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Cluster-induced local distortion in ionic conductors — negative static expansion
Solid State Communications, Vol. 108, No. 8, pp. 513–517, 1998 䉷 1998 Elsevier Science Ltd. All rights reserved 0038–1098/98 $ - see front matter
Pergamon
PII: S0038–1098(98)00402-5
CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS ¹ NEGATIVE STATIC EXPANSION Tadao Ishii Faculty of Engineering, Okayama University, Okayama 700-0082, Japan (Received 9 June 1998; accepted 14 August 1998 by H. Akai) A model theory of the local distortion in the low-temperature phase of superionic conductors is proposed. The model is based on the coupling between an ion-cluster and a static distortion and is investigated in terms of the pair approximation of the cluster variation method. This cluster-induced local distortion drives a first-order phase transition of order–disorder type. The negative and positive local expansions of Cu–Br distance experimentally found for crystal g-CuBr are quantitatively discussed by use of the present theory. The results show a good overall agreement with the experiments, implying that the present model is applicable to ionic conductors. 䉷 1998 Elsevier Science Ltd. All rights reserved Keywords: D. phase transitions.
1. INTRODUCTION
2. MODEL HAMILTONIAN
A local distortion of ionic conductors has sometimes been discussed. The existence of it has been pointed out by a molecular dynamics simulation of the superionic conductor a-AgI in which the volume of the tetrahedron including a Ag þ ion becomes smaller than the empty tetrahedron without a Ag ion [1]. On the other hand, the local distortion of ionic conductors b-AgI [2] and g-CuBr [3] has also been reported experimentally by EXAFS (Extended X-ray Absorption Fine Structure) measurements, in which the pair-potential minimum position of Ag–I or Cu–Br decreases as the temperature is increased. In addition to these local distortions, it was found by X-ray diffraction analysis that even the lattice constant along the c-axis in b-AgI decreases with increasing temperature in the range about 100–300 K, rather consistent with EXAFS [4]. The mechanism of these phenomena has not been known so far. In this Communication, we propose a mechanism of the decrease of the pair-potential minimum position in terms of the coupling between a pair-cluster of mobile ions (Ag þ or Cu þ) and the local static distortion and show how the negative expansion, found in b-AgI [2] or g-CuBr [3], can be induced.
Now consider M primitive cells consisting of two species of ions, immobile and mobile ions, which is applicable to g (zincblende structure)- and b (wurtzite structure)-CuBr, AgI and CuI et al. At the ground state in the low temperature phase, an immobile ion such as Br ¹ and I ¹ in the above materials occupies, say, a C site, which we call C-sublattice, while a mobile ion such as Cu þ and Ag þ stays at an A site, called as A-sublattice (Fig. 1). In addition to these sublattices, we have an equivalent dual lattice B to A which is empty in the ground state. At an elevated temperature, mobile ions can be excited on this dual lattice: we occasionally call it as B-sublattice and ions on this B-sublattice as defects. Then a cluster formed of an excited ion and adjacent ions on A effectively has an extracharge to have the attractive interaction with the adjacent immobile ions on C. This may induce a local static distortion (Fig. 1). Since A and B sublattices are equivalent, the interaction energy should be symmetric with respect to A and B. The Hamiltonian for mobile ions on A and B is thus written as X ela;l⬘a nla nl⬘a⬘ ; (1) HLG ¼
