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Anion reorientation in an ion conducting plastic crystal – coherent quasielastic neutron scattering from sodium ortho-phosphate
Physica B 266 (1999) 60—68
Anion reorientation in an ion conducting plastic crystal — coherent quasielastic neutron scattering from sodium ortho-phosphate D. Wilmer *, K. Funke , M. Witschas , R.D. Banhatti , M. Jansen, G. Korus, J. Fitter, R.E. Lechner Institut fu( r Physikalische Chemie, Schlo}platz 4/7, D-48149 Mu( nster, Germany Institut fu( r Anorganische Chemie, Domagkstr. 1, D-53121 Bonn, Germany Hahn-Meitner-Institut, Glienicker Str. 100, D-14109 Berlin, Germany
Abstract A number of crystalline solids with polyatomic anions like SO\ or PO\ have high-temperature modifications which are both cation conductors and plastic phases, i.e., their anions exhibit dynamic rotational disorder. During the past decade, there has been a vigorous discussion about a possible cation mobility enhancement by dynamic coupling of cation migration and reorientation of the polyanions. Proponents of a strong interaction have coined the term “paddle—wheel” mechanism. We have performed a quasielastic neutron scattering study to investigate rapid anion reorientation in the high-temperature phase of sodium ortho-phosphate, a-Na PO , a plastic-phase fast-ion conductor. Here, the quasielastic scattering is dominated by oxygen scattering and is, therefore, almost entirely coherent. In the examined Q range, the quasielastic linewidth does not depend on Q, indicating the localized character of the observed motion. Our experimental results are compared with the predictions of four different models for the quasielastic part of the coherent scattering function for the oxygen atoms, S (Q, u). According to our results, the reorientational dynamics can be described with a single time constant which is thermally activated with 0.184 eV. Our analysis shows that only three oxygen atoms per anion are participating in the rapid reorientational motion. 1999 Elsevier Science B.V. All rights reserved. PACS: 78.70.Nx; 66.10.Ed Keywords: Coherent quasielastic neutron scattering; Paddle—wheel mechanism; Rotator phases; Fast ion conduction
1. Introduction
* Corresponding author. Tel.: #49-251-832-3436; fax: #49251-832-9138; e-mail: [email protected].
A number of crystalline solid electrolytes with polyatomic anions like SO\ or PO\ have high temperature modifications which are both good cation conductors and plastic phases, i.e., the
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 4 9 4 - X
D. Wilmer et al. / Physica B 266 (1999) 60—68
anions are rotationally disordered and perform some kind of rotational diffusion. In 1973, the unusually high diffusivity of large cation impurities in a-Li SO (the “prototype” of plastic-phase fast-ion conductors) led Kvist and Bengtzelius to assume that the transport of cations is significantly enhanced by a dynamic coupling to the reorientational motion of the translationally fixed polyatomic anions [1]. There is an ongoing debate between the proponents of this “paddle— wheel” mechanism who emphasize the importance of a dynamic interplay of anions and cations [2—6] and those who have been favoring an explanation in terms of a percolation-type mechanism [7—11]. In order to determine the relevance of the dynamic coupling in plastic-phase fast-ion conductors, we examine, for the first time, the dynamics of cations and anions in entirely dynamic experiments. In this paper, we present the results of a coherent quasielastic neutron scattering study probing oxoanion reorientation in the hightemperature phase of Na PO . Sodium ortho-phosphate has been chosen for this study as it exhibits its phase transition at 598 K, some 250 K below the transition temperature of Li SO . The high-temperature modifica tion (a-Na PO ) is both a plastic phase and a good sodium ion conductor [12,13]. The orientationally disordered PO\ anions form an FCC lattice in which the sodium cations occupy all the tetrahedral and octahedral interstices [12]. a-Na PO exhibits a considerable cationic conductivity (about 10\ )\ cm\ at 600 K). Since anionic rotational disorder and high cationic mobility are both observed above the phase transition temperature, it has been suggested that the rotational motion of the translationally fixed tetrahedral PO\ anions enhances the sodium ion transport [14]. Due to the complete occupation of all available cation sites, percolationtype mechanisms are hard to visualize, while a “paddle—wheel” effect may not appear improbable.
2. Experimental Powder samples of pure Na PO were obtained by solid state reaction of Na CO and Na P O [13]. Sample purity was confirmed by X-ray dif-
61
fraction. Samples were kept dry during all stages of handling and measurement. QNS experiments were performed at the coldneutron time-of-flight spectrometer NEAT [15], located at the Berlin Neutron Scattering Center (Hahn-Meitner-Institut, Germany). The powder was contained in a closed ceramic sample holder made of Alsint (99.7% Al O ). The sample was placed at an angle of 45° to the incoming beam. Spectra have been taken at nine temperatures between 300 and 973 K. In a first experiment, we used an incident wavelength of 5.1 As which resulted in an elastic resolution of about 100 leV (FWHM), slightly varying in the Q range of 0.3 up to 2.3 As \. In order to extend the accessible Q range up to about 4 As \, we have performed a second experiment at j "2.9 As with a lower elastic resolution ranging from 270 to 360 leV (FWHM). The data were normalized and corrected for selfabsorption, detector efficiency and detailed balance using the standard NEAT program package for data analysis. Q ranges exhibiting Bragg peaks have been excluded from further analysis.
3. Results In this section, we will first concentrate on the experimental data obtained from the high-resolution measurements using a neutron wavelength of 5.1 As . The low-resolution (i.e., low-wavelength) data are much more influenced by Bragg peaks (especially at higher Q) both from the sample itself, the sample holder and the furnace. Due to the lack of a radial collimator, the data contain multiple Bragg scattering and are, therefore, more complicated to analyze. In the high-resolution experiment, only three Bragg peaks were detected (two from the sample, one from the sample holder) which could easily be removed from the data set. Fig. 1 shows the corrected total scattering S(Q,u), for Q"2.04 As \ at 613 and 473 K, i.e., above and below the phase transition. In the low-temperature phase, the scattering is purely elastic. Its line shape corresponds to a d-function, broadened by the instrumental resolution. In the a-phase, quasielastic energy broadening is an essential feature of the observed scattering.
