- Email: [email protected]
Derivation of the coherent and incoherent scattering laws associated with Na+ motions in β aluminas
Physica B 156 & 157 (1989) North-Holland, Amsterdam
DERIVATION ASSOCIATED
346-349
OF THE COHERENT AND INCOHERENT WITH Na+ MOTIONS IN p ALUMINAS
J.F. BOCQUET’,
A. CHAHID’,
‘Lab. chimie physique. ‘Institut Laue-Langevin,
CSP, Univ. Paris-Nord. Grenoble, France
The derivation of the coherent aluminas is presented; it is based calculations.
G. LUCAZEAU’
and A.J.
LAWS
DIANOUX’
France
and incoherent scattering laws associated with collective on a jump model in which jump rates were determined
1. Introduction The Na B aluminas are known as the best super ionic conductors and are used as solid electrolytes in Na/S batteries. Their two dimensional conductivity has stimulated a large number of theoretical and structural studies. This work is the continuation of the study of the dynamics of Na’ ions in B and B” aluminas [l-4]. The interpretations of our previous neutron scattering data [l] and [2] were based on the Skold incoherent approximation consisting in writing the coherent law from the incoherent one. Moreover no specific mode of diffusion for sodium was considered, although the fraction of mobile ions and the jump constants which were derived were in agreement with the so-called interstitial pair model [5]. Since, using atomatom potentials and dipole-induced dipole interactions, we have derived the relative variations of the crystal energy for different correlated Naf motions in B alumina [3]; however, these motions were limited to 1 or 2 cells, congruent displacements being generated in the whole crystal. Recently [6] we have shown that for two particles jumping over three sites the coherent intensity dominates generally and that for a pair (strong correlations) depending on their position relative to the minima of the host lattice potential one can obtain very different halfwidths (HWHM) for the corresponding S( Q, w) laws. The aim of the present work is to: - describe the correlated motions of Na’ ions by 0921-4526/89/$03.50 0 (North-Holland Physics
SCATTERING
Elsevier Science Publishers Publishing Division)
motions of Na’ from atom-atom
ions in p potential
transitions between configurations - this avoids treating the particles as independent and averaging the jump rates; - derive the corresponding potential barriers by using a large number of atoms and including the defects belonging to the conducting plane; -from these curves extract transition rates between configurations _ solve the rate equations for all the possible initial configurations and derive the coherent and the incoherent scattering laws In this paper we present only preliminary results, the main objective being to demonstrate that our numerical method for deriving the scattering laws for different diffusion modes is possible and gives results which can be compared with experiments. 2. Structure
and Na+ jump model
Na ,+x+~MgyAl,,-y0,7+~,~ is the general formula of this family of compounds; for y = 0 and x = 0.2 it describes the B form where the excess of Na’ ions is compensated by interstitial oxygens noted Oi in fig. 1 while for x = 0 and y = 0.6 the compensation mechanism is ensured by an Al/Mg substitution and leads to the B” form. The structure comprises spinel-like blocks interleaved with conduction planes containing the charge carrying ions and the so-called bridging oxygen (0,). Three possible sites are available for Na’ ions: the Beever-Ross (BR), the antiBR (aBR) and the midoxygen (m0) sites. The B.V.
J. F. Bocquet
et al.
I Scattering
laws associated
with Na’motions
in p aluminas
347
one, Y,, see fig. 2; in such a diffusion mode the Na transport is ensured by correlated jumps of paired Na’ ions. 3. Potential calculations
Fig. 1. Conductivity path and numbering of the 17 sites visited by 5 Na’ mobile ions, 4Na’ ions trapped by an interstitial oxygen are also represented. Open circles are for 0, bridging oxygens. 1 =aBR, 2=mO, 3=BR, 4=mO, 5 = aBR . This conductivity path is not unique but from plain steric considerations is highly probable, moreover it minimizes the energy variations.
BR and aBR sites form the well-known Honeycomb, fig. 1 represents 3 x 3 unit cells in which the actual stoichiometry is respected. As reported in [3] four Na’ ions can be trapped by the interstitial Oi ion and five Na+ ions allowed to move over 17 sites. In the present study, because several studies consider that it gives rise to the lowest activation energy, only the interstitial pair model was investigated. It consists in calculating the energy of the system when it evolves from one initial configuration, e.g. Y1, to the final
y,k. V3 Exl. V4 . V5 . .
