i.e., Va ) mean atom or ion volume); Sahara et al.:18 ln B versus ln d; Ross and Chaplin19 and Liu et al.:20 B versus Vm-1; Shein and Ivanovskii:21 B versus Vcell; Verma:22 log B versus log dn, for n from 3 to 5; and Srivastava and Gaur:23 B versus Vm. The fact that these various forms of correlation are generally linear follows from the rather small range of values over which data for the correlations are available, and also the rather large error range (about 10% in some experimental thermoelastic values) so that any deviations from linearity which might occur are largely obscured. While each of these plots may be individually acceptable, it is necessary to use a common procedure, with

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Figure 2. Compressibility, β (in GPa-1), versus ion-pair volume, Vpr (in nm3), of a group of perovskites (blue diamonds). The line shown has been fitted to the full set of data in Table 1 yielding a slope, constrained through the origin with w ) 0.236 GPa-1 nm-3 (R2 ) 0.674). The red squares represent the data for cubic manganites, not included in the fitting, from Table 2.

normalized volume (as applied in the current work), in order to observe the consistent thermoelastic relations among different materials. Figure 1 shows linear correlations of isothermal compressibility versus volume per ion pair for six sets of materials in four groupings. These have widely different volumes per ion pair, Vpr, widely differing compressibilities, β, and widely different dependence on volume, w. The data used are solely experimentally observed quantities, in order to avoid any bias that might be introduced by theoretical interpretation and interpolation. A consistent pattern of behavior observed is that the compressibilities increase with ion-pair volume, that is, w is always positive; this is not unexpected since compressibility is dependent on the interactions between the ions, and the larger ions are themselves generally considered to be more compressible.16 We discuss details of the thermoelastic behavior of each group of materials in turn below. We have recently reported7 on the compressibilities per ion-pair volume of many binary materials, as follows. For alkali halides (except Cs), w ) 0.908 GPa-1 nm-3; for Cs halides, w ) 0.807 GPa-1 nm-3; for metal oxides, w ) 0.312 GPa-1 nm-3; for metal chalcogenides (S, Se, Te), w ) 0.325 GPa-1 nm-3; and for metal pnictides (N, P, As, Sb), w ) 0.288 GPa-1 nm-3. We here extend our analysis to more complex materials. Among the materials considered here are “framework” structures,15,17 which are characterized by being composed of relatively rigid polyhedral units (often tetrahedral such as SiO4, AlO4, BO4, or PO4, or octahedral such as NaO6 or AlO6) which share corners and edges with one another to create threedimensional networks. These materials generally compress by simple rotation of the rigid units (which we here describe as “crumpling”), with only small amounts of compression of the rigid units. The silicate clinopyroxenes (ABSi2O6) have framework structures which contain chains of edge-sharing octahedra and chains of corner-linked SiO4 tetrahedra.9 These have already been insightfully classified into two groups by McCarthy et al.:9 those with “antipathetic” M2-O3 bonds “whose lengths decreases are inhibited by pressure-induced tetrahedral rotation”, thus imposing rigidity by opposing rotation of the SiO4 tetrahedra (that is, these “clinopyroxenes I” have small compressibilities), and those “clinopyroxenes II” without such bonds,

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Glasser

TABLE 1: Experimental Ion-Pair Volumes, Vpr ()2Vm/5), and Compressibilities, β, of Some Perovskites perovskite

Vpr ) 2Vm/5 (nm3)

β (GPa-1)

MgSiO3 FeSiO3 SrSiO3 CaGeO3 ScAlO3 YAlO3 BiAlO3 SmAlO3 EuAlO3 GdAlO3 FeTiO3 CdTiO3 CaTiO3 SrTiO3 BaTiO3 CaSnO3 CdSnO3 CaZrO3 SrZrO3 BaHfO3

0.0162 0.0169 0.0297 0.0206 0.0185 0.0203 0.0196 0.0209 0.0208 0.0207 0.0210 0.0219 0.0223 0.0238 0.0256 0.0246 0.0243 0.0258 0.0277 0.0315

