LATTICE DYNAMICS OF THE CUBIC-TETRAGONAL PHASE TRANSITION IN KNb03

LATTICE DYNAMICS OF THE CUBIC-TETRAGONAL PHASE TRANSITION IN KNb03

KNb03 continues t o be of a p a r t i c u l a r i n t e r e s t among f e r r o e l e c t r i c (FE) materials because it undergoes three successive phase t r a n s ~ t i o n s , whose mechanism is t h e object of many controversies. I n e l a s t i c neutron scattering measurements1 performed i n the tetragonal phase (T = 245OC) reveal the presence of a low-frequency E(T0) zone center mode together with an high anisotropy in the lowest phonon branches which can be related t o the existence of strong correlations along the <100> axes between t h e l a r g e motions associated t o the soft modes. Recent infrared r e f l e c t i v i t y measurements clearly show the softening of the lowest-frequency Flu phonon with decreasing temperature in the e n t i r e range of the cubic phase (425"C

Lattice-dynamical calculations are c a r r i e d out for the cubic (0; space group) and the tetragonal ( c : ~sp ace group) phases in order t o explain the main experimental features. The aim of t h i s paper is to show, f o r t h e f i r s t time, that the dispersion curves for both phases can be described s a t i s f a c t o r i l y by the same harmonic model.

Description of the model. - In t h i s work for the cubic phase we use the model developped by cowley3 for SrTi03 with axially symmetric short-range force constants A and B between nearest neighbours (K-0, Nb-0, 0-0). In addition we allow an anisotropy in the oxygen core-shell coupling constant k(O) as previously suggested by Migoni e t a l . for both SrTi03 and KTa03. Therefore the components of the tensor 5 (0) in the directions of neighbouring K and Nb ions are denoted k(0-K) and k(0-Nb) respectively. This leads to a 15 parameters model f o r t h e cubic phase (Table 1) .

Since the symmetry is lower in the tetragonal phase (Figure I ) , the number of parameters is, i n principle, larger than in the cubic phase. Using simple geometry arguments we are able to derive the parameters of the tetragonal phase from those of the cubic structure. Consequently, the short-range constants of the tetragonal phase are expressed as functions of the l a t t i c e parameters c and a, of the spontaneous ionic displacements 6, measured by ~ e w a t ~an,d of the s e t of parameters A and B, already defined i n the cubic phase. Furthermore, in the tetragonal phase, we distinguish two kinds of oxygen ions, 0(1) (located on the polar axis passing through the Nb ion) and 0(2) or 0(3) ( i n t h e plane perpendicular) . Consequently we allow the
parameter k (0(1) -Nb) to be different from the parameters k(0 -Nb) and k(0(3)-Nb) .



Results and discussion. - For the tetragonal phase, the dispersion curves are calculated with parameters adjusted to the results of i n e l a s t i c neutron1 and ama an^ scattering measurements. The experimental data are q u i t e s a t i s f a c t o r i l y reproduced by the model calculations (Fig.2) except for the A2 (TOhpranchnearthe zoneboundary This f a i l u r e may be due to the large anharmonic coupling between acoustic and s of toptic phonons in t h i s direction. The strong anisotropy in the lowest dispersion branches is interesting to notice together with an anticrossing and eigenvector exchange between optic and acoustic A2 branches, which take place approximately a t 1/3 of the Brillouin zone edge. The s o f t E(T0) phonon a t qZO is essentiallyckiaracterized by a large vibrationnal amplitude of the niobium ion relative to the oxygen ion located along the mode polarization direction. Moreover the Einstein o s c i l l a t o r response of t h i s mode in the whole Brillouin zone confirms the presence of dynamical correlation chains Nb-0-Nb directed along the [loo] and [010] axes1. This shows t h e v a l i d i -
ty of t h e l i n e a r chain model for describing the s o f t mode behaviour in perovskites, as emphasized by Bilz e t a1.7 for KTa03.



For the cubic phase, the phonon branches are obtained with the parameter values previously used in the tetragonal phase, except for the coupling k(0-Nb) (Table 1). In figure 2 the calculations are compared with the experimental data of Nunes e t a1.8 In addition to t h e l a r g e anisotropy in the phonon dispersion surface, we can also note the softening of the lowest Flu phonon with a r e l a t i v e l y s l i g h t change in the value of k(0-Nb). The other zone-centre modes are nearly insensitive to this variation.

