- Email: [email protected]
Dipolar configuration in the ordered state of alkali cyanides
Solid State Communications, Vol. 39, pp. 181-184. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-1098/81/010181-04502.00/0
DIPOLAR CONFIGURATION IN THE ORDERED STATE OF ALKALI CYANIDES R. Pirc and I. Vilfan J. Stefan Institute, 61001 Ljubljana, Yugoslavia (Received 29 December 1980 by 8. Miihlschlegel) The electrostatic ground state energy of the fully ordered state in KCN and NaCN is evaluated. The main contribution is shown to arise from an indirect interaction of CN- dipoles via induced dipoles due to lattice displacements. The resulting effective interaction has a maximum at wave vector q = (0, 27r/b, 0), corresponding to an antiferroelectric state such that the dipole at the centre of the orthorhombic unit cell is antiparallel to the dipoles at the comers. The predicted configuration agrees with neutron scattering data. I. INTRODUCTION THE CONFIGURATION of CN- dipoles in the ordered phase of alkali cyanides has been the subject of several experimental and theoretical studies. From Raman scattering experiments Dultz [1,2] found evidence of an antiferroelectric ground state configuration. Fontaine [3] and Rowe e t al. [4, 5] carried out a neutron scattering analysis of NaCN and KCN observing a superlattice reflection peak in the [010] orthorhombic direction, which is paratlel to the dipolar axis. This suggested an antiferroelectric structure such that the dipole at the centre of the orthorhombic unit cell is antiparallel to the dipoles at the comers. Preliminary theoretical investigations of the ordered phase of KCN and NaCN were made by Siegel [6], Suga et al. [7], and Matsuo et al. [8]. These authors have pointed out that purely electrostatic forces between CN- dipoles on a rigid lattice would lead to a ferroelectric ground state. Dos Santos et al. [9] reported a detailed analysis of the dipole-dipole interactions for various configurations and again came to the conclusion that electrostatic forces alone favoured a ferroelectric state. The same authors argued that elastic dipolar interactions might be responsible for the actual antiferroelectric configuration. They predicted a state in which the (101) planes of dipoles oriented along the [010] direction alternate with parallel planes of [010] dipoles. In this paper we present a theoretical analysis of the possible dipolar arrangements in the ground state of KCN and NaCN, assuming that the lattice formed by the CN- ions and the K÷ or Na ÷ cations, respectively, is deformable. Thus in addition to the direct CN--CNdipolar interaction we have to consider interactions between CN- dipoles and the induced dipoles created by the displacements of the charges of CN- ions and cations
from their equilibrium positions. It is known from the work of Kanamori [10] on Jahn-Teller systems, Sakurai et al. [11 ] on ferroelectric NaNO2, and Yamada et al. [ 12] on NH4Br that the above mechanism leads to an indirect interaction, which may alter the ground state dipolar configuration. A recent review of phase transitions in these systems is given by Cowley [13]. In the following we propose a model Hamiltonian for the low temperature phase of alkali cyanides and evaluate the effective dipolar interaction energy using the available data for the model parameters of KCN and NaCN. 2. THE MODEL The structure of KCN and NaCN in both lower phases is body-centred orthorhombic. Above the transition temperature To, the CN- ions are aligned along the b-axis corresponding to the [010] direction, the left-fight distribution of the dipoles being still completely random. At T -- Tc, where Tc(KCN) = 83 K and Tc(NaCN) -- 172 K, a second-order phase transition of order-disorder type to an electrically ordered phase OCCUrs.