0.0040 0.0036 0.0059 0.0051 0.0043 0.0049 0.0046 0.0056 0.0049 0.0056 0.0059 0.0047 0.0056 0.0057 0.0056 0.0062 0.0054 0.0065 0.0068 0.0068

TABLE 2: Experimental Ion-Pair Volumes, Vpr ()2Vm/5), and Compressibilities, β, of Some Cubic Manganite Perovskites in Their Insulating Paramagnetic Phase manganite Vpr ) 2Vm/5 (nm3) β (expt) (GPa-1) β (fitted) (GPa-1)c SrMnO3 CaMnO3 LaMnO3 SmMnO3

0.0246 0.0207 0.0238 0.0240

0.0060a 0.0047b 0.0055b 0.0059b

0.0058 0.0049 0.0056 0.0057

a Søndenå, R.; Ravindran, R.; Stølen, S. Phys. ReV. B 2006, 144102 (12 pp). b As listed by: Srivastava, A.; Gaur, N. K. J. Magn. Magn. Mater., 2009, 321, 3854-3865. c The fitted compressibility values with w ) 0.236 GPa-1 nm-3, with a mean absolute deviation (MAD) of 0.0004 GPa-1 for the data in Table 1.

termed “sympathetic”, which are correspondingly less rigid (and so have larger compressibilities). It is striking that these clinopyroxenes have rather small volumes per ion pair, with little variation in volume, yet have compressibilities which are very dependent on volume because of their tendency to crumple, and divide quite clearly into this pair of groups. The perovskites (ABO3) and chalcopyrites (ABX2) have rather similar “crumple” coefficients (that is, compressibility/ion-pair volume relations, w) which are comparable with those of the general run of diatomic materials,7 MX, with w ≈ 0.3 GPa-1 nm-3. More detailed analysis of the compressibilities of the perovskites will be found below. The transition metal diborides, MB2, of layered hexagonal structure are generally dense, with compressibilities which are rather independent of volume. This corresponds with the pattern of behavior of close-packed (“dense”24) oxides, MO2 and M2O3, as well as spinels, AB2O4 ()AO + B2O3). The spinels are not included in Figure 1 since they have almost constant compressibilities25 (∼0.005 GPa-1) and a very small range of ion-pair volumes (Vpr ≈ 0.02 nm3). We examine the data on perovskites contained in Table 1 in more detail, with reference to Figure 2. We note that the compressibilities of these perovskites increase with ion-pair volume at a rate of 0.236 GPa-1 nm-3. From Table 1 and Figure 2 it can be observed that the compressibilities of those materials with the same anion (e g., alkaline earth titanates) tend to be rather similar, within the rough limits of the current data. Furthermore, from the sequence of compressibilities displayed in the table, we suggest that the octahedral units vary in both volume and compressibility in the following sequence: SiO6 < AlO6 < TiO6 < SnO6 < ZrO6.

