- Email: [email protected]
Dynamical magnetic properties of the random one-dimensional anisotropic antiferromagnet
Solld State Communications, Vol. 72, No. 10, pp. 1047-1050, 1989. Printed in Great Britain.
0038-1098/89 $3.00 + .00 Pergamon Press plc
D Y N A M I C A L M A G N E T I C P R O P E R T I E S OF T H E R A N D O M O N E - D I M E N S I O N A L ANISOTROPIC ANTIFERROMAGNET J.D. Cremasco Departamento de Fisica, Universidade Federal de Ouro Preto, 35 400 - Ouro Preto, MG, Brazil and A.S.T. Pires Departamento de Fisica, Universidade Federal de Minas Gerais, 30 161 - Belo Horizonte, MG, Brazil
(Received 11 August 1989 by P. Burlet) In this paper we carry out a spin-wave perturbation calculation of the spin-dynamical properties of the dilute classical anisotropic antiferromagnetic Heisenberg chain at T = 0. We compare the effects of one applied magnetic field with the effects due to the anisotropy. For (CD3)4NMn~CuI_~CI3, in an external field, no Ising resonances associated with states at chain ends were found in our calculation. R E C E N T L Y there has been a great deal of interest in where S = 5/2 for Mn and Pn is 1 if site n is occupied disordered magnetic systems [1]. As has been pointed by a Mn ion and 0 otherwise. They have carried out out by Nagler et al. [2] the effects of the randomness spin-wave perturbation calculations and computer are most dramatic in one-dimensional systems. The simulations of the dynamical properties at T = 0 and host spin waves can flow around a vacancy in systems arbitrary values of J/I. They have found that when of higher dimensionality, but in one dimensional their J = 0, the Ising Hamiltonian has given localized motion is completely blocked by vacancies to create states at the chain ends at a frequency IS. This is just an assembly of finite chains. half the band-edge energy of 2IS and arises because In a magnetic model we have finite-frequency the magnetic ions at the chain ends have only one localised modes associated with spins at the ends o f neighbour rather than two. However the bound state the finite chains. The spin-flip frequency o f the end at co = IS was pushed down to zero frequency as the spin is half that of an interior spin and is expected to transverse term J was increased from 0 to L occur as a resonance within the host band. In oneIn this paper we will consider a dilute anisotropic dimension, however, the host modes push the reson- one-dimensional antiferromagnet in the presence of ances of the end spins, or 1D surface modes, outside an external field, described by the Hamiltonian the band to become localised modes. In an Ising-like system the surface modes are predicted to be local- / t = 2 J ~ ene~+l(S~ .s~+ l - t~s~s~+l) n ised and to occur at a finite frequency within the anisotropy gap. In a gapless Heisenberg system the surface -- lzgH ~,, PnS;. (2) n modes are pushed to zero frequency where they cannot be distinguished from other quasielastic processes. Here P~ is a random variable assuming the values Neutron scattering studies [3-5] and theoretical 1 or 0 depending on whether the site at n is or is works [5-8] o f the random Heisenberg antiferromag- not occupied by a magnetic ion. The non magnetic net (CD3)4NMncCu~_cCI3 ( T M M C : C u ) have contri- i o n s create vacancies randomly distributed on the buted to a better understanding of excitations in chain. random systems. We can use this Hamiltonian to study the magnetic Endoh et aL [5] have studied this system using the elementary excitations of the TMMC: Cu [(CD3)4N following Hamiltonian for a dilute classical one- Mn,Cut_cCl3] with a magnetic field applied perpendimensional antiferromagnet dicularly to the chain beause the M n - C u coupling in this compound is weak. x x y y At T = 0 and for low a concentration of impurY~P~Po+,[/s~sL, + J(s~sL, + s~'sL,)l n ities the classical ground state is the N6el ground state (1) ~with the spin aligned along the y axis. In terms of the 1047
1048
D Y N A M I C A L MAGNETIC PROPERTIES ~¢Z(09) •
spin annihilation and creation operator s.+ = s ~ + IS~, " x
s ~ = s~ - i s~x
(3)
the Hamiltonian (2) becomes
=
"~ P~.(09),
Vol. 72, No. 10 a ( f l ) = 1,2
(8)
and from the dynamical equation for the matrix P and using Tyablicov's decoupling we find
n
ih
where
1 6(s+Sff+ ' + S~Sff-+,) + -~ ( S + -
Sff)}
4
I = -- ~ ~ q,{SYSY+a + ~ I
1 --
(St + St+a
A
P
l
~
[,o] o = Eo,, o,2] ["] E' 0] 0
1
p2t
+ s,- s,+~) - ~ ~(s, + s,+, + s,- s,- s,-+~)
D 21 D z2
p22
0
(lO)
- 1
and ih + ~-(St + - St-)}
(4)
where /t = /t/2J, h = ltgh/2J, A = _+1 and q l = 1 - Pt is an independent random variable that is 1 if a vacancy is present on the site I and 0 otherwise. The Hamiltonian (4) is equivalent to (2) except when two or more vacancies are nearest neighbours of one another in which case spurious vacancy-vacancy interactions are introduced. However, since these interactions do not affect the eigenvalue spectrum of the magnetic spin system, (4) and (2) are equivalent if the magnetic sites can be properly projected out. The first term on the right hand side of (4) is the Hamiltonian nc of the pure crystal and the terms in q in (4) give the impurity Hamiltonian due to vacancies, which we write as
D3'=-Dy,
/~L = Z qt/~,.