VS
l
v7
e
VI vo
.
. .
.
.
.
lczl
.
k
l
.
no .
.
n* .
.
.
.
D
lb) aIn
The energy of an infinite crystal of p alumina made of the repetition of the model of fig. 1 including spine1 blocks has been calculated following the same procedure as in ref. [3]. The following terms were calculated: - Coulombian energy V, = $ Ci, qiq,lrii. -Van der Waals V, = i C,, C,l(r,)“. - Repulsion energy V, = 5 Ctj B,j exp(- P,r,). -Polarization V, = + CiF,(a,Ej). The parameters were taken equal to those of [3]. The curve representing the variation of the energy E(Y) of the system as function of the different distributions Yi’s of Na’ ions has been obtained. However, the potential barriers of about 1 eV which were derived are too large to be representative of the high conductivity of the compound. The role of the defect will be revised in the future. 4. Determination of the scattering laws Jump rates: In principle it is possible to obtain jump rates between configurations by taking the second derivative of the E(Y) function. In the present case only ten points were calculated for determining E(Y), thus the jump rates from Y(i) to Y(j) were estimated from the following Arrhenian relationship:
F’(ij)
= nil
,
(1)
where V(ij) is the barrier separating the i and j configurations; the F’(ji) backward jump rate being defined in a similar way. At 400°C we have chosen the following values: F’(12)
= ~~~(34) = /(56)
~~~(21) = ?(23) Fig. 2. The 9 possible distributions denoted as Y, (i = l-9) describing the propagation of a pair of Na’ ions in a solitonlike regime, also called interstitial model [7].
exp(-V(ij)IRT)
= 7-‘(43).
The pre-exponential 70’(i) to the oscillation frequency in the configuration i.
. . = O.O6ps-’ , . . = 0.1 ps-* ,
term is taken equal of the system when
348
J. F. Bocquet et al. I Scattering laws associated with Na’motions
in f3 aluminas
where the r,(t) vectors give the position of the sites involved in the configuration n having the probability Y:(t) to be occupied at time t and r(0) those of the sites involved in the initial configuration Y,(O) taken equal to the crystallographic occupation factors in order to perform the thermal average represented by the brackets in eq. (2). Using Roth’s data [7] the following Y, (0) values were obtained: Y,(O) = Yj(O) = Y,(O) = Y,(O) = Y,(O) = 0.0296, Y*(O) = Y,(O) = YJO) = Y,(O) = 0.213 .
-1
The L( j, k) is a label for the site k involved in the initial configuration j and L(n, m) for site m involved in configuration IE. The incoherent law is simply obtained with m = k:
.
1 meV
0
Fig. 3. Scattering laws calculated for Q = 2 A-’ (a) Coherent, (b) Incoherent and (c) experimental results for the same Q at 400°C for a polycrystalline p alumina [l].
Jump rate equations: nine systems (one for each initial condition) of nine equations were solved using the Runge-Kutta method for a time interval of 90 ps with a time resolution of 0.001 ps. 81 solutions Y:(t) where the index k designates the initial configuration have been derived. These solutions give the probability of the configuration j at a time t, or alternatively the time dependence of the occupation of the sites defining this configuration. Intermediate
scattering functions:
study was limited to the calculation I
Na-Na( Q,
The of
present
t) = { nNanNa}-I’*
X
(Eexp{i&. [r:(t) - (W)ll) T
(2) which is the coherent
law and can be written
I Na-Na( Q, t) = { nNanNa}-I’*
x
c
I.n,k.m
. rL(l,k),L(n.mj)
’ exp{-iQ
,Fk Y,(O)Y;(t) ’
rl.(l.k),L(n.k))
7
(3)
.
Scattering laws: The quasi-elastic
scattering outside of Bragg peaks can be written
law
S,,,( Q, w> = a;cSNa( Q, w>
+ o-C:$SNaNa(Q, w) + { cr:$$,}
“2SNaox( Q, w)
+ ~cNo’;lSNat,xrdNa,,,hlle( Q, w)
.
