Structural similarities between Ti metal and titanium oxides: implications on the high-pressure behavior of oxygen in metallic matrices

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Solid State Sciences 6 (2004) 809–814www.elsevier.com/locate/sssc

Structural similarities between Ti metal and titanium oxides: implicationon the high-pressure behavior of oxygen in metallic matrices

A. Vegasa,∗, J. Mejía-Lópezb, A.H. Romeroc, M. Kiwi d, D. Santamaría-Péreza, V.G. Baonzae,∗

a Instituto de Química Física Rocasolano, CSIC, Serrano 119, 28006 Madrid, Spainb Departamento de Física, Universidad de Santiago de Chile, Av. Ecuador 3493, Santiago, Chile

c Advanced Materials Department, Av. VenustianoCarranza 2425-A, 78270 San Luis Potosí, SLP, Mexicod Facultad de Física, Pontificia Universidad Católica, Casilla 306, Santiago 22, Chile

e Departamento de Química Física I, Facultad de Ciencias Químicas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Received 16 January 2004; received in revised form 2 April 2004; accepted 6 April 2004

Available online 28 May 2004

Abstract

The stabilities of thebody-centered-tetragonal and distorted-diamond phases of titanium are investigated by first-principles methods. Ouresults, together with previous experimental and theoretical work, confirm two interesting observations. First, that the metal arrayscorrespond to stable or metastable phases of the parent metal; and second, that oxygen provides the pressure medium that stabilizes thphases. In addition, we have confirmed that the bulk modulus of oxygen matrices follows a nearly universal behavior with presthat pressure-induced phase transitions tendto occur when the compressibility of the oxygen matrix reaches the compressibility of thehigh-pressure phase of the oxide. 2004 Elsevier SAS. All rights reserved.

Keywords:Titanium; Titanium dioxide; High pressure; Phase transformations; Density functional theory calculations

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1. Introduction

O’Keeffe and Hyde[1] put forward a model that relatethe cation arrays in oxides to either the crystal strucof elements or of simple alloys. This led to the suggesthat oxides could be described as oxygen-stuffed allRecently, Vegas and Jansen[1] reported more than a hundreexamples of oxides with cation arrays showing structuidentical to those of the corresponding binary alloys formby the cations. Thus, the alloys proposed by O’Keeand Hyde can be considered asreal stuffed alloys, inthe sense that the metallic structure is maintained woxygen is notionally inserted. Interestingly, in many oxidthe cation substructure resembles the high-pressure pof the corresponding alloy; thus, the equivalence betwoxygen insertion and relative compression (expansion)proposed[2,3]. These results raised two interesting ide

* Corresponding authors.E-mail addresses:[email protected] (A. Vegas),

[email protected] (V.G. Baonza).

1293-2558/$ – see front matter 2004 Elsevier SAS. All rights reserved.doi:10.1016/j.solidstatesciences.2004.04.004

e

First, that the cation array should correspond to a stphase or metastable phases of the metal[4], and, secondthat the oxygen anions in oxides might behave universalterms of compression, regardless of the structure and ometal considered. In this work we shall use first-principcalculations to lend support to these ideas. We stuTi and TiO2, because both systems have been extensinvestigated.

Some structural similarities between titanium oxidestheir Ti subarrays can be rationalized in terms of the equlence between oxidation and pressure (positive or negaFor instance, inδ-titanium oxide (Ti2O) [5], the Ti subar-ray is topologically identical to the high-pressure hexanal ω-Ti phase[6,7]. On the other hand, in the superhacotunnite-like high-pressure phase of TiO2 stabilized above60 GPa[8], the Ti atoms form a distortedhcp structure.This structure is equal to that ofγ -Ti stabilized at around120 GPa[6,7]. In view of these similarities one may speulate whether or not the body centered tetragonal (bct orγ -Sn) and distorted diamond (dd) structures, derived fromthe Ti subarrays in rutile and anatase, respectively, are

810 A. Vegas et al. / Solid State Sciences 6 (2004) 809–814

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Fig. 1. The structure ofδ-Ti rendered from data given by Akahama et al.[6].The unit cell is drawn with thin lines. In the bottom right side, the unit celbct-Ti is drawn with thick lines. This unit cell is derived from the distortbccstructure (δ-phase).

