Static and dynamic structure in solid state ionics

Transcription

1 Research Signpost 37/661 (2), Fort P.O., Trivandrum , Kerala, India Physics of Solid State Ionics, 2006: ISBN: Editors: Takashi Sakuma and Haruyuki Takahashi 11 Static and dynamic structure in solid state ionics T. Sakuma 1, K. Basar 1, T. Shimoyama 1, D. Hosaka 1, Xianglian 1 and M. Arai 2 1 Institute of Applied Beam Science, Ibaraki University, Mito , Japan 2 Japan Atomic Energy Research Institute, Tokai, Ibaraki , Japan Abstract The static and dynamic structure in solid state ionics by X-ray diffraction, anomalous X-ray scattering (AXS), neutron diffraction and neutron scattering measurements were discussed. The structure and phase transition of typical solid state ionics were explained. The high temperature phase of cation conductors showed disordered arrangements of atoms with large anharmonic thermal vibrations. The expression of the diffuse scattering intensity including the correlations among the thermal displacements of atoms was given and applied to disordered and ordered structures of solid state ionics. The main Correspondence/Reprint request: Dr. T. Sakuma, Institute of Applied Beam Science, Ibaraki University, Mito , Japan. sakuma@mx.ibaraki.ac.jp

2 324 T. Sakuma et al. contribution to the oscillatory part of the diffuse scattering was from the correlation among the thermal displacements of nearest neighboring cation and anion. The low-energy excitations in crystalline solid state ionics was investigated by neutron inelastic scattering measurements. The relation between the excitation energy and the mass of the cations was discussed. Anomalously large thermal vibration of atoms was vital to realize the static and dynamic structure in solid state ionics. 1. Introduction Solid state ionics exhibit high ionic conductivities at fairly low temperatures below their melting points [1-9]. There are numerous practical applications of solid state ionics to fuel cells, gas sensors, solid-state batteries, energy storage devices and electrolyzers. The high conductivity of solid state ionics has been understood with a disordered structure of atoms. In this article we review the static and dynamic structure in solid state ionics by X-ray diffraction [10,11], AXS [12], neutron diffraction and neutron scattering methods [13,14]. There have been so many works reported on this subject that the works introduced in this paper was limited mainly to those on the substances having simple structures. In section 2, the theoretical treatment of scattering intensity of solid state ionics by X-ray and neutron diffraction measurement is discussed. The treatment of the diffuse scattering intensity including the correlations among the thermal displacements of atoms could be applied to the background function in Rietveld analysis. In section 3, typical crystal structures of solid state ionics are introduced; the structures of the highest temperature phases of AgI, Ag 2 Se and Ag 3 SI having a famous bcc structure of α-agi type and copper ion conductors Cu 2 Se and CuI having a fcc type structure are shown. New results of the structures for copper halide-chalcogen compounds are explained. In section 4, the oscillatory diffuse scattering from ordered and disordered crystals is discussed. The theoretical expression of the diffuse powder scattering intensity in section 2 was applied to α-agi, β-ag 3 SI and α-cu 2 Se and other solid state ionics. The temperature dependence of the oscillatory diffuse scattering was explained by the correlation effect between the thermal displacements of atoms. The inelastic neutron scattering is an effective method to clarify the dynamical properties of the conduction ions. The low-energy dispersionless excitations [15-17] which are characteristic to solid state ionics have been explained in section 5. The relation between the excitation energy and the mass of the cations has been discussed.

3 Static and dynamic structure in solid state ionics Theory of scattering intensity from solid state ionics The structure factor F in a crystal is expressed as: (1) where f is a scattering factor in the case of X-ray diffraction and shown as f and f '' are X-ray anomalous scattering terms. f = f + f ' + i '' [18-21]. ' 0 f A coherent scattering intensity which corresponds to Bragg line intensity I Bragg and diffuse scattering intensity I diffuse are shown as: (2) (3) where k is a function depending on the experimental conditions. In the case of X-ray diffraction k includes the polarization factor. G is Laue function and F n defined as the deviation of the structure factor at the nth site from the mean structure factor, namely. (4) Diffuse scattering is due to the disordered structure and the thermal vibration of atoms. F s ( i ) at s(i) site is defined as: f s(i) is either f i or 0 in the case of disordered arrangement of atoms. The Debye- Waller factor exp { M i } is equal to exp { B ( sinθ / λ) } 2 i. The probability of finding the atom in any site p i is equal to the ratio of the number of the i atoms Q = Q is equal to 4πsinθ/λ. The to the number of the i sites in the crystal. ( ) (5)