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D. Wilmer et al. / Physica B 266 (1999) 60—68
Fig. 1. The neutron total scattering function, S(Q, u), at " 112° (Q"2.04 As \), above and below the phase transition temperature.
Here, the high-resolution spectra consist of two contributions: 1. Incoherent elastic scattering from sodium. The elastic intensity decreases slightly with increasing Q. On the timescale of our experiment (selected by the energy resolution of about 100 leV) the sodium hopping motion is too slow to be distinguished from the elastic scattering. 2. Quasielastic scattering. According to the values of the scattering cross-sections in Na PO , the quasielastic component is dominated by coherent oxygen scattering, reflecting the reorientation of the phosphate ions. Up to 2.3 As \, the quasielastic intensity generally increases with increasing wave-vector transfer. The high-resolution spectra from a-Na PO are well fitted by a sum of a d-function and a single Lorentzian (representing the quasielastic scattering), both broadened by the instrumental resolution function. Fig. 2 is a logarithmic plot of the quasielastic line width versus inverse temperature at Q"2.16 As \. Evidently, the line width increases with temperature in an Arrhenius-type fashion, the activation energy being (0.184$0.006) eV. The Lorentzian line width does not depend on Q, indicating the localized character of the observed rotational motion.
Fig. 2. Arrhenius plot for the quasielastic neutron scattering linewidth, expressed as q\, measured at Q"2.16 As \. The activation energy is (0.184$0.006) eV.
The low-resolution data exhibit an additional contribution whose width is broader than about 2 meV FWHM at all temperatures in the a-phase. This became evident when we tried to fit the data to a resolution-convoluted sum of a d-function and a Lorentzian whose width had been fixed to the value found in the previous experiment. We, therefore, allowed for a second (broader) Lorentzian in the fit. The elastic intensity decreases slightly with increasing Q, while the weight of the fixed-width Lorentzian increases up to a maximum at about 3 As \ with a tendendy to decrease at higher Q. The intensity of the broader line increases monotonically in the whole Q range of the present experiment.
4. Discussion The characteristic features of the neutron-scattering results confirm that we are, indeed, monitoring the dynamic rotational disorder in a-Na PO . In the following, we present model calculations for different reorientational modes of tetrahedral anion motion in a cubic environment. Comparison of the model predictions with our experimental results will then provide a view of phosphate reorientation in space and time. Most investigations of molecular reorientations in solids involve hydrogen nuclei for which the
D. Wilmer et al. / Physica B 266 (1999) 60—68
scattering is predominantly incoherent. Decisions between alternative models are usually made on the basis of the elastic incoherent structure factor. Oxygen, however, is an almost entirely coherent scatterer. Therefore, anion reorientation in aNa PO does not determine the Q-dependent elastic intensity. Different modes of reorientational motion have to be discerned by examining the Q-dependent intensity of the quasielastic line. In the present study, we restrict ourselves to correlations of oxygen atoms of the same phophate unit, assuming that correlations between different anions may be neglected. A similar procedure has already proved successful in other coherent studies of rotationally disordered solids [16,17]. Therefore, all models presented here consider the time-dependent behavior of individual phosphate units in order to describe the quasielastic scattering. An X-ray structural investigation of a-Na PO using a quenched single crystal found one of the P—O bonds preferentially oriented along 11 0 02, while the scattering density of the remaining three oxygen atoms was smeared out [12]. This indicates a circular rotational motion involving three oxygen atoms. Within the temperature range of a-Na PO , we do, however, also consider the possibility of isotropic continuous rotational motion on a sphere and jump diffusion around the C axis (90° jumps) with four mobile oxygen ions.
63
and 2 a "(2m#1)j (QR) ) P (cos h ). (2) K K K IJ I J Here, j , P , D and h denote spherical Bessel K K 0 IJ functions, Legendre polynomials, the rotational diffusion constant and the angle between atoms k and l, respectively. R"1.56 As is the radius of the sphere, viz. the P—O bond length [12]. Calculation of the intensity factors a (Q) shows that in the K Q range of the present study only the terms with m"3, need to be considered (see Fig. 3). Moreover, the m"4 component is of minor importance (at least for the high-resolution data). 4.2. Model (ii): rotational jump diffusion around the tetrahedral C -axis This model assumes that all P—O bonds of a phosphate unit are preferentially oriented along the 11 1 12 directions of the cubic lattice, i.e., they are directed towards four of the eight nearest tetrahedral sodium ions (Fig. 4). Here, the anions may perform reorientational 90° jumps around their C -axes between only two distinct configurations (in contrast to the incoherent case, see Refs. [19—21]). Again, all oxygen atoms are mobile and contribute to the quasielastic scattering.
4.1. Model (i): isotropic rotational diffusion (four mobile ions) In the framework of this model, individual phosphate units without preferred orientation undergo rotational diffusion about a fixed center. Since all oxygen ions are involved in the reorientation, they all contribute to the quasielastic scattering. The model was first developed by Sears [18], see also Ref. [17]. The quasielastic part of the coherent neutron scattering, S (Q, u), is written as q 1 K S (Q, u)" a ) ) with K p 1#uq K K q\"m(m#1)D K 0
(1)
Fig. 3. Quasielastic intensity factors for the different models presented in the text.
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D. Wilmer et al. / Physica B 266 (1999) 60—68
#2 cos[(Q !Q )b]#2 cos[(Q #Q )b] V X V X #2 cos[(Q !Q )b]#2 cos[(Q #Q )b], W X W X (7) where b"2R/(3 is the length of the jump vector. If the anion changes from A to B, we have
Fig. 4. The two possible configurations for the tetrahedral anion in model (ii). b is the length of the jump vector and corresponds to the length of the cube.