High-pressure structure transformations of Ti are a maof interest, because the relative stabilities and structof several phases are still the subject of debate[9–13].Recent ultrahigh-pressure structure studies of titanium[6,7]motivated a number of first-principles calculations[10–13]since structure changes exhibit a rather puzzling behaOn general grounds, it is predicted that titanium shouldmimic the phase transition sequence found in other grouIVb transition metals[14], therefore transforming intobcc phase at ultrahigh pressures, although it has not bobserved at room temperature, at least up to 220 GPa[6,7].Instead, the observed phases (ω-Ti [7] and δ-Ti [6]) aredistortedbcc-phases. The stability of theδ-phase has beerecently questioned in Refs.[10,13]. These authors arguthat nonhydrostatic components might preclude observaof truly hydrostatic structures. We question whether treason applies to the bcc phase, which appears to be unwith respect to shuffling the atoms rather than to shear[15].

A comparison betweenbct-Ti and δ-Ti structures isdepicted inFig. 1. The bct-Ti derived from rutile is thus apotential high-pressure metastable phase, so the first puof this work is to study the relative stability ofbct-Ti by first-principles.

2. Computational details

Our first-principles calculations were performed withthe density-functional theory (DFT) framework[16,17]us-ing the fully relativistic potential linearized augmentplane wave (FL-PLAPW) approach (WIEN2K implemention [18]). For the exchange-correlation energy of the eltrons we used both local-density (LDA)[19], and the generalized gradient (GGA)[20] approximations. In the muffintin (MT) spheres, thel-expansion of the nonspherical ptential and charge density was carried out up tolmax = 10.

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Table 1Unit cell parameters (Å) calculated within the LDA. Experimental resare given in parenthesis

a c/a

TiO2-rutile 4.558(4.5937)a 0.6441(0.64408)a

TiO2-anatase 3.754(3.7845)b 2.514(2.5140)b

Ti-hcp 2.864(2.9506)c 1.588(1.5873)c

Ti-bct 3.337 0.8562Ti-dd 2.690 3.389

a Ref. [40]; b Ref. [41]; c Ref. [42].

in plane waves, with a cutoff ofRMTkmax= 7.5 for rutile andanatase, andRMTkmax= 8 for Ti. We have adopted MT radof 1.8 and 1.6 a.u. for Ti and O, respectively. Thek integra-tion over the Brillouin zone (BZ) was performed using ttetrahedron method[21]. We used a regular mesh with 21and 256k-points in the irreducible BZ for rutile andhcp-Ti,respectively, and 240k-points for other structures. Energconvergence better than 0.1 mRy was achieved when ving the number ofk-points. Minimization of the total energ(E) as function of volume (V ) and of the ratioc/a, achievedstructural optimization. This process requires that eachconsistent calculation be converged for a fixedc/a, and iter-ated until the calculated total energy of the crystal convergeto within 0.1 mRy.

3. Results and discussion

3.1. Relative stability of titanium phases

Structural parameters obtained from our calculationslisted inTable 1. LDA calculations were performed to makour results comparable to existing studies on Ti[10–12,14],TiO2 polymorphs[22–24] and oxygen matrices[25]. Asexpected in LDA calculations, we obtained smaller lattparameters than the experimental ones, and we overestthe bulk moduli (seeTable 2); however, we expect consisterelative variations in parameters of these TiO2 polymorphsand their respective phases of Ti. GGA calculations wperformed forhcp-Ti and bct-Ti to discuss the relativestability of the hcp, ω and bct phases in terms of thresults of Ref.[9]. In all the cases, lattice volumes and bumoduli were determined by fitting theE(V ) curves to theMurnaghan form[26]

E(V ) = E0(V0) + B0V

B ′0(B

′0 − 1)

(1)×[B ′

0

(1− V0

V

)+

(V0

V

)B ′0 − 1

],

whereB0 is the zero-pressure bulk modulus,B ′0 its pressure

derivative, andE0 and V0 are the equilibrium energand volume, respectively. No significant changes in

A. Vegas et al. / Solid State Sciences 6 (2004) 809–814 811

Table 2Parameters of Murnaghan fits to total energy curves calculated under static conditions. Energies andvolumes are per unit formula. We include correctedquantities at 300 K into parenthesis. Experimental results are given in Ref.[7]

