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Spin configuration of two-dimensional orthorhombic lattice
J. Phys. Chem. SolidsVol. Printed in Great Britain.
54, No. II, pp. 159!%1602, 1593
OOZZ-3697/93 $6.00 + 0.00 0 1993 Pergamon Press Ltd
SPIN CONFIGURATION OF TWO-DIMENSIONAL ORTHORHOMBIC LATTICE and
H. S. DARENDEL~O~LU Physics Department,
H. Y~~KSEL
Faculty of Sciences and Arts, SelGuk University,
(Received 23 January 1991; accepted in revisedform
42049, Konya, Turkey
10 May 1993)
Abstract-We have applied the so-called macroscopic and microscopic methods of &-taut to the two-dimensional problem of four spins in an orthorhombic lattice in order to investigate the spin configurations and spin-Hamiltonians for a two-dimensional orthorhombic lattice. We have determined the propagation vectors of the modes and possible spin-Hamiltonians and concluded that the four collinear modes are along the z-axis and the non-collinear modes are in the xy-plane of the two-dimensional orthorhombic lattice within the approximation of the model as long as second order coupling of the modes is present. Keywords: microscopic
Spin configuration, method.
exchange
interaction,
1. INTRODUCTION We
have a two-dimensional
orthorhombic
exchange
constant,
macroscopic
method,
C,, i, G,, G,, G, [4], where E is the identity element, is the inversion operation, C,, C,, C, are rotations of 180” with respect to the x-, y-, and z-axis, respectively, and finally G,, GY,and G, are symmetry planes perpendicular to the x-, y-, and z-axis, respectively. Generating elements of this group (or “independent symmetry elements” as called by crystallographers and neutron diffractionists) are C,, C, and i. As seen from Fig. 1, C, sends the point 1 to 2 and 3 to 4; C, sends the point 1 to 3 and 2 to 4 and finally inversion operator i sends the point 1 to 3 and 2 to 4. Under the operation of C,, C,, and i, the four spin vectors S, , S,, S, and S, transform in a complicated way, so it is more practical to look for linear combinations of the four vectors so that the components of the vectors transform into the same components disregarding the changing of sign. We have the following linear combinations i
lattice
is decorated by spins as seen in Fig. 1. The spins are located at the points: x, y( 1); x, p(2); f, y(3); f,y(4). We assume that all of the spins are axial vectors and have the same values. We assume also that the spins interact with each other by means of exchange interactions only. We exclude super-exchange in which, for instance, there are Mn2+ cations on sites (l)-(4) and an oxygen anion at the origin, the .I, integral between (1) and (3) or between (2) and (4) would be strongly negative as being a 180” super-exchange integral and we would have IJJI > IJ, 1 and IJjI > &I. As the exchange constants decrease very quickly with increasing distance, it is generally quite enough to consider just the first and second neighbour spins [l]. On the other hand, as the distance between spin 1 and spin 2 is shorter than the distance between spin 1 and spin 4, this implies that J, > J2. We suppose that the exchange integral J3 between spin 1 and spin 4 is equal to the exchange integral J2 between spin 1 and spin 3. We will investigate the possible spin-Hamiltonians, spin configurations and also their stability conditions from the point of view of the symmetry arguments for the system defined by Fig. 1. Our approach is mainly the matrix formalism proposed and developed by Bertaut [2,3]. which
F = Sl + S, + Ss + S, G = s, - s* + s3 - s, c=s,+q-s,-sq A = Sl - S2 - S, + S,.
(1)
These F, G, C and A vectors, which we call base vectors, represent the possible magnetic modes. (A) Construction of invariants and spin-Hamiltonians
2. THEORY
The symmetry elements of the point group of a two-dimensional orthorhombic lattice are E, C,, C,,
We apply the generating elements C,, C, and i to the x-, y-, and z-components of the base vectors F,
1599
H. $. DARENDELidLU
1600
and H. Yi_itcsa~
A Y 3 .4
.l
A
.3
3
.3’
.4”
.l’
.4”
x
Fig. 1. Two-dimensional orthorhombic lattice and its independent symmetry operations.