Denoting the time-dependent probabilities to find the tetrahedron in a certain configuration by P (t) and P (t)"1!P (t) with P (R)" P (R)", we can consider (without loss of gener ality) the situation where the polyanion is in configuration A at time t"0, i.e., P (0)"1: 1 2 1 PQ (t)" [!P (t)#P (t)]" !P (t) . (3) q q 2
Here, q\ denotes the jump rate from one configuration into the other. Solving this rate equation yields an expression for the probability of finding a certain configuration at time t, if the anion was in the same configuration at time 0:
1 1 2t P(A, t; A, 0)"P(B, t; B, 0)" # exp ! . 2 2 q
(4)
Analogously, we get
2t 1 1 P(A, t; B, 0)"P(B, t; A, 0)" ! exp ! . q 2 2
(5)
In order to obtain the intermediate scattering function
exp[iQ(R (t)!R (0))] , (6) I J I J we consider two possible situations. If the polyanion was in configuration A at time 0 and stays there at t, we obtain F(Q, t)"
2
f (Q)"f (Q)"4#2 cos[(Q !Q )b] V W #2 cos[(Q #Q )b] V W
f (Q)"fH (Q)"4 cos(Q b)#4 cos(Q b) V W # 4 cos(Q b) X # exp[#i(Q #Q #Q )b] V W X # exp[!i(Q #Q !Q )b] V W X # exp[#i(Q !Q !Q )b] V W X # exp[!i(Q !Q #Q )b]. V W X
(8)
Performing the orientational average then leads to f (Q)"4#12j ((2Qb),
(9)
f (Q)"12j (Qb)#4j ((3Qb), (10) where j (x)"sin(x)/x. The intermediate scattering function can now be written as
1 2t F(Q, t)" 1#exp ! 2 q
1 ) +f (Q)#f (Q), 2 (11)
1 2t # 1!exp ! 2 q
1 ; + f (Q)#f (Q), 2
(12)
and, after rearrangement, 1 F(Q,t)" [ f (Q)#f (Q)] 2
(13)
1 2t # [ f (Q)!f (Q)]exp ! . 2 q
(14)
Fourier transformation yields the dynamic structure factor, 1 q/2 S(Q,u)"a (Q) ) d(u)#a (Q) ) ) , (15) p 1#u(q/2)
D. Wilmer et al. / Physica B 266 (1999) 60—68
65
with
times t and 0, respectively. We obtain
a (Q)"2 ) +1#3j (Qb)#3j ((2Qb)#j ((3Qb), (16)
p 2 ( p F(Q, t)" d
d ) P( , ; t) ) P( ) I J 2p ;exp iQR sin h cos #k ) !u J
and a (Q)"2 ) +1!3j (Qb)#3j ((2Qb)!j ((3Qb),. (17)
2p !cos #l ) !u J
,
(20)
For the phosphate ions in Na PO , b"1.80 As . In the accessible Q range of the present study, the quasielastic structure factor (i.e., a (Q)) and the m"3 contribution of model (i) are nearly identical, see Fig. 3.
where h is the angle between Q and the rotator axis z and u is the angle between x and Q, see Fig. 5. Using
4.3. Model (iii): rotational jump diffusion on a circle (three mobile ions, N equally distributed orientations)
we can rewrite Eq. (20) as
In this case, we consider one of the P—O bonds to be the fixed axis of a circular motion performed by the remaining three oxygen atoms. The model describes their jump diffusion on a circle of radius R "sin h ) "P—O""0.943 ) "P—O", where h is the tet rahedral angle. The accessible phosphate orientations are distributed among N equally spaced positions, i.e., the jump angle is 2p/N. Our derivation here is a coherent extension of the incoherent treatment by Barnes [22], see also Refs. [23—25]. For J mobile coherent scatterers, equally distributed on a circle and moving with a fixed phase relation, F(Q,t) can be written as
2 ( F(Q, t)" 1exp[iQ(R (t)!R (0))]2. I J I J
cos a!cos b"!2 sin[(a#b)/2]sin[(a!b)/2], (21)
p 2 ( p F(Q, t)" d
d P( , ; t) ) P( ) I J
#
;exp i ) 2QR sin h sin u! 2 !(k#l) )
p p
!
#(k!l) ) sin J J 2
.
(22) The orientational average over all possible Q orientations is evaluated by performing the integration
p 1 p F(Q, t)" sin h dh F(Q, t) du, 4p
(23)
(18)
The thermal average is expressed as a double integral,
2 ( F(Q, t)" dR dR exp[iQ(R (t)!R (0))] I J I J I J ;P(R , R ; t) ) P(R ), I J J
(19)
where P(R , R ; t) denotes the probability distribuI J tion for finding atom k at R at time t if atom l was I at R at time 0. P(R ) describes the initial distribuJ J tion. Now let and be the angular positions of the nucleus labeled J on a circle of radius R at
Fig. 5. Schematic representation of the coordinates used in model (iii).
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D. Wilmer et al. / Physica B 266 (1999) 60—68
which yields
p
F(Q, t)"
;j
d
p
d ) P( , ; t) ) P( )
2QR sin
p
!
#(k!l) ) 2 J
.
(24)
The solution for P( , ; t) for the rotation on a circle among N sites is [23] 1 ,\ P( , ; t)" exp(!t/q )exp[in( ! )] L N L > ) d( ! !2pp/N), (25) N\ where q . q" L 1!cos(2pn/N)
(26)
Here q denotes the mean time spent in a certain configuration. The initial distribution among the sites can be written as 1 ,\ P( )" d( !2pq/N). (27) 2pN O Inserting Eqs. (25) and (27) in Eq. (24) yields ,\ F(Q, t)" a (Q) exp(!t/q ), L L L with
(28)
pp p 1 2 ( ,\ #(k!l) a (Q)" j 2QR sin L J N N I J N 2pp ;cos n ) . (29) N
The dynamic structure factor is then obtained by Fourier transforming Eq. (28): ,\ q L S(Q, u)"a (Q) ) d(u)#p\ a (Q) . L 1#(uq ) L L (30) In the case of a-Na PO (with J"3), structural data [12] indicate that the C -axis of the anion coincides with the C -axis of the cubic cell (i.e. the 11 0 02 direction); this might suggest N"12. In this particular case, only terms with n"3 contribute in the Q range of the present study, with
q "q "q. Fig. 3 shows the result from a calcu lation for R "1.47 As which, in fact, coincides with the result from model (iv), see below. Closer inspection of the predictions of Eq. (29) shows, however, that with the exception of N"3 (where no quasielastic scattering is present), the coherent quasielastic structure factor does not noticably depend upon selection of a specific value for N. In contrast to the incoherent case, it is impossible to determine the jump angle from the quasielastic scattering.