Application of the Correlations and Error Analysis. The manganites are materials of particular interest because of their collosal magnetoresistance (CMR) and other significant magnetic and electronic properties.26 We use the data in Table 2 below, for a group of pseudocubic perovskite manganites which were not included in the fitted set in Table 1, to test the reliability of the correlations that we have established here. The experimental data for the four manganites in Table 2 are plotted as (red) squares in Figure 1; as can be seen, these data fit comfortably into the general perovskite correlation. It is interesting to consider the reliability of the data involved. Srivastava and Gaur23 note that eight different values of the compressibility of LaMnO3 have been reported in the literature, ranging from a low of 0.0055 GPa-1 to a high of 0.0118 GPa-1. We may also examine how our correlation results compare with those of the considerably more sophisticated and complex quantum calculations. El Hannani, et al.27 have reported density functional (DFT) values for the compressibility of LaMnO3 between the rather wide limits of 0.004 and 0.008 GPa-1. Fuks et al.26 report DFT compressibilities of 0.0069 GPa-1 for LaMnO3 and 0.0062 GPa-1 for SrMnO3. These are not dissimilar from the correlation values that we find. We conclude that the correlations provide compressibility values within the same range of precision as do DFT calculations, with little effort, and also provide important simple checks of literature values. Compressibility measurement is difficult and demanding, placing high demands on the quality of the experimental materials (apart from purity requirements, there is a need for material of excellent crystallinity with minimal defects). As a consequence, literature data are difficult to come by, and not always reliable (as already noted23 with respect to LaMnO3). Consequently, it is not surprising that the correlations here reported a fall in quality below those of the well-defined alkali halides (R2 ) 0.989). Nevertheless, the coefficients of determination (R2 values) lie well above 0.8, showing that some reliance can be placed on the correlation coefficients in either confirmation of reported values or prediction of unknown values within a given class of materials. The Supporting Information contains graphs of the errors of the fitted against the literature compressibility data. The mean absolute difference (MAD) within a structural class generally lies within about 5% to 10% of the fitted value, although there may be much larger outliers. It is not possible to eliminate such errors because of the paucity (and occasional unreliability) of experimental data. Volume Data Sources. The data and their sources for this paper are listed in the Supporting Information. Where the ionpair volumes could not be obtained from the listed primary source, or needed to be checked, they have been calculated from crystallographic values or densities available from a number of invaluable free databases;National Institute for Materials Science;Pauling File;28 The American Mineralogist Crystal Structure Database;29 Crystallography Open Database (COD);30 WWW-MINCRYST;31 Mineralogy Database,32 Donnay and Ondik33;or estimated by summing ion volumes.4h Discussion: “Crumple” Coefficients It is to be noted that the groupings here introduced are quite natural, and have not been imposed on the data in any way. Indeed, Sirdeshmukh and Subhadra34 have already noted that the simple procedure used by Anderson and Nafe6a of lumping all oxides together is inappropriate. The distinguishing feature among the groupings is the slope of their correlations, w, here termed “crumple” coefficients since the steepness of the slope

Compressibility of Mineral-Structured Materials corresponds to the tendency of the material to increase in compressibility as the ion-pair volume increases. This is clearly evident in the case of the two groupings of clinopyroxenes whose framework structures compress to different extents by restrictions to the rotation of the SiO4 tetrahedra within the silicate chains. In broad terms, we note that materials with framework structures15,17 are most compressible, alkali halides are moderately compressible, while other diatomic solids are less compressible. Close-packed solids,24 such as oxide spinels36 and transition metal diborides, are, by contrast, only slightly compressible. This pattern of behavior has been quite thoroughly explored in recent years by Recio and colleagues.16,25,35-37 They have partitioned the unit cell volume of crystalline solids into separate volume regions by analysis of the topology of the electron density and the electron localization function (ELF). These separate volumes correspond essentially to either the ion constituents or their valence electron shells, each contributing additively to the bulk compressibility;the larger the volume of a component, the greater its compressibility. The bulk compressibility is simply the sum of the individual ion contributions, and so an ion with a dominant volume will dominate the compressibility, independent of the compressibilities of the remaining atoms.37 Oxide ions in materials such as spinels16,36 are found to be much more voluminous (and thus compressible) than the accompanying more rigid cations. Thus, the summed (bulk) compressibility is dominated by that of the oxide ions;resulting in very similar compressibilities (and, indeed, molar volumes) for the oxide spinels. By contrast, the volume in the alkali halides16 is shared more equally between cations and anions, resulting in compressibilities (and molar volumes) which depend on the particular combination of cation and anion (that is, on the chemical composition), with compressibilty increasing as the formula volume increases. Acknowledgment. I acknowledge the helpful advice of Dr. J. M. Recio (Oviedo, Spain) in commenting on this work. Studies such as this, which rely on the collation of data from multiple sources, were previously impractical but now are feasible, thanks to ready online access to search engines, databases, and journals (both free and commercial) and speedy document delivery. I thank the Library and IT services of Curtin University for access to these resources. Supporting Information Available: Tables S1-S5 listing experimental values of ion-pair volumes, Vpr, and compressibilities, β, for respectively clinopyroxenes I and II, for chalcopyrites I and II, for perovskites, for transition metal diborides, and for spinels and Figures S1-S4 showing the deviations of the fitted compressibilities from the experimental values for all the above, except for the diborides (where the differences are constant at 0.0001 GPa-1). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Anderson, O. L. Equations of State of Solids for Geophysics and Ceramic Science; Oxford University Press: Oxford, UK, 1995. (2) Grimvall, G. Thermophysical Properties of Materials, enlarged and revised ed.; North-Holland: Amsterdam, The Netherlands, 1999. (3) Poirier, J.-P. Introduction to the Physics of the Earth’s Interior, 2nd ed.; Cambridge University Press: Cambridge, UK, 2000. (4) (a) Jenkins, H. D. B.; Roobottom, H. K.; Passmore, J.; Glasser, L. Inorg. Chem. 1999, 38, 3609–3620. (b) Glasser, L.; Jenkins, H. D. B. J. Am. Chem. Soc. 2000, 122, 632–638. (c) Jenkins, H. D. B.; Tudela, D.; Glasser,