(5)
The third term has not yet been calculated, it is expected to be very similar to the incoherent part because oxygen atoms including Oi are nonmobile on the ps time scale and thus the timedependent scattering is only due to the motions of individual Na’ ions. Moreover, because the oxygen coherent cross section is larger than sodium (uoX = 4.23 b, aNa = 1.63 b) and because there are 4 times more oxygen than sodium atoms this term is expected to dominate. It can be written ~~~~~~‘~~~~~~~“~SNa(Q, w),
yjcwI,(t)
x exd-iQ
I Na-Na(Q, t) = {nNanNa}yZ
(6)
where the S( Q, w) are the Fourier transforms of the corresponding intermediate scattering laws. For 3 x 3 unit cells the formula is approximately
J. F. Bocquet et al. / Scattering laws associated with Na ‘motions in P aluminas
A11980308Na22, for a single conducting plane, the three terms are proportional to 1 (incoherent), 5 (no. of mobile Na+ on the IN, time scale) and to (5 x 154)“2 = 27.7, respectively. Moreover beNa for the total one can write cause cr,“, = ginc
structure factor S(Q) corresponding to the 3 x 3 unit cells of fig. 1 has been obtained by summing the diagonal terms of the Wnj tables over the initial configurations. 5. Conclusions and perspectives
&,,(Q, w) = c N”{51.1 SNa( Q, w) + 5 SNaNa(Q, w)} ,
349
(7)
51.1 = 1 + (4.23/1.5)“2 x 27.7 + (5 x where 6) “*. Thus it can be seen that the incoherent approximation made in [l] is quite justified, however it is interesting to try to exploit the information contained in Scoh( Q, t). Numerical
calculation of S( Q, w) laws: Instead of performing a numerical Fourier transform of the intermediate Z( Q, t), the Y(t) solutions were fitted with a linear combination of exponential functions. In a first stage two terms were sufficient for the set of jump rates adopted and the general expression for the intermediate scattering law can be written
It has been shown that it is possible to reproduce the partial scattering laws for the coherent and incoherent laws associated with collective motions of Na+ ions corresponding to the interstitial pair model. Before concluding on the Q dependence of the HWHM of the total QE signal, or of its intensity more has to be done: the basic unit will be extended and the nonmobile atoms (oxygen plus trapped Na’ ions) along with a convolution with a resolution will also be introduced in the calculation in order to perform a numerical fitting of the powder and single crystal data obtained for p and p” aluminas [l-4]. In parallel, similar calculations are performed on the data obtained from molecular dynamics [8].
Z(Q, t> = C w,j{a,j exP(b,jt) + C,j exP(d,$)) 7 n,j
References
where
[l] G. Lucazeau, J.R. Gavarri and A.J. Dianoux, J. Phys. Chem. Solids 48 (1987) 57. (21 G. Lucazeau, D. Dohy, N. Fanjat and A.J. Dianoux, Solid State Ionics (1988), in press. [3] J.F. Bocquet and G. Lucazeau, Solid State Ionics 24 (1987) 235. [4] N. Fanjat, G. Lucazeau, J. Bates and A.J. Dianoux, these Proceedings, Physica B 156 & 157 (1989) 342. [5] J.C. Wang, J. Chem. Phys. 73 (1980) 5786. [6] A. Chahid, ILL report, Grenoble, France (1988). [7] W.L. Roth, F. Reidinger and S. Laplaca, in: Superionic Conductors, G.D. Mahan and W.L. Roth, eds. (Plenum, New York, 1976), p. 223. [8] M. Zendejas, J.O. Thomas and G. Lucazeau, to be published.
sin{Qr,(j,t)l(n,k)l lQrqj,kw+,k) k.n
J+‘“j= C
(8)
is the coefficient giving the weight of the different solutions Y(t) after powder average. Thus S(Q, w) is given by a sum of Lorentzian functions. Fig. 3 gives a typical result for the partial coherent and incoherent laws SNaNa(Q, w) and SFa(Q, w). Because the calculations were limited to a time scale of 90 ps the region comprised between -0.1 and 0.1 meV, i.e. the elastic peak, was not properly reproduced. Finally, the static
DERIVATION ASSOCIATED
346-349
OF THE COHERENT AND INCOHERENT WITH Na+ MOTIONS IN p ALUMINAS
J.F. BOCQUET’,
A. CHAHID’,
‘Lab. chimie physique. ‘Institut Laue-Langevin,
CSP, Univ. Paris-Nord. Grenoble, France
The derivation of the coherent aluminas is presented; it is based calculations.