As shown by Holden et al. [14], near the phase transition the system behaves effectively as an Ising model. Therefore, to describe the dipolar orientation, we can introduce a pseudo-spin S t = ± 1, where Si -- + 1 for a dipole oriented parallel, and Sl -- - 1 for a dipole oriented antiparallel to the [010] direction, respectively, i being the cell index. The operator for the ith dipole moment is then D~ = PSi, where p is the dipolar moment of CN-. We shall further denote by Z~ the charge of the ion K, and let K = 0 for a CN- ion and K -- 1 for a cation. When the charge Z~ is displaced from its equilibrium position by an amount u~(K), a = x , y , z, the resulting induced dipole moment is D~i(K) = Z~u~(~). Expressing
181
182
DIPOLAR CONFIGURATION IN ALKALI CYANIDES
the displacements u~'(r) in terms of normal modes Q~(X), ~, being the branch index, the Hamiltonian of the system can be written in the form
H =
x
--
½p2 Z Jc~YfOO)SqS_q - - p Z Jg e~fOK)Z~cM;1/2 q qh
Eq(K~ IX)Qq(X)S_q + ½ ~ [¢%(X)2Qq(X)Q_q(X)
qh
+ ?~(x)e_,(X)l,
(l)
where the first term represents the dipole-dipole interactions between all CN- ions, the second term a crossinteraction between all permanent and induced dipoles, and the last term the vibrational energy of the lattice. As usual, ¢oq(h) are the phonon frequencies, e,(Ka IX) the polarisation vectors, Pq(~,) the normal momenta and M~ the ion masses. J~a(0K) is the Fourier transformed dipolar interaction matrix J~a(0K) defined as the interaction energy of two unit dipoles, the one at R i + r K-o pointing along the a-axis, where rK is the basis vector, and the other at R / + rK along the ¢]-axis: J~af0u) = ~. [3 (RO +lr°~)afRiiR,l + ro~ I+s r°~)~
] -- I Rij + roK ]-3~ot/~](l --~oJ¢~i./)e iq'sL/. (2) J ltere R~j = R i -- R I and ro~ - re - r~. The cross-interaction gives rise to lattice distortions, a q ( x ) = Qq(X) + AQq(X), where 0q(X) are new instantaneous equilibrium coordinates, and AQq(~,) describe vibrations around the distorted positions. The distortions Qq(X) can be eliminated from H by a canonical transformation [13], leading to a new equivalent Hamiltonian
n : - ½p2 y s U s : _ , q
+ ½E q
× [¢.Oq(~.)2AQq(X) AQ_q(~k) -t- ?q(~.)e_q(~x)].
(3)
Now J~ff is the effective dipolar interaction between CN- ions,
Vol. 39, No. 1
CN- dipole with itself; it is determined by the condition that the sum over q o f J ~ tf must vanish [13]. Notice that C >~ 0. The Hamiltonian (3) has the form of an Ising model with modified dipolar interactions in d = 3 dimensions. The ground state configuration is characterised by a wave vector q = qo such that lim j~fr = ~[te~f~ v ~ zmax is
q--'qo
the absolute maximum with respect to all possible values of qo and directions of q on approaching q0. In a meanfield approximation, the transition temperature Tc is determined by the condition k a T c = oP 2(Y~o)m,~, e~t where the parameter o < 1 includes the effect of dipolar directions other than [010] or [0i0] [14]. For KCN, o = 0.92 [14], whereas o = 1 for a purely Ising system. Clearly, to determine T e one needs the value of the constant C in equation (4) which could only be calculated if one knew the frequencies and polarisation vectors for all q and h, say, from a shell-model analysis. Such an analysis has so far been made only for the high-temperature phase of alkali cyanides [15]. One can, however, evaluate j~rf relative to C for certain symmetry points qo in the first Brillouin zone at which the transition is likely to occur. The largest relative value o f J ~ ~t will then determine the wave vector qo and the order parameter ( S ~ >, and in this way provide information on the structure of the ordered phase. Below, we evaluate z"qo ett for a number of symmetry points qo2.1. qo = (0, 0, 0) Due to the anisotropy of dipolar interactions, j~ft depends on the direction o f q as q ~ 0 [8, 16]. In our case, the maximum o f J ~ ~f occurs in the direction q .L [010]. Since only one transverse optic mode is polarised parallel to the b.axis, and the contribution of all acoustic branches vanishes as q -~ 0, the sum over X contains only one mode, ~ = TO. The polarisation vectors for this mode are such that the ions K = 0 and K = 1 vibrate out of phase, the centre of mass being at rest [17] : g ~ / 'eo(Oy ITO) = -- MI / 2eo(ly ITO).
(5)
Together with orthonormality relation
1
Jg," = s~,(oo)+ ~ '°~x) 2
l eo(0y I TO) 12+ I eo(ly I TO) 12 = 1 2
(4)
this leads to the result:
(:~"),,,~ = where the first part is as in equation (I), and the additional term can be interpreted as an indirect C N - - C N - interaction due to the polarisation of the medium consisting of point charges on a deformable lattice. The constant C has been subtracted in order to eliminate from the second term the interaction of a
(6)
Z2 Jo"(oo)
x JoYY(O0)+ Here Z
= Z~ = - - Z o .