G, C and A. Each component will transform into itself with or without a change of sign. We consider here the spins as axial vectors which have quite different transformation properties from polar vectors. For example when C, acts on A, we have: C,A,=C,(S,,-S,,-S,,+S,,) = S,, - S,, - S,, + S,, = -A,,
(2)
which changes the sign of A,. However, when C, acts on F, we have: (7,
= US,,
line. Because of the fact that Hamiltonians must be invariant under spin reversal, the invariant Hamiltonians are simply constructed by pair multiplication of base vector components which belong to the same representation. For example, the square of the terms in parentheses in the representation F, are F:, G: , F, Gy. Each one of these terms is invariant. In the same way we can go on to write the invariants for the other representations. Invariant terms do not change the sign under the symmetry operations. The physical meaning of the term F,G, can be explained as the x-component of F-mode being compatible with the Y-component of the G-mode. We obtain
+ S,, + S,, + S,,)
= S,, + S, + S,, + S.,, = F,,
H,=k,F;+k,G;+k,F,G,
(3)
H2=k4F;+ksG;+k,G,F,,
which does not change the sign of F,. Proceeding as above for all base vectors and all operators, we obtain Table 1 which summarizes the transformation properties of the base vectors. In order to have invariants, we just rearrange the results in Table 1 by choosing the components which transform in the same way. We call a set of definite transformation properties of the mode F, under generating elements C,, C, and i a representation. The results may be simply specified by (+ - +), which signifies that F, does not change sign under the operations C, and i but does change sign under the operation C,. We call F2 the (+ - +) representation as shown in Table 2. Continuing in a similar way we have eight different representations F, (i = 1 . . .8). As seen from Table 2, the mode components which belong to the same representation are on the same
Vectors F G C A
H,=k,A’:+k,C;+k,A,C, H,=k,,CL:+k,,A:+k,2CXAy,
(4)
where k, . . . k,, are constants which may be determined from the experimental results. (B) Determination
of k-vectors
It is possible that the k-vectors can be determined from a neutron diffraction experiment, i.e. from the positions of the diffraction lines. The essential point of the method is that the generalized Fourier transform of the exchange integral must be a maximum [5]. The Fourier coefficient of the exchange integral is defined as follows ei, = ~&,a, exp[2nik(Ri - R,)], Ri
Table 1. Transformation properties of the base vectors Operations i C, c, Z z X X x Y Y Y +
-
-
-
-
+ -
+ +
+ +
+ +
+ +
+ + -
+ + -
+ + -
Z + + -
(5)
Two-dimensional orthorhombic lattice Table 2. Representation of the base vectors i c, c.
which has the following eigenvalue solutions
+
+)
-
-
G,
r2
:=
-
z;
(-
+
Gy -
;
r3
F, -
G,
Fy C,
2 -
l-1
+)
r4
r5 r, r7 r,
;: (-
; + -
I:
A,
&=2.&+4J,-2J2
&=W,-4J,-2JZ
A,= -2J,--4J3+2J,.
It is evident that there is only ferromagnetism if J, , J2 and Jj are positive. If J, and J2 are positive, and J3 is negative, then the G-mode is stabIe. If J2 and J3
R,
Ri and R,. This integral is positive for parallel spins (ferromagnetism) and is negative for antiparallel spins (antiferromagnetism). Some authors use a different convention [I], and some have introduced an exchange integral which has the opposite sign from ours and twice the magnitude. Following E&taut’s matrix formalism [6], the above expression is evaluated by fixing Ri and by summing on all spins of the atoms Rl of lattice j which are equivalent to first neighbours R, and have the same Jpla, values. Therefore from eqn (5) we have the following coefficients of the interaction matrix in the case of exchange for the orthorhombic two-dimensional lattice. The spin at x, v( 1) has the following two neighbours on lattice 2 : x, y’(2) and x, 1 -y(2’). The coefficient E,~ of the matrix, according to eqn (5) is J, exp(2xik2y) + J, exp(2&(2y
= J, exp[2xik2y(l
+ exp - 2nik)].
