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Magnetic phase diagrams for three coupled magnetic moments
Brommer, P.E.
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Journal of Magnetism and Magnetic Materials
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Citation for published version (APA):
Brommer, P. E. (1996). Magnetic phase diagrams for three coupled magnetic moments. Journal of Magnetism
and Magnetic Materials, 157&158, 349-350.
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Download date: 18 Jun 2017
Journal of Magnetism and Magnetic Materials 157/158 (1996) 349-350
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Magnetic phase diagrams for three coupled magnetic moments
P.E. Brommer
*
Van der Waals-Zeeman Institute Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
Abstract
For three arbitrary coupled magnetic moments (sublattices), the stable magnetic structures have been determined. Three
classes do exist: three (all), two and no equal coupling parameters, respectively. At any applied field, only one, collinear or
non-collinear, structure is stable. Transitions are smooth: the magnetization curves show no jumps.
Keywords: Magnetic phase diagram; Non-collinear structures
In many lanthanide-transition metal (RT) compounds,
the magnetic structures can be described, in a two-sublattice model, as a coupling between the total R-moment
and the total T-moment [1]. In applied fields, non-collinear
structures can be formed [2], corresponding, in the absence
of crystalline anisotropy, to a linear dependence, M =
ixoH/[n]. In this way, for antiferromagnetic coupling, the
molecular-field constant n ( < 0) can be measured directly.
The influence of crystalline anisotropy is minimized by
performing the experiments on an assembly of small,
freely orientable, particles [3]. A two-sublatfice model is
not always adequate. For instance, in the case of two
different R-moments o n one sublattice, a three-sublattice
model is necessary. Recently, RMn6_xCrxSn 6 was described [4] as a three-sublattice system, simplified because
the Mn-atoms occupy two equivalent sublattices (the Rmoments form the third one). In Ref. [5], a complete
treatment is presented for three inequivalent coupled moments, with constant magnitude. Here, the method is explained and the main results are presented.
The (free) energy of a system of magnetic moments
M 1, M 2, M 3, placed in an external field /~o//, coupled
either fen'o- ( n i j > 0) or antiferromagnefically ( n i j < 0) is
given by
E = -(nl2M1M
-/ZoH(M
2 + n23M2M 3 + n13MIM3)
1 + M 2 + M3).
(1)
The possible stable structures are found by deriving the
equilibrium conditions:
a i M 1 -- n i 2 M 2 --
n 1 3 M 3 = /£0 H ,
(2a)
- n l 2 M 1 + a 2 n 2 - n23M 3 = / z 0 H ,
(2b)
- n l 3 M 1 - n23M 2 + a 3 g 3 = IXo H .
(2c)
* Fax: +31-20-525-5788; email: [email protected].
The coefficients aj and the determinant D of the
coefficient matrix of this set of equations should be nonnegative, mathematically in order to have an energy minimum, from a physical point of view, because the effective
field acting on a moment Mj should be oriented along M j ,
i.e. equals a j M j with positive aj. For a non-collinear state,
D must vanish, since non-zero components perpendicular
to / / should exist. Moreover, considering a fixed moment
M a and variable moments M 2 and M 3, we find that the
corresponding minor (subdeterminant) mll = a 2 a 3 - n 223,
should also be non-negative. Using m12 = a3n12 + n13n23
(cycl), we can derive
m a i m 2 = ( a 3 + n23)/XoH + m 1 2 M 1.
(4)
An analogous relation holds for M 3 (exchange indices
2 and 3). Hence, by addition:
m l l M t = ( a 2 + a 3 + 2n23)/~0 H + (mi1 + m12 + m a 3 ) M 1.
(~)
For vanishing field, these equations reveal that either
all minors have non-zero values (collinear structure) or all
minors do vanish (non-collinear structure). In the latter
case"
a a = _ n12 ni3/n23 ; a2 = _ n12 n23/n13;
(6)
Since aj should be positive (all j), a non-collinear
zero-field phase can only exist in case all molecular field
constants are negative (antiferromagnetic) or in case only
one such constant is negative, the other two being positive
(ferromagnetic).
For non-zero fields, we may exchange, in Eqs. ( 2 a ) (2c), the first colon and the right hand side. Then, analogously, we find that a non-collinear solution may exist
provided the sum mll + m12 + ma3 vanishes. Looking in
more detail, we find a natural classification of the coupled
systems. Suppose, for instance, that mll vanishes. Then,
a 3 = --n13nz3/n12.