E0 (Ry) V0 (a.u.) B0 (GPa) B ′0

TiO2-rutile −2003.6508 205.79 (206.2) 268(266) 4.58 (4.59)TiO2-anatase −2003.6533 224.18 (224.6) 238(236) 4.31 (4.32)Ti-hcp(LDA) −1704.0091 109.03 (109.5) 127(125) 3.49 (3.50)Ti-hcp(GGA) −1707.6362 119.36 (119.9) 111(109) 3.02 (3.03)Ti-hcp(exp. 300 K) – 119.8 102 9.1Ti-bct (LDA) −1704.0038 107.28 (107.8) 124(122) 3.58 (3.59)Ti-bct (GGA) −1707.6256 117.58 (118.1) 102(100) 3.96 (3.97)Ti-dd −1073.9832 111.31 (111.9) 110(108) 3.04 (3.05)

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Energy differences indicate that thebct-Ti phase ismetastable with respect tohcp-Ti. In comparison,bct-Tiis slightly more stable thanfcc-Ti, and substantially morestable thanbcc-Ti [11–13]. However, since the energdifference between theω and hcp phases is about 1 mRper atom[10–13], ω-Ti becomes energetically favorabat high-pressures[6,7]. Additional confirmation thatbct-Tiis metastable at high-pressures can be obtained compthe transition pressures forhcp-Ti under static conditionsWe evaluated the enthalpy (H = E + PV ) between thecorresponding phases, and we obtained a transition preof 37 GPa at 0 K for thehcp–bct transition. Thehcp-distorted diamond transition occurs at negative pressueven at 0 K, which means that thehcp lattice has tobe expanded. Unfortunately, we cannot give an accutransition pressure, because the transition takes place ivicinity of the mechanical instability of the three phas(around−30 GPa). In any case, our results suggest thatdd-Ti is a candidate for a metastable phase at high tempera

To compare our results with available experimental dthermal corrections were incorporated into the equatiostate (EOS), since our original results are rigorously vonly for absolute zero temperature. We used the MGrüneisen formalism[27], with Grüneisen parameters anDebye temperatures taken from GGA results of Ref.[9].The correction was applied to each phase separately,the corrected quantities at 300 K are listed inTable 2.The quality of the correction can be inferred by comparthe calculated thermal expansion coefficient forhcp-Ti,αp(300 K) = 2.2 × 10−5 K−1, with the experimental valuof 2.6× 10−5 K−1. Relative volumes for thehcp-Ti andbct-Ti are compared, inFig. 2, with high-pressure experimentThe quality of our calculations is apparent from the excelagreement obtained forhcp-Ti, even in the extrapolatehigh-pressure regime. The 300 K isotherm forbct-Ti followsclosely that forω-Ti, but departs from the experimentresults around 30 GPa. It seems that the stabilizaof ω-Ti at moderately pressures precludes observatioother competitive phases likebct-Ti. At 300 K, entropycontributions are quite negligible[28], and thus we estimatthat thehcp–bct transition should take place around 45 GPGGA calculations lower thehcp–bct transition pressure in

g

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e

.

Fig. 2. Equation of state for the different phases of Ti calculated withe LDA. Symbols represent experimental results taken from Refs.[6,7].Dotted and dashed lines represent EOS derived from shock compremeasurements[9]. Present calculations are plotted as continuous lines.

thebct-phase. This again indicates thatbct-Ti is metastableand not accessible in truly hydrostatic experiments at typicatemperatures.

Before proceeding further in our discussion, we memphasize that during the course of this investigationwere aware of a recent study on tetragonal phasezirconium [29]. Both LDA and GGA calculations yieldebct-Zr phases located at energy minima positions withc/a

ratios ranging from 0.80 to 0.85, in excellent agreement wour results ofTable 1. In view of this results, it appearthat the present approach can be extended to rationthe observed behavior in other group IVb metals and toxides.

The previous discussion allows us to give an alternadescription for the structural behavior of titanium, aperhaps other group IVb metals, according to the structurasimilarities from group IVa elements and their respeccation arrays in binary oxides. Thus, cristobalite retathe diamond-like structure of Si, tridymite preservesstructure of londsdaleite, and in the rutile phase of Sn2

the tin atoms have the same topology than the high pres

812 A. Vegas et al. / Solid State Sciences 6 (2004) 809–814

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to relate the array of titanium (and other group IVb metain rutile to that of an element of group IVa thanrelate it to the structure ofβ-Hg, as O’Keeffe and Hydedid [1]. This view is supported by the fact that the titaniusubarray in anatase is a distorted diamond-like strucwith four-connected atoms. In fact, during the thinkarutile to anatase phase transition we might recoverexpanded phase of the cation array, so that one may conthat some internal compressionis released with increasintemperature. Furthermore, if we assume that cationoxides follow their own phase transition, in spite of beiembedded in an oxygen matrix[2], the opposite pathbct todiamond-like is expected at high temperatures, althoughconjecture has to be verified. In this regard, it is interestinpoint out that this transition path is well known in the casetin (a group IVa metal), in which the diamond-like structuat ambient conditions transforms tobct at high pressure.