- 1)) (6)
The spin at xy(1) has the following four nearest neighbours on lattice 3 : Z,?(3), (1 -x), y’(3’), (1 -x), (1 -y)(3”) and I, (1 --y)(3”). The coefficient E,~ of the interaction matrix, according to eqn (5) is g13= Jj exp(2lri(2hx f 2ky)) x (1 + exp - 2rtih)(l +exp - 2nik).
(7)
Finally the spin at x, y(l) has the following two nearest neighbours 4 : 2, y(4) and (1 - x), ~(4’). s14= J2 exp[2&2x(l
+ exp - 2nih)],
(8)
where J, , J, and Jj are the exchange integrals defined above. When the propagation vector is strictly zero, one finds the following determinant to solve -1
&=2J,+4JZ+2J,
(10)
where JR, is the exchange integral between spins at
cl2 =
1601
w,
4J3
2J,
U
-A
ZJz 4J3
4JJ
2J,
-A2J,
2J,
SJ,
W,
=. ’
-1
(9)
are negative, and J, is positive, then the C-mode is stable. If J, and J3 are negative, and Jz is positive, then the A-mode is stable. Supposing that the difference in distance between the diagonal and the longer side is negligible, we assume that the exchange integral JJ between third neighbours (or between spins on the diagonal) is equal to Jz. So if we assume that J2 = J3, the above equations reduce to A,= 2J, + 6J*; A,= -2J,-6J,;
& = -2J,
+ 2J,;
AA= -2J,-W,.
(11)
If J,, J2 and J3 are positive, it is again evident that there is only ferromagnetism. If J, is positive and Jz is negative, then the C-mode is stable. Such a case might happen if the so-called RKKY exchange mechKittel, Kasuya, anism operates (Rudermann, Yoshida). In order to find the possible k-propagation vectors for eqn (5) we have to equate the differentials of eqn (5) with respect to It and k to zero,
CY& - 0. zz-
In order to obtain the stability conditions of the modes we simply write that i, must be a maximum (-1 = H, exchange energy), i.e. the quadratic form the coefficients which are second derivatives of -A, must be definite positive. We can obtain the inequalities between the exchange constants as stability conditions for all of the modes.
3. RESULTS AND DISCUSSION We have simply constructed the second order invariants by pair multiplication of components which belong to the same representation. For example, in the representation Fz of Table 2 the three products Fz, Gz and I;, Gy are invariants. That is, they do not change sign in the symmetry operations of the
1602
H. $. DARENDEL~&LU and H. Y~~KSEL
crystal. The meaning of this is that as long as isotropic exchange forces dominate, only one of the “pure” modes F, G, C, A may appear and the Hamiltonian will reduce to the square of one of base vectors multiplicated by a scalar. In the presence of anisotropy forces, there will be admixtures of at most two other modes. For example, if there is predominantly a G-mode, it can only be coupled with F-mode in the xy-plane (see Tz and r, in Table 2). The effect of mixing off different modes will occur only in the presence of anisotropy forces such as dipoledipole interaction, crystal field anisotropy, etc. We have obtained four collinear modes along the z-axis and the four non-collinear modes in the xy plane by using symmetry arguments. Although the exact solution of the spin-Hamiltonians in eqn (2) is very difficult, sometimes impossible, approximate solutions are convenient for studying magnetic properties such as magnetic susceptibility, magnetization, phase transitions, magnetic heat capacity, etc. As time goes on, new techniques and methods are developing steadily, in order to obtain better solutions of spin-Hamiltonians. The constant in the spin-Hamiltonian equations can be determined from the experimental data of magnetic properties by using the least-squares fit technique. In the approximation of our model we have assumed that the J,-interaction is equal to the J,-interaction between the third neighbour spins. On the other hand, we have assumed that this conditions is justified when we try to find the solutions of the normal eqn (5) including the J3 term, since we find again the same solutions. Among the possible modes which are described by eqn (1) there are
G,, F,, C,, A, modes as pure modes. The coupling modes are F,G,, G, F,, A, C, and C, A,. Harada and Takasaki [7] have also found a F,-mode for the two-dimensional infinite lattice by making use of the interface method of Mtiller-Hartmann and Zittartz. In the general case, say the existence of a propagation k-vector different from zero, one expects again four solutions (except if there are degeneracies present). Such a case could be solved by the phase-comparison method [8] which does not need an explicit knowledge of eigenvectors or eigenvalues and only uses the fact that the linear equations under consideration and belonging to a same eigenvalue can be made either conjugate or identical. We are planning to investigate this problem in the near future using the phase-comparison method of Bertaut, including the next-nearest neighbor interactions and the third neighbor interactions. Acknowledgement--The patient and constructive paper.