4 326 T. Sakuma et al. deviation from an equilibrium position is shown by r. When we calculate the { } diffuse scattering intensity, the thermal average of exp i.( j k) written as: Q r r is The correlation among the thermal displacements of atoms as follows; λ jk (6) is defined At high temperature crystalline superionic conductors have a disordered structure in which the number of available atomic sites is greater than that of atoms. The diffuse scattering intensity I diffuse from a powder samples including the correlations among the thermal displacements of atoms is expressed as follows; (7) (8) Z is the number of sites belonging to the s'th j type neighbor around a sth r s ( i ) s '( j ) S is equal to ( ) Qr i type site. r sin Qr. The probability p i of finding the atom in any site is equal to the ratio of the number of i atoms to the number of the sites in the crystal. Two sites s(i) and s'(j) are apart by the distance r. The probability function α r gives the probability of finding an atom at a site apart by a distance r from a site occupied by an atom, and β r the probability of finding an atom at a site apart by r from a vacant site. u i corresponds to the number of i atoms per unit cell. The prime added to the summation symbol

5 Static and dynamic structure in solid state ionics 327 means to omit the term of r s( i) s'( j) = 0. The equation (8) can be applied as a profile shape function of a background in Rietveld analysis. The expression of the diffuse scattering intensity can be used in ordered and disordered arrangements of atoms (table 1). Table 1. Atomic sites and values of and 3. Phase transition and crystal structure The highest temperature phases of AgI, Ag 2 S and Ag 3 SI having a famous α-agi type have been widely investigated [22-32]. The α-agi type structure has body centered cubic symmetry with the space group Im3m. The cations are distributed randomly over many sites among the anion bcc lattice. The number of cations in the unit cell are two for α-agi and α-cubr, three for α-ag 3 SI, and four for α-ag 2 S and α-ag 2 Se, respectively. AgI, CuI, CuBr and CuCl exhibit the transition to the β-phase from the zincblende-type γ-phase. The low temperature phase of solid state ionics is often of tetrahedral structure such as zincblende or wurtzite which have an ordered structure of cations. The high temperature phases of solid state ionics α-cu 2 Se, α-cuagse and α-ag 2 Te show fcc lattice with disordered arrangement of the cations [33-35]. The intermediate phase of Ag 3 SI belongs to disordered structure and that of CuBrTe to ordered structure. The most difficult point in the structure analysis of the intermediate phase of solid state ionics is to distinguish whether it belongs to ordered or disordered structure. To resolve this problem the electrical conductivity measurement is valuable. 3.1 Ag + ion conductors Above 420 K AgI transforms from the ordered β-phase having wurtzite structure (P6 3 mc) to the disordered α-phase. Many structural studies have been reported on the disordered arrangement of cations in the α-agi since the famous Strock s model was proposed in 1934 [22]. The X-ray and neutron diffraction data of Bragg lines of α-agi were well explained by the structural

6 328 T. Sakuma et al. model in which two silver ions are distributed over 12(d) sites of the space group O 9 h Im3m with large asymmetric anharmonic thermal vibrations [25]. The 12(d) sites are thought to be the positions with the lowest potential energy for the cations. The site symmetry of 12(d) is 42m and the effective anharmonic potential is expressed as follows;. (9) The isotropic temperature factor was used for I atoms because the site symmetry of the iodine position is m3m which does not have third order term. The residence time of Ag in these sites would be longer than the flight time. A similar structural model in which the cations are distributed over 48(j) sites surrounding the 12(d) sites with isotropic temperature factors was also applied to explain the α-agi structure. It was reported that α-ag 2 Se had so-called averaged structure, that is, anions formed bcc lattice and cations were distributed with some probabilities to 6(b), 12(d) and 24(h) sites for the space group O 9 h Im3m. The intensity of Bragg lines of α-ag 2 Se was first explained by a structure model where liquidlike Ag distribution among the Se lattice was included [27]. The average of the structure factor is written as follows; (10) where ρ Ag ( r) is a average density distribution of Ag atoms. The total number of Ag atoms in a unit cell is 4. The unit cell is divided into n 3 meshes (n; integer). The following function for ρ Ag ( r) gave a fairly good result; (11) where d=(the distance between a meshed point r and its nearest anion sites)- (ionic radius of Se (1.98 Å)). The value of ρ Ag ( r) is zero in the space where Se atoms occupy. The average distribution of Ag atoms has bcc symmetry. For the case of n=16, the number of sites of Ag atoms where ρ ( r) is not zero is Ag

