Emergent charge ordering in near half doped Na 0.46 CoO 2
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Emergent charge ordering in near half doped Na . CoO D. N. Argyriou, ∗ O. Prokhnenko, K. Kiefer, and C. J. Milne
Hahn-Meitner-Institut, Glienicker Str. 100, Berlin D-14109, Germany (Dated: November 19, 2018)We have utilized neutron powder diffraction to probe the crystal structure of layered Na x CoO near the half doping composition of x =0.46 over the temperature range of 2 to 600K. Our mea-surements show evidence of a dynamic transition in the motion of Na-ions at 300K which coincideswith the onset of a near zero thermal expansion in the in-plane lattice constants. The effect of theNa-ordering on the CoO layer is reflected in the octahedral distortion of the two crystallograph-ically inequivalent Co-sites and is evident even at high temperatures. We find evidence of a weakcharge separation into stripes of Co +3 . ǫ and Co +3 . − ǫ , ǫ ∼ e below T C O =150K. We arguethat changes in the Na(1)-O bond lengths observed at the magnetic transition at T m =88K reflectchanges in the electronic state of the CoO layer. I. INTRODUCTION
The alkali cobaltates Na x CoO have been the subjectof intense interest as they are a rare example of compet-ing interactions on a triangular lattice that can be easilytuned by chemical means. Varying the amount of Na ( x )produces a rich phase diagram which exhibits spin de-pendent thermopower ( x =0.75)[1], metal-insulator tran-sitions ( x =0.5)[2, 3], antiferromagnetism and 5K su-perconductivity at x =0.3 for a hydrated compound[4]. More recently it has been realized both experimentallyand theoretically that the role of the Na ions goes beyondproviding a simple means to electronically dope the CoO layer[5, 6, 7]. Rather, the ordering of Na-ions leads to apotential that perturbs the CoO layer to produce strongelectronic correlations[5]. The role of these correlationsis still under investigation but it demonstrates that thesematerials can exhibit frustration in two different ways,one by the triangular topology of the CoO layer and theother by the Na induced potential.This double frustration is best exhibited at half-doping. Here the Na ordering results in a relativelysimple orthorhombic distortion of the parent hexagonalphase in sharp contrast to the complex incommensu-rate structures found for higher x compounds[5]. For x =0.5 Na-ions order as to form stripes as shown in fig.1while the magnetic susceptibility shows two abrupt de-creases (see for example inset in fig.2) at T m =88K and at T m =52K[2, 3]. The first transition is associated with theonset of a long range antiferromagnetic ordering[2, 3, 8]while the second transition coincides with a sharp risein the resistivity[2, 3]. This second transition has beenascribed to be driven by charge ordering (CO) of a t g electron to form distinct LS Co ( t g , S = 0) and LSCo ( t g , S = 1 /
2) ions [2]. Recent µ SR and neutrondiffraction measurements[8, 9, 10] propose a magneticstructure consistent with this picture, as the magneticlattice comprises of stripes of magnetically inactive Co ∗ Email of corresponding author: [email protected] and antiferromagnetic (AF) coupled Co (see fig. 1(a)).What is striking in this cobaltate is that the se-quence of charge ordering and N`eel transitions is re-versed ( T m Polycrystalline samples were prepared using standardsolid state synthesis techniques. The starting stoichiom-etry for these samples was Na . CoO . In order todeintercalate Na from the lattice to achieve a x ∼ x =0.75 sample wasimmersed in a bromine-acetonitrile solution with a 1:1Na to Br ratio, stirred in solution for 7-14 days andwashed. The Na/Co ratio of the product was measuredusing neutron activation analysis (NAA) giving a compo-sition x =0.46(1). Magnetic susceptibility as a functionof temperature ( χ ( T )) was measured using a Quantumdesign MPMS and was found to be identical to the pub-lished literature as shown in the inset of fig. 2 [2, 3].Rapid measurements of high resolution neutron powderdiffraction data were collected from the x =0.46(1) sampleusing the HRPD diffractometer (∆ d/d ∼ × − ) at theISIS-facility, Rutherford Appleton Laboaratory. Higherstatistics data suitable for Rietveld refinement were mea-sured between 2 to 600K using