authors criticism
thank the referee for his of an earlier version of the
REFERENCES 1. Samuel Smart J., Exchange Interactions from Experimenfal Data. Academic Press, New York (1969). 2. Bertaut E. F., J. Phys. Chem. Solids 21, 295 (1961). 3. Bertaut E. F.. Ann. Phvs. 7. 203 (1972). 4. Bhagavantam S. and Vknkataryu T., Theory of Groups and its Applications to Physical Problems. Academic Press, New York (1969). 5. Bertaut E. F.. Co&pr. Rend Acad. Sci. 250, 85 (1960). 6. Bertaut E. F.. J. Phvs. Chem. Solids 21, 256 (1961). 7. Harada I. and Takasaki K., J. Phys. Soc.‘Jpn 54, 2210 (1985). 8. Bertaut E. F., J. de Phys 35, 659 (1974).
54, No. II, pp. 159!%1602, 1593
OOZZ-3697/93 $6.00 + 0.00 0 1993 Pergamon Press Ltd
SPIN CONFIGURATION OF TWO-DIMENSIONAL ORTHORHOMBIC LATTICE and
H. S. DARENDEL~O~LU Physics Department,
H. Y~~KSEL
Faculty of Sciences and Arts, SelGuk University,
(Received 23 January 1991; accepted in revisedform
42049, Konya, Turkey
10 May 1993)
Abstract-We have applied the so-called macroscopic and microscopic methods of &-taut to the two-dimensional problem of four spins in an orthorhombic lattice in order to investigate the spin configurations and spin-Hamiltonians for a two-dimensional orthorhombic lattice. We have determined the propagation vectors of the modes and possible spin-Hamiltonians and concluded that the four collinear modes are along the z-axis and the non-collinear modes are in the xy-plane of the two-dimensional orthorhombic lattice within the approximation of the model as long as second order coupling of the modes is present. Keywords: microscopic
Spin configuration, method.