J. Phys. Chem. C, Vol. 114, No. 25, 2010 11251 L. Inorg. Chem. 2002, 41, 2364–2367. (d) Jenkins, H. D. B.; Glasser, L. Inorg. Chem. 2003, 42, 8702–8708. (e) Glasser, L.; Jenkins, H. D. B. Thermochim. Acta 2004, 414 (2), 125–130. (f) Glasser, L.; Jenkins, H. D. B. Chem. Soc. ReV. 2005, 34 (10), 866–874. (g) Jenkins, H. D. B.; Glasser, L. Inorg. Chem. 2006, 45, 1754–1756. (h) Glasser, L.; Jenkins, H. D. B. Inorg. Chem. 2008, 47 (4), 6195–6202. (5) See: Wang, H. F. Phys. Chem. Miner. 1978, 3, 251–261. Birch, F. J. Geophys. Res. 1960, 65, 1083–1102. Birch, F. J. Geophys. Res. 1961, 66, 2199–2224. Birch, F. Geophys. J. 1961, 4, 295–311. (6) (a) Anderson, O. L.; Nafe, J. E. J. Geophys. Res. 1965, 70, 3951– 3963. (b) Anderson, D. L.; Anderson, O. L. J. Geophys. Res. 1970, 75, 3494–3500. (7) Glasser, L. Inorg. Chem. 2010, 49, 3424–3427. (8) (a) Bridgman, P. W. Proc. Am. Acad. Arts Sci. 1923, 58, 165–242. (b) Ledbetter, J.; Kim. S. In Handbook of Elastic Properties of Solids, Liquids and Gases; Levy, M., Furr, L., Eds.; Academic Press: San Diego, CA, 2001; Vol. II, Chapter 4. (9) McCarthy, A. C.; Downs, R. T.; Thompson, R. M. Am. Mineral. 2008, 93, 198–209. (10) Zimmer, H. G.; Winzen, H.; Syassen, K. Phys. ReV. B 1985, 32, 4066–4070. (11) Jayaraman, A.; Batlogg, B.; Maines, R. M. Phys. ReV. B 1982, 26, 3347–3351. (12) Neumann, H. Phys. Status Solidi A 1986, 96, K121–K125. Neumann, H. Cryst. Res. Technol. 1988, 23, 531–536. (13) Fischer, G. J.; Wang, Z.; Karato, S.-i. Phys. Chem. Miner. 1993, 20, 97–103. (14) Webb, S. R.; Dingwell, D. B. Contrib. Mineral. Petrol. 1994, 118, 157–168. (15) Ross, N. L.; Angel, R. J. Am. Mineral. 1999, 84, 277–281. (16) Penda´s, A. M.; Costales, A.; Blanco, M. A.; Recio, J. M.; Luanˇa, V. Phys. ReV. B 2000, 62, 13970–13978. (17) Ross, N. L. ReV. Mineral. Geochem. 2000, 41, 257–287. Angel, R. J.; Ross, N. L.; Zhao, J. Eur. J. Mineral. 2005, 17, 193–199. (18) Sahara, R.; Shishido, T.; Nomura, A.; Kudou, K.; Okada, S.; Kumar, V.; Nakajima, K.; Kawazoe, Y. Comput. Mater. Sci. 2006, 36, 12–16. (19) Ross, N. L.; Chaplin, T. D. J. Solid State Chem. 2003, 172, 123– 126. (20) Liu, W.; Kung, K.; Wang, L.; Li, B. Am. Mineral. 