where
(5)
t
The system is conveniently studied using the Green function formalism [9-1 1]. We define (¢'2.(09) •
=
((sd ; s; ))
~Z(09)
=
((&+ ; s. + ) )
~'.(09)
=
((sf. ; s; ))
~.(09)
=
((s#. ; s. + ) )
(6)
where <
We define the spatial Fourier transform p~,(q, 09) =
1 ~ p:~(09)eiq,,._.,
(12)
where the summation is made over all elements of the block P'P, a(fl) = 1, 2. So, from (9) we find e(q, o9) =
1
X2-
4 + d 2cos2q
x[ x+2 cosq 1
(13)
209S'
(14)
-dcosq
X =
S
-X+
d = 2 1 -
2
.
To study the diluted system we use a perturbation theory, based on the small number of vacancies, developed by Harris et al. [9]. Using a dynamic equation for the Green function for the random diluted system we find a Dyson equation f~,..(09) = Pro.(09) + ~ Pmk(09)qiV~(09)f~j.(09) tkj
=
-i
j O(t) ([A(t), B(0)]) eic°' dt. -o0
(7)
(15)
where
We will consider a model of two sublattices such ~(09) = [(¢,,(09)fq,z(09)] that in a classical ground state Sy is always along the ~2~ (09) ~22(09) positive y direction in each sublattice. So r ~ and (~22 are non null if the sites rn and n belongs to the same V'(09) = [ V'l(09) V'2(09)] (16) sublattice and f f ~ and f ~ l are non null if m and n V2'(09) V22(09) belong to different sublattices. and the blocks of V'a(09) with non null elements are For the pure crystal we write
VoL 72, No. 10
_V"(09) =
-
DYNAMICAL MAGNETIC PROPERTIES
1
0
0
2 + ~
0
0
lJ
70
1 0
'
01 0
1049
~t~/~/,
(17.a)
(
2,o) O )l°l,o,o 1/21(09)
1/22(09) = V"(--09).
1/12(09),
(17.c)
The spatial Fourier transform is (b)
G(q, 09) = 1 ~ eiq(m_n)(ffmn(09) =
Z eiq(m-n)<(ffmn(09)>' ra
N --. ~
(18.a)
1
Fig. 1. Frequency to in units of 2JS vs the anisotropy parameter b with no magnetic field (h = 0), showing how the impurity state is pushed to zero frequency as b in increased from 0 (XY-model) to 1 (Heisenbergmodel).
G,.. (09) = ~ ~q e-iq("-") G(q, 09) = ((q,..(09)> (18.b) (7(q, 09) =
where
[1 - L(q, 09)]G(q, 09)
and from (15) and (20) we obtain [10]
l_
<(q,,,(09)) = ~ f f , . , ( t o ) .