7 Static and dynamic structure in solid state ionics in a unit cell. If we choose ρ Ag ( r) =const, the agreement between the calculated and observed intensities of Bragg lines is not satisfactory. The room temperature β-phase of Ag 3 SI has a structure of simple cubic 1 perovskite type ( Oh Pm3m) in which two kinds of anions I and S occupy the cube corner and the body center, respectively, while three cations are distributed over 12(h) positions deviated about 0.5 Å from the face centered position toward every [100] directions (figure 1). The 12(h) positions occupied by Ag ions are the center of distorted tetrahedron formed by two S and two I ions and the site symmetry is mm2. At 519 K it transforms to the α-phase with a disordered arrangement of both anions and cations. Τhe β-phase was found to undergo a low temperature phase transition at 157 K and the lowest temperature γ-phase has a rhombohedral symmetry of the space group C 4 3 R3 in which both the anions and the cations show the ordered arrangements [30]. γ-ag 3 SI showed no evidence of a ferroelectric but the pyroelectricity has been found. The β to γ phase transition of Ag 3 SI may be caused by the condensation at zone boundary phonon mode. The α-phase of Ag 3 SI could be quenched to low temperature [36]. β-ag 3 SBr has a similar structure as that of β-ag 3 SI. However, β-ag 3 SBr does not show a phase transition to the α-phase but decomposes at high temperature. The structure of the γ-phase of Ag 3 SBr below 145 C is orthorhombic. γ-ag 3 SBr shows an antiferroelectric structure with a superstructure of the lattice periods twice along every main axes [31]. 5 The space group of Oh Fm3m with disordered arrangement of copper and silver ions was plausible for the high temperature phase of CuAgSe above 200 C. All copper and silver ions are statistically distributed on the 8(c) and 157K 519K : Ag : S : I : Ag/4 : (S+I)/2 Figure 1. Structure and phase transition of Ag 3 SI. 32(f) sites in the unit cell. The number of atoms and the atomic positions are as follows;

8 330 T. Sakuma et al. The atomic positions in the unit cell are illustrated in figure 2. The shortest interatomic distance between cation sites is ~1.56 Å. The path connecting two nearest neighboring sites of 8(c) and 32(f) could be suggested as an easiest channel of cation movement relevant to the fast ion diffusion in the high temperature phase of CuAgSe. Figure 2. Atomic arrangements in the unit cell of CuAgSe at 523 K. 3.2 Cu + ion conductors Copper iodide (CuI) exhibits phase transitions at 369 C (γ β) and 407 C (β α). The high temperature β- and α-phase are known as having a high ionic conductivity. The structure of β-cui belonged to the trigonal system with the space group C 1 3v P3m1 [37]. In regard to the arrangement of Cu atoms in the γ-phase, the structural models with the ordered and the disordered arrangement of cations have been given. In the ordered arrangement Cu atoms occupied 4(c) sites. In the disordered arrangement Cu atoms were statistically distributed over the 16(e) sites. The arrangement of Cu atoms in CuI was investigated by the AXS measurement at room temperature. The incident X-ray energies of 8.05, 8.30, 8.70 kev monochromated with the 220 reflection of germanium, 17.48, 21.00, 24.00, kev with the 440 reflection and 32.50, kev with the 660 reflection were used. The energy dependence for several Bragg

9 Static and dynamic structure in solid state ionics 331 lines was studied. The energy dependence in the observed intensities of 111, 200, 220, 311, 400, 331, 511, 440, 531, 600, 620, 533, 444, 551 and 642 Bragg lines was explained well by both the ordered and disordered arrangements of Cu atoms. However, the observed intensities for 222, 420 and 622 lines could be explained only by the ordered model. The cubic system with the ordered arrangement of copper atoms would be feasible for the crystal structure of γ- CuI at room temperature. The anomalous X-ray scattering method is a powerful tool for a research in the atomic distribution in solid state ionics. It would be interesting to resolve the structure of high temperature α-phase of CuI whether it has a ZnS structure or a CaF 2 structure. From the ordinary powder X-ray diffraction measurement of α-cu 2 Se the space group of fcc system O 5 h Fm3m with disordered arrangement of all copper ions was obtained [33]. Se atoms occupy 4(a) sites. All copper ions are statistically distributed on the two 32(f) sites in the unit cell. The path connecting two nearest neighboring sites of 32(f) (x=0.297) and 32(f) (x=0.471) is suggested as an easiest channel of cation movement. The AXS measurements with powder sample were made to confirm the crystal structure at the energies of about 300 ev and 25 ev away from the K-absorption edges of Cu (9.979 kev) and Se ( kev). In these energy regions, the scattering intensity indicates the distinct energy dependence, arising from the so-called anomalous dispersion phenomena. The energy dependence of I A (A=Cu or Se) is taken from the difference between the intensity at 300 ev away from the absorption edge of A atom and that at 25 ev away from the absorption edge. The signs of the energy dependence of I A for the following Bragg lines were recognized to be positive (+), although they are rather weak, compared with those of the 111, 220, 311 and 422 lines. 400, , 440, 531 and 620 at Cu edge 400, 331, , 440, 531 and 620 at Se edge On the other hand, the signs of the energy dependence in intensity for the 222 and 420 lines at Cu edge and the 200 line at the Se edge were found to be negative ( ). The signs of the energy dependence in intensity obtained by the AXS measurements agree with those of the model calculation based on the above crystal structure. Copper halide-chalcogen compounds exhibit essentially ionic conductivities that may be one or two orders of magnitude greater than those of the parent copper halides [38-42]. CuBrTe, for example, possesses a room temperature ionic conductivity of 10-5 (Ωcm) -1 compared with a CuBr conductivity of 10-7 (Ωcm) -1. Below room temperature new phase transitions in