513
hla僆A;l⬘a⬘僆Bi
514
CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS
Vol. 108, No. 8
tensors and the colon defines the tensor product. The sum of {l; a} in equation (3) is taken over the nearest neighbors of each C site and the sum of {l⬘; a⬘} is over the neighbor sites adjacent to the CB or CA pair. The factor Sl⬘a⬘ð⫽laÞ僆A;B 1 is a normalization factor which is now 3 since the number of nearest neighbor sites adjacent to the CB or CA pair is 3. When the potential energy of the system is approximated by nearest neighbor pair potentials between mobile and immobile ions, it can be expanded in terms of the small static distortion DR around a complete lattice R 0 and displacement u around R0 þ DR as Vðu; R0 þ DRÞ ⬇ Vð0; R0 Þ 1 2 Vðu; R0 Þ 1 3 Vðu; R0 Þ þ : uu þ 2! uu 3! uuu u¼0
Fig. 1. Definitions of sublattices A, B and C for zincblende structure CuBr. When Cu þ at an A site gets excited to its adjacent B site (arrow), then Br-s at the B’s nearest neighbor sites C distort towards the B. la;l⬘a⬘
having the nearest-neighbor interaction energy e between A{l; a} and B{l⬘; a⬘}, where n 1a is the lattice gas operator taking nla ¼ 1 or 0 at ath site on A of the lth cell. Let us introduce the static distortion DR now considered to occur only when a mobile ion cluster is produced and thus this quantity is completely different from the displacement u of lattice vibration around the lattice. Therefore DR should be written as a function of the position where the cluster exists. Considering the above, we define the position vector R between nearest neighbor mobile (m) and immobile (im) ions by R ¼ R0 þ DR þ u;
(2)
where R 0 is defined as R0 ¼ R0 ðmÞ ¹ R0 ðimÞ for the complete lattice and DR ¼ DRðmÞ ¹ DRðimÞ and u ¼ uðmÞ ¹ uðimÞ, respectively, are the static and dynamic distortions. Equation (2) is interpreted as R ¼ R0 þ u at the position where no cluster exists, while it is R ¼ R0 þ DR þ u when cluster exists. Thus the interaction Hamiltonian between nearest neighbor pair clusters nla nl⬘a⬘ and distortions is taken as X 1X X 1 X Hint ¼ 2 C la僆A;B 1 l⬘a⬘ð⫽laÞ僆A;B l⬘a⬘ð⫽laÞ僆A;B
⫻ ðhC;la;l⬘a⬘ : DRC;la þ lC;la:l⬘a⬘ : uC;la Þnla nl⬘a⬘ ;
þ
: uuu þ … u¼0
1 2 Vð0; R0 Þ : DRDR þ …; u ¼ R ¹ ðR0 þ DRÞ; 2! R0 R0 ð4Þ
neglecting the coupling term of the form uDR. Since we are interested only in the elastic energy stored by the cluster induced static distortion, it is enough to consider the harmonic terms as simply extracted from equation (4) as X X 1 (5) HSD ¼ 2 mC;la : DRC;la DRC;la : C la僆A
From EXAFS analysis, we can know the average nearest neighbor radial distance hRi ¼ R¯ of AgI or CuBr, which is formed of static distortion DR and dynamic distortion hui ¼ u¯ such that R¯ ¼ R0 þ DR þ u¯ . This u¯ is the mean relative displacement between Ag and I, and originates from the anharmonic potential. So we consider the anharmonic vibrations (phonons) and simply take the following approximate Einstein Hamiltonian from equation (4): 1 2 1 2 2 3 p þ m0 q0 u þ zu ; (6) Hvib ¼ M 2m0 2 where m 0 is the bare reduced mass, q 0 is the bare energy and z is the coefficient of cubic anharmonic term [5]. Therefore we have the total Hamiltonian H as H ¼ HLG þ Hint þ HSD þ Hvib :
(7)
Corresponding to the Hamiltonian (7), we have the free energy. 3. FREE ENERGY
ð3Þ to the lowest-order in DR and u, where hC;la;l⬘a⬘ and lC;la;l⬘a⬘ defined in equation (3) are coupling constant
For the free energy consisting of the lattice gas terms, we use the pair approximation of the cluster variation method that is appropriate to strong coupling cases [6].