4.4. Model (iv): continuous rotational diffusion on a circle (three mobile ions) As in the incoherent treatment, the continuous rotational diffusion is the limiting case of jump diffusion (model (iii)) for small jump angles, i.e., for large N and short q [23—25]. In this limit, the rotational diffusion constant is evaluated using (see [23]) 1!cos(2p/N) D " . 0 q
(31)
As mentioned before, model (iii) with N"12 and model (iv) both predict a single Lorentzian with the same Q-dependent intensity. In the accessible Q range of the high-resolution experiment (0.322.3 As \), all models predict single Lorentzians with intensity maxima at Q'2.5 As \. On the basis of the existing data, it is hardly possible to make a distinction between models (i) and (ii) or (iii) and (iv). There is, however, a clear difference in the absolute quasielastic intensity between the two sets of models due to the different number of oxygen atoms involved in the reorientational motion, see Fig. 3. For direct comparison with the measured quasielastic structure factors, the calculated intensity factors a (Q) are multiplied with an appropriate L scaling factor. For a purely coherent scatterer, one would not expect any elastic scattering with the exception of Bragg peaks; the disorder is regarded as entirely dynamic. Sodium, however, is a partly incoherent scatterer (p, "1.62 barn [25]) and its hopping motion is too slow to cause quasielastic
D. Wilmer et al. / Physica B 266 (1999) 60—68
67
Fig. 6. Measured quasielastic structure factors (613, 973 K) from the high-resolution measurement as compared to the models presented in the text. Model curves have been scaled according to the incoherent elastic scattering for QP0 at 773 K.
Fig. 7. Measured quasielastic structure factors at 773 K from the low-resolution measurement as compared to the models presented in the text. Model curves have been scaled according to the incoherent elastic scattering for QP0.
broadening. We, therefore, assume that the experimentally detected elastic intensity may be written as
a circular motion involving three oxygen atoms per anion. We may conclude that in agreement with the X-ray data of quenched samples of a-Na PO [12], the disorder and rotational reorientation of the phosphate ions is mainly represented by a circular motion of three oxygen atoms. This does not exclude other types of anion reorientation, which, if existing, are simply too slow to be detected in our study. Interestingly, both experiments exhibit additional quasielastic intensity at lower Q, viz., up to about 1.5 As \, see Figs. 6 and 7. Its position on the Q scale suggests that sodium ions, further away from the center of rotation, might be involved in the reorientational motion of some of the anions.
I (Q)"c ) 3 p, ) S (Q), (32) where S (QP0)"1 while the factor 3 reflects the ratio of cations to anions. From I (QP0) (found to be independent of temperature) we obtain the numerical value of c. If we now assume that the quasielastic scattering is essentially due to the reorientational motion of the polyanions, we may write for the quasielastic intensity I (Q)"c ) p- ) S (Q) (33) and thus scale our calculated S (Q), i.e., the inten sity factors a (Q), to the measured quasielastic L intensity. The results are presented in Figs. 6 and 7 for the high and low-resolution experiments, respectively. Considering that there is no adjustable parameter for the quasielastic structure factor, the agreement between models (iii), (iv) and the experimental data in Fig. 6 is surprising; models (i) and (ii) do not give a satisfactory fit. Fig. 7 shows the same trend for the low-resolution measurement: the quasielastic intensity is closer to the predictions of the models describing
5. Conclusion Coherent quasielastic neutron scattering has been used as a powerful tool for determining the dynamics and geometry of the anion reorientation in the plastic-phase fast-ion conductor a-Na PO . The predictions of four simple models for the quasielastic intensity as a function of wave vector transfer show that in the present case of a highly
68
D. Wilmer et al. / Physica B 266 (1999) 60—68
symmetric coherent scatterer, geometric details like jump angles are less easily accessible than in the incoherent case. Our experiments, however, show that in agreement with structural information from a quenched single crystal, anion reorientation in a-Na PO is restricted to a circular motion around one of the C -axes, performed by only three of the four oxy gen atoms. The reorientational anion dynamics can be described by a single time constant q which is thermally activated with an energy of 0.184 eV.
Acknowledgements Two of the authors (RDB, DW) are grateful for fellowships granted by the Alexander von Humboldt Foundation. Financial help from the Fonds der Chemischen Industrie is gratefully acknowledged.
References [1] A. Kvist, A. Bengtzelius, in: W. van Gool (Ed.), Fast Ion Transport in Solids, North-Holland, Amsterdam, 1973, p. 193. [2] A. Lunde´n, Solid State Ionics 28—30 (1988) 163.