Mi
+ - ~ o o ( T O ) 2 Mo(Mo
J~'(Ol)
-- C.
+ M,) (7)
Vol. 39, No. 1
DIPOLAR CONFIGURATION IN ALKALI CYANIDES
183
Table 1. Parameters u ~ d in calculation o f J~tt
Quantity
Symbol
KCN
NaCN
Lattice parameters of the ordered phase (in rim)s*b
a b
0.418 0.524 0.607
0.363 0.485
180 185
220
Lattice frequency (era- l) q = 0e q = (n/a, O, n/c) c
q = (o, n/b, n/c) ~
q
= (0,2n/b, O)d
C ~oq(TO) toq(TO) ¢oq(TA) toq(TO) ~oq(LO) ~q(TA) ~oq(LA) ~q(LA) ~q(LO)
b~:omponent of the polarisation vector q = (0, 2n/b, 0) e
eq(Oy I l_A) eq( 1Yl LA) eq(0y I LO) eq( l y l LO )
Ionic charge (units of eo)e
Z
0.545
100 185 195 80
250 120 170
150 190
- 0.43 0.90 0.90 0.43
0.78 - 0.63 0.63 0.78
0.86
1.09
• Ref. [4]. b Ref. [19]. e Estimated from the data of the cubic phase [15]. d Experimental data from Raman scattering [2, 18, 19]. • Representative figure, estimated from the rigid ion model. 2.2. q'o = (± n/a, 0, ± n/c) and q; = (0, ± n/b , ± n/c)
e(Oyl L~) = c/(b 2 + c2)1/2;
These vectors lie at the zone boundary. From symmetry considerations it follows that in the optic modes only the lighter ions vibrate and the heavier ones are at rest, whereas the opposite is true for the acoustic branches. In the longitudinal modes, the polarisation vectors are parallel to q, while in the transverse modes they are oriented along [010] and [101] for q = q~, and along [I00] and [011] for q = q~, respectively. Another feature of these symmetry points is that the dipolar interaction between different sublattices vanishes, i.e.
¢(0y ITIT,) = -- b/(b 2 + c2) 1/2.
~J(0t)=0. For q = q~ we obtain the result:
The effective interaction becomes in this case
: ' " = J~(00) + z2 [J~(0o)] 2 q~
Mo
.[e(OY L X ) 2I ~ ×/
e(0y [TIX)2 ]
~t
Here ~oq;(TtX) is the frequency of the TIO mode polafiaed along [010] for KCN, whereas in the case of NaCN this is a T~A mode of the same polarisation. For q = q~ one longitudinal branch (IX) and one transverse branch (TIX), where X = O for KCN and X = A for NaCN, can have the y-component of their polarisation vectors different from zero. It follows
o0)
rt
2.3. q0 = (± 2n/a, O, 0), q0 = (0, ± 2n/b, 0), qo = (0, 0, ± 2n/c) These wave vectors correspond to the symmetry point X at the zone boundary [18] and are all equivalent since they differ by reciprocal lattice vectors. The polarisation vectors are parallel to the crystal axes. Therefore, for q - qo the polarisations of both longitudinal modes, LA and LO, axe parallel to the b-axis, whereas the polarisations of TA and TO modes axe perpendicular to the b-axis and thus do not ftt
(8)
-c.
Since at this symmetry point longitudinal vibrations have higher frequencies than the corresponding transverse ones, j~!t is larger the, l - J~'etf" r
Z2 = J ~ ( 0 0 ) + t~q;(T,X)2M0 [J~o~(00)12 - C.