10 332 T. Sakuma et al. CuBrTe, CuITe and CuITe 2 were observed at 230 K, 281 K and 286 K, respectively. In the low temperature phase silver halide-chalcogen compounds are found to exhibit an ordered arrangement of silvers [30, 31]. The new phases of copper halide-chalcogen compounds would be related to an ordered arrangement of copper atoms. The fast ion conductor CuBrTe exhibits structural phase transitions at 230 K and 351 K. The crystal structure of β-cubrte at room temperature belongs 24 to the orthorhombic system 2a 2b 3c with the space group D2h Fddd, where the crystal system of the α-phase is assumed as a a c. The phase diagram of CuBrTe-CuBrSe system was investigated. The electric conductivity vs. temperature curves of this system are shown in figure 3. The slope of the conductivity curves with the composition in the β-phase is large compared to that of α-phase. The large slope would be related to the ionic diffusion process through interstitial positions. The ordered structure of the β-phase was inferred from this conductivity measurement [43]. Atomic arrangements in β-cubrte are displayed in figure 4. Te atoms form an infinite helix along the c axis. The distance between Te-Te atoms is Å, which is almost the same as that in tellurium metal (2.86 Å). The tellurium helices are alternately right- and left- handed and related by mirror planes parallel to the c-axis. The bromine atoms form tetrahedra. These tetrahedra are dispersed among the tellurium helices. Figure 3. Conductivity versus composition parameter for CuBrTe x Se 1-x solid solutions at 223, 294 and 373 K.

11 Static and dynamic structure in solid state ionics 333 Figure 4. Crystal structure of β-cubrte. In silver halide-chalcogen compounds chalcogen atoms keep off other chalcogen atoms, and the nearest neighbors of the chalcogen atoms are metal atoms because of electric forces between them. In the case of Ag 3 SI, the formal charges of Ag, S and I are +1, 2 and 1, respectively. The sulfur atoms keep off other sulfur atoms in the silver halide-chalcogen. On the contrary, it is possible that the atomic types of the nearest neighbors of the tellurium atoms are tellurium atoms in the copper halide-chalcogen. The formal charges of Cu, Te and Br in CuBrTe are +1, 0 and 1, respectively. The high temperature tetragonal phase including a statistical distribution of Cu atoms in CuClTe can be maintained even at 10 K [44]. The space group 19 of CuClTe is D4h I4 1/amd. The crystal structure of CuClTe is shown in figure 5. A unit cell includes 16 Cu atoms, 16 Cl atoms and 16 Te atoms. There are 40 sites for 16 Cu atoms in the unit cell of CuC1Te; two 4-fold sets designated 4(a), 4(b) and two 16-fold sets designated 16(b) and 16(h). Temperature change of the values of Te-Cu 16(g) and Te-Te bond lengths in CuClTe is small. The tellurium atoms in CuClTe form infinite helical chains that are known in metallic tellurium. The inter atomic distances between Cu 4(b) and Cu 16(h) sites, Cu 16(h) and the first nearest Cu 16(h) sites, and Cu 16(h) and second nearest Cu 16(h) sites have a tendency to decrease with the increase of temperature. This shows that the volume of tetrahedron formed by four Cu 16(h) sites decreases although the volume of unit cell increases with temperature. At 10 K the volume of tetrahedron which surrounds 4(b) site (2.874 Å 3 ) is greater than that of the tetrahedron for 4(a) site (2.332 Å 3 ). The volume of tetrahedron for 4(b) site (1.464 Å 3 ) is almost same as that of 4(a) site (1.430 Å 3 ) at 633 K. From the distances between neighboring atoms Cu atoms would primarily diffuse through the route 4(b) 16(h) 4(a) 16(h) 4(b) sites along the c-axis at high temperature.

12 334 T. Sakuma et al. As the crystal structure of CuClTe belongs to tetragonal system, the electrical conductivity would show anisotropic properties. c b 10 K 633 K a Figure 5. Crystal structure of CuClTe. It had been reported that the crystal structure of CuITe above 281 K belonged to the tetragonal system with the space group I4 1 /amd and the copper atoms were in disordered arrangement. The low temperature phase belonged to the orthorhombic system. Recently the phase transition and the crystal structures of CuITe were investigated by differential thermal analysis and X- ray powder diffraction measurements in a high temperature region. A new phase transition in CuITe was observed at 592 K [45]. In the room temperature phase some Bragg reflections which can not be explained by the space group I4 1 /amd were observed. We need to investigate the crystal structure of the room temperature phase. The new phase above 592 K belongs to tetragonal system with the space group I4 1 /amd. The existence of a similar high temperature phase transition at 592 K in CuITe has been reported in CuBrTe. The space group and the detailed crystal structure of CuBrSe 3 were derived by the Rietveld method with X-ray powder diffraction measurement. The obtained results show that the structure of CuBrSe 3 belongs to the orthorhombic system with the space group Pnc2. If the space group of 6 CuBrSe 3 is C2v Pnc2, its pyroelectricity and the piezolelectricity are expected to be observed along the 001 direction. Their measurements with single crystals are desirable. The selenium atoms in CuBrSe 3 form infinite helical chains. The nature of a bond between copper and bromine atoms would be ionic, and that between selenium atoms covalent.