(19)
L(q, 09) = (1 - c) ~_, Mtke iq(t-k) We can formally write a Dyson equation for the Green function G, which is a 2 × 2 matrix equation, G(09) =
P(09) + P(09) Y~(09)G(09)
(20.a)
[1 - P(09) Y. (09)]-'P(09)
(20.b)
or
G(09) =
where Y. is the self energy. In the limit of low concentration of vacancies widely separated we take the average of (14) and we find for the selfenergy [9] (q, 09) =
(1 - c) ~ T ~ (09) e iqt"- ")
(2 1)
mn
here T~,a, a(fl) = 1, 2, is an element of the block matrix T ~a for a single vacancy on the site I and has non null elements for re(n) = l - 1, l, l + I. The scattering matrix is
T,..(09) = ~ V~k(09)ML' (09)
(22)
k
where Mm.(09)
=
(24)
a,nn -- Z Pmk(09)Vl(09)"
(23)
k
The Green function for the random system G(q, 09) contains features representing the spin excitations and spurious excitations on the vacancies. The physical Green function restricted to magnetic sites G(q, 09) is related to G(q, 09) by the relation
(25)
k
where l is a site occupied by a vacancy and M is given by (23) and L(q, 09) is the vacancy projector. The resonant states are now obtained from the poles of tT(q, 09). We show our results, first for h = 0, in Fig. 1. We present the position of the bound state as a function of the anisotropy parameter b = 1 - 5. We also show the bulk spin-wave band given by co =
x/4 -- (1 + b) 2 COS2 q.
(26)
The bottom edge of the spin-wave band corresponds to q = 0 and the top edge to q = re/2. For b = 0, we have the X Y model and for b = 1 the isotropic Heisenberg model. When b = 0, we have localized bound state at a frequency equals to half of the frequency of the bottom of the spin-wave band. The bound state is pushed down to zero frequency as the anisotropic term b goes to 1. This part of our work complements the calculation of Endoh et aL [5] who have studied the cross-over from the Ising to the Heisenberg model. Now we consider the effect of an applied magnetic field. From the thermodynamical point of view, magnetic field along the x direction is equivalent to an anisotropic term in the Hamiltonian A Y. (S,X)~ with [13], /~292H2
A
=
16JS 2 "
(27)
DYNAMICAL MAGNETIC PROPERTIES
1050
Vol. 72, Nov. 10
Acknowledgements - One of the authors (A.S.T.P.) gratefully acknowledges useful discussions with Dr J.P. Boucher. This work was partially supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnolrgico, Financiadora de Estudos e Projetos and Coordena~o de Aperfei~oamento de Pessoal de Nivel Superior.
0.4-
REFERENCES 1.
2. 3. 4.
o.0 --¢'--0.00
(h)
0.3,2
Fig. 2. Frequency ~o in units of 2JS vs the scaled applied magnetic field h = I~gh/2JS. In particular if A = D the system behaves thermodynamically like an Ising model. In Fig. 2 we present our results for TMMC :Cu where b = 0.9914. As we can see, when the magnetic field, starting from zero, increases the frequency of the excitations increases also, but there is no Ising resonance at half of the band-edge energy, even for the case A = D, which means h = 0.2624. So we conclude that the magnetic field and an anisotropic term has different effects in the frequency of the bound state of the system.
5. 6. 7. 8. 9. 10. 11. 12. 13.
See, for example, R.A. Cowley, R.J. Birgeneau & G. Shirane, in Proceedings of the NATO Advanced Study Institute on "Strongly - Fluctuating Condensed Matter Systems", Geilo Norway, 1979, (Edited by T. Riste), Plenum, New York and London (1980), and references therein. S.E. Nagler, W.J.L. Buyers, RoL. Armstrong & R.A. Ritchie, J. Phys. C. 17, 4819 (1984). Y. Endoh, G. Shirane, R.J. Birgeneau & Y. Ajiro, Phys. Rev. BI9, 1476 (1979). J.P. Boucher, C. Dujas, W.J. Fitzgerald, K. Knorr & J.P. Renard, J. Phys. (Paris) 39, L-86 (1978). Y. Endoh, I.U. Heilmann, R.J. Birgeneau, G. Shirane, A.R. McGurn & M.F. Thorpe, Phys. Rev. B23, 4582 (1981). M.F. Thorpe, J. Phys. (Paris) 36, 1177 (1975). T. Tonegawa, M. Shida & P. Pincus, Phys. Rev. BI1, 4683 (1975). T. Tonegawa, J. Phys. Soc. Jpn. 47, 1471 (1979). A.B. Harris, P.L. Leath, B.G. Nickel & R.J. Elliot, J. Phys. C. 7, 1693 (1974). R.A. Cowley & W.J.L. Buyers, Rev. Mod. Phys. 44, 406 (1972). R.J. Elliot, J.A. Krumhansl & P.L. Leath, Rev. Mod. Phys. 46, 465 (1974). D.N. Zubarev, Soy. Phys - Usp. 3, 320 (1960). A.S.T. Pires & M.E. Gouvea, J. Phys. C. 17, 4009 (1984).