[3] A. Lunde´n, Solid State Commun. 65 (1988) 1237. [4] A. Lunde´n, M.A.K.L. Dissanayake, J. Solid State Chem. 90 (1991) 179. [5] A. Lunde´n, J. Solid State Chem. 107 (1993) 296. [6] A. Lunde´n, Solid State Ionics 68 (1994) 77. [7] E.A. Secco, Solid State Ionics 28—30 (1988) 168. [8] E.A. Secco, Solid State Commun. 66 (1988) 921. [9] E.A. Secco, Solid State Ionics 45 (1991) 335. [10] E.A. Secco, J. Solid State Chem. 96 (1992) 366. [11] E.A. Secco, Solid State Ionics 60 (1993) 233. [12] D.M. Wiench, M. Jansen, Z. anorg. allg. Chem. 461 (1980) 101. [13] H. Hruschka, E. Lissel, M. Jansen, Solid State Ionics 28—30 (1988) 159. [14] M. Jansen, Angew. Chem. 103 (1991) 1574. [15] R.E. Lechner, R. Melzer, J. Fitter, Physica B 226 (1996) 86. [16] D.L. Price, M.-L. Saboungi, Phys. Rev. B 44 (1991) 7289. [17] D.A. Neumann et al., Phys. Rev. Lett. 67 (1991) 3808. [18] V.F. Sears, Can. J. Phys. 45 (1967) 237. [19] K.H. Michel, J. Chem. Phys. 58 (1973) 1143. [20] R.C. Livingston, J.M. Rowe, J.J. Rush, J. Chem. Phys. 60 (1974) 4541. [21] J. To¨pler, D.R. Richter, T. Springer, J. Chem. Phys. 69 (1978) 3170. [22] J.D. Barnes, J. Chem. Phys. 58 (1973) 5193. [23] A.J. Dianoux, F. Volino, H. Hervet, Mol. Phys. 30 (1975) 1181. [24] A.J. Leadbetter, R.E. Lechner, Neutron scattering studies, in: J.N. Sherwood (Ed.), The Plastically Crystalline State, Wiley, Chichester, 1979, p. 285—320. [25] M. Be´e, Quasielastic Neutron Scattering, Adam Hilger, Bristol, 1988.
Anion reorientation in an ion conducting plastic crystal — coherent quasielastic neutron scattering from sodium ortho-phosphate D. Wilmer *, K. Funke , M. Witschas , R.D. Banhatti , M. Jansen, G. Korus, J. Fitter, R.E. Lechner Institut fu( r Physikalische Chemie, Schlo}platz 4/7, D-48149 Mu( nster, Germany Institut fu( r Anorganische Chemie, Domagkstr. 1, D-53121 Bonn, Germany Hahn-Meitner-Institut, Glienicker Str. 100, D-14109 Berlin, Germany
Abstract A number of crystalline solids with polyatomic anions like SO\ or PO\ have high-temperature modifications which are both cation conductors and plastic phases, i.e., their anions exhibit dynamic rotational disorder. During the past decade, there has been a vigorous discussion about a possible cation mobility enhancement by dynamic coupling of cation migration and reorientation of the polyanions. Proponents of a strong interaction have coined the term “paddle—wheel” mechanism. We have performed a quasielastic neutron scattering study to investigate rapid anion reorientation in the high-temperature phase of sodium ortho-phosphate, a-Na PO , a plastic-phase fast-ion conductor. Here, the quasielastic scattering is dominated by oxygen scattering and is, therefore, almost entirely coherent. In the examined Q range, the quasielastic linewidth does not depend on Q, indicating the localized character of the observed motion. Our experimental results are compared with the predictions of four different models for the quasielastic part of the coherent scattering function for the oxygen atoms, S (Q, u). According to our results, the reorientational dynamics can be described with a single time constant which is thermally activated with 0.184 eV. Our analysis shows that only three oxygen atoms per anion are participating in the rapid reorientational motion. 1999 Elsevier Science B.V. All rights reserved. PACS: 78.70.Nx; 66.10.Ed Keywords: Coherent quasielastic neutron scattering; Paddle—wheel mechanism; Rotator phases; Fast ion conduction
1. Introduction
* Corresponding author. Tel.: #49-251-832-3436; fax: #49251-832-9138; e-mail: [email protected].
A number of crystalline solid electrolytes with polyatomic anions like SO\ or PO\ have high temperature modifications which are both good cation conductors and plastic phases, i.e., the
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 4 9 4 - X
D. Wilmer et al. / Physica B 266 (1999) 60—68
anions are rotationally disordered and perform some kind of rotational diffusion. In 1973, the unusually high diffusivity of large cation impurities in a-Li SO (the “prototype” of plastic-phase fast-ion conductors) led Kvist and Bengtzelius to assume that the transport of cations is significantly enhanced by a dynamic coupling to the reorientational motion of the translationally fixed polyatomic anions [1]. There is an ongoing debate between the proponents of this “paddle— wheel” mechanism who emphasize the importance of a dynamic interplay of anions and cations [2—6] and those who have been favoring an explanation in terms of a percolation-type mechanism [7—11]. In order to determine the relevance of the dynamic coupling in plastic-phase fast-ion conductors, we examine, for the first time, the dynamics of cations and anions in entirely dynamic experiments. In this paper, we present the results of a coherent quasielastic neutron scattering study probing oxoanion reorientation in the hightemperature phase of Na PO . Sodium ortho-phosphate has been chosen for this study as it exhibits its phase transition at 598 K, some 250 K below the transition temperature of Li SO . The high-temperature modifica tion (a-Na PO ) is both a plastic phase and a good sodium ion conductor [12,13]. The orientationally disordered PO\ anions form an FCC lattice in which the sodium cations occupy all the tetrahedral and octahedral interstices [12]. a-Na PO exhibits a considerable cationic conductivity (about 10\ )\ cm\ at 600 K). Since anionic rotational disorder and high cationic mobility are both observed above the phase transition temperature, it has been suggested that the rotational motion of the translationally fixed tetrahedral PO\ anions enhances the sodium ion transport [14]. Due to the complete occupation of all available cation sites, percolationtype mechanisms are hard to visualize, while a “paddle—wheel” effect may not appear improbable.
2. Experimental Powder samples of pure Na PO were obtained by solid state reaction of Na CO and Na P O [13]. Sample purity was confirmed by X-ray dif-
61
fraction. Samples were kept dry during all stages of handling and measurement. QNS experiments were performed at the coldneutron time-of-flight spectrometer NEAT [15], located at the Berlin Neutron Scattering Center (Hahn-Meitner-Institut, Germany). The powder was contained in a closed ceramic sample holder made of Alsint (99.7% Al O ). The sample was placed at an angle of 45° to the incoming beam. Spectra have been taken at nine temperatures between 300 and 973 K. In a first experiment, we used an incident wavelength of 5.1 As which resulted in an elastic resolution of about 100 leV (FWHM), slightly varying in the Q range of 0.3 up to 2.3 As \. In order to extend the accessible Q range up to about 4 As \, we have performed a second experiment at j "2.9 As with a lower elastic resolution ranging from 270 to 360 leV (FWHM). The data were normalized and corrected for selfabsorption, detector efficiency and detailed balance using the standard NEAT program package for data analysis. Q ranges exhibiting Bragg peaks have been excluded from further analysis.