(9)
m
tw
184
DIPOLAR CONFIGURATION IN ALKALI CYANIDES
Table 2. Dipolar lattice sums J~(00), J~Y(01)and effective dipolar interaction j~ft for symmetry points in the reciprocal space o f KCN and NaCN. In units o f nm-3
q
Jg'(0o)
Jg'(01)
Jg" + c
54.44 -- 44.72 41.97 -- 5.91
65.47 0 0 -- 153.83
74.1 -- 38.6 47.7 96.7
72.16 --63.10 57.23 - 20.37
95.60 0 0 - 225.88
KCN (0, 0, 0) (0, ± n/b, +-n/c) (± rr/a, O, ± ~r[c) (0, ± 2n/b, 0) NaCN (0,0,0) (O,±n[b,+-Tr/c) (+- n]a, O, +-Ir/c) (0, ± 2~r/b, 0)
-
128.0 43.0 115.0 321.0
contribute to ~r~ ff. For q = qo and q = qoFFP the polarisations of T~A and TtO modes are parallel to the b-axis, and the other modes, T2A, T20, LA and LO make no contribution. No other simplifications are possible in this case, and J~off is given by the general result (4), where, however,
The effective interaction J~ff has a pronounced maximum at wave vectors qo = (0, 0, ± 21r/c), qo = O, +-2*r/b, 0), or qg' = (± 27r/a, 0, 0), that is in the case 2.3 above. Each of these wave vectors corresponds to a modulation of the order parameter
Acknowledgement - This work has been supported by the Research Council of Slovenia through Grant No. G-106[2822-80. REFERENCES 1. 2. 3. 4.
~, = TIA, T10 for q = q~ and q = qo" 5.
and ~, = LA, L O f o r q = qo. It should again be noted that J~!f = J ~ f = J~!,f,. 3. RESULTS The dipolar interaction matrices J~Y(00) and J ~ ( 0 1 ) have been calculated using a generalised Ewald theta function transformation method [ 17]. The physical parameters for KCN and NaCN used in evaluating J~ ft are listed in Table 1. The polarisation vectors needed in the case 2.3 were estimated on the basts of a rigid ion model. It should be stressed, however, that variations in the relative magnitudes of the polarisation vectors have little effect upon the value of J~ff, provided that the vectors are properly normalised. The results are summarised in Table 2. The bare electrostatic dipole--dipole interaction J ~ ( 0 0 ) has its maximum at q = 0 with q approaching zero perpendicular to the [010] direction. Our values agree with DOs Santos et al. [9], though minor discrepancies exist because we use slightly different lattice constants. Obviou.qly, the bare dipolar interaction favours a ferroelectric ordering of the CN- ions in both KeN and NaCN.
Vol. 39, No. 1
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
18. 19.
W. Dultz, Solid State Commun. 15,595 (1974). W. Dultz,J. Chem. Phys. 65, 2812 (1976). D. Fontaine, C.R. Acad. Sci. Paris 281, B-443 (1975). J.M. Rowe, J.J. Rush & E. Prince, J. Chem. Phys. 66, 5147 (1977). J.M. Rowe, J.J. Rush, E. Prince & N.J. Chesser, Ferroelectrics 16,107 (1977). L.A. Siegel, J.. Chem. Phys. 17, 1146(1949). H. Suga, T. Matsuo & S. Seki, Bull Chem. Soc. Japan 38, 1115 (1965). T. Matsuo, H. Suga & S. Seki, Bull. Chem. Soc. Japan 41,583 (1968). R.R. Dos Santos, B. Koiller, J.P. yon der Weid, S. Costa Ribeiro, A.S. Chaves & F.C. Sa Barreto, J. Phys. C: Solid State Phys. 11,4557 (1978). J. Kanamori,J. Appl. Phys. 31, 14S (1960). J. Sakurai, R.A. Cowley & G. Dolling, J. Phys. Soc. Japan 28, 1426 (1970). Y. Yamada, M. Mori & Y. Noda, J. Phys. Soc. Japan 32, 1565 (1972). R.A. Cowley,Adv. Phys. 29, 1 (1980). A.J. Holden, V. Heine, J.C. lnkson, C.M. Varma & M.A. Bosch, J. Phys. C: Solid State Phys. 12, 1035 (1979). H. Jex & C.J. Maetz, Phys. Status Solidi (b) 85, 511 (1978). M.H. Cohen & F. Keffer, Phys. Rev. 99, 1128 (1955). A.A. Maradudin, E.W. Montroll, G.H. Weiss & I.P. Ipatova, Theory o f Lattice Dynamics in the Harmonic Approximation. Academic Press, New York (1971). W. Dultz, H. Krause & L.W. Winchester, Jr., Z Che~ Phys. 67, 2560 (1977).