At the cubic-tetragonal phase transition, therefore, the splitting and the frequency shift6 of these phonons depend only on the geometrical effect in the force constants. On the contrary, the large separation of the FE soft Flu phonon into a soft E component (1.55 THz) and a hardened A1 (8.35 Mz) essentially originates from a k (0 -Nb) value larger than k (0 -Nb) (Table 1) . (1) (2) All these results emphasize the role of the intraionic oxygen polarizability in the phase transition mechanism of KNb03. This polarizability is dynamically enhanced by the hybridization of oxygen 2p states with niobium 4d states9 in the directdn of the chain like coupling Nb-0-Nb. This leads to a softening of the lowest cubic Flu(TO) and tetragonal E(TO1 modes. At the phase transition, the disappearance of the [001] correlation due to the asymmetry of the Nb-0-Nb bound is related to the abrupt change in the value of k(0-Nb) along the polar direction, and therefore to the stabilization of the ferroelectric Al(T0) component.

In order to specify the behaviour of the oxygen polarizability, calculations with a model including the temperature dependence of the soft mode are actually in progress.

References :
M.D. Fontana, G Dolling, G.E. Kugel and C. Carabatos, Phys. Rev. *, 3850 (1979)
M.D. Fontana, G. Metrat, J.L. Servoin and F. Gervais, Fifth International Meeting on Ferroelectricity, submitted to Ferroelectrics (1981)
R.A. Cowley, Phys. Rev. E, 981 (1964)
R. Migoni, H. Bilz and D. BHuerle, Phys. Rev. Lett. 37, 1155 (1976)
A.W. Hewat, J. Phys. C 6, 2559 (1973)
M.D. Fontana, G.E. Kugel, G. Metrat and C. Carabatos, Phys. Stat. Sol. (b) -103, 211 (1981)
H. Bilz, A. Bussmann, G. Benedek, H. Biittner and D. Strauch, Ferroelectrics 25 , 339 (1980)
A.C. Nunes, J.D. Axe and G. Shirane, Ferroelectrics, -2, 231 (1971)
A. Bussmann, H. Bilz, R. Roenspiess and K. Schwarz, Ferroelectrics, 25, 343 (1980)

Pressure Dependence of Phonon Anomalies in Molybdenum

Pressure Dependence of Phonon Anomalies in Molybdenum

A collaborative group of researchers from Lawrence Livermore National Laboratory and the ESRF have been able to pin down the high-pressure lattice dynamics of the transition metal molybdenum by mapping its phonon energies under extremely high pressure. Using the inelastic X-ray scattering beamline ID28 at the European Synchrotron Radiation Facility (ESRF) and theoretical calculations, the team tracked the pressure evolution of a dynamical anomaly within molybdenum that has challenged scientists for over 40 years.

Much of the interest in the H-point phonon is derived from its anomalous increase in energy with increasing temperature. This observation stimulated numerous theoretical atte mpts to explain this strange behaviour. Changing the temperature or pressure is helpful in that it allows one to probe systems in different thermodynamic configurations. Indeed, the study of mater ials at high pressure is very useful for gaining insight into the nature of the chemical bonds in materials. Notably, the study of lattice dynamics at high pressures in general cannot be performed with neutrons due to the requirement of relatively large samples.

The group developed a new technique for preparing extremely small single Mo crystals of high crystalline quality [1]. These samples (40 micrometres in diameter by 20 micrometres thick) were placed into diamond anvil cells and taken to pressures as high as 40 GPa (400,000 atmospheres) to observe the evolution of the anomaly.

Fig. 1: A small molybednum single crystal loaded in the helium pressure medium. The photomicrograph was taken of the sample in situ at high pressure in the diamond anvil cell.

The researchers observed strong changes in the phonon dispersions at high pressure [2]. The most significant was a large difference in the Gruneisen parameter of modes at the H-point and those around q=0.65 along [001]. These differences lead to a dramatic decrease in the magnitude of the H-point anomaly with increasing pressure. Using theoretical codes developed to model molybdenum, the group showed that there is strong sensitivity of the H-point phonon on the electronic band structure. In fact, the decrease in the H-point anomaly required significant pressure induced broadening to match the experimental data. This implied a strong coupling between electronic states and phonons. With compression, the combination of an increase in the Fermi energy together with a broadening of the electronic states, leads to a significant decrease in this electron-phonon coupling. Thus, molybdenum becomes a much more 'normal' bcc metal at high pressures possibly explaining it's extraordinary stability in the bcc structure to pressures in excess of 400 GPa.

Fig. 2: Phonon dispersions in molybdenum at high pressure. The filled symbols show IXS data taken at 17 GPa at ID28, the open symbols are inelastic neutron scattering results at one atmosphere. Circles are longitudinal acoustic modes; squares transverse acoustic modes. Along [0] the triangles and squares show the two non-degenerate transverse acoustic modes TA[110]<-110> and TA[110]<001> respectively. The dashed lines show the calculations performed at one atmosphere, and the solid lines the calculations at 17 GPa.