the high resolution pow-der diffractometer E9 (∆ d/d ∼ × − , λ =1.7973˚A),located at the Berlin Neutron Scattering Center, at theHahn-Meitner-Institut (HMI). Supplementary tempera-ture dependent data were also measured from the x =0.75sample between 5-300K. All NPD data were analyzedusing the Rietveld method which allowed us to measurelattice parameters, atomic positions and atomic displace-ment parameters as a function of temperature. A typicalRietveld refinement of the NPD data is shown in fig. 2. III. EVIDENCE OF WEAK CHARGEORDERING The validity of the reported orthorhombic P nmm structure for x = [3, 15] was tested by considering spacegroups that arise from distortions of the parent hexago-nal structure P6 /mmc which are consistent with thereported orthorhombic unit cell. The program ISODIS-PLACE was used for this purpose[17]. This approach forthe symmetry analysis resulted in space groups that wereeither centered or primitive monoclinic, in both cases in-compatible with the diffraction data. Alternately, us-ing as a starting point the reported space group P nmm ,and testing for related space groups that are compat-ible with the diffraction data resulted in primitive non-centrosymmetric solutions such as P mm mm . Ri-etvelds analysis on the basis of these space groups re-sulted in somewhat poorer fits than the reported struc-ture. Our best modeling of the NPD data between 2-450K were obtained using the P nmm model [3, 15], pro-ducing refinements with wR p of approximately 4% (be-tween 2- 450K). Contrary to the observation for higher x compositions, we find no evidence of incommensurabilityin the NPD data[5].The crystallography of the P nmm structure has beenpublished elsewhere[3, 15], however for clarity we illus-trate it in fig. 1 and remind the reader that here there aretwo symmetry in-equivalent sites for Na (labeled Na(1)and Na(2)) and two for Co (labeled Co(1) and Co(2)).The local symmetry for the Co(1) and Co(2) sites dif-fers in that the Na(1)-atom lies directly above or be-low the Co(1)-atom while the Na(2)-atom resides abovea space formed between CoO octahedra as illustrated infig. 1(b).The effect of the Na-ion potential correlates with thedistortion of the CoO octahedra. This distortion isshown in fig. 1(b) where we illustrated the differenceon the CoO -Na coordination and selected Co-O bondlengths. Here the Na-ion potential results in a distor-tion of the Co(1)O octahedron, so at 1.8K the six equalCo-O bonds found in metallic x = 0 . 75 distort to formthree long bonds (1.90-1.93 ˚A) and three shorter bonds(1.84-1.88 ˚A). In sharp contrast the Co(2)O octahedronis more regular with bond lengths varying between 1.88to 1.91 ˚A. This indicates that the Co(1) ions experienceessentially a different crystalline potential than the Co(2)ions, as a direct consequence of the Na ordering. Infig. 3(a,b) we show the temperature dependence of theCo-O bond lengths for the two Co sites. From these datait is evident that the larger distortion of the Co(1)O octahedron is maintained from low until high tempera-ture, while the spread of bond lengths for the Co(2)O issmaller and relatively temperature invariant. These oc- FIG. 3: (Color online) (a,b) Temperature dependence of Co-Obond lengths computed from Rietveld analysis of NPD datameasured on the E9 diffractometer.(c) Bond valence sums forthe Co(1) and Co(2) sites (filled circles) as a function of tem-perature computed from Co-O bond lengths determined fromRietveld analysis of the NPD data. The dashed line throughthe data is the average BVS for the two Co sites. BVS valuesobtained from ref. 15 are also shown. On the same figure weplot the octahedral distortion parameter ∆ for the CoO oc-tahedra centered in the Co(1) and Co(2) sites. Dashed linesare guides to the eye. tahedral distortions can be quantified by the parameter∆, where ∆ = q Σ( h Co − O i− ( Co − O ) i ) h Co − O i , h Co − O i is the av-erage Co-O bond length and the summation is done overthe 6 Co-O bonds. Here ∆ would be zero for a regularoctahedron with 6 equivalent Co-O bonds. We find thatthe Co(1)O octahedron is more distorted by a factor of3 at high temperatures (see fig. 3(c)), compared to theoctahedron centered on the Co(2) site. With decreasingtemperature however this difference increases to a factorof 5 