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Biquadratic exchange interaction between Ru5+ ions in (Ru2O9) clusters
"Solid State Communications Vol. 21, pp: 425--427, 1977.
Pergamon Press.
Printed in Great Britain
BIQUADRATIC EXCHANGE INTERACTION BETWEEN Ru s+' IONS IN (Ru2 09) CLUSTERS M. Drillon Laboratoire de Chimie du Solide du CNRS, Universit6 de Bordeaux I, 351 cours de la Liberation, 33405 Talence-Cedex, France
(Received 28 September 1976 by E.F. Bertaut) The energy levels of a Ru209 cluster have been calculated, including a higher order spin interaction. The RuS+-Ru s÷ coupling is described by the Hamiltonian ~f = -- 2JS1 -$2 --/(St .$2) 2 • The temperature dependence of the magnetic susceptibility is used to determine the values of the bilinear J and biquadratic/exchange integrals: d/k = -- 161 K and ]/k = 6.6 K. The second term in the Hamiltonian corresponds to a fourth order perturbation involving low spin states. THE CATIONIC ordering in the BaCal/a Ruz/aOa oxide which crystallizes with the 6H perovskite derived structure gives rise to RuS+-Ru s+ pairs in (Ru209) clusters (Fig. 1). In an earlier paper, the magnetic properties have been analyzed using a vector model of spin S = 3/2, where a quenching of orbital angular momentum by cristalline field was assumed. 1 We have proposed an exchange Hamiltonian of the usual form: ~Tex = -- 2JSt. $2
(1)
where St = $2 = 3/2 and the isotropic exchange interaction J is negative (antiferromagnetic). The eigenvalues of this Hamiltonian are given by:
E = --J.S(S + 1)
(2)
where S = Sl + $2 (S = O, 1, 2, 3). Using J/k = -- 170 K, the calculation of the magnetic susceptibility gives a good agreement with the experimental values at high temperature, but slight discrepancies appear when temperature decreases (Fig. 2). In this Note we report the results of an investigation of the interactions which take account of the temperature dependance of the susceptibility, particularly at low temperature. The spin Hamiltonian for a binuclear cluster may be written: ~f = ~Ct + ~2 + ~ex. (3) In this expression ~ft and ~f2 are single-ion spin Hamiltonians and ~ex is the coupling term, which is given by: ~fex = -- 2dSt. S2 - - / ( S t - $2) 2
+ c.
sl.s2-~(S,rt~)(S~r~)
+ D . ( s , x s~).
The last term, postulated by Dzialoshinsky is the antisymmetrical spin-spin coupling. 2 This interaction 425
Ru5+ Fig. 1. Geometry of the RuS*--Ru s÷ pairs in (Ru2Og) clusters. tends to cant the spins, since the energy is the lowest when the spinsare perpendicular to each other. This term can have an appreciable effect when both ions are chemically dissimilar, or in the absence of a center of symmetry between the magnetic sites. In our case we can suppose that its influence will be relatively small. The previous term derives from the familiar pseudodipolar interaction, a The orbital singlet 4/12 ground state of Ru s+ ions being far away below the degenerate excited states, it may be expected to be negligibly small in the present case. In other words the hamiltonian may be written as the sum of an isotropie bilinear exchange and an isotropic biquadratic contribution:
INTERACTION BETWEEN Rus÷IONS IN (Ru=09) CLUSTERS
426
t XM (uera/mole)
E/J
12
//
4ool J=-,7oK (5)
The higher order spin coupling may arise either from fourth-order transfer processes4 or from exchange strictions. 5 In view of the obtained results, we only shall take account of the first mechanism.
Ngu~ X = - kT
S=2
S=I
S=I
S:O
S=O
j / k =OK
j / J = 0,05
j2 En
Fig. 3. The realtive energies and possible spin states of a binuclear cluster: Without the biquadratic exchange (A); Including the biquadratic exchange (B). with the energy levels calculated with the classical Hamiltonian (1). The magnetic susceptibility is easily determined using Van Vleck's formula, which is simplified in this case since the matrix elements of Sz between states with different S vanish, s The expression obtained for × (per ion) is the following:
exp ( 2 x - - 6.5y) + 5 exp ( 6 x - - 13.5y) + 14 exp ( 1 2 x - - 9y) 1 + 3 exp (2x -- 6.5y) + 5 exp (6x ~ 13.5y) + 7 exp (12x -- 9y)
The contributions due to transfer processes are stemed from fourth-order perturbation terms which involve low-spin states at an anergy E n (Hund's rule energy) above the ground state. The resulting coefficient of the biquadratic exchange / has been approximatively calculated by van Stapele :4 (6)
where b is the transfer integral. From equation (6) it follows that the j value is positive and of the order of few percent o f £ The eigenvalues of the new exchange Hamiltonian are given by: E = -J.S(S+
S=3
800
~fex = -- 2JSt- S2 --/(St. S:) 2.
b4
S= 3
4
Fig. 2. Magnetic susceptibilities of BaCal/aRu2/aO a. The broken and solid lines represent the best fit to the dimer equation: (1) Without the biquadratic exchange; (2) Ineluding the biquadratic exchange.
