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Magnetic phase transitions in Fe1−xCoxCl2: A computer simulation
~
0038-1098/81/210803-03502.00/0
Solid State Communications, Vol.3S, pp.803-805. Pergamon Press Ltd. 1981. Printed in Great Britain.
MAGNETIC
PHASE TRANSITIONS
IN Fel_xCOxCl2:
A COMPUTER SIMULATION
M.C.K. Wiltshire Department of Solid State Physics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 2600. Australia. (Received 19 January 1981 by J. Kanam0ri)
The equation of motion method has been used to calculate the ~ = 0 magnon in Fel_xCoxCl 2. For a certain range of concentrations, a response at negative frequency is found, which implies that there is a new magnetic phase intermediate between FeCI 2 and CoCI 2 in this composition region. The boundaries of this phase are in good agreement with the available experimental data.
(1-3) The equation of motion method has been used in the past to calculate the s2in dynamics of a variety of randomly dilute and mixed magnetic systems. In the case of the dilute magnet, assuming that the exchange forces remain constant as a function of dilution, there are no variable parameters in the calculation. Nevertheless an excellent representation of experimental results has been achieved by this method (2,4-6). The treatment of mixed magnets is somewhat less certain because the exchange interaction between different species of ion is unknown. It is usual to assume JAB = (JAAJBB)~ for the exchange between ions of type A and B a n ~ using this approach, randomly mixed magnets have also been successfully treated. (l,z) In recent years, considerable interest has centred on the properties of mixed magnets where the anisotropies of the two components differ, i.e. there exists competition in the direction of order of the mixture. In particular, theoretical work (7,8) using renormalisation group methods has shown that such a system may possess a tetracritical point. Moreover, it has been shown experimentally in this type of system that there exists a new magnetic phase for a particular concentration range (e.g. K2(Mn,Fe)F4(9); (Fe,Co)CI2.2H20 (10-12);Cs(Mn,Co)CI3.2H20(13,14); and (Fe,Co) C12(15-17)). The mean field approximation has been used (18,19) to investigate this phase and a Monte-Carlo calculation (20) has been carried out on a model system with competing anisotropy. This work has shown that in the new phase, described by the term "oblique antiferromagnet" (OAF), the spins order in directions intermediate between those characteristic of the two pure components. In this paper, we apply the equation of motion method to a model of the (Fe,Co)CI 2 system. The model exchange parameters are obtained using the geometric mean approximation and the q = 0 magnon of the mixture has been investigated as a function of concentration. It is found that, at certain critical concentrations, a response appears at negative frequencies. This indicates that the assumed ground state is no longer
the lowest lying state and that a different ground state is appropriate. Hence there is a transition to a new magnetic phase. Both FeCI 2 and CoCI 2 have the CdCI 2 structure (space group R3m) and become antiferromagnetic below their Neel temperatures of 23.5K and 24.9K (or 25.3K(21)), respectively. The magnetic order in both materials is that of ferromagnetic layers antiferromagnetically stacked. However in FeCI 2 the anisotropy is such as to order the spins parallel to the c-axis while in CoCI 2 they lie in the basal plane. Neutron diffraction measurements have been made on both materials (22,23) and the exchange parameters of both systems have been determined. More recently the present author (4,5) used values obtained from these measurements to calculate the dynamics of the diluted materials. In this note the Hamiltonians and exchange parameters used previously (4,5) are again adopted. For FeCI2, we have ~ = - ~ [JijS/.Sj + KijsiZsjZ}-Z D(SiZ) 2 {i) ij i where Oz lies along the crystal c-axis, and for CoCI 2 ~ = - ~ {Jij~/'~j + KijSixSjx} (2)
ij where Oz lies along the spin direction in the basal plane and Ox is the crystal c-axis. We rewrite these Hamiltonians in terms of a cartesian set of crystal axes (a,b and c) as (FeCl2)=?~{~ijsiasj a + nijsibsjb + 13 ~ijsiCsj c} -Z D(siC)2 X
(CoCl2)=?~{~ijsiasja 13
(3)
+ qijsibsjb +
~ijsiCsjC}. Here ~ij=ni4=Jij while ~ij=(Jij+Ki9 ) . If we now conside~ each component of the exchange separately, we can obtain the Fe-Co parameters by the geometric-mean approximation, eg. ~ij(Fe-Co)=(~ij(Fe-Fe). ~ij(Co-Co))~" Hence we can deduce the values of Jl, J2 and K 1 appropriate to the Fe-Co interaction,
MAGNETIC PHASE TRANSITIONS IN Fel_xCOxCl2:
804
where Jl and J2 are the isotropic exchange parameters for nearest (in-plane) and nextnearest (out of plane) neighbours respectively and K I is the anisotropic nearestneighbour exchange. There remain two problems. First, ~2(Fe-Co), the next nearest neighbour exchange in the c-direction, is imaginary in the geometric mean approximation. We have adopted, for simplicity, the arithmetic mean ~2(Fe-Co)=~(~2(Fe-Fe)+ ~2(Co-Co)), instead and hence have obtained a value for K2(Fe-Co), the out-of-plane anisotropic exchange. Second, the single ion anisotropy of Fe 2+ in COC12, D(Fe:CoCI2), is unknown. Since it arises from the trigonal component of the crystal field, we write it as -~D(S~) 2 and consider the effect of such a i term in the Hamiltonian of CoCI 2 at a later stage. The resulting set of J and K parameters is given in Table I. Table i.