while the distortion saturates below ∼ and Co -O bond lengths of 1.93and 1.83˚A respectively would suggest that there is no ev-idence for integer charge separation between Co(1) andCo(2) sites. The analysis of the experimentally deter-mined Co-O bond lengths using the bond valence sum(BVS) method allows us to estimate the difference inthe valance of the two Co-ions. For the calculation ofthe BVS we used b =0.37 and R o =1.70 [24][16]. TheBVS as a function of temperature for the two Co sites isshown in fig. 3(c). For high temperatures we find thatthe BVS shows some scatter that reflects changes in themobility and ordering of Na atoms between CoO sheets(see below) and possibly to geometrical differences thatarises from the Na-ion potential imposed on the CoO sheet over the same temperature range.[25] However, be-low ∼ ǫ ∼ e in the BVS for the two Co atoms.Although the difference is comparably smaller than whatis found in conventional charge ordered systems, the sep-aration of the data into two values (one low and one high)below 150K is statistically significant. These data wouldsuggest that below T C O =150K there is a separation ofcharge into Co . ǫ and Co . − ǫ stripes running along the b − axis as shown in fig. 1(a). This charge ordered struc-ture is in agreement with the magnetic neutron diffrac-tion measurements, where the magnetically active Co-site would correspond to the Co(1) site with the slightlyhigher BVS and octahedral distortion. The values of theBVS obtained are in good agreement with both theoreti-cal predictions[6] and recently reported values at 10 and300K respectively[15] which are also plotted on fig. 3(c)for comparison. The weak charge ordering found here isconsistent with the relatively low resistivity of this mate-rial at low temperatures ( ∼ m Ω cm at 2K)[2, 3, 8].That the average BVS is approximately 3.3 e reflects themixed valent nature of this compound.[26]. IV. NA ORDERING AND BEHAVIOR OF NA-OBOND LENGTHS We now turn our attention to the behavior of the Na-layer for this composition. In fig. 4 we plot the tempera-ture dependence of the atomic displacement parameters U iso (Debye-Waller factor) determined from our Rietveldanalysis. We find that the U iso values of the O- and Co-atoms to be in general of the expected amplitude andshow a linear behavior with temperature. The behaviorof U iso for the Na-ions however is unusual in that thereis a clear change in slope at 300K separating a low and FIG. 4: (Color online) Isotropic atomic displacementparameters ( U iso ) for the Na, Co and O-atoms determinedfrom the Rietveld analysis of the NPD data. In the analysisthe following constraints were used U iso (Co(1))= U iso (Co(2)), U iso (Na(1))= U iso (Na(2)), U iso (O(1))= U iso (O(2))= U iso (O(3)).Lines through the data are guides to the eye. A slope changein the U iso (Na) is evident around 300K. In the inset weshow the temperature dependence of the (111) reflectionin the orthorhombic P nmm setting. This reflection is asuperlattice reflections with respect to the parent P6 /mmc crystal structure and arises from the ordering of Na ions.The temperature T s ∼ x CoO and Co O . a high temperature behavior, while for T > U iso values for Na become large. Such behavior is indicative ofa dynamical transition occurring at 300K involving onlythe motion of Na-ions, as similar signatures are absentfor the Co- and O-atoms[18]. Indeed such large vales of U iso suggest that Na-ions may become mobile betweenCoO layers above 300K.For higher temperatures we find that our sample de-composes at T s ∼ x CoO and Co O . Thistransition is quantified by tracking the intensity of the(111) reflection as shown in the inset of fig. 4. This re-flection is a superlattice reflection with respect to theparent P6 /mmc structure and arises from the orderingof Na-ions [3]. Our neutron powder data measured at475K indicate the loss of this and other superstructurereflections and a return to P6 /mmc symmetry with theaddition of Co O reflections.Within this perspective we now look more closely tothe temperature dependence of the Na-O bond lengthsshow in fig. 5(a,b). For both Na sites the Na-O bond FIG. 5: Temperature dependence of and Na-O bond lengthscomputed from Rietveld analysis of NPD data measured onthe E9 diffractometer. Dashed lines are