Vol. 108, No. 8
CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS
Assuming the translational symmetry of the lattice and using the pair-cluster density matrix ra;a⬘ ð{j}Þ with the relation na ¼ 12ð1 þ ja Þ for the spin variables ja ¼ ⫾ 1, the free energy can be obtained from Kikuchi [6] and Ducastella [7] cluster variation methods as FðrÞ ¼ Tr ra;a⬘ H þ b ¹ 1 " X ⫻ Tr ra;a⬘ ln ra;a⬘ ha;a⬘i
¹
X
#
ð8Þ
ð2ga ¹ 1ÞTr ra ln ra ;
Tra⬘ ra;a⬘ ¼ ra ; with b ¼ 1=kB T and 2g a the coordination number of a site. The site probability of finding a mobile ion on að¼ A or B) is defined by xai ¼ rai , in which i ¼ 1 or v corresponding to ion (j ¼ 1) or vacancy ðj ¼ ¹1Þ, respectively and the pair probability of ions {i; j} at a;a⬘ sites {a; a⬘} by yaa⬘ ij ¼ rij , both of which can be evaluated self-consistently later in Section 4. The corresponding free energy per pair ions on A and B to equations (1) and (3) can then be written as 1 ðFLG þ Fint Þ M ¯ þ luÞyAB ¼ 2geAB yAB 11 þ 2gðhDR 11 ¹ kB T ln G;
ln G ¼
X
ð2g ¹ 1Þ X
i;j¼1;v
ð9Þ Lðxai Þ
X
2g
Lðyaa⬘ ij Þ;
a;a⬘ða⫽a⬘Þ¼A;B
LðxÞ ¼ x ln x ¹ x, 2gBA ¼ 2gAB ¼ 2g ¼ 6 and 2gAC ¼ 2gBC ¼ 2g¯ ¼ 4, where 2ga⬘ a is the coordination number of a site to a⬘ sublattice, hnla nl⬘a⬘ i ¼ yaa⬘ 11 for nearest neighbor pair ions on {l; a; l⬘; a⬘}. Here we simply consider the static and dynamic distortions by their radial components DR and u. The elastic energy caused by the static distortion, on the other hand, is given from equation (5) by fSD ¼
hHSD i 2 : ¼ gmDR ¯ M
where m, q and hui ¼ u¯ are the variational parameters determined by the minimum principle. Thus the displacement u in f int of equation (9) is also replaced by u¯ . Then we have the total free energy per pair: (13) fpair ¼ fLG þ fint þ fSD þ fvib : 4. COUPLED EQUATIONS Dropping the suffixes as e ¼ eAB and y ¼ yAB 11 and minimizing the free energy (13) with respect to DR, the order parameter y defined by y ¼ xA1 ¹ xB1
(14) xA1
xB1
with the condition þ ¼ 1 and y, we have a set of equations h (15) DR ¼ ¹ y m and 1 þ y 2ð2g ¹ 1Þ 1 ¹ y ¹ 2y 2g ¼ 1; 1¹y 1 þ y ¹ 2y 4y2 ð1 ¹ y ¹ 2yÞð1 þ y ¹ 2yÞ g¯ g¯ ¼ exp ¹ b e þ l¯u þ hDR : g g
a¼A;B
i¼1;v
¹
X
we have the free energy of Fvib fvib ¼ M 1 2 hp i0 þ 12m0 q20 hu2 i0 þ zhu3 i0 ¼ 2m0 bÉq Éq bÉq ¹1 coth þ b ln 2sinh ¹ ; 2 2 2 ð12Þ
hai
fLG þ fint ¼
515
(10)
Finally as to the phonons, we utilize the variational principle as in our previous paper [8]. On taking the 0thorder trial Hamiltonian as 1 2 1 2 2 p þ mq ðu ¹ u¯ Þ ; (11) H0 ¼ M 2m 2
ð16Þ
Equation (15), indicating that DR ⫽ 0 only when the cluster exists y ⫽ 0, leads to the negative energy given by g¯ (17) 2gðhyDR ¯ þ 12mDR2 Þ ¼ ¹ h2 y2 ⬍ 0; m which is effectively an attractive energy between pairclusters, implying that the static distortion prefers to occur. On putting q ¼ u ¹ u¯ and again minimizing eqaution (13) with respect to variational parameters, we have m ¼ m0 and q2 ¼ q20 þ
6z u¯ ; m0
3zhq2 i0 þ 2gly ¯ ; 2 m0 q0 þ 3z u¯ É bÉq 2 coth : hq i0 ¼ 2m0 q 2 u¯ ¼ ¹
ð18Þ
516
CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS
These equations (15), (16) and (18) give us the static and dynamic distortions DR and u¯ , the order parameter y, the pair probability y and the normalized phonon energy Éq. As previously known from equation (17), the free energy of the system gets lowered due to the clustering of mobile ions as the temperature increases. Thus it accelerates the instability of the low-temperature phase to transform into a disordered phase at critical temperature T C. In the present equivalent A, B-lattice system, the type of phase transition is mainly the firstorder type, although the lattice gas system only with f LG would