Exchange parameters (cm-I) in the Hamiltonians (i) and (2) for (Fe, Co)C12 mixtures Jl
Fe-Fe Fe-Co Co-Co
2.36 4.82 9.84
K1 0.765 -1.15 -5.52
J2 -0.12 -0.30 -0.75
K2 -0.035 0.41 1.13
The equation of motion simulation method has been fully described for FeCI 2 and COC12 (4,5) and will only be outlined here. The usual Green function Gij(t)=-i@(t ) <01Si+ (t) Sj- (0)10> / 2(SiSj)~ (4) has been used to calculate the spectral response function at zero wavevector. We first transform to quasi-reciprocal space by writing Giq(t)=N-~ ~ exp(-iK.Rj)Sij(t) (5) J The equation of motion of this function is obtained as ih GiK(t)=[Si~(t),'~ (6) and used to update Gig(t) by setting SiK(t+~t)=GiK(t-~t)+26 t GiK(t) After a finite time T the spectral response function is calculated as
(7)
S(K,~)=Im I~exp (i~t) ~{exp(iK.Ri)Giq(t) } x i exp(-lt2)dt (8) including Gaussian damping to remove termination ripples. The calculation is initialised at t = 0 using equations (4) and (5). We need, therefore, to make assumptions about the configuration of the spins and their interactions. It is known that, for small concentrations of one material in the other, the spins all align in the direction characteristic of the major component (15,24,25). However, the effect of this alignment on the exchange parameters is unknown. We have made the simplest possible assumption, namely that the reorienration does not affect them. Thus, in FeCI2:Co, we take the Co-Co exchange parameters as for pure CoCI 2 whilst of course taking into account the new spin direction. The ground state of
A COMPUTER SIMULATION
Vol. 38, No. 9
the system is therefore assumed to be the N~el state with the spins ordered in the direction characteristic of the major component. We now consider the single ion anisotropy for Fe 2+ in CoCl 2. Since this gives rise only to an energy shift in the excitation spectrum, it can be treated in isolation. The term of interest is then -~ D(S~) '2 (9) i in the coordinate system appropriate to CoCI 2. We take the Green function (4) and calculate its equation of motion under this Hamiltonian. The higher order Green functions thus generated are decoupled in the random phase approximation (RPA) and it can then be shown that the energy shift is zero. Since the remainder of the calculation also relies on the RPA decoupling method, we use it here as a first approximation and set the effective anistropy D(Fe:CoCI 2) = 0. Calculations were carried out on relatively small meshes, of the order of i0 x 12 x 14 ions arranged in a hexagonal close packed array, at a resolution of ~3cm -I. A few calculations were performed on much larger meshes (20 x 22 x 24) at 0.25cm -I resolution. However, for the present purpose, the higher resolution calculations, obtained at the expense of considerably greater computing time, did not reveal any significant details not found in the smaller computations. The concentration of CoCI 2 in FeCI 2 and likewise of FeCI 2 in CoCl 2 was varied for the parameters given in table 1 and the spectral response function was examined for ~ = O. In the pure materials a sharp response is obtained at the antiferromagnetic resonance (AFM-R) frequency which represents the lowest frequency of the magnon density of states. As the concentration of the minor component is increased in the model, a localised magnetic mode appears below the AFMR frequency, and this localised mode softens with increasing minority concentration in either