0038-1098/89 $3.00 + .00 Pergamon Press plc
D Y N A M I C A L M A G N E T I C P R O P E R T I E S OF T H E R A N D O M O N E - D I M E N S I O N A L ANISOTROPIC ANTIFERROMAGNET J.D. Cremasco Departamento de Fisica, Universidade Federal de Ouro Preto, 35 400 - Ouro Preto, MG, Brazil and A.S.T. Pires Departamento de Fisica, Universidade Federal de Minas Gerais, 30 161 - Belo Horizonte, MG, Brazil
(Received 11 August 1989 by P. Burlet) In this paper we carry out a spin-wave perturbation calculation of the spin-dynamical properties of the dilute classical anisotropic antiferromagnetic Heisenberg chain at T = 0. We compare the effects of one applied magnetic field with the effects due to the anisotropy. For (CD3)4NMn~CuI_~CI3, in an external field, no Ising resonances associated with states at chain ends were found in our calculation. R E C E N T L Y there has been a great deal of interest in where S = 5/2 for Mn and Pn is 1 if site n is occupied disordered magnetic systems [1]. As has been pointed by a Mn ion and 0 otherwise. They have carried out out by Nagler et al. [2] the effects of the randomness spin-wave perturbation calculations and computer are most dramatic in one-dimensional systems. The simulations of the dynamical properties at T = 0 and host spin waves can flow around a vacancy in systems arbitrary values of J/I. They have found that when of higher dimensionality, but in one dimensional their J = 0, the Ising Hamiltonian has given localized motion is completely blocked by vacancies to create states at the chain ends at a frequency IS. This is just an assembly of finite chains. half the band-edge energy of 2IS and arises because In a magnetic model we have finite-frequency the magnetic ions at the chain ends have only one localised modes associated with spins at the ends o f neighbour rather than two. However the bound state the finite chains. The spin-flip frequency o f the end at co = IS was pushed down to zero frequency as the spin is half that of an interior spin and is expected to transverse term J was increased from 0 to L occur as a resonance within the host band. In oneIn this paper we will consider a dilute anisotropic dimension, however, the host modes push the reson- one-dimensional antiferromagnet in the presence of ances of the end spins, or 1D surface modes, outside an external field, described by the Hamiltonian the band to become localised modes. In an Ising-like system the surface modes are predicted to be local- / t = 2 J ~ ene~+l(S~ .s~+ l - t~s~s~+l) n ised and to occur at a finite frequency within the anisotropy gap. In a gapless Heisenberg system the surface -- lzgH ~,, PnS;. (2) n modes are pushed to zero frequency where they cannot be distinguished from other quasielastic processes. Here P~ is a random variable assuming the values Neutron scattering studies [3-5] and theoretical 1 or 0 depending on whether the site at n is or is works [5-8] o f the random Heisenberg antiferromag- not occupied by a magnetic ion. The non magnetic net (CD3)4NMncCu~_cCI3 ( T M M C : C u ) have contri- i o n s create vacancies randomly distributed on the buted to a better understanding of excitations in chain. random systems. We can use this Hamiltonian to study the magnetic Endoh et aL [5] have studied this system using the elementary excitations of the TMMC: Cu [(CD3)4N following Hamiltonian for a dilute classical one- Mn,Cut_cCl3] with a magnetic field applied perpendimensional antiferromagnet dicularly to the chain beause the M n - C u coupling in this compound is weak. x x y y At T = 0 and for low a concentration of impurY~P~Po+,[/s~sL, + J(s~sL, + s~'sL,)l n ities the classical ground state is the N6el ground state (1) ~with the spin aligned along the y axis. In terms of the 1047
1048
D Y N A M I C A L MAGNETIC PROPERTIES ~¢Z(09) •
spin annihilation and creation operator s.+ = s ~ + IS~, " x
s ~ = s~ - i s~x
(3)
the Hamiltonian (2) becomes
=
"~ P~.(09),
Vol. 72, No. 10 a ( f l ) = 1,2
(8)
and from the dynamical equation for the matrix P and using Tyablicov's decoupling we find
n
ih
where
1 6(s+Sff+ ' + S~Sff-+,) + -~ ( S + -
Sff)}
4
I = -- ~ ~ q,{SYSY+a + ~ I
1 --
(St + St+a
A
P
l
~
[,o] o = Eo,, o,2] ["] E' 0] 0
1
p2t
+ s,- s,+~) - ~ ~(s, + s,+, + s,- s,- s,-+~)
D 21 D z2
p22
0
(lO)
- 1
and ih + ~-(St + - St-)}
(4)
where /t = /t/2J, h = ltgh/2J, A = _+1 and q l = 1 - Pt is an independent random variable that is 1 if a vacancy is present on the site I and 0 otherwise. The Hamiltonian (4) is equivalent to (2) except when two or more vacancies are nearest neighbours of one another in which case spurious vacancy-vacancy interactions are introduced. However, since these interactions do not affect the eigenvalue spectrum of the magnetic spin system, (4) and (2) are equivalent if the magnetic sites can be properly projected out. The first term on the right hand side of (4) is the Hamiltonian nc of the pure crystal and the terms in q in (4) give the impurity Hamiltonian due to vacancies, which we write as
D3'=-Dy,
/~L = Z qt/~,.