3.2. General behavior of oxygen in metallic matrices

We noted above that our study is also devoted toalyze the effect of pressure on oxygen, using rutileanatase polymorphs as examples. The central problemis whether or not we can construct an effective EOS for ogen in a crystal. That goal cannot be achieved by directtraction of the corresponding titanium EOS at each presssince it measures the behavior of Ti cores embedded inelectron gas as whole. To a first approximation, the voluof the core could be obtained from the effective ionic radat zero pressure[30], but its pressure behavior is unknowso an alternative approach is required.

We first searched the literature for previous studiesvolving oxygen matrices, which—we mention in passingare few and scarce. In particular, we found only ofirst-principles study for an oxygen array derived fromcotunnite-like structure[25]. The EOS of this hypotheticaphase is plotted inFig. 3, together with averaged valueof topological volumes of oxygen in spinels, obtained froBader’s atoms in molecules (AIM) formalism[31], and thevolume of interstitial oxygen in vanadium, obtained fromshock-wave experiments[32]. It is concluded fromFig. 3that, contrary to previous suggestions[32], the behavior ofoxygen in a crystal is rather different from that observedthe molecular phase[33]. In addition, it is clear that the zeropressure atomic volumes of oxygen strongly differ betwthe different sources. Obviously, these differences are linto the problem of partitioning atomic/ionic volumes in crytals. This issue precludes definition of a universal EOSoxygen in terms of volume, so we followed a different aproach.

Thus, regardless of the zero pressure volume of oxyin a given crystal application of an external pressure mincrease its bulk modulus,B. If we analyze the studiedescribed above, we find surprisingly consistent values oB0andB ′

0 for oxygen (seeTable 3). This suggests thatB is the

r

,

Fig. 3. Atomic volume of oxygen as a function of pressure. Open symboare results obtained in this work by subtracting the contribution of Tthe EOS of TiO2 rutile and anatase (see text for explanation). The dasline represents the EOS of the hypothetical oxygen array derived frocotunnite-like structure[25]. Averaged topological volumes of oxygenspinels[31] are represented by a continuous line. The volume of interstoxygen in vanadium, as obtainedfrom shock-wave experiments[32], isplotted as a dotted line. Closed circles are LMTO calculations under Ecore conditions[36]. Filled squares are experimental results on molecoxygen[33].

Table 3Zero-pressure bulk modulus,B0 (GPa), and its pressure derivative,B ′

0, foroxygen according to different studies

B0 B ′0

Cotunnite subarray of oxygens[25] 181–195 4.15–4.29AIM topological volumes in spinels[31] 196–203 3.5–3.8AIM topological volumes in TiO2 anatase[35] 192 –

oxygen in the crystal. Now, the difficulty arises in knowithe bulk modulus of oxygen in the crystal.

Geophysicists[34] have used phenomenological aproaches to correlate the properties of oxides and relcompounds. The central idea is that, if an ion or atom dinates the volume of a crystal, the compressibility ofwhole system will be dominated by that contribution, as loas the compressibility of the core atoms is sufficiently smThis intuitive concept hasbeen only very recently quantfied in microscopic terms[31,35]. Following this schemeCalatayud et al.[35] calculated topological volumes of boTi and O as a function of pressure within the AIM formaism. Their results allow us to estimate an absolute volumecontribution for Ti in both anatase and rutile at any pressThe resulting atomic volumes for oxygen from our LDA cculations are illustrated inFig. 3. Murnaghan fits to oxygevolumes yieldedB0 = 215 GPa andB ′

0 = 4.6 for anataseand B0 = 245 GPa andB ′

0 = 4.5 for rutile. These valueare about 10–20% larger than those compiled inTable 3, asexpected from the LDA over-binding of the system. In a

A. Vegas et al. / Solid State Sciences 6 (2004) 809–814 813

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Table 4Oxygen pressureP O calculated from experimental bulk moduliBT ofseveral high-pressure TiO2 polymorphs (see text for explanation) antransition pressuresPT. All quantities in GPa