guides to the eye. lengths show a set of short bonds ( ∼ ∼ a − axis. At 150K we find thatthe small charge disproportionation in the CoO layer isnot reflected in the Na-O bonds. Surprisingly however wefind that at T m a decrease of ∼ layer at T m . The nature of any coupling betweenchanges in the electronic state of the CoO and Na canarises from the orbital configuration of the Co-ion it-self. It is argued by Kroll et al. [19] that the edge-sharingCo O octahedra are compressed along the c − axis re-duces the point group symmetry to D d . The t g orbitalof the Co is split in D d as t g = a ′ g + e ′ g , givinga fully occupied e ′ g and a half filed a ′ g . The latter or-bital looks like a 3 d z − r orbital and points along the c − axis[19]. For the case of the higher valent Co(1)-ion(nominally Co ), this orbital would point in betweenthe O-atoms and towards the Na(1)-atom as shown inthe inset of fig. 5(a). Since the charge disproportionationhere is small each Co ions will have a similar electronicand orbital configuration.Therefore the orbital configuration of the Co-ions pro-vides a means to couple electronically the CoO and Nalayers. The nature of the coupling is electrostatic andwould arise from the occupation of the e ′ g orbital. Wewould expect that changes in the electronic configura-tion of the CoO sheet to be reflected also in the relative positions of the Na-ions as indicated by the Na-O bondlengths. More precisely changes in the electronic stateof Co should be clearest for the Na(1)-O bonds as theNa(1)-ion sits directly above (or below) a Co(1)-ion. Thesame argument would suggest a less pronounced effect forthe Na(2)-O bond lengths as the Na(2) ion resides aboveand between CoO octahedra. This is indeed reflected inthe bond length data where a strong response is foundin the Na(1)-O bonds and a less clear responce in theNa(2)-O bonds[27]. At T m we find no clear evidence ofchanges in the Na-O bond lengths. This is expected asthe changes in charge separation at this lower transitionare computed to be much smaller than those at T m [6]. V. ANOMALOUS BEHAVIOR OF LATTICECONSTANTS In fig. 6(a-b) we show the temperature dependenceof the lattice constants determined from Rietveld refine-ment of the NPD data. These data show a positive ther-mal expansion (TE) for the c − axis between 2 and 450K,but for the a − and b − axis we find an almost constant TEbetween 2 and 300K; here for T < − × − /K and 1 . × − /K for a and b respectively. Such small TE was also discussed inref. 15 for a much more limited number of temperaturesand smaller range in temperature. For T > U iso for Na.The behavior of the lattice constants for this x =0.46sample is in sharp contrast to our x =0.75 sample (shownin the inset fig. 6), were we find a positive TE for both a − and c − axes between 5 and 300K. Assuming that TE isdominated by acoustic phonons below 300K for Na x CoO materials and whose frequency is relatively invariant be-tween x =0.75 to x =0.46, the T − dependence of thelattice constants for the x =0.75 sample (see inset infig. 6(f)) can be used to quantify the difference in the in-plane TE between these two samples. Here we define theterm α = ( a ′ − a hex ) /a ′ where a ′ = ( √ a + b/ / a and b are the orthorhombic lattice constants of the x =0.46compound and a hex is the hexagonal lattice constant ofthe x = 0 . 75 compound.[28] For the higher temperaturedata (T > a hex was assumed to vary linearly withtemperature. Here α represents an additional temper-ature dependent contribution to the expected in-planelattice constants (as defined by a hex ) and its tempera-ture dependence in shown is fig 6(c). For T > α is near zero, although the data in this region are morelimited. However for T < α increases with decreas-ing temperature and reaches a value of ∼ × − %at 1.8K. The maximum value of α is ∼ × − % at150K, close to T C O . The decrease below this tempera-ture is due to an increase in the a hex for the x =0.75as seen in the inset of fig. 6(c). We speculate that the FIG. 6: (Color online) (a,b) Lattice constants and unit cellvolume (inset in panel (a)) obtained from the Rietveld refine-ment of NPD data measured on HRPD at ISIS (filled circles)and E9 (open circles) over the temperature range of 2 to 450K.In panel (a) and inset the solid lines represent the thermalexpansion obtained from a fit to the data using the secondorder Gr¨uneisen approximation.