D. Fontaine & H. Poulet, Phys. Status Solidi (b) 58, K9 (1973).
0038-1098/81/010181-04502.00/0
DIPOLAR CONFIGURATION IN THE ORDERED STATE OF ALKALI CYANIDES R. Pirc and I. Vilfan J. Stefan Institute, 61001 Ljubljana, Yugoslavia (Received 29 December 1980 by 8. Miihlschlegel) The electrostatic ground state energy of the fully ordered state in KCN and NaCN is evaluated. The main contribution is shown to arise from an indirect interaction of CN- dipoles via induced dipoles due to lattice displacements. The resulting effective interaction has a maximum at wave vector q = (0, 27r/b, 0), corresponding to an antiferroelectric state such that the dipole at the centre of the orthorhombic unit cell is antiparallel to the dipoles at the comers. The predicted configuration agrees with neutron scattering data. I. INTRODUCTION THE CONFIGURATION of CN- dipoles in the ordered phase of alkali cyanides has been the subject of several experimental and theoretical studies. From Raman scattering experiments Dultz [1,2] found evidence of an antiferroelectric ground state configuration. Fontaine [3] and Rowe e t al. [4, 5] carried out a neutron scattering analysis of NaCN and KCN observing a superlattice reflection peak in the [010] orthorhombic direction, which is paratlel to the dipolar axis. This suggested an antiferroelectric structure such that the dipole at the centre of the orthorhombic unit cell is antiparallel to the dipoles at the comers. Preliminary theoretical investigations of the ordered phase of KCN and NaCN were made by Siegel [6], Suga et al. [7], and Matsuo et al. [8]. These authors have pointed out that purely electrostatic forces between CN- dipoles on a rigid lattice would lead to a ferroelectric ground state. Dos Santos et al. [9] reported a detailed analysis of the dipole-dipole interactions for various configurations and again came to the conclusion that electrostatic forces alone favoured a ferroelectric state. The same authors argued that elastic dipolar interactions might be responsible for the actual antiferroelectric configuration. They predicted a state in which the (101) planes of dipoles oriented along the [010] direction alternate with parallel planes of [010] dipoles. In this paper we present a theoretical analysis of the possible dipolar arrangements in the ground state of KCN and NaCN, assuming that the lattice formed by the CN- ions and the K÷ or Na ÷ cations, respectively, is deformable. Thus in addition to the direct CN--CNdipolar interaction we have to consider interactions between CN- dipoles and the induced dipoles created by the displacements of the charges of CN- ions and cations
from their equilibrium positions. It is known from the work of Kanamori [10] on Jahn-Teller systems, Sakurai et al. [11 ] on ferroelectric NaNO2, and Yamada et al. [ 12] on NH4Br that the above mechanism leads to an indirect interaction, which may alter the ground state dipolar configuration. A recent review of phase transitions in these systems is given by Cowley [13]. In the following we propose a model Hamiltonian for the low temperature phase of alkali cyanides and evaluate the effective dipolar interaction energy using the available data for the model parameters of KCN and NaCN. 2. THE MODEL The structure of KCN and NaCN in both lower phases is body-centred orthorhombic. Above the transition temperature To, the CN- ions are aligned along the b-axis corresponding to the [010] direction, the left-fight distribution of the dipoles being still completely random. At T -- Tc, where Tc(KCN) = 83 K and Tc(NaCN) -- 172 K, a second-order phase transition of order-disorder type to an electrically ordered phase OCCUrs.