species. At a critical Composition, a mode appears at negative frequency. This implies that the assumed ground state is no longer the lowest energy state of the model and that the system undergoes a transition to a new phase, with the spins ordered neither parallel to the c-axis nor in the basal plane. The new phase, which we identify as the OAF phase, has composition limits with the parameters in table 1 of Fe0.79Co0.21C12 and Fe0.57Co0.43CI 2. Because of the previously mentioned uncertainty in K2(Fe-Co), this param.eter has been varied and its effect on the phase boundaries examined. K2(Fe-Co) was varied between the values for K2(Fe-Fe) and K2(Co-Co) but the variation of the phase boundary is slight, so the original value of K2(Fe-Co) = 0.41cm -I is retained. The best experimental estimate of the position of the phase boundary at zero temperature is obtained from the neutron scattering work of Wong et al (17) which gives Fe0.77Co0.23CI 2 and Fe0.57Co0.43CI 2 as the critical compositions. These are in excellent agreement with the results of the present calculation. Experiments are in progress using =
Vol. 38, No. 9
MAGNETIC PHASE TRANSITIONS IN Fel_xCOxCl2: A COM25~ER SIMLq.ATION
M~ssbauer spectroscopy in an attempt to verify directly the position of the low temperature phase boundary. Far infrared measurements are also being used to observe the predicted localised magnetic modes at small minority concentrations. In conclusion, the equation of ,otion method of simulation, using a single
set of exchange parameters obtained by standard methods, has been applied to the (Fe,Co) Cl 2 mixed system. It has demonstrated the existence of a phase transition to a mixed phase where the spin ordering direction is neither parallel to the c-axis nor in the basal plane.
References I. 2. 3. 4. 5. 6.
7. 8. 9. i0. ii. 12. 13.
R.Alben and M.F. Thorpe, J.Phys.C 8,L275 (1975). M.F.Thorpe and R.Alben, J.Phys. C 2,2555 (1976). R.Alben, S.Kirkpatrick and D.Beeman, Phys. Rev. B15,346(1977). M.C.K.Wiltshire and W.Hayes, J.Phys.C ii, 3701(1978). M.C.K. wiltshire, J.Phys.C 12,3571(1979). R.A.Cowley, R.J.Birgeneau, G.Shirane, H.J. Guggenheim and H.Ikeda, Phys. Rev. B21,4038 (1980). A.Aharony and S.Fishman, Phys. Rev. Lett.37, 1587(1976). S.Fishman and A.Aharony, Phys.Rev.B18,3507 (1978). L.Bevaart, E.Frikee, J.V.Lebesque and L.J. de Jongh, Phys.Rev.B18,3376(1978). K. Katsumata, M.Kobayashi and H.Yoshizawa, Phys.Rev.Lett.43,960(1979). K.Katsumata and T.Tawaraya, J.Magn.Magn. Mater.15-18,253(1980). K.Katsumata, M.Kobayashi, T.Sato and Y. Miyako, Phys.Rev.B19,2700(1979). I.Yamamoto, K.Nagata, T.Sato and Y.Miyako, J.Magn.Magn.Mater.15-18,251(1980).
14. I.Yamamoto, J.Phys. Soc.Jpn.49,74(1980). 15. T.Tawaraya and K.Katsumata, Sol.State Commun.32,377(1979). 16. T.Tawaraya, K.Katsumata and H.Yoshizawa, J.Phys.Soc.Jpn.49,1299(1980). 17. P-z.Wong, P.M.Horn, R.J.Birgeneau, C.R. Safinya and G.Shirane, Phys.Rev.Lett. 45,1974(1980). 18. F.Matsubara and S.Inawashiro, J.Phys. Soc. JDn.42,1529 (1977). 19. T.Oguchi and T.Ishikawa, J.Phys.Soc.Jpn. 45,1213 (1978) . 20. S.Inawashiro, T.Togawa and R.Kurosaka, J.Phys.Cl2, 2351(1979). 21. D.Moses, J.E.Kardontchik, R.Brener and H.Shechter, J.Phys.C13,3903(1980). 22. R.J.Birgeneau, W.B.Yelon, E.Cohen and J.Makovsky, Phys.Rev.B5,2607(1972). 23. M.T.Hutchings, J.Phys.C6,3143(1973). 24. T.Fujita, A.Ito and K.Ono, J.Phys.Soc.gpn. 27,1143(1969). 25. T.Oguchi and T.Ishikawa, J.Phys.Soc.Jpn. 42,1513(1977).