(c) Temperature dependenceof the additional component to the thermal expansion α forNa . CoO obtained by subtracting the temperature behav-ior of the in-plane lattice constant of the x =0.75 compound.In the inset of panel (c) we show lattice parameters measuredas a function of temperature from a hexagonal x =0.75 sam-ple. anomalous TE below 300K may be driven by electroniccorrelations induced by Na-ions. VI. DISCUSSION The measurement we present here suggest a picture ofincipient charge ordering for near half doped Na x CoO .At high temperatures the ordering of Na-ions defines twodifferent CoO octahedra, one that is relatively undis-torted and one that more distorted. On cooling thedifferences in octahedral distortions between these twodifferent Co-sites becomes larger and may reflect an in-creasing influence of the Na-ion potential on the CoO sheet, as Na ions become more localized around their mean crystallographic positions. Indeed the near zeroTE for the in-plane lattice constants coincides with a dy-namical transition in the displacement parameter U iso of the Na-ion. Further the physical meaning of α canbe interpreted as a measure of the Na-induced electroniccorrelations onto the CoO layer which in this view sat-urate at T C O =150K.While our NPD work can correlate the distortion of theCoO octahedra and the Na-ordering even at high tem-peratures, it is not until T C O that we find evidence for aweak charge separation into stripes. Indeed it is possiblethat this pattern of charge ordering is present in the lat-tice from the onset as a direct result of the Na-orderingbut it in effect is hidden by the dynamic behavior ofthe Na-ions. Therefore we argue that the charge order-ing emerges at low temperatures as Na-motion becomesmore confined.More critically in terms of the physics of these ma-terials we demonstrate that charge ordering occurs at ahigher temperature that the magnetic ordering and elec-tronic transitions at T m and T m respectively. This isconsistent with recent theoretical models that suggestthat the Na potential imposes a degree of charge or-dering to the lattice[6, 7, 14]. Indeed Zhou and Wang[6], suggest that as much as half of the expected chargedisproportionation would occur at a temperature above T m [29], consistent with our observations.At lower temperature the changes in the CoO layermay be inferred indirectly by monitoring the Na(1)-Obonds. Here the orbital configuration of the Co pro-vides for charge density pointing directly to the Na(1)-ions thus providing a sensitive parameter to electronicchanges in the CoO layer. Indeed changes in the Na-O bond lengths may be more sensitive than changes inCo-O bonds as e g axial orbitals are empty. Our mea-surements find that that changes in the Na(1)-O bondlengths correlate with the magnetic transition at T m suggestive of further changes in the electronic state ofthe CoO layer. It is predicted that charge separationis enhanced gradually below T m [6] but the changes hereare overall again small and may fall outside the limits ofour sensitivity. At T m we find no evidence of changesin the lattice or changes in the lattice symmetry. Theprediction of a modulation of the amplitude of antiferro-magnetically coupled spins as well as the charge within aCo +3 . ǫ stripe is much smaller than our detection limit( ∼ e )[6].The structural observations at T C O we report here cor-relate with features in the charge dynamics. For exampleQuian et al. [20] report from ARPES measurements thatwith increasing T from the insulating region (were a cleargap is found) the size of the gap and the spectral weightaround the gap decrease. Although the gap closes at T m the spectral weight does not completely vanish until ap-proximately 120K, a behavior that is attributed to theformation of quasiparticles that gain significant weightdue to coupling along the c − axis. For similar temper-atures optical spectroscopy measurements find a broadfeature that is associated with fluctuating charge order-ing or a CDW in both anhydrous[21, 22] and hydrated su-perconducting samples[23]. These observation togetherwith our structural measurements point towards a pic-ture where at 150K an incipient