As shown by Holden et al. [14], near the phase transition the system behaves effectively as an Ising model. Therefore, to describe the dipolar orientation, we can introduce a pseudo-spin S t = ± 1, where Si -- + 1 for a dipole oriented parallel, and Sl -- - 1 for a dipole oriented antiparallel to the [010] direction, respectively, i being the cell index. The operator for the ith dipole moment is then D~ = PSi, where p is the dipolar moment of CN-. We shall further denote by Z~ the charge of the ion K, and let K = 0 for a CN- ion and K -- 1 for a cation. When the charge Z~ is displaced from its equilibrium position by an amount u~(K), a = x , y , z, the resulting induced dipole moment is D~i(K) = Z~u~(~). Expressing
181
182
DIPOLAR CONFIGURATION IN ALKALI CYANIDES
the displacements u~'(r) in terms of normal modes Q~(X), ~, being the branch index, the Hamiltonian of the system can be written in the form
H =
x
--
½p2 Z Jc~YfOO)SqS_q - - p Z Jg e~fOK)Z~cM;1/2 q qh
Eq(K~ IX)Qq(X)S_q + ½ ~ [¢%(X)2Qq(X)Q_q(X)
qh
+ ?~(x)e_,(X)l,
(l)
where the first term represents the dipole-dipole interactions between all CN- ions, the second term a crossinteraction between all permanent and induced dipoles, and the last term the vibrational energy of the lattice. As usual, ¢oq(h) are the phonon frequencies, e,(Ka IX) the polarisation vectors, Pq(~,) the normal momenta and M~ the ion masses. J~a(0K) is the Fourier transformed dipolar interaction matrix J~a(0K) defined as the interaction energy of two unit dipoles, the one at R i + r K-o pointing along the a-axis, where rK is the basis vector, and the other at R / + rK along the ¢]-axis: J~af0u) = ~. [3 (RO +lr°~)afRiiR,l + ro~ I+s r°~)~
] -- I Rij + roK ]-3~ot/~](l --~oJ¢~i./)e iq'sL/. (2) J ltere R~j = R i -- R I and ro~ - re - r~. The cross-interaction gives rise to lattice distortions, a q ( x ) = Qq(X) + AQq(X), where 0q(X) are new instantaneous equilibrium coordinates, and AQq(~,) describe vibrations around the distorted positions. The distortions Qq(X) can be eliminated from H by a canonical transformation [13], leading to a new equivalent Hamiltonian
n : - ½p2 y s U s : _ , q
+ ½E q
× [¢.Oq(~.)2AQq(X) AQ_q(~k) -t- ?q(~.)e_q(~x)].
(3)
Now J~ff is the effective dipolar interaction between CN- ions,
Vol. 39, No. 1
CN- dipole with itself; it is determined by the condition that the sum over q o f J ~ tf must vanish [13]. Notice that C >~ 0. The Hamiltonian (3) has the form of an Ising model with modified dipolar interactions in d = 3 dimensions. The ground state configuration is characterised by a wave vector q = qo such that lim j~fr = ~[te~f~ v ~ zmax is
q--'qo
the absolute maximum with respect to all possible values of qo and directions of q on approaching q0. In a meanfield approximation, the transition temperature Tc is determined by the condition k a T c = oP 2(Y~o)m,~, e~t where the parameter o < 1 includes the effect of dipolar directions other than [010] or [0i0] [14]. For KCN, o = 0.92 [14], whereas o = 1 for a purely Ising system. Clearly, to determine T e one needs the value of the constant C in equation (4) which could only be calculated if one knew the frequencies and polarisation vectors for all q and h, say, from a shell-model analysis. Such an analysis has so far been made only for the high-temperature phase of alkali cyanides [15]. One can, however, evaluate j~rf relative to C for certain symmetry points qo in the first Brillouin zone at which the transition is likely to occur. The largest relative value o f J ~ ~t will then determine the wave vector qo and the order parameter ( S ~ >, and in this way provide information on the structure of the ordered phase. Below, we evaluate z"qo ett for a number of symmetry points qo2.1. qo = (0, 0, 0) Due to the anisotropy of dipolar interactions, j~ft depends on the direction o f q as q ~ 0 [8, 16]. In our case, the maximum o f J ~ ~f occurs in the direction q .L [010]. Since only one transverse optic mode is polarised parallel to the b.axis, and the contribution of all acoustic branches vanishes as q -~ 0, the sum over X contains only one mode, ~ = TO. The polarisation vectors for this mode are such that the ions K = 0 and K = 1 vibrate out of phase, the centre of mass being at rest [17] : g ~ / 'eo(Oy ITO) = -- MI / 2eo(ly ITO).
(5)
Together with orthonormality relation
1
Jg," = s~,(oo)+ ~ '°~x) 2
l eo(0y I TO) 12+ I eo(ly I TO) 12 = 1 2
(4)
this leads to the result:
(:~"),,,~ = where the first part is as in equation (I), and the additional term can be interpreted as an indirect C N - - C N - interaction due to the polarisation of the medium consisting of point charges on a deformable lattice. The constant C has been subtracted in order to eliminate from the second term the interaction of a
(6)
Z2 Jo"(oo)
x JoYY(O0)+ Here Z
= Z~ = - - Z o .