cause the second-order type transition at a higher critical temperature than the present T C. 5. RESULTS The local static distortion DR(T) is proportional to the pair probability y as in equation (15). Therefore the distance of the pair-potential minimum of Ag–I or Cu–Br has a temperature dependence of h (19) Rst ðTÞ ¼ R0 þ DRðTÞ ¼ R0 ¹ y; m where R 0 is a constant value at some reference temperature. For h ⬎ 0, the distance of the pair-potential minimum decreases with temperature since the pair probability y increases. In order to examine the present result for the static distortion R st(T), we quantitatively apply it to g-CuBr by choosing the parameter values such that m0 ¼ 35:4 amu, ˚ and Éq0 ¼ 18:0 meV as the optical phonon R0 ¼ 2:44 A
Vol. 108, No. 8
˚ , e ¼ 62:0 meV, z ¼ energy [9], m ¼ 1:70 eV A ˚ ¹1 ˚ ¹3 and coupling constants h ¼ 450 meV A ¹ 1:85 eV A ¹1 ˚ . The theoretical result indicated and l ¼ ¹ 9:00 meV A by the solid line shows a reasonable agreement with the experiments as shown in Fig. 2. Thus the negative static distortions found both in CuBr and AgI are due to the clustering of mobile ions. On the other hand, the average distance of nearest neighbor ions ¯ (20) hRðTÞi ¼ RðTÞ ¼ Rst ðTÞ þ u¯ ðTÞ; ¹2
is also evaluated with the same parameters and shown together in Fig. 2. The hRðTÞi mostly increases with temperature. These two theoretical curves are quite consistent with experimental results [3]. It is also seen that this model reproduces the negative expansion ¯ dRðTÞ=dT ⬍ 0 at around T C in Fig. 2. The same is true for AgI case as well although the result is not shown here. 6. DISCUSSION The theoretical model of the static and dynamic distortions in terms of the coupling between pair-clusters of lattice gas ions and distortions is proposed for defects system. It is found that the positive or negative expansion of Cu–Br and Ag–I distances with temperature possibly occurs. In the positive local expansion, the dynamic distortion is larger than the static distortion, while in the negative local expansion, the dynamic distortion is smaller than the static distortion. In Ge case [10], however, static distortion does not occur, since the crystal is comparatively rigid and defects are few.
¯ Fig. 2. Theoretical results of temperature dependence of Rst ðTÞ ¼ R0 þ DRðTÞ and hRðTÞi ¼ RðTÞ with Tc ¼ 658:5 K for g-CuBr, compared with experimental results X [3].
Vol. 108, No. 8
CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS
The ionic conductor g-CuBr crystal, having the ˚ in nearest neighbor distance of Cu–Cu of 4.02 A the ideal zincblende structure, shows rather large ionic conductivity attributed to the hopping of mobile Cu ions [11]: the Bragg and diffuse scattering measurement ˚ of Cu–Cu reveals in this case that the peak at 2.4 A distribution function exists [12]. Thus it is considered ˚ that thermally generated pairs of Cu–Cu ions nearly 2.4 A apart couple with the local distortion of Cu–Br, which is exactly consistent with the present model. Finally, we comment on the negative expansion of the c-axis in b-AgI. This expansion is believed to be due to the same mechanism. The wurtzite structure tends to yield defects along the c-axis and so the negative local expansion occurs in this direction. It has been recently confirmed that the MD simulation reproduces defectinduced c-axis negative expansion and a-axis positive expansion in b-AgI [13]. Acknowledgements—The author would like to thank O. Kamishima for the calculation of Fig. 2. This work was partially supported by the Grant-in-Aid for Scientific Research (#07239101) from the Ministry of Education, Science, Sports and Culture.