References

[1] D.L. Farber, D. Antonangeli, C. Aracne, J. Benterou, High Pressure Research 26, 1 (2006).[2] D.L. Farber, M. Krisch, D. Antonangeli et al., Physical Review Letters 96, 115502 (2005).
Principal Publication and Authors
D.L. Farber (a), M. Krisch (b), D. Antonangeli (a,b), A. Beraud (b), J. Badro (a,c), F. Occelli (a), D. Orlikowski (d), Physical Review Letters 96, 115502 (2005).(a) Earth Science Division, Lawrence Livermore National Laboratory and Department of Earth Sciences, University of California (USA)(b) ESRF(c) Département de Minéralogie, Institut de Minéralogie et de Physique des Milieux Condensés, Institut de Physique du Globe de Paris, Université Paris (France)(d) Physics and Advanced Technology Directorate, Lawrence Livermore National Laboratory, California (USA)

Lattice dynamics of rubidium-IV, an incommensurate host-guest system

In recent years, a number of surprisingly complex crystal structures have been discovered in the elements at high pressures, in particular incommensurately modulated structures and incommensurate host-guest composite structures (see [1] for a review). The crystal structure of the high-pressure phase rubidium-IV shown in Figure 11 belongs to the group of incommensurate host-guest structures that have also been observed in the elements Na, K, Ba, Sr, Sc, As, Sb, and Bi. The structure comprises a framework of rubidium host atoms with open channels that are occupied by linear chains of rubidium guest atoms, and the periodicities of the host and guest subsystems are incommensurate with each other (i.e., they have a non-rational ratio). Although considerable progress has been made in determining the detailed crystal structures of the complex metallic phases at high pressure, little is known about their other physical properties, and the mechanisms that lead to their formation and stability are not yet fully understood.

Fig. 11: Inelastic X-ray scattering spectrum of Rb-IV at 17.0 GPa, with the scattering vector q = (0 0 3.2)h referring to the host lattice. The inset shows the composite crystal structure of Rb-IV with the rubidium host and guest atoms in blue and red, respectively.

We investigated the lattice dynamics in incommensurate composite Rb-IV by inelastic X-ray scattering (IXS) on beamline ID28. The focus was on the longitudinal-acoustic (LA) phonons along the direction of the incommensurate wavevector (parallel to the guest-atom chains). Calculations on simpler model systems predict these phonons to reflect the incommensurability most clearly. Phase IV of Rb is stable at pressures of 16 to 20 GPa at room temperature, and a high-quality single crystal of Rb-IV was grown in a diamond anvil high pressure cell. In the IXS experiment, the incident radiation was monochromatised at a photon energy of 17.8 keV, and two grazing-incidence mirrors focussed the X-rays onto the sample with a focal size of 25 x 60 µm. The spectrum of the scattered radiation was analysed by a high-resolution silicon crystal analyser to yield an overall energy resolution of 3 meV.

Figure 11 shows a typical IXS spectrum of Rb-IV along with its decomposition into the elastic line, the phonon excitation peaks and a constant background, which were obtained by least-squares fitting using the FIT28 software. From a series of IXS spectra collected for different momentum transfers Q, phonon dispersion curves were obtained as shown in Figure 12a. A central result of this study is the observation of two well-defined longitudinal-acoustic (LA)-type phonon branches along the chain direction. They are attributed to separate LA excitations in the host and the guest sublattices, which is a unique feature of an incommensurate composite crystal.

A series of dispersion curves was measured at different pressures, and from this the sound velocities of the host and guest excitations and their pressure dependences were determined (Figure 12b). While the absolute values of the sound velocities in the host and the guest are rather similar, their pressure dependences differ notably. A simple ball-and-spring model of Rb-IV with only one spring constant reproduces these observations semi-quantitatively. This suggests that the difference in the pressure dependences is determined largely by geometrical factors, i.e., by the spatial arrangement of the atoms rather than differences in the chemical bonding in the two subsystems.

There is only very weak coupling between the incommensurate host and the guest in Rb-IV, which raises a rather interesting question. Can the 1D chains of guest atoms in Rb-IV be considered a manifestation of the “monatomic linear chain” treated in solid-state physics textbooks to introduce the concepts of crystal lattice dynamics? The pressure dependence of the interatomic spacing in the guest-atom chains was measured in earlier structural studies and enables the spring constant in the linear chain model to be determined, and also its pressure dependence. On this basis, the sound velocity in the linear chains and its pressure dependence were modelled as shown in Figure 12b. The results are in excellent agreement with the IXS data for the guest-atom chains in the composite Rb-IV structure, which can thus be regarded as a manifestation of the monatomic linear chain model with regard to the LA phonons.