Mi
+ - ~ o o ( T O ) 2 Mo(Mo
J~'(Ol)
-- C.
+ M,) (7)
Vol. 39, No. 1
DIPOLAR CONFIGURATION IN ALKALI CYANIDES
183
Table 1. Parameters u ~ d in calculation o f J~tt
Quantity
Symbol
KCN
NaCN
Lattice parameters of the ordered phase (in rim)s*b
a b
0.418 0.524 0.607
0.363 0.485
180 185
220
Lattice frequency (era- l) q = 0e q = (n/a, O, n/c) c
q = (o, n/b, n/c) ~
q
= (0,2n/b, O)d
C ~oq(TO) toq(TO) ¢oq(TA) toq(TO) ~oq(LO) ~q(TA) ~oq(LA) ~q(LA) ~q(LO)
b~:omponent of the polarisation vector q = (0, 2n/b, 0) e
eq(Oy I l_A) eq( 1Yl LA) eq(0y I LO) eq( l y l LO )
Ionic charge (units of eo)e
Z
0.545
100 185 195 80
250 120 170
150 190
- 0.43 0.90 0.90 0.43
0.78 - 0.63 0.63 0.78
0.86
1.09
• Ref. [4]. b Ref. [19]. e Estimated from the data of the cubic phase [15]. d Experimental data from Raman scattering [2, 18, 19]. • Representative figure, estimated from the rigid ion model. 2.2. q'o = (± n/a, 0, ± n/c) and q; = (0, ± n/b , ± n/c)
e(Oyl L~) = c/(b 2 + c2)1/2;
These vectors lie at the zone boundary. From symmetry considerations it follows that in the optic modes only the lighter ions vibrate and the heavier ones are at rest, whereas the opposite is true for the acoustic branches. In the longitudinal modes, the polarisation vectors are parallel to q, while in the transverse modes they are oriented along [010] and [101] for q = q~, and along [I00] and [011] for q = q~, respectively. Another feature of these symmetry points is that the dipolar interaction between different sublattices vanishes, i.e.
¢(0y ITIT,) = -- b/(b 2 + c2) 1/2.
~J(0t)=0. For q = q~ we obtain the result:
The effective interaction becomes in this case
: ' " = J~(00) + z2 [J~(0o)] 2 q~
Mo
.[e(OY L X ) 2I ~ ×/
e(0y [TIX)2 ]
~t
Here ~oq;(TtX) is the frequency of the TIO mode polafiaed along [010] for KCN, whereas in the case of NaCN this is a T~A mode of the same polarisation. For q = q~ one longitudinal branch (IX) and one transverse branch (TIX), where X = O for KCN and X = A for NaCN, can have the y-component of their polarisation vectors different from zero. It follows
o0)
rt
2.3. q0 = (± 2n/a, O, 0), q0 = (0, ± 2n/b, 0), qo = (0, 0, ± 2n/c) These wave vectors correspond to the symmetry point X at the zone boundary [18] and are all equivalent since they differ by reciprocal lattice vectors. The polarisation vectors are parallel to the crystal axes. Therefore, for q - qo the polarisations of both longitudinal modes, LA and LO, axe parallel to the b-axis, whereas the polarisations of TA and TO modes axe perpendicular to the b-axis and thus do not ftt
(8)
-c.
Since at this symmetry point longitudinal vibrations have higher frequencies than the corresponding transverse ones, j~!t is larger the, l - J~'etf" r
Z2 = J ~ ( 0 0 ) + t~q;(T,X)2M0 [J~o~(00)12 - C.