517
REFERENCES 1. Hokazono, M., Ueda, A. and Hiwatari, Y., Solid State Ionics, 13, 1984, 151. 2. Dalba, G., Fornasini, P., Gotter, R. and Rocca, F., Phys. Rev., B52, 1995, 149. 3. Kamishima, O., Ishii, T., Maeda, H. and Kashino, S., Solid State Commun., 103, 1997, 141. 4. Yoshiasa, A., Inaba, A., Ishii, T. and Koto, K., Solid State Ionics, 79, 1995, 67. 5. Matsubara, T., J. Phys. Soc. Jpn., 38, 1975, 1076. 6. Kikuchi, R., Phys. Rev., 81, 1951, 988. 7. Ducastella, F., Prog. Theor. Phys., Suppl., No. 115, 1994, 255. 8. Kondo, Y. and Ishii, T., Solid State Ionics, 9&10, 1983, 1433. 9. Hoshino, S., Fujii, Y., Harada, J. and Axe, J.D., J. Phys. Soc. Jpn., 41, 1976, 965. 10. Kamishima, O., doctor thesis. Okayama Univ., 1998. 11. Wagner, J.B. and Wagner, C., J. Chem. Phys., 26, 1957, 1597. 12. Nield, V.M., McGreevy, R.L., Keen, D.A. and Hayes, W., Physica, B202, 1994, 159. 13. Seko, T. and Ishii, T., J. Phys. Soc. Jpn., 67, 1998, 1837.
Pergamon
PII: S0038–1098(98)00402-5
CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS ¹ NEGATIVE STATIC EXPANSION Tadao Ishii Faculty of Engineering, Okayama University, Okayama 700-0082, Japan (Received 9 June 1998; accepted 14 August 1998 by H. Akai) A model theory of the local distortion in the low-temperature phase of superionic conductors is proposed. The model is based on the coupling between an ion-cluster and a static distortion and is investigated in terms of the pair approximation of the cluster variation method. This cluster-induced local distortion drives a first-order phase transition of order–disorder type. The negative and positive local expansions of Cu–Br distance experimentally found for crystal g-CuBr are quantitatively discussed by use of the present theory. The results show a good overall agreement with the experiments, implying that the present model is applicable to ionic conductors. 䉷 1998 Elsevier Science Ltd. All rights reserved Keywords: D. phase transitions.
1. INTRODUCTION
2. MODEL HAMILTONIAN
A local distortion of ionic conductors has sometimes been discussed. The existence of it has been pointed out by a molecular dynamics simulation of the superionic conductor a-AgI in which the volume of the tetrahedron including a Ag þ ion becomes smaller than the empty tetrahedron without a Ag ion [1]. On the other hand, the local distortion of ionic conductors b-AgI [2] and g-CuBr [3] has also been reported experimentally by EXAFS (Extended X-ray Absorption Fine Structure) measurements, in which the pair-potential minimum position of Ag–I or Cu–Br decreases as the temperature is increased. In addition to these local distortions, it was found by X-ray diffraction analysis that even the lattice constant along the c-axis in b-AgI decreases with increasing temperature in the range about 100–300 K, rather consistent with EXAFS [4]. The mechanism of these phenomena has not been known so far. In this Communication, we propose a mechanism of the decrease of the pair-potential minimum position in terms of the coupling between a pair-cluster of mobile ions (Ag þ or Cu þ) and the local static distortion and show how the negative expansion, found in b-AgI [2] or g-CuBr [3], can be induced.
Now consider M primitive cells consisting of two species of ions, immobile and mobile ions, which is applicable to g (zincblende structure)- and b (wurtzite structure)-CuBr, AgI and CuI et al. At the ground state in the low temperature phase, an immobile ion such as Br ¹ and I ¹ in the above materials occupies, say, a C site, which we call C-sublattice, while a mobile ion such as Cu þ and Ag þ stays at an A site, called as A-sublattice (Fig. 1). In addition to these sublattices, we have an equivalent dual lattice B to A which is empty in the ground state. At an elevated temperature, mobile ions can be excited on this dual lattice: we occasionally call it as B-sublattice and ions on this B-sublattice as defects. Then a cluster formed of an excited ion and adjacent ions on A effectively has an extracharge to have the attractive interaction with the adjacent immobile ions on C. This may induce a local static distortion (Fig. 1). Since A and B sublattices are equivalent, the interaction energy should be symmetric with respect to A and B. The Hamiltonian for mobile ions on A and B is thus written as X ela;l⬘a nla nl⬘a⬘ ; (1) HLG ¼