13 Static and dynamic structure in solid state ionics Diffuse scattering The detailed study of the diffuse scattering of solid state ionics was started by CuKα X-ray radiation on the α-agi type compounds which had disordered arrangements of cations. After several works it is found that the oscillating diffuse scattering consists of two major parts, one is the disorder scattering and the other is the thermal scattering. Most of the intensities of the oscillatory diffuse scattering corresponds to the thermal scattering except the first peak. We could expect to observe an oscillatory diffuse scattering even from ordered crystals due to the correlation effect between the thermal displacements of atoms. In order to inspect the existence of the oscillatory diffuse scattering, X- ray and neutron diffraction measurements of ordered crystals, AgBr, CuX (X=Cl, Br, I) and GaAs were performed. From these crystals the oscillating diffuse scattering were observed [47,48]. Rietveld analysis method is a fundamental technique of characterizing polycrystalline materials [49]. The profile-shape function for Bragg intensities in the Rietveld method was extensively modified to obtain better fits between observed and calculated patterns. The background function for Rietveld analysis was approximated by a finite sum of Legendre polynomials which had no physical meaning. Diffuse scattering has a lot of information about crystal structure and thermal vibration of atoms. By use of the diffuse scattering theory in the background function of Rietveld analysis new physical information could be obtained. 4.1 Diffuse scattering from disordered crystals An anomalously strong diffuse scattering from the α-agi type structure has been studied by X-ray and neutron scattering experiment. The second peak of the diffuse scattering in the α-phase of AgI is clearly observed with MoKα radiation. Contrary to this, the second peak in the diffuse background is not so clear with CuKα radiation. The theoretical treatment of the diffuse scattering including the correlations among the thermal displacements of atoms was applied to the analysis of these diffuse scattering of α-agi. The disordered distribution of two Ag atoms in the unit cell into 48(j) sites with isotropic thermal vibration of space group O 9 h Im3m was used in the calculation [18]. At present it is a difficult problem to calculate the diffuse scattering including anharmonic thermal vibrations. The calculated intensity of the second peak of the diffuse scattering at Q~5 Å -1 produce the characteristics by CuKα and MoKα. The first peak in the diffuse background of α-agi consists of the disorder scattering among silver atoms and the thermal scattering of silver and iodine atoms. Most of the intensities of the diffuse scattering in the range of Q greater than that of the first peak corresponds to the thermal scattering.

14 336 T. Sakuma et al. Neutron diffuse scattering data of the single crystal α-agi taken with the E 0 scan method were also collected. The feature of the observed intensity of the diffuse scattering is the existence of a strong intensity region that spreads around the 110 reciprocal lattice point. The calculation of the diffuse scattering intensity of α-agi single crystal with E 0 scan method was carried out based on the disordered distribution of two Ag atoms in the unit cell into 48(j) sites [46]. All the positional and thermal parameters for the E 0 scan method are the same as those for X-ray powder scattering. The qualitative feature of the observed intensity of α-agi single crystal is well explained by these parameters. The oscillatory diffuse scattering was observed even in β-agi where silver and iodine atoms show ordered arrangements. The thermal scattering was strong in the ordered β-phase and show an oscillatory scheme [50, 51]. The oscillation form is expressed as sin(qr)/qr. The hexagonal system with the space group P6 3 mc was applied to the calculation of diffuse scattering of β- AgI. The positions of the first and second peak of β AgI are almost the same as those of α-agi. The parameter B of Debye-Waller temperature factor of silver and iodine atoms increases with the temperature. The deviation from the linear increase of B Ag and B I at high temperature suggests the presence of the anharmonic effect. The large B values and the value of correlation parameter contribute to the oscillatory diffuse scattering intensity. The estimated diffuse scattering intensities of the correlation effect of Ag-I, Ag-Ag and I-I atoms are shown in figure 6. The value of the correlation λ A,B (A, B= Ag, I) was assumed to be 0.7. The estimated diffuse scattering intensity of Ag-I component corresponds to the oscillatory characteristic of the observed diffuse scattering of AgI. It is found that the main contribution to the observed oscillatory diffuse scattering is from the correlation among the thermal displacements of nearest silver and iodine atoms. Intensity (arb. unit) Q (Å -1 ) Figure 6. Estimated diffuse scattering intensities of the correlation effect of Ag-I, Ag- Ag and I-I atoms. The value of the correlation λ A,B (A, B= Ag, I) was assumed to be 0.7.