/ -- U 2 E n -
(B)
S=2
i/2, - JJk=_,6,Kijlk= t K, 400
(A)
8
0
0
Vol. 21, No. 5
1 ) - - 1 / 4 ] . S ( S + 1){S(S+ 1)--15}. (7)
The relative energies and the multiplicity of the possible spin states are illustrated by Fig. 3 and compared
wheie x = J / k T and y = ffkT. The best fit to the experimental data, holding g = 2 (determined by ESR) and letting J and j vary, is shown in the Fig. 2 for the final v a l u e s J / k = -- 161 K a n d j / K = 6.6K. It may be seen that the comparison with observed data is significantly improved at low temperature. These results lead to a singlet-triplet splitting &E ~ 365 K slightly different from that previously expected without taking account of the biquadratic exchange term (AE = 340 K). ESR measurements could give a peremptory answer to this point. Finally the value of/does not seem inconsistent with Anderson's theory of superexchange which predicts a / / J ratio of few percent. 7 The obtained result appears thus to support the hypothesis that there is an appreciable biquadratic exchange between Ru s+ ions in (Ru209) clusters.
VoW.21, No. 5
INTERACTION BETWEEN Ru s+ IONS IN (Ru2Og) CLUSTERS
427
REFERENCES 1.
DARRIET J., DRILLON M., VILLENEUVE G. & HAGEN-MULLERP., Z Solid State Chem. (to be published).
2. 3.
DZIALOSHINSKI I., Phys. Chem. Solids 4, 241 0958). STEVENS K.W.H., in Magnetism (Edited by RADO G. & SUHL H.). Academic Press, New York (1963).
4.
HENNING J.C.M., den BOEF J.H. & VANGORKOM G.G.P.,Phys. Rev. B7, 1825 0973).
5.
KITTEL C., Phys. Rev. 120, 35 0960).
6.
MABBSF.E. & MACHIN D.J., Magnetism and Transition Metal Complexes. Chapman and Hall (1973).
7.
ANDERSON P.W., Solid State Phys. 14, 99 (1963).
Pergamon Press.
Printed in Great Britain
BIQUADRATIC EXCHANGE INTERACTION BETWEEN Ru s+' IONS IN (Ru2 09) CLUSTERS M. Drillon Laboratoire de Chimie du Solide du CNRS, Universit6 de Bordeaux I, 351 cours de la Liberation, 33405 Talence-Cedex, France
(Received 28 September 1976 by E.F. Bertaut) The energy levels of a Ru209 cluster have been calculated, including a higher order spin interaction. The RuS+-Ru s÷ coupling is described by the Hamiltonian ~f = -- 2JS1 -$2 --/(St .$2) 2 • The temperature dependence of the magnetic susceptibility is used to determine the values of the bilinear J and biquadratic/exchange integrals: d/k = -- 161 K and ]/k = 6.6 K. The second term in the Hamiltonian corresponds to a fourth order perturbation involving low spin states. THE CATIONIC ordering in the BaCal/a Ruz/aOa oxide which crystallizes with the 6H perovskite derived structure gives rise to RuS+-Ru s+ pairs in (Ru209) clusters (Fig. 1). In an earlier paper, the magnetic properties have been analyzed using a vector model of spin S = 3/2, where a quenching of orbital angular momentum by cristalline field was assumed. 1 We have proposed an exchange Hamiltonian of the usual form: ~Tex = -- 2JSt. $2
(1)
where St = $2 = 3/2 and the isotropic exchange interaction J is negative (antiferromagnetic). The eigenvalues of this Hamiltonian are given by:
E = --J.S(S + 1)
(2)
where S = Sl + $2 (S = O, 1, 2, 3). Using J/k = -- 170 K, the calculation of the magnetic susceptibility gives a good agreement with the experimental values at high temperature, but slight discrepancies appear when temperature decreases (Fig. 2). In this Note we report the results of an investigation of the interactions which take account of the temperature dependance of the susceptibility, particularly at low temperature. The spin Hamiltonian for a binuclear cluster may be written: ~f = ~Ct + ~2 + ~ex. (3) In this expression ~ft and ~f2 are single-ion spin Hamiltonians and ~ex is the coupling term, which is given by: ~fex = -- 2dSt. S2 - - / ( S t - $2) 2
+ c.
sl.s2-~(S,rt~)(S~r~)
+ D . ( s , x s~).