G. LUCAZEAU’
and A.J.
LAWS
DIANOUX’
France
and incoherent scattering laws associated with collective on a jump model in which jump rates were determined
1. Introduction The Na B aluminas are known as the best super ionic conductors and are used as solid electrolytes in Na/S batteries. Their two dimensional conductivity has stimulated a large number of theoretical and structural studies. This work is the continuation of the study of the dynamics of Na’ ions in B and B” aluminas [l-4]. The interpretations of our previous neutron scattering data [l] and [2] were based on the Skold incoherent approximation consisting in writing the coherent law from the incoherent one. Moreover no specific mode of diffusion for sodium was considered, although the fraction of mobile ions and the jump constants which were derived were in agreement with the so-called interstitial pair model [5]. Since, using atomatom potentials and dipole-induced dipole interactions, we have derived the relative variations of the crystal energy for different correlated Naf motions in B alumina [3]; however, these motions were limited to 1 or 2 cells, congruent displacements being generated in the whole crystal. Recently [6] we have shown that for two particles jumping over three sites the coherent intensity dominates generally and that for a pair (strong correlations) depending on their position relative to the minima of the host lattice potential one can obtain very different halfwidths (HWHM) for the corresponding S( Q, w) laws. The aim of the present work is to: - describe the correlated motions of Na’ ions by 0921-4526/89/$03.50 0 (North-Holland Physics
SCATTERING
Elsevier Science Publishers Publishing Division)
motions of Na’ from atom-atom
ions in p potential
transitions between configurations - this avoids treating the particles as independent and averaging the jump rates; - derive the corresponding potential barriers by using a large number of atoms and including the defects belonging to the conducting plane; -from these curves extract transition rates between configurations _ solve the rate equations for all the possible initial configurations and derive the coherent and the incoherent scattering laws In this paper we present only preliminary results, the main objective being to demonstrate that our numerical method for deriving the scattering laws for different diffusion modes is possible and gives results which can be compared with experiments. 2. Structure
and Na+ jump model
Na ,+x+~MgyAl,,-y0,7+~,~ is the general formula of this family of compounds; for y = 0 and x = 0.2 it describes the B form where the excess of Na’ ions is compensated by interstitial oxygens noted Oi in fig. 1 while for x = 0 and y = 0.6 the compensation mechanism is ensured by an Al/Mg substitution and leads to the B” form. The structure comprises spinel-like blocks interleaved with conduction planes containing the charge carrying ions and the so-called bridging oxygen (0,). Three possible sites are available for Na’ ions: the Beever-Ross (BR), the antiBR (aBR) and the midoxygen (m0) sites. The B.V.
J. F. Bocquet
et al.
I Scattering
laws associated
with Na’motions
in p aluminas
347
one, Y,, see fig. 2; in such a diffusion mode the Na transport is ensured by correlated jumps of paired Na’ ions. 3. Potential calculations
Fig. 1. Conductivity path and numbering of the 17 sites visited by 5 Na’ mobile ions, 4Na’ ions trapped by an interstitial oxygen are also represented. Open circles are for 0, bridging oxygens. 1 =aBR, 2=mO, 3=BR, 4=mO, 5 = aBR . This conductivity path is not unique but from plain steric considerations is highly probable, moreover it minimizes the energy variations.
BR and aBR sites form the well-known Honeycomb, fig. 1 represents 3 x 3 unit cells in which the actual stoichiometry is respected. As reported in [3] four Na’ ions can be trapped by the interstitial Oi ion and five Na+ ions allowed to move over 17 sites. In the present study, because several studies consider that it gives rise to the lowest activation energy, only the interstitial pair model was investigated. It consists in calculating the energy of the system when it evolves from one initial configuration, e.g. Y1, to the final
y,k. V3 Exl. V4 . V5 . .