805
0038-1098/81/210803-03502.00/0
Solid State Communications, Vol.3S, pp.803-805. Pergamon Press Ltd. 1981. Printed in Great Britain.
MAGNETIC
PHASE TRANSITIONS
IN Fel_xCOxCl2:
A COMPUTER SIMULATION
M.C.K. Wiltshire Department of Solid State Physics, Research School of Physical Sciences, Australian National University, Canberra, A.C.T. 2600. Australia. (Received 19 January 1981 by J. Kanam0ri)
The equation of motion method has been used to calculate the ~ = 0 magnon in Fel_xCoxCl 2. For a certain range of concentrations, a response at negative frequency is found, which implies that there is a new magnetic phase intermediate between FeCI 2 and CoCI 2 in this composition region. The boundaries of this phase are in good agreement with the available experimental data.
(1-3) The equation of motion method has been used in the past to calculate the s2in dynamics of a variety of randomly dilute and mixed magnetic systems. In the case of the dilute magnet, assuming that the exchange forces remain constant as a function of dilution, there are no variable parameters in the calculation. Nevertheless an excellent representation of experimental results has been achieved by this method (2,4-6). The treatment of mixed magnets is somewhat less certain because the exchange interaction between different species of ion is unknown. It is usual to assume JAB = (JAAJBB)~ for the exchange between ions of type A and B a n ~ using this approach, randomly mixed magnets have also been successfully treated. (l,z) In recent years, considerable interest has centred on the properties of mixed magnets where the anisotropies of the two components differ, i.e. there exists competition in the direction of order of the mixture. In particular, theoretical work (7,8) using renormalisation group methods has shown that such a system may possess a tetracritical point. Moreover, it has been shown experimentally in this type of system that there exists a new magnetic phase for a particular concentration range (e.g. K2(Mn,Fe)F4(9); (Fe,Co)CI2.2H20 (10-12);Cs(Mn,Co)CI3.2H20(13,14); and (Fe,Co) C12(15-17)). The mean field approximation has been used (18,19) to investigate this phase and a Monte-Carlo calculation (20) has been carried out on a model system with competing anisotropy. This work has shown that in the new phase, described by the term "oblique antiferromagnet" (OAF), the spins order in directions intermediate between those characteristic of the two pure components. In this paper, we apply the equation of motion method to a model of the (Fe,Co)CI 2 system. The model exchange parameters are obtained using the geometric mean approximation and the q = 0 magnon of the mixture has been investigated as a function of concentration. It is found that, at certain critical concentrations, a response appears at negative frequencies. This indicates that the assumed ground state is no longer
the lowest lying state and that a different ground state is appropriate. Hence there is a transition to a new magnetic phase. Both FeCI 2 and CoCI 2 have the CdCI 2 structure (space group R3m) and become antiferromagnetic below their Neel temperatures of 23.5K and 24.9K (or 25.3K(21)), respectively. The magnetic order in both materials is that of ferromagnetic layers antiferromagnetically stacked. However in FeCI 2 the anisotropy is such as to order the spins parallel to the c-axis while in CoCI 2 they lie in the basal plane. Neutron diffraction measurements have been made on both materials (22,23) and the exchange parameters of both systems have been determined. More recently the present author (4,5) used values obtained from these measurements to calculate the dynamics of the diluted materials. In this note the Hamiltonians and exchange parameters used previously (4,5) are again adopted. For FeCI2, we have ~ = - ~ [JijS/.Sj + KijsiZsjZ}-Z D(SiZ) 2 {i) ij i where Oz lies along the crystal c-axis, and for CoCI 2 ~ = - ~ {Jij~/'~j + KijSixSjx} (2)
ij where Oz lies along the spin direction in the basal plane and Ox is the crystal c-axis. We rewrite these Hamiltonians in terms of a cartesian set of crystal axes (a,b and c) as (FeCl2)=?~{~ijsiasj a + nijsibsjb + 13 ~ijsiCsj c} -Z D(siC)2 X
(CoCl2)=?~{~ijsiasja 13
(3)
+ qijsibsjb +
~ijsiCsjC}. Here ~ij=ni4=Jij while ~ij=(Jij+Ki9 ) . If we now conside~ each component of the exchange separately, we can obtain the Fe-Co parameters by the geometric-mean approximation, eg. ~ij(Fe-Co)=(~ij(Fe-Fe). ~ij(Co-Co))~" Hence we can deduce the values of Jl, J2 and K 1 appropriate to the Fe-Co interaction,
MAGNETIC PHASE TRANSITIONS IN Fel_xCOxCl2:
804
where Jl and J2 are the isotropic exchange parameters for nearest (in-plane) and nextnearest (out of plane) neighbours respectively and K I is the anisotropic nearestneighbour exchange. There remain two problems. First, ~2(Fe-Co), the next nearest neighbour exchange in the c-direction, is imaginary in the geometric mean approximation. We have adopted, for simplicity, the arithmetic mean ~2(Fe-Co)=~(~2(Fe-Fe)+ ~2(Co-Co)), instead and hence have obtained a value for K2(Fe-Co), the out-of-plane anisotropic exchange. Second, the single ion anisotropy of Fe 2+ in COC12, D(Fe:CoCI2), is unknown. Since it arises from the trigonal component of the crystal field, we write it as -~D(S~) 2 and consider the effect of such a i term in the Hamiltonian of CoCI 2 at a later stage. The resulting set of J and K parameters is given in Table I. Table i.