0304-8853/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved.
SSDI 0 3 0 4 - 8 8 5 3 ( 9 5 ) 0 1 0 3 2 - 7
P.E. Brommer/ Journal of Magnetism and Magnetic Materials 157/158 (1996) 349-350
350
all minors should vanish. This can only happen in case all
molecular-field constants are equal: n12 = n13 = n23 = n,
referred to as Class I. Then, for n < 0, we find, from any
of Eqs. (2a)-2(c), the relation M = IxoH/In] in a non-collinear structure. Moreover, at a given field, all solutions
(possibly non-coplanar) are degenerate.
In case mll does not vanish, Eq. (4) shows that M 2
(and by analogy M 3) is a linear combination of H and
M1. Consequently, if a non-collinear structure exists, with
non-collinear H and M 1, it is a coplanar one. Moreover,
inserting ml~ + m12 + m13 = 0 in Eq. (5), we see that M t
points in the direction of H. Furthermore, in case two
molecular-field constants are equal (referred to as Class II,
with say hi3 = n23), only two distinct types of non-collinear structures may exist: either M 3 is oriented along H
(parallel or antiparallel), or M 1 and M 2 behave as one
sublattice (oriented either parallel or antiparallel, °pseudotwo-sublattice model'). In these non-collinear states, the
magnetization depends linearly on H. The magnetization
curve consists of straight lines and constant parts, without
any jump (see Fig. 1). Finally, if all molecular-field constants are different (Class III), such non-collinear structures cannot exist: no moment is directed along another
moment or along the field. The magnetization curve consists of constant parts (collinear states) connected by possi-
=
1
,
n12 =
-2
M2 =
4
n13
--0.999
M3 =
12
n23
--1.00~,/
25
MI
20
,
'/ [
[+++]
/
~¢:: 1 5
,,D
L
ine3+_] . . ~ 2"
.~u10
/
/
IncA]
[+-+1
IIAA<c
/
0
i
i
i
(Gdo.3Ndo.7)Niz
3.5
• .............. ~.~-• ..-"
@~'"
iI"
3.0
~
i
2.5
.~ 2.0
~t
1.5
0"
0"
w
t"
Q.
~ i ~ . " e,"
nGdNd =
--4.0
nGdNi =
--20
nNdNi =
1.0 "
MNd
MNi
0
40 T / / ~ s
MGd = Z.Z5 / z B / f . u .
0.5
0.0
T//£ B
T///3 B
= 1.05 /ZB/f.u.
= 0 . 2 /£B//f.u.
'
'
'
'
'
'
'
5
10
15
20
25
30
35
40
B(T)
Fig. 2. Class HI; application to (Nd,Gd) Ni 2 (see Ref. [5]).
bly strongly curved parts (the non-collinear states). Fig. 2
shows, as an example, the result of a fitting procedure to
some preliminary data observed on (Nd,Gd)Ni 2. The Ni
sublattice is assumed to have a non-zero (constant) moment. In this way, a way is opened to determine directly
the ferromagnetic interaction between a light lanthanide
and a transition metal.
In conclusion, the relationships between the coupling
parameters appear to form a natural basis for classification.
Class I, 'three equal coupling parameters', is rather simple,
but has the remarkable property that 'non-coplanar' structures can occur. Class II, 'only two equal coupling parameters', yields interesting magnetization 'curves', still consisting of a combination of linear parts and constants.
Class HI, 'three different coupling parameters', allows a
non-linear relation between H and M.
References
--+1 /
/
0
i
4.0
I
I,
I
I
5
10
15
20
B (orb.
units)
25
Fig. 1. Class II (the difference between hi3 and n23 must be
ignored). See Ref. [5] for more details.
[1] J.J.M. Franse and R.J. Radwanski, Ferromagnetic Materials,
vol.7, ed. K.H.P. Buschow (North-Holland, Amsterdam, 1993)
p. 307.
[2] S.V. Tyablikov, Methods in the Quantum Theory of Magnetism (Plenum Press, New York, 1967).
[3] R. Verhoef, Thesis, Amsterdam (1990).
[4] J.H.P. Colpa and J.H.V.J. Brabers, Physica B 205 (1994) 29.
[5] P.E. Brommer, Physica B 225 (1996) 143.