BTa P O PT

b PTc

Anatase 178–179 −5 – –Rutile 211–230 5–7 – –α-PbO2 253–206 13–14 8–12 4.5–7Baddeleyite 290–304 22–26 25 13–1Cotunnite 431 58 55 –

a Experimental sources quoted in Ref.[24].b Obtained in decompression runs from the cotunnite phase at r

temperature in Ref.[8].c Single-crystal Raman measurements of Ref.[43].

vious results, even in the megabar range, as confirmethe linear–muffin–tin–orbital calculations for oxygen undEarth core conditions[36] included inFig. 3. In general, theresults ofFig. 3andTable 3confirm that oxygen presentsnearly universal behavior in terms ofB.

Using the above arguments, we can now approximthe valueB for oxygen by the bulk modulus of the crystaBCR, since oxygen is the dominant component in any T2polymorph, and thus we can estimate the pressure actinoxygen,P O, in the crystal. We used a simple MurnaghEOS, so we may write:P O = (BCR − B0)/B

′0. The results

for several TiO2 polymorphs are listed inTable 4. Theseresults have been calculated from experimental bulk moand the approximate quantities for oxygen (seeTable 3):B0 = 200 GPa, andB ′

0 = 4. Not surprisingly, at roomconditions, many oxygen-based compounds exhibit vaof B0 around 200 GPa, with pressure derivativesB ′

0∼= 4 (for

instance, see Ref.[34]).If one compares the calculatedP O with experimental

transition pressures of TiO2 polymorphs, one finds a surpriingly good correlation. To analyze whether that correlatis fortuitous or not, we have also analyzed the high-presphases of ZrO2 and HfO2. It is found experimentally thaboth dioxides transform from the baddeleyite structurePbcastructure at moderate pressures, and to the cotustructure at higher pressures. From experimental valueBCR in the cotunnite phase: (306–332 GPa for ZrO2 [37]and 340 GPa for HfO2 [38]) we derive transition pressureof 27–33 GPa for ZrO2 and 35 GPa for HfO2, in good agree-ment with the experiment: 25 and 32 GPa, respectively,[38].If we consider the recent results of Ohtaka et al.[39] forZrO2, the cotunnite phase shows a bulk modulus betw265 and 296 GPa, soP O = 16–24 GPa. These authors rported a transition pressure of 12.5 GPa, which confirmsa nice correlation still holds when data from the same soare considered. In general, theprediction that metal dioxidein the cotunnite structure have zero-pressure bulk modclustering around 300 GPa[25] provides additional suppoto our observations, since transition pressures are expto occur at 20–30 GPa, as found in the experiments.

Taking into consideration the approximations involved

f

d

stabilized when the compressibility of the oxygen matrix iscomparable to that expected for the new phase. Thus, hpressure phases can be viewed as having internal pres(at room conditions) similar to those needed to inducphase transition.

4. Conclusion

Our results provide support to the ideas that metal sarrays in oxides correspond to stable or metastable phof the parent metal, and that oxygen seems to prothe “pressure medium” needed to stabilize these phaFurthermore, we have found that oxygen matrices fola nearly universal high-pressure behavior in terms of bmodulus, and that high-pressure phase transitions tenoccur when the compressibility of oxygen is close to thecompressibility expected for the new phase. We believethe present observations open new ways to interpret hpressure transition phenomena, which might have impoimplications to synthesize potentially superhard phasestatic compression.

Acknowledgements

We acknowledge financial support from MillenniuInitiative, CONACYT-Mexico, under Grant W-8001 anProject G-25851-E, FONDECYT under Projects 10200and 1030957 and MCyT, Spain, under Projects BQU201695 and BFM2002-01992. One of us, D.S.-P. is indebto the Residencia de Estudiantes and to the AyuntamienMadrid for their support.

References

[1] M. O’Keeffe, B.G. Hyde, Struct. Bonding 61 (1985) 77.[2] A. Vegas, M. Jansen, Acta Crystallogr. B 58 (2002) 38.[3] A. Vegas, A. Grzechnik, K. Syassen, I. Loa, M. Hanfland, M. Jans

Acta Crystallogr. B 57 (2001) 151.[4] A. Vegas, L.A. Martínez-Cruz, A. Ramos-Gallardo, A. Romero,

Kristallogr. 210 (1995) 574.[5] S. Andersson, Acta Chem. Scand. 13 (1959) 415.[6] Y. Akahama, H. Kawamura, T. Le Bihan, Phys. Rev. Lett. 87 (20

275,503.[7] Y.K. Vohra, P.T. Spencer, Phys. Rev. Lett. 86 (2001) 3068.[8] L.S. Dubrovinsky, Nature 410 (2001) 653.[9] C.W. Greeff, D.R. Trinkle, R.C. Albers, J. Appl. Phys. 90 (2001) 22

[10] K.D. Joshi, G. Jyoti, S. Gupta, S.K. Sikka, Phys. Rev. B 65 (20052,106.