where
(5)
t
The system is conveniently studied using the Green function formalism [9-1 1]. We define (¢'2.(09) •
=
((sd ; s; ))
~Z(09)
=
((&+ ; s. + ) )
~'.(09)
=
((sf. ; s; ))
~.(09)
=
((s#. ; s. + ) )
(6)
where <
We define the spatial Fourier transform p~,(q, 09) =
1 ~ p:~(09)eiq,,._.,
(12)
where the summation is made over all elements of the block P'P, a(fl) = 1, 2. So, from (9) we find e(q, o9) =
1
X2-
4 + d 2cos2q
x[ x+2 cosq 1
(13)
209S'
(14)
-dcosq
X =
S
-X+
d = 2 1 -
2
.
To study the diluted system we use a perturbation theory, based on the small number of vacancies, developed by Harris et al. [9]. Using a dynamic equation for the Green function for the random diluted system we find a Dyson equation f~,..(09) = Pro.(09) + ~ Pmk(09)qiV~(09)f~j.(09) tkj
=
-i
j O(t) ([A(t), B(0)]) eic°' dt. -o0
(7)
(15)
where
We will consider a model of two sublattices such ~(09) = [(¢,,(09)fq,z(09)] that in a classical ground state Sy is always along the ~2~ (09) ~22(09) positive y direction in each sublattice. So r ~ and (~22 are non null if the sites rn and n belongs to the same V'(09) = [ V'l(09) V'2(09)] (16) sublattice and f f ~ and f ~ l are non null if m and n V2'(09) V22(09) belong to different sublattices. and the blocks of V'a(09) with non null elements are For the pure crystal we write
VoL 72, No. 10
_V"(09) =
-
DYNAMICAL MAGNETIC PROPERTIES
1
0
0
2 + ~
0
0
lJ
70
1 0
'
01 0
1049
~t~/~/,
(17.a)
(
2,o) O )l°l,o,o 1/21(09)
1/22(09) = V"(--09).
1/12(09),
(17.c)
The spatial Fourier transform is (b)
G(q, 09) = 1 ~ eiq(m_n)(ffmn(09) =
Z eiq(m-n)<(ffmn(09)>' ra
N --. ~
(18.a)
1
Fig. 1. Frequency to in units of 2JS vs the anisotropy parameter b with no magnetic field (h = 0), showing how the impurity state is pushed to zero frequency as b in increased from 0 (XY-model) to 1 (Heisenbergmodel).
G,.. (09) = ~ ~q e-iq("-") G(q, 09) = ((q,..(09)> (18.b) (7(q, 09) =
where
[1 - L(q, 09)]G(q, 09)
and from (15) and (20) we obtain [10]
l_
<(q,,,(09)) = ~ f f , . , ( t o ) .