Exchange parameters (cm-I) in the Hamiltonians (i) and (2) for (Fe, Co)C12 mixtures Jl
Fe-Fe Fe-Co Co-Co
2.36 4.82 9.84
K1 0.765 -1.15 -5.52
J2 -0.12 -0.30 -0.75
K2 -0.035 0.41 1.13
The equation of motion simulation method has been fully described for FeCI 2 and COC12 (4,5) and will only be outlined here. The usual Green function Gij(t)=-i@(t ) <01Si+ (t) Sj- (0)10> / 2(SiSj)~ (4) has been used to calculate the spectral response function at zero wavevector. We first transform to quasi-reciprocal space by writing Giq(t)=N-~ ~ exp(-iK.Rj)Sij(t) (5) J The equation of motion of this function is obtained as ih GiK(t)=[Si~(t),'~ (6) and used to update Gig(t) by setting SiK(t+~t)=GiK(t-~t)+26 t GiK(t) After a finite time T the spectral response function is calculated as
(7)
S(K,~)=Im I~exp (i~t) ~{exp(iK.Ri)Giq(t) } x i exp(-lt2)dt (8) including Gaussian damping to remove termination ripples. The calculation is initialised at t = 0 using equations (4) and (5). We need, therefore, to make assumptions about the configuration of the spins and their interactions. It is known that, for small concentrations of one material in the other, the spins all align in the direction characteristic of the major component (15,24,25). However, the effect of this alignment on the exchange parameters is unknown. We have made the simplest possible assumption, namely that the reorienration does not affect them. Thus, in FeCI2:Co, we take the Co-Co exchange parameters as for pure CoCI 2 whilst of course taking into account the new spin direction. The ground state of
A COMPUTER SIMULATION
Vol. 38, No. 9
the system is therefore assumed to be the N~el state with the spins ordered in the direction characteristic of the major component. We now consider the single ion anisotropy for Fe 2+ in CoCl 2. Since this gives rise only to an energy shift in the excitation spectrum, it can be treated in isolation. The term of interest is then -~ D(S~) '2 (9) i in the coordinate system appropriate to CoCI 2. We take the Green function (4) and calculate its equation of motion under this Hamiltonian. The higher order Green functions thus generated are decoupled in the random phase approximation (RPA) and it can then be shown that the energy shift is zero. Since the remainder of the calculation also relies on the RPA decoupling method, we use it here as a first approximation and set the effective anistropy D(Fe:CoCI 2) = 0. Calculations were carried out on relatively small meshes, of the order of i0 x 12 x 14 ions arranged in a hexagonal close packed array, at a resolution of ~3cm -I. A few calculations were performed on much larger meshes (20 x 22 x 24) at 0.25cm -I resolution. However, for the present purpose, the higher resolution calculations, obtained at the expense of considerably greater computing time, did not reveal any significant details not found in the smaller computations. The concentration of CoCI 2 in FeCI 2 and likewise of FeCI 2 in CoCl 2 was varied for the parameters given in table 1 and the spectral response function was examined for ~ = O. In the pure materials a sharp response is obtained at the antiferromagnetic resonance (AFM-R) frequency which represents the lowest frequency of the magnon density of states. As the concentration of the minor component is increased in the model, a localised magnetic mode appears below the AFMR frequency, and this localised mode softens with increasing minority concentration in either