3. Results In this section, we will first concentrate on the experimental data obtained from the high-resolution measurements using a neutron wavelength of 5.1 As . The low-resolution (i.e., low-wavelength) data are much more influenced by Bragg peaks (especially at higher Q) both from the sample itself, the sample holder and the furnace. Due to the lack of a radial collimator, the data contain multiple Bragg scattering and are, therefore, more complicated to analyze. In the high-resolution experiment, only three Bragg peaks were detected (two from the sample, one from the sample holder) which could easily be removed from the data set. Fig. 1 shows the corrected total scattering S(Q,u), for Q"2.04 As \ at 613 and 473 K, i.e., above and below the phase transition. In the low-temperature phase, the scattering is purely elastic. Its line shape corresponds to a d-function, broadened by the instrumental resolution. In the a-phase, quasielastic energy broadening is an essential feature of the observed scattering.
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D. Wilmer et al. / Physica B 266 (1999) 60—68
Fig. 1. The neutron total scattering function, S(Q, u), at " 112° (Q"2.04 As \), above and below the phase transition temperature.
Here, the high-resolution spectra consist of two contributions: 1. Incoherent elastic scattering from sodium. The elastic intensity decreases slightly with increasing Q. On the timescale of our experiment (selected by the energy resolution of about 100 leV) the sodium hopping motion is too slow to be distinguished from the elastic scattering. 2. Quasielastic scattering. According to the values of the scattering cross-sections in Na PO , the quasielastic component is dominated by coherent oxygen scattering, reflecting the reorientation of the phosphate ions. Up to 2.3 As \, the quasielastic intensity generally increases with increasing wave-vector transfer. The high-resolution spectra from a-Na PO are well fitted by a sum of a d-function and a single Lorentzian (representing the quasielastic scattering), both broadened by the instrumental resolution function. Fig. 2 is a logarithmic plot of the quasielastic line width versus inverse temperature at Q"2.16 As \. Evidently, the line width increases with temperature in an Arrhenius-type fashion, the activation energy being (0.184$0.006) eV. The Lorentzian line width does not depend on Q, indicating the localized character of the observed rotational motion.
Fig. 2. Arrhenius plot for the quasielastic neutron scattering linewidth, expressed as q\, measured at Q"2.16 As \. The activation energy is (0.184$0.006) eV.
The low-resolution data exhibit an additional contribution whose width is broader than about 2 meV FWHM at all temperatures in the a-phase. This became evident when we tried to fit the data to a resolution-convoluted sum of a d-function and a Lorentzian whose width had been fixed to the value found in the previous experiment. We, therefore, allowed for a second (broader) Lorentzian in the fit. The elastic intensity decreases slightly with increasing Q, while the weight of the fixed-width Lorentzian increases up to a maximum at about 3 As \ with a tendendy to decrease at higher Q. The intensity of the broader line increases monotonically in the whole Q range of the present experiment.
4. Discussion The characteristic features of the neutron-scattering results confirm that we are, indeed, monitoring the dynamic rotational disorder in a-Na PO . In the following, we present model calculations for different reorientational modes of tetrahedral anion motion in a cubic environment. Comparison of the model predictions with our experimental results will then provide a view of phosphate reorientation in space and time. Most investigations of molecular reorientations in solids involve hydrogen nuclei for which the
D. Wilmer et al. / Physica B 266 (1999) 60—68
scattering is predominantly incoherent. Decisions between alternative models are usually made on the basis of the elastic incoherent structure factor. Oxygen, however, is an almost entirely coherent scatterer. Therefore, anion reorientation in aNa PO does not determine the Q-dependent elastic intensity. Different modes of reorientational motion have to be discerned by examining the Q-dependent intensity of the quasielastic line. In the present study, we restrict ourselves to correlations of oxygen atoms of the same phophate unit, assuming that correlations between different anions may be neglected. A similar procedure has already proved successful in other coherent studies of rotationally disordered solids [16,17]. Therefore, all models presented here consider the time-dependent behavior of individual phosphate units in order to describe the quasielastic scattering. An X-ray structural investigation of a-Na PO using a quenched single crystal found one of the P—O bonds preferentially oriented along 11 0 02, while the scattering density of the remaining three oxygen atoms was smeared out [12]. This indicates a circular rotational motion involving three oxygen atoms. Within the temperature range of a-Na PO , we do, however, also consider the possibility of isotropic continuous rotational motion on a sphere and jump diffusion around the C axis (90° jumps) with four mobile oxygen ions.
63
and 2 a "(2m#1)j (QR) ) P (cos h ). (2) K K K IJ I J Here, j , P , D and h denote spherical Bessel K K 0 IJ functions, Legendre polynomials, the rotational diffusion constant and the angle between atoms k and l, respectively. R"1.56 As is the radius of the sphere, viz. the P—O bond length [12]. Calculation of the intensity factors a (Q) shows that in the K Q range of the present study only the terms with m"3, need to be considered (see Fig. 3). Moreover, the m"4 component is of minor importance (at least for the high-resolution data). 4.2. Model (ii): rotational jump diffusion around the tetrahedral C -axis This model assumes that all P—O bonds of a phosphate unit are preferentially oriented along the 11 1 12 directions of the cubic lattice, i.e., they are directed towards four of the eight nearest tetrahedral sodium ions (Fig. 4). Here, the anions may perform reorientational 90° jumps around their C -axes between only two distinct configurations (in contrast to the incoherent case, see Refs. [19—21]). Again, all oxygen atoms are mobile and contribute to the quasielastic scattering.