charge ordering forms. VII. SUMMARY In summary this work establishes that ( a ) the Na-ordering on the CoO layer is reflected in the octahedraldistortion of the two crystallographically in-equivalentCo-sites and is evident even at high temperatures; ( b )The charge ordering occurs below T C O =150K, a tem-perature higher than the magnetic ordering found at T m =88K, consistent with theoretical models that sug- gest that the Na potential imposes a degree of chargeordering to the lattice[6, 7, 14]; ( c ) Below T C O we find aweak charge ordering into stripes of Co . ǫ and Co . − ǫ with a ǫ ∼ e , a value in good agreement with thatobtained from a Hubbard model using the Gutzwillerapproximation[6]; ( d ) A dynamic transition in the mo-tion of Na-ions occurs at 300K and coincides with theonset of a near zero thermal expansion for the in-planelattice constants of our Na . CoO sample. Acknowledgments The authors thank P.G. Radaelli, and L.C. Chapon forhelpful discussions and W.S. Howells for assistance in thecollection and reduction of the HRPD data. [1] Y. Wang, N. P. Ong, N. S. Rogado, and R. J. Cava,Nature , 425 (2003).[2] M. L. Foo, S. Watauchi, R. J. Cava, Y. Wang, N. P.Ong, H. W. Zandbergen, and T. He, Phys. Rev. Lett. , 247001 (2004).[3] Q. Huang, J. W. Lynn, B. H. Toby, M. L. Foo, R. J. Cava,H. W. Zandbergen, G. Lawes, Y. Wang, N. P. Ong, andA. P. Ramirez, Journal of Physics Condensed Matter ,5803 (2004),[4] K. Takada, H. Sakurai, E. Takayama-Muromachi,F. Izumi, R. A. Dilanian, and T. Sasaki, Nature ,53 (2003).[5] M. Roger, D. J. P. Morris, D. A. Tennant, M. J.Gutmann, J. P. Goff, J. U. Hoffmann, R. Feyerherm,E. Dudzik, D. Prabhakaran, A. T. Boothroyd, et al., Na-ture , 631 (2007),[6] S. Zhou and Z. Wang, Phys. Rev. Lett. , 226402 (2007),[7] C. A. Marianetti and G. Kotliar, Phys. Rev. Lett. ,176405 (2007),[8] G. Gasparovic, R. A. Ott, J.-H. Cho, F. C. Chou, Y. Chu,J. W. Lynn, and Y. S. Lee, Phys. Rev. Lett. , 046403(2006),[9] P. Mendels, D. Bono, J. Bobroff, N. Blanchard, H. Al-loul, I. Mukhamedshin, F. Bert, G. Collin, D. Colson,A. Amato, et al., Phys. Rev. Lett. , 136403 (2005).[10] M. Yokoi, T. Moyoshi, Y. Kobayashi, M. Soda, Y. Yasui,M. Sato, and K. Kakurai, Journal of the Physical Societyof Japan , 3046 (2005),[11] D. N. Argyriou, H. N. Bordallo, B. J. Campbell, A. K.Cheetham, A. Dos Santos, D. E. Cox, J. S. Gardner,K. Hanif, and G. F. Strouse, Phys. Rev. B , 15269(2000).[12] J. P. Wright, J. P. Attfield, and P. G. Radaelli, Phys.Rev. Lett. , 266401 (2001).[13] J. Bobroff, G. Lang, H. Alloul, N. Blanchard, andG. Collin, Phys. Rev. Lett. , 107201 (2006),[14] T.-P. Choy, D. Galanakis, and P. Phillips, Phys. Rev. B , 073103 (2007), [15] A. J. Williams, J. P. Attfield, M. L. Foo, L. Viciu, andR. J. Cava, Phys. Rev. B , 134401 (2006),[16] N. E. Brese and M. O’Keeffe, Acta Crystallographica Sec-tion B , 192 (1991).[17] B.J. Campbell H.T. Stokes D.E. Tanner and D.M. HatchJ. Appl. Cryst. , 607-614 (2006).[18] Q. Huang, B. Khaykovich, F. C. Chou, J. H. Cho, J. W.Lynn, , and Y. S. Lee, Physical Review B , 134115(2004),[19] T. Kroll, A. A. Aligia, and G. A. Sawatzky, PhysicalReview B (Condensed Matter and Materials Physics) ,115124 (2006),[20] D. Qian, L. Wray, D. Hsieh, D. Wu, J. L. Luo, N. L.Wang, A. Kuprin, A. Fedorov, R. J. Cava, L. Viciu, et al.,Phys. Rev. Lett. , 046407 (2006),[21] N. L. Wang, D. Wu, G. Li, X. H. Chen, C. H. Wang, andX. G. Luo, Phys. Rev. Lett. , 147403 (2004),[22] J. Hwang, J. Yang, T. Timusk, and F. Chou, Phys. Rev.B , 024549 (2005).[23] P. Lemmens, K. Y. Choi, V. Gnezdilov, E. Y. Sherman,D. P. Chen, C. T. Lin, F. C. Chou, and B. Keimer, Phys.Rev. Lett. , 167204 (2006),[24] These values correspond to Co -O bonds. Reliable val-ues of Co -O bonds are not available[25] Overall the validity of the BVS method may not hold inthe case of dynamic effects. At high temperatures Na ismobile which is reflected in a large Debye-Waller factor( U iso ∼ × − ˚ A ) at around 300K which decreasesmothly and rapidly to values similar as those found forCo and O ( U iso ∼ × − ˚ A ) below 100K.[26] The average BVS value lower than 3.5 reflect the absenceof reliable parameters for Co . A similar BVS numberare noted in ref. 15[27] The Na(2)-O response may arise from the repulsion ofNa-ions[28] Here a ′ was normalized to equal a hexhex