513
hla僆A;l⬘a⬘僆Bi
514
CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS
Vol. 108, No. 8
tensors and the colon defines the tensor product. The sum of {l; a} in equation (3) is taken over the nearest neighbors of each C site and the sum of {l⬘; a⬘} is over the neighbor sites adjacent to the CB or CA pair. The factor Sl⬘a⬘ð⫽laÞ僆A;B 1 is a normalization factor which is now 3 since the number of nearest neighbor sites adjacent to the CB or CA pair is 3. When the potential energy of the system is approximated by nearest neighbor pair potentials between mobile and immobile ions, it can be expanded in terms of the small static distortion DR around a complete lattice R 0 and displacement u around R0 þ DR as Vðu; R0 þ DRÞ ⬇ Vð0; R0 Þ 1 2 Vðu; R0 Þ 1 3 Vðu; R0 Þ þ : uu þ 2! uu 3! uuu u¼0
Fig. 1. Definitions of sublattices A, B and C for zincblende structure CuBr. When Cu þ at an A site gets excited to its adjacent B site (arrow), then Br-s at the B’s nearest neighbor sites C distort towards the B. la;l⬘a⬘
having the nearest-neighbor interaction energy e between A{l; a} and B{l⬘; a⬘}, where n 1a is the lattice gas operator taking nla ¼ 1 or 0 at ath site on A of the lth cell. Let us introduce the static distortion DR now considered to occur only when a mobile ion cluster is produced and thus this quantity is completely different from the displacement u of lattice vibration around the lattice. Therefore DR should be written as a function of the position where the cluster exists. Considering the above, we define the position vector R between nearest neighbor mobile (m) and immobile (im) ions by R ¼ R0 þ DR þ u;
(2)
where R 0 is defined as R0 ¼ R0 ðmÞ ¹ R0 ðimÞ for the complete lattice and DR ¼ DRðmÞ ¹ DRðimÞ and u ¼ uðmÞ ¹ uðimÞ, respectively, are the static and dynamic distortions. Equation (2) is interpreted as R ¼ R0 þ u at the position where no cluster exists, while it is R ¼ R0 þ DR þ u when cluster exists. Thus the interaction Hamiltonian between nearest neighbor pair clusters nla nl⬘a⬘ and distortions is taken as X 1X X 1 X Hint ¼ 2 C la僆A;B 1 l⬘a⬘ð⫽laÞ僆A;B l⬘a⬘ð⫽laÞ僆A;B
⫻ ðhC;la;l⬘a⬘ : DRC;la þ lC;la:l⬘a⬘ : uC;la Þnla nl⬘a⬘ ;
þ
: uuu þ … u¼0
1 2 Vð0; R0 Þ : DRDR þ …; u ¼ R ¹ ðR0 þ DRÞ; 2! R0 R0 ð4Þ
neglecting the coupling term of the form uDR. Since we are interested only in the elastic energy stored by the cluster induced static distortion, it is enough to consider the harmonic terms as simply extracted from equation (4) as X X 1 (5) HSD ¼ 2 mC;la : DRC;la DRC;la : C la僆A
From EXAFS analysis, we can know the average nearest neighbor radial distance hRi ¼ R¯ of AgI or CuBr, which is formed of static distortion DR and dynamic distortion hui ¼ u¯ such that R¯ ¼ R0 þ DR þ u¯ . This u¯ is the mean relative displacement between Ag and I, and originates from the anharmonic potential. So we consider the anharmonic vibrations (phonons) and simply take the following approximate Einstein Hamiltonian from equation (4): 1 2 1 2 2 3 p þ m0 q0 u þ zu ; (6) Hvib ¼ M 2m0 2 where m 0 is the bare reduced mass, q 0 is the bare energy and z is the coefficient of cubic anharmonic term [5]. Therefore we have the total Hamiltonian H as H ¼ HLG þ Hint þ HSD þ Hvib :
(7)
Corresponding to the Hamiltonian (7), we have the free energy. 3. FREE ENERGY
ð3Þ to the lowest-order in DR and u, where hC;la;l⬘a⬘ and lC;la;l⬘a⬘ defined in equation (3) are coupling constant
For the free energy consisting of the lattice gas terms, we use the pair approximation of the cluster variation method that is appropriate to strong coupling cases [6].