4.1. Model (i): isotropic rotational diffusion (four mobile ions) In the framework of this model, individual phosphate units without preferred orientation undergo rotational diffusion about a fixed center. Since all oxygen ions are involved in the reorientation, they all contribute to the quasielastic scattering. The model was first developed by Sears [18], see also Ref. [17]. The quasielastic part of the coherent neutron scattering, S (Q, u), is written as q 1 K S (Q, u)" a ) ) with K p 1#uq K K q\"m(m#1)D K 0
(1)
Fig. 3. Quasielastic intensity factors for the different models presented in the text.
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D. Wilmer et al. / Physica B 266 (1999) 60—68
#2 cos[(Q !Q )b]#2 cos[(Q #Q )b] V X V X #2 cos[(Q !Q )b]#2 cos[(Q #Q )b], W X W X (7) where b"2R/(3 is the length of the jump vector. If the anion changes from A to B, we have
Fig. 4. The two possible configurations for the tetrahedral anion in model (ii). b is the length of the jump vector and corresponds to the length of the cube.
Denoting the time-dependent probabilities to find the tetrahedron in a certain configuration by P (t) and P (t)"1!P (t) with P (R)" P (R)", we can consider (without loss of gener ality) the situation where the polyanion is in configuration A at time t"0, i.e., P (0)"1: 1 2 1 PQ (t)" [!P (t)#P (t)]" !P (t) . (3) q q 2
Here, q\ denotes the jump rate from one configuration into the other. Solving this rate equation yields an expression for the probability of finding a certain configuration at time t, if the anion was in the same configuration at time 0:
1 1 2t P(A, t; A, 0)"P(B, t; B, 0)" # exp ! . 2 2 q
(4)
Analogously, we get
2t 1 1 P(A, t; B, 0)"P(B, t; A, 0)" ! exp ! . q 2 2
(5)
In order to obtain the intermediate scattering function
exp[iQ(R (t)!R (0))] , (6) I J I J we consider two possible situations. If the polyanion was in configuration A at time 0 and stays there at t, we obtain F(Q, t)"
2
f (Q)"f (Q)"4#2 cos[(Q !Q )b] V W #2 cos[(Q #Q )b] V W
f (Q)"fH (Q)"4 cos(Q b)#4 cos(Q b) V W # 4 cos(Q b) X # exp[#i(Q #Q #Q )b] V W X # exp[!i(Q #Q !Q )b] V W X # exp[#i(Q !Q !Q )b] V W X # exp[!i(Q !Q #Q )b]. V W X
(8)
Performing the orientational average then leads to f (Q)"4#12j ((2Qb),
(9)
f (Q)"12j (Qb)#4j ((3Qb), (10) where j (x)"sin(x)/x. The intermediate scattering function can now be written as
1 2t F(Q, t)" 1#exp ! 2 q
1 ) +f (Q)#f (Q), 2 (11)
1 2t # 1!exp ! 2 q
1 ; + f (Q)#f (Q), 2
(12)
and, after rearrangement, 1 F(Q,t)" [ f (Q)#f (Q)] 2
(13)
1 2t # [ f (Q)!f (Q)]exp ! . 2 q
(14)
Fourier transformation yields the dynamic structure factor, 1 q/2 S(Q,u)"a (Q) ) d(u)#a (Q) ) ) , (15) p 1#u(q/2)
D. Wilmer et al. / Physica B 266 (1999) 60—68
65
with
times t and 0, respectively. We obtain
a (Q)"2 ) +1#3j (Qb)#3j ((2Qb)#j ((3Qb), (16)
p 2 ( p F(Q, t)" d
d ) P( , ; t) ) P( ) I J 2p ;exp iQR sin h cos #k ) !u J
and a (Q)"2 ) +1!3j (Qb)#3j ((2Qb)!j ((3Qb),. (17)
2p !cos #l ) !u J
,
(20)
For the phosphate ions in Na PO , b"1.80 As . In the accessible Q range of the present study, the quasielastic structure factor (i.e., a (Q)) and the m"3 contribution of model (i) are nearly identical, see Fig. 3.
where h is the angle between Q and the rotator axis z and u is the angle between x and Q, see Fig. 5. Using
4.3. Model (iii): rotational jump diffusion on a circle (three mobile ions, N equally distributed orientations)
we can rewrite Eq. (20) as
In this case, we consider one of the P—O bonds to be the fixed axis of a circular motion performed by the remaining three oxygen atoms. The model describes their jump diffusion on a circle of radius R "sin h ) "P—O""0.943 ) "P—O", where h is the tet rahedral angle. The accessible phosphate orientations are distributed among N equally spaced positions, i.e., the jump angle is 2p/N. Our derivation here is a coherent extension of the incoherent treatment by Barnes [22], see also Refs. [23—25]. For J mobile coherent scatterers, equally distributed on a circle and moving with a fixed phase relation, F(Q,t) can be written as
2 ( F(Q, t)" 1exp[iQ(R (t)!R (0))]2. I J I J
cos a!cos b"!2 sin[(a#b)/2]sin[(a!b)/2], (21)
p 2 ( p F(Q, t)" d
d P( , ; t) ) P( ) I J
#
;exp i ) 2QR sin h sin u! 2 !(k#l) )
p p
!
#(k!l) ) sin J J 2
.
(22) The orientational average over all possible Q orientations is evaluated by performing the integration
p 1 p F(Q, t)" sin h dh F(Q, t) du, 4p
(23)
(18)
The thermal average is expressed as a double integral,
2 ( F(Q, t)" dR dR exp[iQ(R (t)!R (0))] I J I J I J ;P(R , R ; t) ) P(R ), I J J
(19)
where P(R , R ; t) denotes the probability distribuI J tion for finding atom k at R at time t if atom l was I at R at time 0. P(R ) describes the initial distribuJ J tion. Now let and be the angular positions of the nucleus labeled J on a circle of radius R at
Fig. 5. Schematic representation of the coordinates used in model (iii).