species. At a critical Composition, a mode appears at negative frequency. This implies that the assumed ground state is no longer the lowest energy state of the model and that the system undergoes a transition to a new phase, with the spins ordered neither parallel to the c-axis nor in the basal plane. The new phase, which we identify as the OAF phase, has composition limits with the parameters in table 1 of Fe0.79Co0.21C12 and Fe0.57Co0.43CI 2. Because of the previously mentioned uncertainty in K2(Fe-Co), this param.eter has been varied and its effect on the phase boundaries examined. K2(Fe-Co) was varied between the values for K2(Fe-Fe) and K2(Co-Co) but the variation of the phase boundary is slight, so the original value of K2(Fe-Co) = 0.41cm -I is retained. The best experimental estimate of the position of the phase boundary at zero temperature is obtained from the neutron scattering work of Wong et al (17) which gives Fe0.77Co0.23CI 2 and Fe0.57Co0.43CI 2 as the critical compositions. These are in excellent agreement with the results of the present calculation. Experiments are in progress using =
Vol. 38, No. 9
MAGNETIC PHASE TRANSITIONS IN Fel_xCOxCl2: A COM25~ER SIMLq.ATION
M~ssbauer spectroscopy in an attempt to verify directly the position of the low temperature phase boundary. Far infrared measurements are also being used to observe the predicted localised magnetic modes at small minority concentrations. In conclusion, the equation of ,otion method of simulation, using a single
set of exchange parameters obtained by standard methods, has been applied to the (Fe,Co) Cl 2 mixed system. It has demonstrated the existence of a phase transition to a mixed phase where the spin ordering direction is neither parallel to the c-axis nor in the basal plane.
References I. 2. 3. 4. 5. 6.
7. 8. 9. i0. ii. 12. 13.
R.Alben and M.F. Thorpe, J.Phys.C 8,L275 (1975). M.F.Thorpe and R.Alben, J.Phys. C 2,2555 (1976). R.Alben, S.Kirkpatrick and D.Beeman, Phys. Rev. B15,346(1977). M.C.K.Wiltshire and W.Hayes, J.Phys.C ii, 3701(1978). M.C.K. wiltshire, J.Phys.C 12,3571(1979). R.A.Cowley, R.J.Birgeneau, G.Shirane, H.J. Guggenheim and H.Ikeda, Phys. Rev. B21,4038 (1980). A.Aharony and S.Fishman, Phys. Rev. Lett.37, 1587(1976). S.Fishman and A.Aharony, Phys.Rev.B18,3507 (1978). L.Bevaart, E.Frikee, J.V.Lebesque and L.J. de Jongh, Phys.Rev.B18,3376(1978). K. Katsumata, M.Kobayashi and H.Yoshizawa, Phys.Rev.Lett.43,960(1979). K.Katsumata and T.Tawaraya, J.Magn.Magn. Mater.15-18,253(1980). K.Katsumata, M.Kobayashi, T.Sato and Y. Miyako, Phys.Rev.B19,2700(1979). I.Yamamoto, K.Nagata, T.Sato and Y.Miyako, J.Magn.Magn.Mater.15-18,251(1980).
14. I.Yamamoto, J.Phys. Soc.Jpn.49,74(1980). 15. T.Tawaraya and K.Katsumata, Sol.State Commun.32,377(1979). 16. T.Tawaraya, K.Katsumata and H.Yoshizawa, J.Phys.Soc.Jpn.49,1299(1980). 17. P-z.Wong, P.M.Horn, R.J.Birgeneau, C.R. Safinya and G.Shirane, Phys.Rev.Lett. 45,1974(1980). 18. F.Matsubara and S.Inawashiro, J.Phys. Soc. JDn.42,1529 (1977). 19. T.Oguchi and T.Ishikawa, J.Phys.Soc.Jpn. 45,1213 (1978) . 20. S.Inawashiro, T.Togawa and R.Kurosaka, J.Phys.Cl2, 2351(1979). 21. D.Moses, J.E.Kardontchik, R.Brener and H.Shechter, J.Phys.C13,3903(1980). 22. R.J.Birgeneau, W.B.Yelon, E.Cohen and J.Makovsky, Phys.Rev.B5,2607(1972). 23. M.T.Hutchings, J.Phys.C6,3143(1973). 24. T.Fujita, A.Ito and K.Ono, J.Phys.Soc.gpn. 27,1143(1969). 25. T.Oguchi and T.Ishikawa, J.Phys.Soc.Jpn. 42,1513(1977).
805