(9)
m
tw
184
DIPOLAR CONFIGURATION IN ALKALI CYANIDES
Table 2. Dipolar lattice sums J~(00), J~Y(01)and effective dipolar interaction j~ft for symmetry points in the reciprocal space o f KCN and NaCN. In units o f nm-3
q
Jg'(0o)
Jg'(01)
Jg" + c
54.44 -- 44.72 41.97 -- 5.91
65.47 0 0 -- 153.83
74.1 -- 38.6 47.7 96.7
72.16 --63.10 57.23 - 20.37
95.60 0 0 - 225.88
KCN (0, 0, 0) (0, ± n/b, +-n/c) (± rr/a, O, ± ~r[c) (0, ± 2n/b, 0) NaCN (0,0,0) (O,±n[b,+-Tr/c) (+- n]a, O, +-Ir/c) (0, ± 2~r/b, 0)
-
128.0 43.0 115.0 321.0
contribute to ~r~ ff. For q = qo and q = qoFFP the polarisations of T~A and TtO modes are parallel to the b-axis, and the other modes, T2A, T20, LA and LO make no contribution. No other simplifications are possible in this case, and J~off is given by the general result (4), where, however,
The effective interaction J~ff has a pronounced maximum at wave vectors qo = (0, 0, ± 21r/c), qo = O, +-2*r/b, 0), or qg' = (± 27r/a, 0, 0), that is in the case 2.3 above. Each of these wave vectors corresponds to a modulation of the order parameter
Acknowledgement - This work has been supported by the Research Council of Slovenia through Grant No. G-106[2822-80. REFERENCES 1. 2. 3. 4.
~, = TIA, T10 for q = q~ and q = qo" 5.
and ~, = LA, L O f o r q = qo. It should again be noted that J~!f = J ~ f = J~!,f,. 3. RESULTS The dipolar interaction matrices J~Y(00) and J ~ ( 0 1 ) have been calculated using a generalised Ewald theta function transformation method [ 17]. The physical parameters for KCN and NaCN used in evaluating J~ ft are listed in Table 1. The polarisation vectors needed in the case 2.3 were estimated on the basts of a rigid ion model. It should be stressed, however, that variations in the relative magnitudes of the polarisation vectors have little effect upon the value of J~ff, provided that the vectors are properly normalised. The results are summarised in Table 2. The bare electrostatic dipole--dipole interaction J ~ ( 0 0 ) has its maximum at q = 0 with q approaching zero perpendicular to the [010] direction. Our values agree with DOs Santos et al. [9], though minor discrepancies exist because we use slightly different lattice constants. Obviou.qly, the bare dipolar interaction favours a ferroelectric ordering of the CN- ions in both KeN and NaCN.
Vol. 39, No. 1
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
18. 19.
W. Dultz, Solid State Commun. 15,595 (1974). W. Dultz,J. Chem. Phys. 65, 2812 (1976). D. Fontaine, C.R. Acad. Sci. Paris 281, B-443 (1975). J.M. Rowe, J.J. Rush & E. Prince, J. Chem. Phys. 66, 5147 (1977). J.M. Rowe, J.J. Rush, E. Prince & N.J. Chesser, Ferroelectrics 16,107 (1977). L.A. Siegel, J.. Chem. Phys. 17, 1146(1949). H. Suga, T. Matsuo & S. Seki, Bull Chem. Soc. Japan 38, 1115 (1965). T. Matsuo, H. Suga & S. Seki, Bull. Chem. Soc. Japan 41,583 (1968). R.R. Dos Santos, B. Koiller, J.P. yon der Weid, S. Costa Ribeiro, A.S. Chaves & F.C. Sa Barreto, J. Phys. C: Solid State Phys. 11,4557 (1978). J. Kanamori,J. Appl. Phys. 31, 14S (1960). J. Sakurai, R.A. Cowley & G. Dolling, J. Phys. Soc. Japan 28, 1426 (1970). Y. Yamada, M. Mori & Y. Noda, J. Phys. Soc. Japan 32, 1565 (1972). R.A. Cowley,Adv. Phys. 29, 1 (1980). A.J. Holden, V. Heine, J.C. lnkson, C.M. Varma & M.A. Bosch, J. Phys. C: Solid State Phys. 12, 1035 (1979). H. Jex & C.J. Maetz, Phys. Status Solidi (b) 85, 511 (1978). M.H. Cohen & F. Keffer, Phys. Rev. 99, 1128 (1955). A.A. Maradudin, E.W. Montroll, G.H. Weiss & I.P. Ipatova, Theory o f Lattice Dynamics in the Harmonic Approximation. Academic Press, New York (1971). W. Dultz, H. Krause & L.W. Winchester, Jr., Z Che~ Phys. 67, 2560 (1977).
D. Fontaine & H. Poulet, Phys. Status Solidi (b) 58, K9 (1973).