exchange
interaction,
1. INTRODUCTION We
have a two-dimensional
orthorhombic
exchange
constant,
macroscopic
method,
C,, i, G,, G,, G, [4], where E is the identity element, is the inversion operation, C,, C,, C, are rotations of 180” with respect to the x-, y-, and z-axis, respectively, and finally G,, GY,and G, are symmetry planes perpendicular to the x-, y-, and z-axis, respectively. Generating elements of this group (or “independent symmetry elements” as called by crystallographers and neutron diffractionists) are C,, C, and i. As seen from Fig. 1, C, sends the point 1 to 2 and 3 to 4; C, sends the point 1 to 3 and 2 to 4 and finally inversion operator i sends the point 1 to 3 and 2 to 4. Under the operation of C,, C,, and i, the four spin vectors S, , S,, S, and S, transform in a complicated way, so it is more practical to look for linear combinations of the four vectors so that the components of the vectors transform into the same components disregarding the changing of sign. We have the following linear combinations i
lattice
is decorated by spins as seen in Fig. 1. The spins are located at the points: x, y( 1); x, p(2); f, y(3); f,y(4). We assume that all of the spins are axial vectors and have the same values. We assume also that the spins interact with each other by means of exchange interactions only. We exclude super-exchange in which, for instance, there are Mn2+ cations on sites (l)-(4) and an oxygen anion at the origin, the .I, integral between (1) and (3) or between (2) and (4) would be strongly negative as being a 180” super-exchange integral and we would have IJJI > IJ, 1 and IJjI > &I. As the exchange constants decrease very quickly with increasing distance, it is generally quite enough to consider just the first and second neighbour spins [l]. On the other hand, as the distance between spin 1 and spin 2 is shorter than the distance between spin 1 and spin 4, this implies that J, > J2. We suppose that the exchange integral J3 between spin 1 and spin 4 is equal to the exchange integral J2 between spin 1 and spin 3. We will investigate the possible spin-Hamiltonians, spin configurations and also their stability conditions from the point of view of the symmetry arguments for the system defined by Fig. 1. Our approach is mainly the matrix formalism proposed and developed by Bertaut [2,3]. which
F = Sl + S, + Ss + S, G = s, - s* + s3 - s, c=s,+q-s,-sq A = Sl - S2 - S, + S,.
(1)
These F, G, C and A vectors, which we call base vectors, represent the possible magnetic modes. (A) Construction of invariants and spin-Hamiltonians
2. THEORY
The symmetry elements of the point group of a two-dimensional orthorhombic lattice are E, C,, C,,
We apply the generating elements C,, C, and i to the x-, y-, and z-components of the base vectors F,
1599
H. $. DARENDELidLU
1600
and H. Yi_itcsa~
A Y 3 .4
.l
A
.3
3
.3’
.4”
.l’
.4”
x
Fig. 1. Two-dimensional orthorhombic lattice and its independent symmetry operations.
G, C and A. Each component will transform into itself with or without a change of sign. We consider here the spins as axial vectors which have quite different transformation properties from polar vectors. For example when C, acts on A, we have: C,A,=C,(S,,-S,,-S,,+S,,) = S,, - S,, - S,, + S,, = -A,,
(2)
which changes the sign of A,. However, when C, acts on F, we have: (7,
= US,,
line. Because of the fact that Hamiltonians must be invariant under spin reversal, the invariant Hamiltonians are simply constructed by pair multiplication of base vector components which belong to the same representation. For example, the square of the terms in parentheses in the representation F, are F:, G: , F, Gy. Each one of these terms is invariant. In the same way we can go on to write the invariants for the other representations. Invariant terms do not change the sign under the symmetry operations. The physical meaning of the term F,G, can be explained as the x-component of F-mode being compatible with the Y-component of the G-mode. We obtain
+ S,, + S,, + S,,)
= S,, + S, + S,, + S.,, = F,,
H,=k,F;+k,G;+k,F,G,
(3)
H2=k4F;+ksG;+k,G,F,,
which does not change the sign of F,. Proceeding as above for all base vectors and all operators, we obtain Table 1 which summarizes the transformation properties of the base vectors. In order to have invariants, we just rearrange the results in Table 1 by choosing the components which transform in the same way. We call a set of definite transformation properties of the mode F, under generating elements C,, C, and i a representation. The results may be simply specified by (+ - +), which signifies that F, does not change sign under the operations C, and i but does change sign under the operation C,. We call F2 the (+ - +) representation as shown in Table 2. Continuing in a similar way we have eight different representations F, (i = 1 . . .8). As seen from Table 2, the mode components which belong to the same representation are on the same
Vectors F G C A
H,=k,A’:+k,C;+k,A,C, H,=k,,CL:+k,,A:+k,2CXAy,
(4)
where k, . . . k,, are constants which may be determined from the experimental results. (B) Determination
of k-vectors
It is possible that the k-vectors can be determined from a neutron diffraction experiment, i.e. from the positions of the diffraction lines. The essential point of the method is that the generalized Fourier transform of the exchange integral must be a maximum [5]. The Fourier coefficient of the exchange integral is defined as follows ei, = ~&,a, exp[2nik(Ri - R,)], Ri
Table 1. Transformation properties of the base vectors Operations i C, c, Z z X X x Y Y Y +
-
-
-
-
+ -
+ +
+ +
+ +
+ +
+ + -
+ + -
+ + -
Z + + -
(5)
Two-dimensional orthorhombic lattice Table 2. Representation of the base vectors i c, c.