[11] M.J. Mehl, D.A. Papaconstantopoulos, Europhys. Lett. 60 (2002) 24[12] A. Aguayo, G. Murrieta, R. de Coss, Phys. Rev. B 65 (2002) 092,1[13] A.L. Kutepov, S.G. Kutepova, Phys. Rev. B 67 (2003) 132,102.[14] R. Ahuja, J.M. Wills, B. Johansson, O. Eriksson, Phys. Rev. B

(1993) 16,269.[15] M. Sanati, A. Saxena, T. Lookman, Phys. Rev. B 64 (2001) 092,1

814 A. Vegas et al. / Solid State Sciences 6 (2004) 809–814

drties,

96)

94)

12.

53

58.

-

ña,

58

mo-

d

.

.

toni,

91)

[17] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133.[18] P. Blaha, K. Schwarz, J. Luitz, WIEN2K, A full potential linearize

augmented plane wave package for calculating crystal propeTechnical University of Vienna, Vienna, 2000.

[19] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13,244.[20] J.P. Perdew, S. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (19

3865.[21] P.E. Böchl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (19

16,223.[22] K.M. Glassford, J.R. Chelikowsky, Phys. Rev. B 46 (1992) 1284.[23] J.K. Dewhurst, J.E. Lowther, Phys. Rev. B 54 (1996) R3673.[24] J. Muscat, V. Swamy, N.M. Harrison, Phys. Rev. B 65 (2002) 224,1[25] J.K. Dewhurst, J.E. Lowther, Phys. Rev. B 64 (2001) 014,104.[26] F.D. Murnaghan, Proc. Acad. Sci. USA 30 (1944) 5390.[27] V.G. Baonza, M. Taravillo, M. Cáceres, J. Núñez, Phys. Rev. B

(1996) 5252.[28] E.G. Moroni, G. Grinvall, T. Jarlborg, Phys. Rev. Lett. 76 (1996) 27[29] X.Z. Ji, F. Jona, P.M. Marcus, Phys. Rev. B 68 (2003) 075,421.[30] N.W. Alcock, Bonding and Structure: Structural Principles in Inor

ganic and Organic Chemistry, Ellis Horwood, New York, 1990.

[31] A. Martín Pendás, A. Costales, M.A. Blanco, J.M. Recio, V. LuaPhys. Rev. B 62 (2000) 13,970.

[32] M. Nitta, S. Iino, Y. Fukai, K. Fukuota, Y. Syono, Phys. Rev. B(1998) 8301.

[33] Y. Akahama, H. Kawamura, D. Häusermann, M. Hanfland, O. Shimura, Phys. Rev. Lett. 74 (1995) 4690.

[34] O.L. Anderson, Equations ofState of Solids for Geophysics anCeramic Science, Oxford Univ. Press, New York, 1995.

[35] M. Calatayud, P. Mori-Sánchez, A. Beltrán, A. Martín Pendas, EFrancisco, J. Andrés, J.M. Recio, Phys. Rev. B 64 (2001) 184,113

[36] D.A. Boness, J.M. Brown, J. Geophys. Res. 95 (1990) 21,712.[37] J. Haines, J.M. Leger, S. Hull, J.P. Petitet, A.S. Pereira, C.A. Perot

J.A.H. da Jornada, J. Amer. Ceram. Soc. 80 (1997) 1910.[38] S. Desgreniers, K. Lagarec, Phys. Rev. B 59 (1999) 8467.[39] O. Ohtaka, et al., Phys. Rev. B 63 (2001) 174,108.[40] W. Gonschorek, Z. Kristallogr. 160 (1982) 187.[41] C.J. Howard, T.M. Sabine, F. Dickson, Acta Crystallogr. B 47 (19

462.[42] W.T. Roberts, J. Less-Common Met. 4 (1962) 345.[43] K. Lagarec, S. Desgreniers, Solid State Commun. 94 (1995) 519.