Principal publication and authors
I. Loa (a), L.F. Lundegaard (a), M.I. McMahon (a), S.R. Evans (a), A. Bossak (b), and M. Krisch (b), Phys. Rev. Lett. 99, 035501 (2007).(a) The University of Edinburgh (UK)(b) ESRF
Reference
[1] M.I. McMahon and R.J. Nelmes, Chem. Soc. Rev. 35, 943 (2006)

Referencia Bibliografica :

http://www.esrf.eu/UsersAndScience/Publications/Highlights/2008/HRRS/hrrs4

Lattice dynamics of lithium oxide

Lattice dynamics of lithium oxide

Abstract. Li2O fnds several important technological applications, as it is used in solid- state batteries, can be used as a blanket breeding material in nuclear fusion reactors, etc. Li2O exhibits a fast ion phase, characterized by a thermally induced dynamic disorder in the anionic sub-lattice of Li+, at elevated temperatures around 1200 K. We have car- ried out lattice-dynamical calculations of Li2O using a shell model in the quasi-harmonic approximation. The calculated phonon frequencies are in excellent agreement with the reported inelastic neutron scattering data. Thermal expansion, speci¯c heat, elastic con- stants and equation of state have also been calculated which are in good agreement with the available experimental data.
Introduction

Lithium oxide (Li2O) belongs to the class of superionics, which allow macroscopic movement of ions through their structure. This behavior is characterized by the rapid di®usion of a significant fraction of one of the constituent species within an essentially rigid framework of the other species. In Li2O, Li is the di®using species, while oxygens constitute the rigid framework [1,2].

This material finds several technological applications ranging from lightweight high power-ensity lithium-ion batteries to being a possible candidate for blanket material in future fusion reactors [3,4]. Li2O crystallizes in the anti-fuorite structure with a face-centered cubic lattice and belongs to the Fm3m (O5 h) space group [1,2], lithium being in the tetrahedral coordination. Like other fuorites [5], this also shows a decrease in the elastic constant C11 [6] with emperature around the transition the fast ion phase.In several other fuorites, the fast ion phase is characterized by a specific heat anomaly [7], a Schottky hump in the speci¯c heat. However, no such anomaly has been observed in Li2O [6,8,9].

This paper reports the lattice dynamics calculations done to understand dynamics of anti-fuorite Li2O. Lattice dynamics calculations have been done to calculate the phonon spectrum, specific heat and elastic constants of the oxide. These results are in very good agreement with the available experimental data [6,9{11].

Lattice dynamics calculations

Calculations have been carried out in the quasi-harmonic approximation using inter-atomic potential consisting of both long and short-range terms, using DISPR [12].The form of the potential is given below:

where a and b are empirical parameters [13], and a = 1822 eV and b = 12:364. Oxygen ions have been modeled using a shell model [13,14]. Group theoretical considerations classify the phonons in the entire Brillouin zone into the following representations:

The phonon dispersion relation at ambient conditions is given in figure 1. The zone center modes and the phonons in the entire Brillouin zone are in very good agreement with the available inelastic neutron scattering data [10,11]. The phonon density of states at ambient conditions along with the partial densities of lithium and oxygen is given in figure 2. Both lithium and oxygen contribute almost in the entire Brillouin zone. Lithium's contribution is higher on the higher energy side, with a prominent peak in the region between 50 and 75 meV. Oxygen contribution is greater on the lower energy side with prominent peaks below 60 meV. The specisic heat, CP(T) can be calculated from the knowledge of the phonon density of states. The calculated ratio CP(T)=T has been compared with the ex-perimental result in ¯gure 3. The variation of the Debye temperature (µD) with temperature is given in ¯gure 4. Table 1 gives the calculated values of the elas-tic constants and equilibrium lattice parameters using the model calculations as compared with the experimental results.

Conclusions
A shell model has been successfully used to study the phonon properties of Li2O. The interatomic potential is able to reproduce the equilibrium lattice constant

elastic constants (except C12) and phonon frequencies, which are in unison with the experimental data [1,6]. The phonon dispersion in the entire Brillouin zone agrees well with reported experimental data. The calculated specific heat is in good agreement with the experimental data. The interatomic potential formulated for Li2O oxide may be transferred to other similar °uorites and anti°uorites like Na2O, K2O, UO2, ThO2 etc., with suitable modifications.

References
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