which has the following eigenvalue solutions
+
+)
-
-
G,
r2
:=
-
z;
(-
+
Gy -
;
r3
F, -
G,
Fy C,
2 -
l-1
+)
r4
r5 r, r7 r,
;: (-
; + -
I:
A,
&=2.&+4J,-2J2
&=W,-4J,-2JZ
A,= -2J,--4J3+2J,.
It is evident that there is only ferromagnetism if J, , J2 and Jj are positive. If J, and J2 are positive, and J3 is negative, then the G-mode is stabIe. If J2 and J3
R,
Ri and R,. This integral is positive for parallel spins (ferromagnetism) and is negative for antiparallel spins (antiferromagnetism). Some authors use a different convention [I], and some have introduced an exchange integral which has the opposite sign from ours and twice the magnitude. Following E&taut’s matrix formalism [6], the above expression is evaluated by fixing Ri and by summing on all spins of the atoms Rl of lattice j which are equivalent to first neighbours R, and have the same Jpla, values. Therefore from eqn (5) we have the following coefficients of the interaction matrix in the case of exchange for the orthorhombic two-dimensional lattice. The spin at x, v( 1) has the following two neighbours on lattice 2 : x, y’(2) and x, 1 -y(2’). The coefficient E,~ of the matrix, according to eqn (5) is J, exp(2xik2y) + J, exp(2&(2y
= J, exp[2xik2y(l
+ exp - 2nik)].
- 1)) (6)
The spin at xy(1) has the following four nearest neighbours on lattice 3 : Z,?(3), (1 -x), y’(3’), (1 -x), (1 -y)(3”) and I, (1 --y)(3”). The coefficient E,~ of the interaction matrix, according to eqn (5) is g13= Jj exp(2lri(2hx f 2ky)) x (1 + exp - 2rtih)(l +exp - 2nik).
(7)
Finally the spin at x, y(l) has the following two nearest neighbours 4 : 2, y(4) and (1 - x), ~(4’). s14= J2 exp[2&2x(l
+ exp - 2nih)],
(8)
where J, , J, and Jj are the exchange integrals defined above. When the propagation vector is strictly zero, one finds the following determinant to solve -1
&=2J,+4JZ+2J,
(10)
where JR, is the exchange integral between spins at
cl2 =
1601
w,
4J3
2J,
U
-A
ZJz 4J3
4JJ
2J,
-A2J,
2J,
SJ,
W,
=. ’
-1
(9)
are negative, and J, is positive, then the C-mode is stable. If J, and J3 are negative, and Jz is positive, then the A-mode is stable. Supposing that the difference in distance between the diagonal and the longer side is negligible, we assume that the exchange integral JJ between third neighbours (or between spins on the diagonal) is equal to Jz. So if we assume that J2 = J3, the above equations reduce to A,= 2J, + 6J*; A,= -2J,-6J,;
& = -2J,
+ 2J,;
AA= -2J,-W,.