(19)
L(q, 09) = (1 - c) ~_, Mtke iq(t-k) We can formally write a Dyson equation for the Green function G, which is a 2 × 2 matrix equation, G(09) =
P(09) + P(09) Y~(09)G(09)
(20.a)
[1 - P(09) Y. (09)]-'P(09)
(20.b)
or
G(09) =
where Y. is the self energy. In the limit of low concentration of vacancies widely separated we take the average of (14) and we find for the selfenergy [9] (q, 09) =
(1 - c) ~ T ~ (09) e iqt"- ")
(2 1)
mn
here T~,a, a(fl) = 1, 2, is an element of the block matrix T ~a for a single vacancy on the site I and has non null elements for re(n) = l - 1, l, l + I. The scattering matrix is
T,..(09) = ~ V~k(09)ML' (09)
(22)
k
where Mm.(09)
=
(24)
a,nn -- Z Pmk(09)Vl(09)"
(23)
k
The Green function for the random system G(q, 09) contains features representing the spin excitations and spurious excitations on the vacancies. The physical Green function restricted to magnetic sites G(q, 09) is related to G(q, 09) by the relation
(25)
k
where l is a site occupied by a vacancy and M is given by (23) and L(q, 09) is the vacancy projector. The resonant states are now obtained from the poles of tT(q, 09). We show our results, first for h = 0, in Fig. 1. We present the position of the bound state as a function of the anisotropy parameter b = 1 - 5. We also show the bulk spin-wave band given by co =
x/4 -- (1 + b) 2 COS2 q.
(26)
The bottom edge of the spin-wave band corresponds to q = 0 and the top edge to q = re/2. For b = 0, we have the X Y model and for b = 1 the isotropic Heisenberg model. When b = 0, we have localized bound state at a frequency equals to half of the frequency of the bottom of the spin-wave band. The bound state is pushed down to zero frequency as the anisotropic term b goes to 1. This part of our work complements the calculation of Endoh et aL [5] who have studied the cross-over from the Ising to the Heisenberg model. Now we consider the effect of an applied magnetic field. From the thermodynamical point of view, magnetic field along the x direction is equivalent to an anisotropic term in the Hamiltonian A Y. (S,X)~ with [13], /~292H2
A
=
16JS 2 "
(27)
DYNAMICAL MAGNETIC PROPERTIES
1050
Vol. 72, Nov. 10
Acknowledgements - One of the authors (A.S.T.P.) gratefully acknowledges useful discussions with Dr J.P. Boucher. This work was partially supported by Conselho Nacional de Desenvolvimento Cientifico e Tecnolrgico, Financiadora de Estudos e Projetos and Coordena~o de Aperfei~oamento de Pessoal de Nivel Superior.
0.4-
REFERENCES 1.
2. 3. 4.
o.0 --¢'--0.00
(h)
0.3,2
Fig. 2. Frequency ~o in units of 2JS vs the scaled applied magnetic field h = I~gh/2JS. In particular if A = D the system behaves thermodynamically like an Ising model. In Fig. 2 we present our results for TMMC :Cu where b = 0.9914. As we can see, when the magnetic field, starting from zero, increases the frequency of the excitations increases also, but there is no Ising resonance at half of the band-edge energy, even for the case A = D, which means h = 0.2624. So we conclude that the magnetic field and an anisotropic term has different effects in the frequency of the bound state of the system.
5. 6. 7. 8. 9. 10. 11. 12. 13.
See, for example, R.A. Cowley, R.J. Birgeneau & G. Shirane, in Proceedings of the NATO Advanced Study Institute on "Strongly - Fluctuating Condensed Matter Systems", Geilo Norway, 1979, (Edited by T. Riste), Plenum, New York and London (1980), and references therein. S.E. Nagler, W.J.L. Buyers, RoL. Armstrong & R.A. Ritchie, J. Phys. C. 17, 4819 (1984). Y. Endoh, G. Shirane, R.J. Birgeneau & Y. Ajiro, Phys. Rev. BI9, 1476 (1979). J.P. Boucher, C. Dujas, W.J. Fitzgerald, K. Knorr & J.P. Renard, J. Phys. (Paris) 39, L-86 (1978). Y. Endoh, I.U. Heilmann, R.J. Birgeneau, G. Shirane, A.R. McGurn & M.F. Thorpe, Phys. Rev. B23, 4582 (1981). M.F. Thorpe, J. Phys. (Paris) 36, 1177 (1975). T. Tonegawa, M. Shida & P. Pincus, Phys. Rev. BI1, 4683 (1975). T. Tonegawa, J. Phys. Soc. Jpn. 47, 1471 (1979). A.B. Harris, P.L. Leath, B.G. Nickel & R.J. Elliot, J. Phys. C. 7, 1693 (1974). R.A. Cowley & W.J.L. Buyers, Rev. Mod. Phys. 44, 406 (1972). R.J. Elliot, J.A. Krumhansl & P.L. Leath, Rev. Mod. Phys. 46, 465 (1974). D.N. Zubarev, Soy. Phys - Usp. 3, 320 (1960). A.S.T. Pires & M.E. Gouvea, J. Phys. C. 17, 4009 (1984).