The last term, postulated by Dzialoshinsky is the antisymmetrical spin-spin coupling. 2 This interaction 425
Ru5+ Fig. 1. Geometry of the RuS*--Ru s÷ pairs in (Ru2Og) clusters. tends to cant the spins, since the energy is the lowest when the spinsare perpendicular to each other. This term can have an appreciable effect when both ions are chemically dissimilar, or in the absence of a center of symmetry between the magnetic sites. In our case we can suppose that its influence will be relatively small. The previous term derives from the familiar pseudodipolar interaction, a The orbital singlet 4/12 ground state of Ru s+ ions being far away below the degenerate excited states, it may be expected to be negligibly small in the present case. In other words the hamiltonian may be written as the sum of an isotropie bilinear exchange and an isotropic biquadratic contribution:
INTERACTION BETWEEN Rus÷IONS IN (Ru=09) CLUSTERS
426
t XM (uera/mole)
E/J
12
//
4ool J=-,7oK (5)
The higher order spin coupling may arise either from fourth-order transfer processes4 or from exchange strictions. 5 In view of the obtained results, we only shall take account of the first mechanism.
Ngu~ X = - kT
S=2
S=I
S=I
S:O
S=O
j / k =OK
j / J = 0,05
j2 En
Fig. 3. The realtive energies and possible spin states of a binuclear cluster: Without the biquadratic exchange (A); Including the biquadratic exchange (B). with the energy levels calculated with the classical Hamiltonian (1). The magnetic susceptibility is easily determined using Van Vleck's formula, which is simplified in this case since the matrix elements of Sz between states with different S vanish, s The expression obtained for × (per ion) is the following:
exp ( 2 x - - 6.5y) + 5 exp ( 6 x - - 13.5y) + 14 exp ( 1 2 x - - 9y) 1 + 3 exp (2x -- 6.5y) + 5 exp (6x ~ 13.5y) + 7 exp (12x -- 9y)
The contributions due to transfer processes are stemed from fourth-order perturbation terms which involve low-spin states at an anergy E n (Hund's rule energy) above the ground state. The resulting coefficient of the biquadratic exchange / has been approximatively calculated by van Stapele :4 (6)
where b is the transfer integral. From equation (6) it follows that the j value is positive and of the order of few percent o f £ The eigenvalues of the new exchange Hamiltonian are given by: E = -J.S(S+
S=3
800
~fex = -- 2JSt- S2 --/(St. S:) 2.
b4
S= 3
4
Fig. 2. Magnetic susceptibilities of BaCal/aRu2/aO a. The broken and solid lines represent the best fit to the dimer equation: (1) Without the biquadratic exchange; (2) Ineluding the biquadratic exchange.
/ -- U 2 E n -
(B)
S=2
i/2, - JJk=_,6,Kijlk= t K, 400
(A)
8
0
0
Vol. 21, No. 5
1 ) - - 1 / 4 ] . S ( S + 1){S(S+ 1)--15}. (7)
The relative energies and the multiplicity of the possible spin states are illustrated by Fig. 3 and compared
wheie x = J / k T and y = ffkT. The best fit to the experimental data, holding g = 2 (determined by ESR) and letting J and j vary, is shown in the Fig. 2 for the final v a l u e s J / k = -- 161 K a n d j / K = 6.6K. It may be seen that the comparison with observed data is significantly improved at low temperature. These results lead to a singlet-triplet splitting &E ~ 365 K slightly different from that previously expected without taking account of the biquadratic exchange term (AE = 340 K). ESR measurements could give a peremptory answer to this point. Finally the value of/does not seem inconsistent with Anderson's theory of superexchange which predicts a / / J ratio of few percent. 7 The obtained result appears thus to support the hypothesis that there is an appreciable biquadratic exchange between Ru s+ ions in (Ru209) clusters.
VoW.21, No. 5
INTERACTION BETWEEN Ru s+ IONS IN (Ru2Og) CLUSTERS
427
REFERENCES 1.
DARRIET J., DRILLON M., VILLENEUVE G. & HAGEN-MULLERP., Z Solid State Chem. (to be published).
2. 3.
DZIALOSHINSKI I., Phys. Chem. Solids 4, 241 0958). STEVENS K.W.H., in Magnetism (Edited by RADO G. & SUHL H.). Academic Press, New York (1963).
4.
HENNING J.C.M., den BOEF J.H. & VANGORKOM G.G.P.,Phys. Rev. B7, 1825 0973).
5.
KITTEL C., Phys. Rev. 120, 35 0960).
6.
MABBSF.E. & MACHIN D.J., Magnetism and Transition Metal Complexes. Chapman and Hall (1973).
7.
ANDERSON P.W., Solid State Phys. 14, 99 (1963).