Vol. 108, No. 8
CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS
Assuming the translational symmetry of the lattice and using the pair-cluster density matrix ra;a⬘ ð{j}Þ with the relation na ¼ 12ð1 þ ja Þ for the spin variables ja ¼ ⫾ 1, the free energy can be obtained from Kikuchi [6] and Ducastella [7] cluster variation methods as FðrÞ ¼ Tr ra;a⬘ H þ b ¹ 1 " X ⫻ Tr ra;a⬘ ln ra;a⬘ ha;a⬘i
¹
X
#
ð8Þ
ð2ga ¹ 1ÞTr ra ln ra ;
Tra⬘ ra;a⬘ ¼ ra ; with b ¼ 1=kB T and 2g a the coordination number of a site. The site probability of finding a mobile ion on að¼ A or B) is defined by xai ¼ rai , in which i ¼ 1 or v corresponding to ion (j ¼ 1) or vacancy ðj ¼ ¹1Þ, respectively and the pair probability of ions {i; j} at a;a⬘ sites {a; a⬘} by yaa⬘ ij ¼ rij , both of which can be evaluated self-consistently later in Section 4. The corresponding free energy per pair ions on A and B to equations (1) and (3) can then be written as 1 ðFLG þ Fint Þ M ¯ þ luÞyAB ¼ 2geAB yAB 11 þ 2gðhDR 11 ¹ kB T ln G;
ln G ¼
X
ð2g ¹ 1Þ X
i;j¼1;v
ð9Þ Lðxai Þ
X
2g
Lðyaa⬘ ij Þ;
a;a⬘ða⫽a⬘Þ¼A;B
LðxÞ ¼ x ln x ¹ x, 2gBA ¼ 2gAB ¼ 2g ¼ 6 and 2gAC ¼ 2gBC ¼ 2g¯ ¼ 4, where 2ga⬘ a is the coordination number of a site to a⬘ sublattice, hnla nl⬘a⬘ i ¼ yaa⬘ 11 for nearest neighbor pair ions on {l; a; l⬘; a⬘}. Here we simply consider the static and dynamic distortions by their radial components DR and u. The elastic energy caused by the static distortion, on the other hand, is given from equation (5) by fSD ¼
hHSD i 2 : ¼ gmDR ¯ M
where m, q and hui ¼ u¯ are the variational parameters determined by the minimum principle. Thus the displacement u in f int of equation (9) is also replaced by u¯ . Then we have the total free energy per pair: (13) fpair ¼ fLG þ fint þ fSD þ fvib : 4. COUPLED EQUATIONS Dropping the suffixes as e ¼ eAB and y ¼ yAB 11 and minimizing the free energy (13) with respect to DR, the order parameter y defined by y ¼ xA1 ¹ xB1
(14) xA1
xB1
with the condition þ ¼ 1 and y, we have a set of equations h (15) DR ¼ ¹ y m and 1 þ y 2ð2g ¹ 1Þ 1 ¹ y ¹ 2y 2g ¼ 1; 1¹y 1 þ y ¹ 2y 4y2 ð1 ¹ y ¹ 2yÞð1 þ y ¹ 2yÞ g¯ g¯ ¼ exp ¹ b e þ l¯u þ hDR : g g
a¼A;B
i¼1;v
¹
X
we have the free energy of Fvib fvib ¼ M 1 2 hp i0 þ 12m0 q20 hu2 i0 þ zhu3 i0 ¼ 2m0 bÉq Éq bÉq ¹1 coth þ b ln 2sinh ¹ ; 2 2 2 ð12Þ
hai
fLG þ fint ¼
515
(10)
Finally as to the phonons, we utilize the variational principle as in our previous paper [8]. On taking the 0thorder trial Hamiltonian as 1 2 1 2 2 p þ mq ðu ¹ u¯ Þ ; (11) H0 ¼ M 2m 2
ð16Þ
Equation (15), indicating that DR ⫽ 0 only when the cluster exists y ⫽ 0, leads to the negative energy given by g¯ (17) 2gðhyDR ¯ þ 12mDR2 Þ ¼ ¹ h2 y2 ⬍ 0; m which is effectively an attractive energy between pairclusters, implying that the static distortion prefers to occur. On putting q ¼ u ¹ u¯ and again minimizing eqaution (13) with respect to variational parameters, we have m ¼ m0 and q2 ¼ q20 þ
6z u¯ ; m0
3zhq2 i0 þ 2gly ¯ ; 2 m0 q0 þ 3z u¯ É bÉq 2 coth : hq i0 ¼ 2m0 q 2 u¯ ¼ ¹
ð18Þ
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CLUSTER-INDUCED LOCAL DISTORTION IN IONIC CONDUCTORS
These equations (15), (16) and (18) give us the static and dynamic distortions DR and u¯ , the order parameter y, the pair probability y and the normalized phonon energy Éq. As previously known from equation (17), the free energy of the system gets lowered due to the clustering of mobile ions as the temperature increases. Thus it accelerates the instability of the low-temperature phase to transform into a disordered phase at critical temperature T C. In the present equivalent A, B-lattice system, the type of phase transition is mainly the firstorder type, although the lattice gas system only with f LG would cause the