2008, 93, 745– 750. (21) Shein, I. R.; Ivanovskii, A. L. J. Phys: Condens. Matter 2008, 20, 415218 (9 pp). (22) Verma, A. S. Phys. Status Solidi B 2009, 246, 345–353. (23) Srivastava, A.; Gaur, N. K. J. Magn. Magn. Mater. 2009, 321, 3854–3865. (24) Smyth, J. R.; Jacobsen, S. D.; Hazen, R. M. ReV. Mineral. 2000, 40, 157–186. (25) Wa´skowska, A.; Gerward, L.; Olsen, J. S.; Feliz, M.; Llusar, R.; Gracia, L.; Marque´s, M.; Recio, J. M. J. Phys.: Condens. Matter 2004, 16, 53–63. (26) Fuks, D.; Dorfman, S.; Felsteiner, J.; Bakaleinikov, L.; Gordon, A.; Kotomin, E. A. Solid State Ionics 2004, 173, 107–111. Note: We report values of compressibilities for LaMnO3 and SrMnO3 based on the simple relation B ) -b1, where b1 is a fitted parameter noted by the authors. Their reported value of 118.7 GPa for SrMnO3 does not accord with this calculation. (27) El Hannani, M. D.; Rached, D.; Rabah, M.; Khenata, R.; Benayad, N.; Hichour, M.; Bouhemadou, A. Mater. Sci. Semicond. Process 2008, 11, 81–86. (28) National Institute for Materials Science (NIMS), Materials Database Station (MDBS), Basic Database for Crystal StructuressPauling File: http:// crystdb.nims.go.jp/. (29) Downs, R. T.; Hall-Wallace, M. Am. Mineral. 2003, 88, 247–250: http://rruff.geo.arizona.edu/AMS/amcsd.php. (30) Crystallography Open Database (COD): http://www.crystallography. net/ (31) WWW-MINCRYST, Crystallographic and Crystallochemical Database for Minerals and their Structural Analogues: http://database.iem.ac.ru/ mincryst/. (32) Mineralogy Database: http://www.mindat.org/index.php. (33) Donnay, J. D. H.; Ondik, H. M. Crystal Data: DeterminatiVe Tables, 3rd ed.; National Bureau of Standards: Washington, DC, 1973. (34) Sirdeshmukh, D. B.; Subhadra, K. G. J. Appl. Phys. 1986, 59, 276– 277. (35) Recio, J. M.; Franco, R.; Penda´s, A. M.; Blanco, M. A.; Pueyo, L. Phys. ReV. B 2001, 63, 184101 (7 pp). (36) Contreras-Garcı´a, J.; Mori-Sa´nchez, P.; Silvi, B.; Recio, J. M. J. Chem. Theory Comput. 2009, 5, 2108–2114. (37) Taravillo, M.; del Corro, E.; Contreras-Garcia´, J.; Penda´s, A. M.; Flo´rez, M.; Recio, J. M.; Baonza, V. G. High Pressure Res. 2009, 29, 97–102.

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