15 Static and dynamic structure in solid state ionics 337 Oscillatory diffuse scatterings were found in the α-, β- and γ-phase of Ag 3 SI (figure 7). The first, second and third peak of the diffuse scattering in the β-phase appear at Q~2.7, 4.6, and 6.4 Å -1, respectively. These positions in the β-phase are almost same as those in γ- and α-phase of Ag 3 SI and those of β- and α-agi. The calculation of the diffuse scattering intensity of β-ag 3 SI for a powder sample was based on the disordered distribution of Ag atoms into 12(h) sites with isotropic thermal vibration in the unit cell. Main contribution to the observed oscillatory diffuse scattering from β-ag 3 SI is from the correlation among thermal vibrations of the nearest-neighbor silver and iodine atoms. In order to confirm the correlation effects between the thermal displacement of atoms with an interatomic distance a computer simulation for the calculation of the value rs ( A) r would be useful. s'( B) The diffuse scattering intensity of α-cu 2 Se has been measured at the Cu K-absorption edge by AXS technique using synchrotron radiation. The measurements were made at the energies of about -300 ev and 25 ev away from the K-absorption edge of Cu (8.979 kev). The signs and the values of the difference in intensity I Cu (=I(Q, kev) I(Q, kev)) expected from the structural model with two 32(f) sites for Cu atoms are consistent with those obtained from AXS measurements. The crystal structure of α-cu 2 Se has been supported by the analysis of the energy dependence in diffuse scattering intensity obtained by AXS measurement. 80 α-phase 40 Intensity (arb. unit) β-phase γ-phase θ (deg.) Figure 7. Powder X-ray diffraction curves of Ag 3 SI with CuKα radiation.

16 338 T. Sakuma et al. The diffuse scattering intensity of high temperature phase of CuAgSe was investigated by CuKα X-ray powder diffraction measurements at 523 K. The wide and strong peak of the diffuse scattering appears around 2θ 35 o (figure 8). In addition to the peak, a very weak second peak of the oscillatory diffuse scattering could be observed around 2θ 75 o. The diffuse scattering of α-agi was discussed by the disordered arrangements of Ag and vacancy. In the case of CuAgSe, a more complicated theoretical treatment with disordered arrangement of Cu, Ag and vacancy is needed. There were two contributions to the oscillatory diffuse scattering, one was the short range order of cations and the other the correlation between thermal displacements of anion-cation and cation-cation Intensity (arb. unit) θ (deg.) Figure 8. Observed intensity of CuAgSe at 523 K by CuKα X-ray diffraction measurement. 4.2 Diffuse scattering from ordered crystals Rietveld refinements of a neutron powder diffraction pattern of γ-cui (figure 9) were performed by the two background functions; equation (8) with the correlation effect between the thermal displacements of atoms and the Legendre polynomials. The background function including the correlation between the thermal displacements of atoms gives somewhat lower R factors at 290 K than that with Legendre polynomials does. The background function including the correlation between the thermal displacements of atoms would be available in Rietveld refinements for the angle-dispersive diffraction data above room temperature. The X-ray measurements of AgBr were carried out by the double axis goniometer with an intrinsic pure germanium solid state detector and a multichannel analyzer system [1]. The incident X-ray energy of kev (λ=0.491 Å) monochromated with the 440 reflection of germanium was used.

17 Static and dynamic structure in solid state ionics 339 Intensity (arb. unit) CuI 290 K 8 K sin θ/λ ( -1 Å ) Figure 9. Diffuse neutron scattering intensity of CuI. The diffuse scattering of AgBr obtained is anomalously strong and has an oscillatory form with several peaks. The main contribution to the oscillatory diffuse scattering of AgBr was from the correlation among the thermal displacements of silver and bromine atoms. The observed neutron powder diffraction intensity of AgCl at 260 K is shown by thin solid lines in figure 10. Observed diffuse scattering has an oscillatory form and the peaks appear in the positions around Q~2.5, 3.5 and 5.5 Å -1. Diffuse scattering intensities are calculated with the correlation effect between the thermal displacements of atoms. The peaks of Q~3.3 and 5.0 Å -1 are expected to appear by thermal correlation between the first nearest neighboring atoms (Ag-Cl). The peak of Q~2.2 Å -1 is the influence of the correlation effect between the second nearest neighboring atoms (Ag-Ag, Cl- Cl). The three peaks observed in the diffuse scattering are produced by the superposition of these correlation effects as is shown in thick solid lines in figure 10. The value of correlation between the first nearest neighboring atoms is about 0.7 and that between the second nearest neighboring atoms about 0.5. If the values of correlation become less than 0.1, the influences on diffuse scattering are not observed. The value of thermal correlation decreases rapidly with the interatomic distance. PbF 2 is one of typical anion conducting solid state ionics [52-54]. The room temperature β-phase of PbF 2 was investigated by CuKα X-ray diffraction measurement at 15 K and 300 K. The obtained intensity difference between these temperatures are shown in figure 11. The difference of these intensities would be from the contribution of thermal vibration. There are three peaks in the figure; 2θ ~25 o, 50 o and 80 o. The number of peaks in the diffuse scattering of PbF 2 are greater than that of other solid state ionics by CuKα radiation.