VS
l
v7
e
VI vo
.
. .
.
.
.
lczl
.
k
l
.
no .
.
n* .
.
.
.
D
lb) aIn
The energy of an infinite crystal of p alumina made of the repetition of the model of fig. 1 including spine1 blocks has been calculated following the same procedure as in ref. [3]. The following terms were calculated: - Coulombian energy V, = $ Ci, qiq,lrii. -Van der Waals V, = i C,, C,l(r,)“. - Repulsion energy V, = 5 Ctj B,j exp(- P,r,). -Polarization V, = + CiF,(a,Ej). The parameters were taken equal to those of [3]. The curve representing the variation of the energy E(Y) of the system as function of the different distributions Yi’s of Na’ ions has been obtained. However, the potential barriers of about 1 eV which were derived are too large to be representative of the high conductivity of the compound. The role of the defect will be revised in the future. 4. Determination of the scattering laws Jump rates: In principle it is possible to obtain jump rates between configurations by taking the second derivative of the E(Y) function. In the present case only ten points were calculated for determining E(Y), thus the jump rates from Y(i) to Y(j) were estimated from the following Arrhenian relationship:
F’(ij)
= nil
,
(1)
where V(ij) is the barrier separating the i and j configurations; the F’(ji) backward jump rate being defined in a similar way. At 400°C we have chosen the following values: F’(12)
= ~~~(34) = /(56)
~~~(21) = ?(23) Fig. 2. The 9 possible distributions denoted as Y, (i = l-9) describing the propagation of a pair of Na’ ions in a solitonlike regime, also called interstitial model [7].
exp(-V(ij)IRT)
= 7-‘(43).
The pre-exponential 70’(i) to the oscillation frequency in the configuration i.
. . = O.O6ps-’ , . . = 0.1 ps-* ,
term is taken equal of the system when
348
J. F. Bocquet et al. I Scattering laws associated with Na’motions
in f3 aluminas
where the r,(t) vectors give the position of the sites involved in the configuration n having the probability Y:(t) to be occupied at time t and r(0) those of the sites involved in the initial configuration Y,(O) taken equal to the crystallographic occupation factors in order to perform the thermal average represented by the brackets in eq. (2). Using Roth’s data [7] the following Y, (0) values were obtained: Y,(O) = Yj(O) = Y,(O) = Y,(O) = Y,(O) = 0.0296, Y*(O) = Y,(O) = YJO) = Y,(O) = 0.213 .
-1
The L( j, k) is a label for the site k involved in the initial configuration j and L(n, m) for site m involved in configuration IE. The incoherent law is simply obtained with m = k:
.
1 meV
0
Fig. 3. Scattering laws calculated for Q = 2 A-’ (a) Coherent, (b) Incoherent and (c) experimental results for the same Q at 400°C for a polycrystalline p alumina [l].
Jump rate equations: nine systems (one for each initial condition) of nine equations were solved using the Runge-Kutta method for a time interval of 90 ps with a time resolution of 0.001 ps. 81 solutions Y:(t) where the index k designates the initial configuration have been derived. These solutions give the probability of the configuration j at a time t, or alternatively the time dependence of the occupation of the sites defining this configuration. Intermediate
scattering functions:
study was limited to the calculation I
Na-Na( Q,
The of
present
t) = { nNanNa}-I’*
X
(Eexp{i&. [r:(t) - (W)ll) T
(2) which is the coherent
law and can be written
I Na-Na( Q, t) = { nNanNa}-I’*
x
c
I.n,k.m
. rL(l,k),L(n.mj)
’ exp{-iQ
,Fk Y,(O)Y;(t) ’
rl.(l.k),L(n.k))
7
(3)
.
Scattering laws: The quasi-elastic
scattering outside of Bragg peaks can be written
law
S,,,( Q, w> = a;cSNa( Q, w>
+ o-C:$SNaNa(Q, w) + { cr:$$,}
“2SNaox( Q, w)
+ ~cNo’;lSNat,xrdNa,,,hlle( Q, w)
.