(11)
If J,, J2 and J3 are positive, it is again evident that there is only ferromagnetism. If J, is positive and Jz is negative, then the C-mode is stable. Such a case might happen if the so-called RKKY exchange mechKittel, Kasuya, anism operates (Rudermann, Yoshida). In order to find the possible k-propagation vectors for eqn (5) we have to equate the differentials of eqn (5) with respect to It and k to zero,
CY& - 0. zz-
In order to obtain the stability conditions of the modes we simply write that i, must be a maximum (-1 = H, exchange energy), i.e. the quadratic form the coefficients which are second derivatives of -A, must be definite positive. We can obtain the inequalities between the exchange constants as stability conditions for all of the modes.
3. RESULTS AND DISCUSSION We have simply constructed the second order invariants by pair multiplication of components which belong to the same representation. For example, in the representation Fz of Table 2 the three products Fz, Gz and I;, Gy are invariants. That is, they do not change sign in the symmetry operations of the
1602
H. $. DARENDEL~&LU and H. Y~~KSEL
crystal. The meaning of this is that as long as isotropic exchange forces dominate, only one of the “pure” modes F, G, C, A may appear and the Hamiltonian will reduce to the square of one of base vectors multiplicated by a scalar. In the presence of anisotropy forces, there will be admixtures of at most two other modes. For example, if there is predominantly a G-mode, it can only be coupled with F-mode in the xy-plane (see Tz and r, in Table 2). The effect of mixing off different modes will occur only in the presence of anisotropy forces such as dipoledipole interaction, crystal field anisotropy, etc. We have obtained four collinear modes along the z-axis and the four non-collinear modes in the xy plane by using symmetry arguments. Although the exact solution of the spin-Hamiltonians in eqn (2) is very difficult, sometimes impossible, approximate solutions are convenient for studying magnetic properties such as magnetic susceptibility, magnetization, phase transitions, magnetic heat capacity, etc. As time goes on, new techniques and methods are developing steadily, in order to obtain better solutions of spin-Hamiltonians. The constant in the spin-Hamiltonian equations can be determined from the experimental data of magnetic properties by using the least-squares fit technique. In the approximation of our model we have assumed that the J,-interaction is equal to the J,-interaction between the third neighbour spins. On the other hand, we have assumed that this conditions is justified when we try to find the solutions of the normal eqn (5) including the J3 term, since we find again the same solutions. Among the possible modes which are described by eqn (1) there are
G,, F,, C,, A, modes as pure modes. The coupling modes are F,G,, G, F,, A, C, and C, A,. Harada and Takasaki [7] have also found a F,-mode for the two-dimensional infinite lattice by making use of the interface method of Mtiller-Hartmann and Zittartz. In the general case, say the existence of a propagation k-vector different from zero, one expects again four solutions (except if there are degeneracies present). Such a case could be solved by the phase-comparison method [8] which does not need an explicit knowledge of eigenvectors or eigenvalues and only uses the fact that the linear equations under consideration and belonging to a same eigenvalue can be made either conjugate or identical. We are planning to investigate this problem in the near future using the phase-comparison method of Bertaut, including the next-nearest neighbor interactions and the third neighbor interactions. Acknowledgement--The patient and constructive paper.
authors criticism
thank the referee for his of an earlier version of the
REFERENCES 1. Samuel Smart J., Exchange Interactions from Experimenfal Data. Academic Press, New York (1969). 2. Bertaut E. F., J. Phys. Chem. Solids 21, 295 (1961). 3. Bertaut E. F.. Ann. Phvs. 7. 203 (1972). 4. Bhagavantam S. and Vknkataryu T., Theory of Groups and its Applications to Physical Problems. Academic Press, New York (1969). 5. Bertaut E. F.. Co&pr. Rend Acad. Sci. 250, 85 (1960). 6. Bertaut E. F.. J. Phvs. Chem. Solids 21, 256 (1961). 7. Harada I. and Takasaki K., J. Phys. Soc.‘Jpn 54, 2210 (1985). 8. Bertaut E. F., J. de Phys 35, 659 (1974).