18 340 T. Sakuma et al. Figure 10. Observed neutron powder diffraction intensity (thin solid lines) and calculated diffuse scattering intensity (thick solid lines) for AgCl at 260 K. Figure 11. Difference of X-ray diffraction intensities of PbF 2 between 15 K and 300 K. Usually the contribution to the oscillatory diffuse scattering is from the correlation effect among thermal vibrations of first nearest neighboring atoms. In the case of PbF 2 the scattering factor f F is very less than that of f Pb. Therefore the intensity from the contribution of first nearest neighboring atoms would be less than that of second nearest Pb-Pb atoms. It is found that the peak positions correspond to the interatomic distance of Pb and Pb by the estimation of the oscillating function sin(qr)/qr. The intensity difference between 300 and 15 K in figure 11 would be from the correlation effects of second nearest neighboring atoms.

19 Static and dynamic structure in solid state ionics Diffuse scattering from Li 2 SO 4 Li 2 SO 4 transforms to a face-centered cubic phase with space group 5 Oh Fm3m at 883 K [55-60]. This phase then persists up to its melting point at K. SO 4 ions are at the origin and the oxygen atoms in a state of rotational disordered around the sulphur atom. Most of Li + ions occupied the ±(1/4 1/4 1/4) positions. Anomalously strong diffuse scattering was observed for a lithium sulphate powder sample. The intensity of the diffuse scattering for Li 2 SO 4 at 908 K has been calculated including the correlation effects 2 between the thermal displacement of atoms and the reorientation of SO 4 ions. The major part of the observed diffuse scattering of Li 2 SO 4 was explained by the reorientational scattering and the thermal scattering of SO ions. 5. Neutron inelastic scattering Neutron scattering gives an information on the motion of atoms through space-time correlation. In order to study the relation between the mobile ions and the lattice vibration, the measurement of phonon is very important. However, there are various difficulties in the measurement of phonons because of the large anharmonicity in the solid state ionics. The typical phonon scattering measurement and peculiar dispersionless low energy excitations of solid state ionics are discussed. The theoretical treatment which could explain the normal phonons and dispersionless modes is not sufficient. 5.1 Phonon scattering Using a single crystal of Ag 3 SI, the temperature dependence of the phonon scattering was measured. At room temperature, the phonon dispersion relation for the TA mode toward [ζ00], [ζζ0] and [ζ ζζ] up to ζ ~0.2 was measured [61]. The optical phonons were highly damped and could not be observed. From the slope of the dispersion curve at the zone center for each TA mode, sound velocities are estimated as listed in table 2. These sound velocities are fairly small, suggesting the elastic constants of the β-ag 3 SI are also small. There are a lot of inelastic neutron scattering measurements which include temperature dependence of phonons in solid state ionics [6]. Table 2. Sound velocities (km/s) in Ag 3 SI at room temperature estimated from the dispersion curve. The subscripts 1 and 2 of T correspond to polarizations parallel to the [110 ] and [001] directions, respectively. 2 4

20 342 T. Sakuma et al. 5.2 Low-lying dispersionless excitation A remarkable feature of the inelastic scattering spectrum of powder samples of β-cu 2 Se is the existence of the low-lying dispersionless excitation near 3~4 mev over the measured range of Q. The low-energy dispersionless excitation of about 3~4 mev was also observed in the copper chalcogenides Cu 1.8 S, and Cu 1.8 Se. A low-lying dispersionless excitation near 3.4 mev was observed in the inelastic scattering spectra of a powder sample of CuI over the wide range of Q. As the temperature is lowered the excitation near 3.4 mev becomes clear and the intensity of the inelastic scattering spectra in the energy range mev decreases. The value of low-energy excitation is almost independent of temperature. The half-width Γ of the low-energy excitation mode increases drastically above 200 C. The half-width shows the degree of anharmonicity of thermal vibration, which is proportional to the inverse of the life time of the mode. As the thermal vibration becomes large, a mobile ion could easily diffuse over the barrier of the activation energy. The ionic conductivity of CuI increases above 200 C [62], indicating the strong anharmonic vibration of the atoms at high temperatures. Similar low-lying dispersionless excitation near 3.4 mev was also observed in CuITe as is shown in figure 12. In CuITe iodine atoms form tetrahedra. These tetrahedra are dispersed among the tellurium helices. 200 I (Q, ω ) (arb. unit) CuITe Q = 2.5 Å Energy Transfer (mev) Figure 12. The observed inelastic neutron scattering spectra of CuITe at room temperature. From the inelastic neutron scattering studies, the low-energy dispersionless excitations in addition to normal phonons have been observed in solid state ionics. The dynamic scattering function S(Q, ω) was derived from