(5)
The third term has not yet been calculated, it is expected to be very similar to the incoherent part because oxygen atoms including Oi are nonmobile on the ps time scale and thus the timedependent scattering is only due to the motions of individual Na’ ions. Moreover, because the oxygen coherent cross section is larger than sodium (uoX = 4.23 b, aNa = 1.63 b) and because there are 4 times more oxygen than sodium atoms this term is expected to dominate. It can be written ~~~~~~‘~~~~~~~“~SNa(Q, w),
yjcwI,(t)
x exd-iQ
I Na-Na(Q, t) = {nNanNa}yZ
(6)
where the S( Q, w) are the Fourier transforms of the corresponding intermediate scattering laws. For 3 x 3 unit cells the formula is approximately
J. F. Bocquet et al. / Scattering laws associated with Na ‘motions in P aluminas
A11980308Na22, for a single conducting plane, the three terms are proportional to 1 (incoherent), 5 (no. of mobile Na+ on the IN, time scale) and to (5 x 154)“2 = 27.7, respectively. Moreover beNa for the total one can write cause cr,“, = ginc
structure factor S(Q) corresponding to the 3 x 3 unit cells of fig. 1 has been obtained by summing the diagonal terms of the Wnj tables over the initial configurations. 5. Conclusions and perspectives
&,,(Q, w) = c N”{51.1 SNa( Q, w) + 5 SNaNa(Q, w)} ,
349
(7)
51.1 = 1 + (4.23/1.5)“2 x 27.7 + (5 x where 6) “*. Thus it can be seen that the incoherent approximation made in [l] is quite justified, however it is interesting to try to exploit the information contained in Scoh( Q, t). Numerical
calculation of S( Q, w) laws: Instead of performing a numerical Fourier transform of the intermediate Z( Q, t), the Y(t) solutions were fitted with a linear combination of exponential functions. In a first stage two terms were sufficient for the set of jump rates adopted and the general expression for the intermediate scattering law can be written
It has been shown that it is possible to reproduce the partial scattering laws for the coherent and incoherent laws associated with collective motions of Na+ ions corresponding to the interstitial pair model. Before concluding on the Q dependence of the HWHM of the total QE signal, or of its intensity more has to be done: the basic unit will be extended and the nonmobile atoms (oxygen plus trapped Na’ ions) along with a convolution with a resolution will also be introduced in the calculation in order to perform a numerical fitting of the powder and single crystal data obtained for p and p” aluminas [l-4]. In parallel, similar calculations are performed on the data obtained from molecular dynamics [8].
Z(Q, t> = C w,j{a,j exP(b,jt) + C,j exP(d,$)) 7 n,j
References
where
[l] G. Lucazeau, J.R. Gavarri and A.J. Dianoux, J. Phys. Chem. Solids 48 (1987) 57. (21 G. Lucazeau, D. Dohy, N. Fanjat and A.J. Dianoux, Solid State Ionics (1988), in press. [3] J.F. Bocquet and G. Lucazeau, Solid State Ionics 24 (1987) 235. [4] N. Fanjat, G. Lucazeau, J. Bates and A.J. Dianoux, these Proceedings, Physica B 156 & 157 (1989) 342. [5] J.C. Wang, J. Chem. Phys. 73 (1980) 5786. [6] A. Chahid, ILL report, Grenoble, France (1988). [7] W.L. Roth, F. Reidinger and S. Laplaca, in: Superionic Conductors, G.D. Mahan and W.L. Roth, eds. (Plenum, New York, 1976), p. 223. [8] M. Zendejas, J.O. Thomas and G. Lucazeau, to be published.
sin{Qr,(j,t)l(n,k)l lQrqj,kw+,k) k.n
J+‘“j= C
(8)
is the coefficient giving the weight of the different solutions Y(t) after powder average. Thus S(Q, w) is given by a sum of Lorentzian functions. Fig. 3 gives a typical result for the partial coherent and incoherent laws SNaNa(Q, w) and SFa(Q, w). Because the calculations were limited to a time scale of 90 ps the region comprised between -0.1 and 0.1 meV, i.e. the elastic peak, was not properly reproduced. Finally, the static