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D. Wilmer et al. / Physica B 266 (1999) 60—68
which yields
p
F(Q, t)"
;j
d
p
d ) P( , ; t) ) P( )
2QR sin
p
!
#(k!l) ) 2 J
.
(24)
The solution for P( , ; t) for the rotation on a circle among N sites is [23] 1 ,\ P( , ; t)" exp(!t/q )exp[in( ! )] L N L > ) d( ! !2pp/N), (25) N\ where q . q" L 1!cos(2pn/N)
(26)
Here q denotes the mean time spent in a certain configuration. The initial distribution among the sites can be written as 1 ,\ P( )" d( !2pq/N). (27) 2pN O Inserting Eqs. (25) and (27) in Eq. (24) yields ,\ F(Q, t)" a (Q) exp(!t/q ), L L L with
(28)
pp p 1 2 ( ,\ #(k!l) a (Q)" j 2QR sin L J N N I J N 2pp ;cos n ) . (29) N
The dynamic structure factor is then obtained by Fourier transforming Eq. (28): ,\ q L S(Q, u)"a (Q) ) d(u)#p\ a (Q) . L 1#(uq ) L L (30) In the case of a-Na PO (with J"3), structural data [12] indicate that the C -axis of the anion coincides with the C -axis of the cubic cell (i.e. the 11 0 02 direction); this might suggest N"12. In this particular case, only terms with n"3 contribute in the Q range of the present study, with
q "q "q. Fig. 3 shows the result from a calcu lation for R "1.47 As which, in fact, coincides with the result from model (iv), see below. Closer inspection of the predictions of Eq. (29) shows, however, that with the exception of N"3 (where no quasielastic scattering is present), the coherent quasielastic structure factor does not noticably depend upon selection of a specific value for N. In contrast to the incoherent case, it is impossible to determine the jump angle from the quasielastic scattering.
4.4. Model (iv): continuous rotational diffusion on a circle (three mobile ions) As in the incoherent treatment, the continuous rotational diffusion is the limiting case of jump diffusion (model (iii)) for small jump angles, i.e., for large N and short q [23—25]. In this limit, the rotational diffusion constant is evaluated using (see [23]) 1!cos(2p/N) D " . 0 q
(31)
As mentioned before, model (iii) with N"12 and model (iv) both predict a single Lorentzian with the same Q-dependent intensity. In the accessible Q range of the high-resolution experiment (0.322.3 As \), all models predict single Lorentzians with intensity maxima at Q'2.5 As \. On the basis of the existing data, it is hardly possible to make a distinction between models (i) and (ii) or (iii) and (iv). There is, however, a clear difference in the absolute quasielastic intensity between the two sets of models due to the different number of oxygen atoms involved in the reorientational motion, see Fig. 3. For direct comparison with the measured quasielastic structure factors, the calculated intensity factors a (Q) are multiplied with an appropriate L scaling factor. For a purely coherent scatterer, one would not expect any elastic scattering with the exception of Bragg peaks; the disorder is regarded as entirely dynamic. Sodium, however, is a partly incoherent scatterer (p, "1.62 barn [25]) and its hopping motion is too slow to cause quasielastic
D. Wilmer et al. / Physica B 266 (1999) 60—68
67
Fig. 6. Measured quasielastic structure factors (613, 973 K) from the high-resolution measurement as compared to the models presented in the text. Model curves have been scaled according to the incoherent elastic scattering for QP0 at 773 K.
Fig. 7. Measured quasielastic structure factors at 773 K from the low-resolution measurement as compared to the models presented in the text. Model curves have been scaled according to the incoherent elastic scattering for QP0.
broadening. We, therefore, assume that the experimentally detected elastic intensity may be written as
a circular motion involving three oxygen atoms per anion. We may conclude that in agreement with the X-ray data of quenched samples of a-Na PO [12], the disorder and rotational reorientation of the phosphate ions is mainly represented by a circular motion of three oxygen atoms. This does not exclude other types of anion reorientation, which, if existing, are simply too slow to be detected in our study. Interestingly, both experiments exhibit additional quasielastic intensity at lower Q, viz., up to about 1.5 As \, see Figs. 6 and 7. Its position on the Q scale suggests that sodium ions, further away from the center of rotation, might be involved in the reorientational motion of some of the anions.
I (Q)"c ) 3 p, ) S (Q), (32) where S (QP0)"1 while the factor 3 reflects the ratio of cations to anions. From I (QP0) (found to be independent of temperature) we obtain the numerical value of c. If we now assume that the quasielastic scattering is essentially due to the reorientational motion of the polyanions, we may write for the quasielastic intensity I (Q)"c ) p- ) S (Q) (33) and thus scale our calculated S (Q), i.e., the inten sity factors a (Q), to the measured quasielastic L intensity. The results are presented in Figs. 6 and 7 for the high and low-resolution experiments, respectively. Considering that there is no adjustable parameter for the quasielastic structure factor, the agreement between models (iii), (iv) and the experimental data in Fig. 6 is surprising; models (i) and (ii) do not give a satisfactory fit. Fig. 7 shows the same trend for the low-resolution measurement: the quasielastic intensity is closer to the predictions of the models describing
5. Conclusion Coherent quasielastic neutron scattering has been used as a powerful tool for determining the dynamics and geometry of the anion reorientation in the plastic-phase fast-ion conductor a-Na PO . The predictions of four simple models for the quasielastic intensity as a function of wave vector transfer show that in the present case of a highly
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D. Wilmer et al. / Physica B 266 (1999) 60—68
symmetric coherent scatterer, geometric details like jump angles are less easily accessible than in the incoherent case. Our experiments, however, show that in agreement with structural information from a quenched single crystal, anion reorientation in a-Na PO is restricted to a circular motion around one of the C -axes, performed by only three of the four oxy gen atoms. The reorientational anion dynamics can be described by a single time constant q which is thermally activated with an energy of 0.184 eV.
Acknowledgements Two of the authors (RDB, DW) are grateful for fellowships granted by the Alexander von Humboldt Foundation. Financial help from the Fonds der Chemischen Industrie is gratefully acknowledged.
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