21 Static and dynamic structure in solid state ionics 343 the observed spectra. From the incoherent scattering cross section we could obtain a generalized density of states G( Q,ω ) [13]. The model scattering function for the generalized density of states consists of two components: the low-energy excitation mode and the phonon modes mainly due to the transverse acoustic branch. The components of the density of state are approximated by the low-lying local vibrational mode at the lower-energy side, and the phonon modes mainly due to the acoustic branch at the higher-energy side; (12) (13) where ω l and Γ are the frequency and the half width of the local mode, respectively. g(ω) is the density of states due mainly to the acoustic branch which is damped quickly. The values of the energy excitations for Ag ions in α-agi type solid state ionics are about 2.0~3.0 mev [3]. The low-energy dispersionless excitation 3.4 mev would be caused by the local vibration of Cu ions. It is found that there is a relation between the excitation energy and the mass of cations; the value of the excitation energy increases as the atomic number of cations decreases. Such a relation could be qualitatively represented by E 1 1 / M, where E 1 and M are the excitation energy and the atomic weight of the cations, respectively. Assuming that the excitation energy of Ag is 2.6 mev, the excitation energies for Cu and Na are calculated as 3.4 mev and 5.7 mev with the above equation. These calculated values almost coincide with the observed values. The result that the excitation energy depends not on the lattice type but on the mass of the cation suggests a similarity in the local potential in several compounds. In the case of Ag ion conducting solid state ionics the excitation would be due to an isolated vibrational mode; AgI 4 structural unit existing in the AgI crystal. The high ionic conduction occurs in various kinds of amorphous or glassy electrolytes. If the short-range order in the typical superionic glasses is very alike with the crystalline material, the low-energy excitation of about mev due to an isolated vibrational mode of AgI 4 structural unit could be observed in the superionic glasses by a neutron scattering experiment. An excitation was observed near E=2 3 mev in the composite glass (AgI) 0.5 (AgPO 3 ) 0.5 [63]. The relation between the value of low-energy excitation mode and the mass of cations may be explained by the following models: (i) a Brownian

22 344 T. Sakuma et al. motion mode, (ii) an optical phonon mode and (iii) an ionic plasma model [64]. The value of the low-energy excitation mode is not sensitive to the crystal structures of solid state ionics. In the case of the Brownian motion model, it is not consistent with the observation because the height of the potential barrier in general depends on the given crystal structure. As we treat the low-energy excitation mode as a branch of optical phonon, the value of M has to be replaced by the reduced mass of the cation and the anion. Further studies of theoretical treatments including the temperature dependence of the low-energy excitation mode are expected. A low-lying dispersionless excitation near 1.8 mev was observed in the inelastic scattering spectra of anion conductor CsPbCl 3 and CsPbBr 3 over the wide range of Q. The value of low-energy excitation is almost independent of temperature. As the temperature is lowered, the excitation near 1.8 mev becomes clear and the intensity of the inelastic scattering spectra in energy around 1 mev decreases. If we apply the relation between the excitation energy and the conducting mass to anion conductor CsPbCl 3, the estimated excitation energy for Cl ion is 4.6 mev. This value does not agree with the observed value. If we adopt the mass of Cs or Pb ions, the excitation energy about 2.0 mev is estimated. The excitation might be due to an isolated vibrational mode of PbCl 6 structural unit. 6. Conclusion In silver halide-chalcogen compounds chalcogen atoms keep off other chalcogen atoms, and the nearest neighbors of the chalcogen atoms are metal atoms because of electric forces between them. In the case of Ag 3 SI, the formal charges of Ag, S and I are +1, 2 and 1, respectively. On the contrary, it is possible that the atomic types of the nearest neighbors of the tellurium atoms are tellurium atoms in the copper halide-chalcogen. The formal charges of Cu, Te and Br in CuBrTe are +1, 0 and 1, respectively. An oscillatory diffuse scattering was observed from the crystals which have ordered structures. Usually the contribution to the oscillatory diffuse scattering is from the correlation effect among thermal vibrations of first nearest neighboring atoms. In the case of PbF 2 the correlation effects of second neighboring atoms could be observed by X-ray diffraction measurement. From the slope of the dispersion curve of Ag 3 SI at the zone center for each TA mode, fairly small sound velocities were obtained. A low-energy dispersionless excitation of 3~4 mev and 2~3 mev was observed in the copper and silver ion conducting solid state ionics, respectively. The low-lying dispersionless excitation near 1.8 mev was observed in the inelastic scattering spectra of anion conductor CsPbCl 3 over the wide range of Q. The excitation would be due to an isolated vibrational modes in solid state ionics.

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