second-order type transition at a higher critical temperature than the present T C. 5. RESULTS The local static distortion DR(T) is proportional to the pair probability y as in equation (15). Therefore the distance of the pair-potential minimum of Ag–I or Cu–Br has a temperature dependence of h (19) Rst ðTÞ ¼ R0 þ DRðTÞ ¼ R0 ¹ y; m where R 0 is a constant value at some reference temperature. For h ⬎ 0, the distance of the pair-potential minimum decreases with temperature since the pair probability y increases. In order to examine the present result for the static distortion R st(T), we quantitatively apply it to g-CuBr by choosing the parameter values such that m0 ¼ 35:4 amu, ˚ and Éq0 ¼ 18:0 meV as the optical phonon R0 ¼ 2:44 A
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˚ , e ¼ 62:0 meV, z ¼ energy [9], m ¼ 1:70 eV A ˚ ¹1 ˚ ¹3 and coupling constants h ¼ 450 meV A ¹ 1:85 eV A ¹1 ˚ . The theoretical result indicated and l ¼ ¹ 9:00 meV A by the solid line shows a reasonable agreement with the experiments as shown in Fig. 2. Thus the negative static distortions found both in CuBr and AgI are due to the clustering of mobile ions. On the other hand, the average distance of nearest neighbor ions ¯ (20) hRðTÞi ¼ RðTÞ ¼ Rst ðTÞ þ u¯ ðTÞ; ¹2
is also evaluated with the same parameters and shown together in Fig. 2. The hRðTÞi mostly increases with temperature. These two theoretical curves are quite consistent with experimental results [3]. It is also seen that this model reproduces the negative expansion ¯ dRðTÞ=dT ⬍ 0 at around T C in Fig. 2. The same is true for AgI case as well although the result is not shown here. 6. DISCUSSION The theoretical model of the static and dynamic distortions in terms of the coupling between pair-clusters of lattice gas ions and distortions is proposed for defects system. It is found that the positive or negative expansion of Cu–Br and Ag–I distances with temperature possibly occurs. In the positive local expansion, the dynamic distortion is larger than the static distortion, while in the negative local expansion, the dynamic distortion is smaller than the static distortion. In Ge case [10], however, static distortion does not occur, since the crystal is comparatively rigid and defects are few.
¯ Fig. 2. Theoretical results of temperature dependence of Rst ðTÞ ¼ R0 þ DRðTÞ and hRðTÞi ¼ RðTÞ with Tc ¼ 658:5 K for g-CuBr, compared with experimental results X [3].
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The ionic conductor g-CuBr crystal, having the ˚ in nearest neighbor distance of Cu–Cu of 4.02 A the ideal zincblende structure, shows rather large ionic conductivity attributed to the hopping of mobile Cu ions [11]: the Bragg and diffuse scattering measurement ˚ of Cu–Cu reveals in this case that the peak at 2.4 A distribution function exists [12]. Thus it is considered ˚ that thermally generated pairs of Cu–Cu ions nearly 2.4 A apart couple with the local distortion of Cu–Br, which is exactly consistent with the present model. Finally, we comment on the negative expansion of the c-axis in b-AgI. This expansion is believed to be due to the same mechanism. The wurtzite structure tends to yield defects along the c-axis and so the negative local expansion occurs in this direction. It has been recently confirmed that the MD simulation reproduces defectinduced c-axis negative expansion and a-axis positive expansion in b-AgI [13]. Acknowledgements—The author would like to thank O. Kamishima for the calculation of Fig. 2. This work was partially supported by the Grant-in-Aid for Scientific Research (#07239101) from the Ministry of Education, Science, Sports and Culture.
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